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Langevin algorithms: ULA bias and MALA exactness

1 · Prerequisites

2 · Summary

ULA applies a noisy gradient step without correction, while MALA uses the same Gaussian proposal with a Hastings acceptance rule and rejected self-transitions. This page proves the exact Gaussian ULA stationary variance and MALA target invariance. For a twice continuously differentiable potential with globally Lipschitz gradient and the stated superlinear radial drift, it proves existence and uniqueness of each ULA invariant law and a weighted total-variation bias bound of order square root of the step size throughout the specified step-size interval.

Local suppliers establish finite-horizon Brownian change of drift and directional relative entropy, a reversible diffusion density and compact minorization, weighted contraction and invariant existence, and the uniform exponential moments needed by the proof. Every quantitative conclusion retains its explicit hypotheses; the Gaussian example alone does not establish the general bias bound.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6-luna)Open item page →

Weighted total variation for probability measures

Definition

Let $d \geq 1$ , let $W : R^{d} \to [1, \infty)$ be Borel measurable, and let $μ, ν$ be Borel probability measures on $R^{d}$ with $\int W d μ < \infty$ and $\int W d ν < \infty$ . Define their $W$ -weighted total-variation distance by $∥ μ - ν ∥_{W} : = \sup_{f B o r e l, ∣ f ∣ \leq W} ∣ \int f d μ - \int f d ν ∣ .$ The moment hypotheses make every integral in the supremum finite. This is the signed-measure test-function norm used in weighted-variation bounds. For $W \equiv 1$ it has the usual factor of two relative to the event-supremum probability distance in Total variation distance for probability laws. Here a Borel probability measure is in the sense of Probability measures and probability spaces, on the Borel sigma-algebra of The Borel sigma-algebra of a topological space.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Radial gradient drift forces potential coercivity

Statement

Let $d \geq 1$ , $U \in C^{2} (R^{d})$ , and suppose $\nabla U$ is $L$ -Lipschitz for some $L > 0$ . Let $x_{⋆}$ be a global minimizer, normalize $U (x_{⋆}) = 0$ , and assume that for some $ρ > 0$ , $α \in (1, 2]$ , and $M_{ρ} \geq 0$ , $⟨ \nabla U (x), x - x_{⋆} ⟩ \geq ρ ∥ x - x_{⋆} ∥^{α} whenever ∥ x - x_{⋆} ∥ \geq M_{ρ} .$ Then every $x \in R^{d}$ satisfies $U (x) \geq \frac{ρ}{α + 1} ∥ x - x_{⋆} ∥^{α} - a_{α}, a_{α} : = \frac{ρ M_{ρ}^{α}}{α + 1} + \frac{L M_{ρ}^{2}}{2} .$

Facts & Assumptions

Given: The hypotheses in the Statement; $\nabla U$ is the Euclidean coordinate gradient and $⟨ \cdot, \cdot ⟩$ and $∥ \cdot ∥$ are the Euclidean inner product and norm.

[A1]

The differentiability, global-minimizer, Lipschitz-gradient, and radial-drift hypotheses are exactly those stated above. The gradient convention is The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case and the inner product is The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ .

[F1]

At an interior differentiable local minimum, the gradient vanishes. Fermat's theorem: an interior differentiable local extremum has zero gradient

[F2]

The derivative of a composition with a differentiable path is given by the total-derivative chain rule. The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$

[F3]

A continuous derivative integrates to the endpoint difference. Newton–Leibniz needs only continuity on $[a, b]$ , differentiability on $(a, b)$ , and a Riemann-integrable extension of the interior derivative

[F4]

$∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ . Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation

Proof

Given: The potential and constants in the Statement.

1.1A1F1F2F3F4

Put $h = x - x_{⋆}$ and $r = ∥ h ∥$ . By [F1], $\nabla U (x_{⋆}) = 0$ . The path $g (t) = U (x_{⋆} + t h)$ is continuously differentiable, and [F2]–[F3] give $U (x) - U (x_{⋆}) = \int_{0}^{1} ⟨ \nabla U (x_{⋆} + t h), h ⟩ d t .$ For $0 \leq t \leq 1$ , Lipschitz continuity gives $∥ \nabla U (x_{⋆} + t h) ∥ \leq L t r$ , hence the integrand is at least $- L t r^{2}$ by [F4].

2.1A1step 1.1algebra

If $r < M_{ρ}$ , integrating the bound from step 1.1 over $[0, 1]$ yields $U (x) - U (x_{⋆}) \geq - L r^{2} / 2 \geq - L M_{ρ}^{2} / 2$ . Since $r^{α} \leq M_{ρ}^{α}$ , this is at least $ρ r^{α} / (α + 1) - a_{α}$ .

2.2A1step 1.1algebra

Suppose $r \geq M_{ρ}$ and $r > 0$ . On $0 \leq t \leq M_{ρ} / r$ , the bound in step 1.1 integrates to at least $- L M_{ρ}^{2} / 2$ . On $M_{ρ} / r \leq t \leq 1$ , the radial assumption gives $t ⟨ \nabla U (x_{⋆} + t h), h ⟩ \geq ρ t^{α} r^{α} .$ For $t \in (0, 1]$ , division by $t$ and $t^{α - 1} \geq t^{α}$ show that the path derivative is at least $ρ t^{α} r^{α}$ .

3.1step 2.2algebra

Integrating the outer-segment bound in step 2.2 and adding the inner-segment bound gives $U (x) - U (x_{⋆}) \geq - \frac{L M_{ρ}^{2}}{2} + \frac{ρ r^{α}}{α + 1} (1 - (M_{ρ} / r)^{α + 1}) .$ Because $r \geq M_{ρ}$ , $M_{ρ}^{α + 1} / r \leq M_{ρ}^{α}$ , so this is at least $ρ r^{α} / (α + 1) - a_{α}$ .

4.1step 2.1step 3.1algebra∎

If $r = 0$ , the claimed inequality reads $0 \geq - a_{α}$ and holds because $a_{α} \geq 0$ . Steps 2.1 and 3.1 cover all other $r$ , proving the bound.

LemmaStatement: AI-adaptedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Exponential Lyapunov drift for the formal Langevin generator

Statement

Let $d \geq 1$ , $U \in C^{2} (R^{d})$ , and suppose $\nabla U$ is $L$ -Lipschitz with $L > 0$ . Let $x_{⋆}$ be a global minimizer with $U (x_{⋆}) = 0$ , and assume the radial condition of Radial gradient drift forces potential coercivity with parameters $ρ > 0$ , $α \in (1, 2]$ , and $M_{ρ} \geq 0$ . For $0 < s < 1$ , set $W_{s} (x) = e^{s U (x)}, L_{U} f (x) : = \sum_{i = 1}^{d} \partial_{i i} f (x) - ⟨ \nabla U (x), \nabla f (x) ⟩,$ where $L_{U}$ is only the displayed formal differential operator here. Define $K_{s} : = \max {M_{ρ}, {(\frac{2 d L}{(1 - s) ρ^{2}})}^{1 / (2 (α - 1))}}, β_{s} : = 2 s d L \exp (s L K_{s}^{2} / 2) .$ Then for every $x \in R^{d}$ , $L_{U} W_{s} (x) \leq - s d L W_{s} (x) + β_{s} 1_{{∥ x - x_{⋆} ∥ \leq K_{s}}} .$

Facts & Assumptions

Given: $U$ , $x_{⋆}$ , and the constants in the Statement.

[A1]

$U \in C^{2}$ , its gradient is $L$ -Lipschitz, $x_{⋆}$ is a global minimizer with $U (x_{⋆}) = 0$ , and the stated radial drift condition holds. The Euclidean inner product and gradient are those of The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ and The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, the Hessian convention is The Hessian matrix and critical points of a scalar field, and the radial condition is from Radial gradient drift forces potential coercivity.

[F1]

A differentiable function at an interior local minimum has zero gradient. Fermat's theorem: an interior differentiable local extremum has zero gradient

[F2]

The chain rule computes derivatives of compositions. The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$

[F3]

[F4]

The fundamental theorem of calculus integrates a continuous derivative along a compact interval. Newton–Leibniz needs only continuity on $[a, b]$ , differentiability on $(a, b)$ , and a Riemann-integrable extension of the interior derivative

Proof

Given: The hypotheses and formal operator in the Statement.

1.1A1F1algebra

By [F1], $\nabla U (x_{⋆}) = 0$ . For each coordinate vector $e_{i}$ , the Lipschitz bound gives $∣ \frac{\partial_{i} U (x + h e_{i}) - \partial_{i} U (x)}{h} ∣ \leq L (h \neq 0) .$ Taking $h \to 0$ yields $∣ \partial_{i i} U (x) ∣ \leq L$ , so $\sum_{i} \partial_{i i} U (x) \leq d L$ .

2.1F2step 1.1algebra

Applying [F2] coordinatewise to $W_{s} = e^{s U}$ gives $\nabla W_{s} = s W_{s} \nabla U, \sum_{i} \partial_{i i} W_{s} = s W_{s} \sum_{i} \partial_{i i} U + s^{2} W_{s} ∥ \nabla U ∥^{2} .$ Substitution in the displayed definition of $L_{U}$ therefore gives $L_{U} W_{s} = s W_{s} (\sum_{i} \partial_{i i} U - (1 - s) ∥ \nabla U ∥^{2}) .$

3.1A1F3step 2.1algebra

If $r = ∥ x - x_{⋆} ∥ \geq M_{ρ}$ and $r > 0$ , radial drift and [F3] imply $∥ \nabla U (x) ∥ \geq ρ r^{α - 1}$ . Thus when $r > K_{s}$ , the formula in step 2.1 and $r > K_{s}$ give $L_{U} W_{s} (x) \leq s W_{s} (x) (d L - (1 - s) ρ^{2} r^{2 (α - 1)}) \leq - s d L W_{s} (x) .$ The same conclusion holds for $r \geq K_{s}$ by the definition of $K_{s}$ .

3.2A1F1F2F4step 1.1step 2.1

For $h = x - x_{⋆}$ and $0 \leq t \leq 1$ , Lipschitz continuity and [F1] give $∥ \nabla U (x_{⋆} + t h) ∥ \leq L t r$ . The chain rule and [F4] therefore yield $U (x) - U (x_{⋆}) \leq L r^{2} / 2$ . Since $U (x_{⋆}) = 0$ , on $r \leq K_{s}$ we have $W_{s} (x) \leq e^{s L K_{s}^{2} / 2}$ . Also $\sum_{i} \partial_{i i} U \leq d L$ by step 1.1, so step 2.1 implies $L_{U} W_{s} + s d L W_{s} \leq 2 s d L e^{s L K_{s}^{2} / 2} = β_{s}$ .

4.1step 3.1step 3.2∎

On $r > K_{s}$ use step 3.1, and on $r \leq K_{s}$ use step 3.2. These two regions cover $R^{d}$ and give the displayed drift inequality.

AlgorithmAlgorithm: Literature-sourcedProof: Not applicablejudge pass (gpt-6-luna)Open item page →

Unadjusted Langevin algorithm (ULA)

Definition

Assume AC (The Axiom of Choice). Let $d \geq 1$ , let $U : R^{d} \to R$ be Borel measurable in the sense of Borel measurable and Lebesgue measurable functions on $R^{n}$ and continuously differentiable with $0 < Z_{U} : = \int_{R^{d}} e^{- U (x)} d x < \infty,$ and define the target probability $π_{U} (d x) : = Z_{U}^{- 1} e^{- U (x)} d x .$ This is the measure with density $Z_{U}^{- 1} e^{- U}$ relative to Lebesgue measure (The measure with density $f$ relative to $μ$ and Lebesgue measurable sets, the family $L (R^{n})$ , and the restricted set function $λ_{n}$ ); the probability-measure convention is Probability measures and probability spaces, and the normalizer makes its total mass one. Given a step size $γ > 0$ , an initial state $X_{0} \in R^{d}$ , and an IID sequence $Z_{1}, Z_{2}, \dots \sim N_{d} (0, I_{d})$ , using the multivariate-normal law of Multivariate normal law, including singular covariance and standard-normal convention of Standard normal and normal laws, repeat for $k \geq 0$ :

Compute the Euclidean gradient $\nabla U (X_{k})$ using the convention of The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case.
Take the fixed-step gradient update ${\tilde{X}}_{k + 1} = X_{k} - γ \nabla U (X_{k})$ , the fixed-step branch of Gradient descent.
Set $X_{k + 1} = {\tilde{X}}_{k + 1} + \sqrt{2 γ} Z_{k + 1} .$

The one-step transition law is $Q_{γ} (x, \cdot) = N_{d} (x - γ \nabla U (x), 2 γ I_{d}) .$ The procedure applies the proposal at every step and has no accept/reject correction. The probability-kernel and density properties of $Q_{γ}$ are proved in The ULA Gaussian proposal is a probability kernel with a positive density ↗.

Remarks

The definition only asserts a well-defined Gaussian update. It asserts no convergence rate, invariant-law existence, or equality of the invariant law with $π_{U}$ ; those are separate mathematical claims.

LemmaStatement: Literature-sourcedProof: AI-adaptedOpen item page →

A weighted total-variation bound from relative entropy

Statement

Assume AC. Let $μ, ν$ be Borel probability measures on $R^{d}$ , and let $W : R^{d} \to [1, \infty)$ be Borel measurable with $\int W^{2} d μ + \int W^{2} d ν < \infty$ . Then $∥ μ - ν ∥_{W} \leq \sqrt{2 (\int W^{2} d μ + \int W^{2} d ν) D (μ ∥ ν)},$ where $μ, ν$ are probability measures in the sense of Probability measures and probability spaces, $∥ \cdot ∥_{W}$ is defined in Weighted total variation for probability measures and $D (μ ∥ ν)$ is the directional relative entropy of Kullback–Leibler divergence. If $D (μ ∥ ν) = + \infty$ , the inequality is understood in the extended-real sense and is immediate.

Facts & Assumptions

Given: AC; probability measures $μ, ν$ ; a Borel weight $W \geq 1$ with finite second moments under both measures.

[A1]

AC supplies the Radon–Nikodym representative used in the definition of directional relative entropy. The Axiom of Choice The Radon-Nikodym derivative as an almost-everywhere equivalence class

[F1]

If $D (μ ∥ ν) < \infty$ , then $μ ≪ ν$ , the density $r = d μ / d ν$ is nonnegative, and $D (μ ∥ ν) = \int (r \log r) d ν$ with $0 \log 0 = 0$ . Kullback–Leibler divergence Integrating against a Radon-Nikodym derivative recovers integration against the measure

[F2]

The directional relative entropy has finite negative part and $\int r d ν = 1$ . Kullback–Leibler divergence Integrating against a Radon-Nikodym derivative recovers integration against the measure

[F3]

Cauchy–Schwarz gives $\int ∣ f g ∣ d ν \leq (\int ∣ f ∣^{2} d ν)^{1 / 2} (\int ∣ g ∣^{2} d ν)^{1 / 2}$ for square-integrable $f, g$ . Cauchy-Schwarz inequality for $L^{2}$

[F4]

For $z > 0$ , $(\log z)^{'} = 1 / z$ and $\log 1 = 0$ , because the integral logarithm $L$ agrees with the natural logarithm of The natural logarithm as the inverse of the exponential function. The integral logarithm satisfies $L^{'} (x) = 1 / x$ for $x > 0$ and $L (1) = 0$ The integral logarithm $L$ is the published natural logarithm The natural logarithm as the inverse of the exponential function

[F5]

Differentiation obeys the sum, scalar-multiple and product rules on $(0, \infty)$ ; squares are products of two identical factors. The mean value theorem applies to the resulting continuously differentiable scalar functions. Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ The integral logarithm satisfies $L^{'} (x) = 1 / x$ for $x > 0$ and $L (1) = 0$ The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$

[F6]

Integrals of integrable real functions are linear. The Lebesgue integral is linear on $L^{1} (μ)$

Proof

Given: The hypotheses in the Statement.

1.1F4F5algebra

Define $ϕ (0) = 1$ and $ϕ (t) = t \log t - t + 1$ for $t > 0$ . For $0 < t \leq 1$ , the function $h (t) = t - 1 - \log t$ satisfies $h (1) = 0$ and $h^{'} (t) = 1 - 1 / t \leq 0$ . The mean value theorem on $[t, 1]$ gives $h (t) \geq 0$ , so $\log t \leq t - 1$ . Now $k (t) = ϕ (t) - (1 - t)^{2} / 2$ satisfies $k (1) = 0$ and $k^{'} (t) = \log t + 1 - t \leq 0$ by [F4]–[F5]; another application of the mean value theorem gives $k (t) \geq 0$ . At $t = 0$ the inequality holds because $1 \geq 1 / 2$ .

2.1F4F5step 1.1algebra

For $t \geq 1$ , put $g (t) = 2 (1 + t) ϕ (t) - (t - 1)^{2}$ . By [F5], $g (1) = g^{'} (1) = 0$ and $g^{''} (t) = 4 \log t + 2 / t \geq 0$ using [F4] and $\log t \geq 0$ for $t \geq 1$ . The mean value theorem applied first to $g^{'}$ and then to $g$ shows $g^{'} (t) \geq 0$ and $g (t) \geq 0$ for $t \geq 1$ . Combining with step 1.1 gives $∣ 1 - t ∣^{2} \leq 2 (1 + t) ϕ (t) (t \geq 0) .$

3.1A1F1F2F3F6step 2.1given∎

It remains to consider finite $D (μ ∥ ν)$ . Let $r = d μ / d ν$ as in [F1]. For every Borel $f$ with $∣ f ∣ \leq W$ , the Radon–Nikodym identity gives $∣ \int f d μ - \int f d ν ∣ \leq \int W ∣ r - 1 ∣ d ν .$ Applying step 2.1 pointwise and [F3] with factors $W \sqrt{1 + r}$ and $\sqrt{2 ϕ (r)}$ bounds this by $\sqrt{\int W^{2} (1 + r) d ν} \sqrt{2 \int ϕ (r) d ν} = \sqrt{(\int W^{2} d ν + \int W^{2} d μ) 2 D (μ ∥ ν)} .$ The moment hypotheses make the first factor finite, and [F1], [F2], and [F6] identify the second integral with the directional relative entropy, using $\int r d ν = \int 1 d ν = 1$ . Taking the supremum over $f$ proves the claim.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Global drift and one-step minorization give weighted contraction and a unique invariant law

Statement

Assume AC. Let $K$ be a Borel probability kernel on $R^{d}$ , $d \geq 1$ , and let $V : R^{d} \to [1, \infty)$ be finite and Borel. Suppose $K V \leq a V + b, 0 < a < 1, 0 \leq b < \infty .$ Suppose that for some $R > \max {2, 2 b / (1 - a)}$ the set $D = {V \leq R}$ is nonempty and there are a probability $ν$ and $ϵ_{0} > 0$ such that $K (x, A) \geq ϵ_{0} ν (A) (x \in D, A Borel) .$ Then $K$ has a unique invariant probability $π$ , uniqueness being among all Borel probabilities, and $π V \leq b / (1 - a) < \infty$ . There are $C < \infty$ and $0 < r < 1$ such that for all finite- $V$ -moment probabilities $μ, λ$ and $n \geq 0$ , $∥ μ K^{n} - λ K^{n} ∥_{V} \leq C r^{n} ∥ μ - λ ∥_{V} .$ In particular, $∥ K^{n} (x, \cdot) - π ∥_{V} \leq C r^{n} (V (x) + π V) .$ The norm is that of Weighted total variation for probability measures. Explicitly, put $ϵ = \min {ϵ_{0}, 1 / 2}, β = \frac{ϵ}{2 (b + 1)}, a^{'} = a + \frac{2 b}{R}, C = \frac{1 + β}{β},$ $r = \max {\frac{2 + β a^{'} R}{2 + β R}, 1 - \frac{2 (ϵ - β b)}{2 + β R}} .$ The same contraction holds for finite signed $η$ with $η (R^{d}) = 0$ and $\int V d ∣ η ∣ < \infty$ , with $μ - λ$ replaced by $η$ .

Facts & Assumptions

Given: The kernel, finite weight, global drift and one-step sublevel minorization in the Statement.

[A1]

AC, The Axiom of Choice, is assumed for the general signed-measure Jordan existence clause below.

[F1]

Jordan decomposition of a signed measure into unique mutually singular positive parts gives a Hahn partition and mutually singular positive parts under AC; For a signed measure, total variation is nu-plus plus nu-minus, finite partitions suffice, and nu-plus and nu-minus are extremal identifies $∣ η ∣ = η^{+} + η^{-}$ .

[F2]

The indefinite integral of a nonnegative measurable function is a measure constructs measures by nonnegative measurable densities; Monotone convergence for the integral extends integration identities from simple functions.

[F3]

Finite-total-variation signed measures are complete gives completeness of finite signed measures in the norm $∣ η ∣ (R^{d})$ .

[F4]

Measure kernel and probability kernel and Measurability of integration against a kernel give measurable kernel integrals.

[F5]

Finite products of probability kernels are probability kernels gives finite probability products; Tonelli's theorem for nonnegative measurable functions on a sigma-finite product gives their marginal integration identities.

[F6]

Invariant and stationary distribution for a Markov kernel defines $μ K$ and invariance. Dominated convergence applies to bounded measurable kernel evaluations under any probability.

[F7]

Weighted total variation for probability measures uses the supremum over all Borel $∣ f ∣ \leq V$ , without the event-distance factor of one half.

Proof

Given: The hypotheses and facts [A1], [F1]–[F7].

1.1A1F1F2F3F7constructalgebra

Put $w = 1 + β V$ and, for finite signed $η$ with $\int w d ∣ η ∣ < \infty$ , define $∥ η ∥_{w} = \sup_{∣ f ∣ \leq w} ∣ η f ∣$ . Obtain a Hahn partition $P, N$ by [F1]. The upper bound $∣ η f ∣ \leq \int w d ∣ η ∣$ and the test $f = w (1_{P} - 1_{N})$ give $∥ η ∥_{w} = \int w d ∣ η ∣$ . Multiplication $T η = w η$ is an isometry onto the ordinary finite-total-variation signed measures: define it by multiplying the Jordan parts, and note that the same partition is Hahn, so $∣ T η ∣ = w ∣ η ∣$ . Conversely, for any finite signed $ζ$ , multiplication by $w^{- 1}$ gives $∣ w^{- 1} ζ ∣ = w^{- 1} ∣ ζ ∣$ and is inverse to $T$ . The density integration identities follow first for indicators and simple functions, then by monotone convergence. Thus [F3] makes the finite- $w$ -moment signed-measure space complete. Also $β V \leq w \leq (1 + β) V$ .

2.1F1F2F4F6step 1.1givenalgebra

Define $η K = η^{+} K - η^{-} K$ . Each positive part acts by integration of the kernel evaluations. Countable additivity follows by monotone convergence. The inequality $∣ η K ∣ \leq ∣ η ∣ K$ follows by bounding the variation sum on every countable measurable partition by the corresponding positive measure $∣ η ∣ K$ . The drift gives $K w \leq 1 + β a V + β b \leq (1 + β b) w$ , so $η K$ has finite $w$ moment whenever $η$ does. Kernel integration is Borel and finite for $∣ f ∣ \leq c w$ , and $(η K) f = η (K f)$ , by the indicator identity and monotone convergence applied separately to positive and negative parts. Total mass is preserved.

3.1givenstep 2.1algebra

The constants satisfy $0 < a^{'} < 1$ and $0 < β b < ϵ$ if $b > 0$ , while $β b = 0 < ϵ$ if $b = 0$ . Both displayed component rates are in $(0, 1)$ . Fix Borel $∣ f ∣ \leq w$ , and write $S = V (x) + V (y)$ . If $S \geq R$ , then $∣ K f (x) - K f (y) ∣ \leq K w (x) + K w (y) \leq 2 + β (a S + 2 b) \leq 2 + β a^{'} S$ . Since $s \mapsto (2 + β a^{'} s) / (2 + β s)$ is decreasing for $s \geq R$ , this gives $∣ K f (x) - K f (y) ∣ \leq r (2 + β S)$ .

4.1F2F4step 2.1step 3.1givenconstructalgebra

If $S < R$ , both states are in $D$ . The residual probabilities $H_{x} = (K (x, \cdot) - ϵ ν) / (1 - ϵ)$ are well defined there. Fixing one point of the nonempty $D$ , the minorization and finite $K V$ give $ϵ ν V \leq K V (x_{D}) < \infty$ . Consequently $ν f$ is finite, and the common term $ϵ ν f$ cancels from $K f (x) - K f (y)$ . Thus $∣ K f (x) - K f (y) ∣ \leq (1 - ϵ) (H_{x} w + H_{y} w) \leq 2 (1 - ϵ) + β (a S + 2 b) \leq (2 + β S) - 2 (ϵ - β b) \leq r (2 + β S)$ . The residual evaluations are Borel on $D$ , so these are legitimate kernel integrations; no selected optimal coupling is involved.

5.1A1F1F5F7step 1.1step 2.1step 3.1step 4.1algebra

Let $η$ have zero mass and finite $w$ norm. If $η \neq 0$ , its Jordan masses agree at some $m > 0$ ; put $p = η^{+} / m$ and $q = η^{-} / m$ . They concentrate on disjoint Hahn sets. Integration of steps 3.1 and 4.1 against $p \otimes q$ gives, for every $∣ f ∣ \leq w$ , $∣ (η K) f ∣ = m ∣ \int (K f (x) - K f (y)) p (d x) q (d y) ∣ \leq m r \int (w (x) + w (y)) p (d x) q (d y) = r ∥ η ∥_{w}$ . Every integral is finite by the finite moments and step 2.1. For $η = 0$ the inequality is immediate. Taking the supremum and iterating, using preservation of mass and moments, proves $∥ η K^{n} ∥_{w} \leq r^{n} ∥ η ∥_{w}$ . Norm comparison in step 1.1 yields $∥ η K^{n} ∥_{V} \leq C r^{n} ∥ η ∥_{V}$ , including differences of the stated probabilities.

6.1F6step 1.1step 2.1step 5.1constructalgebra

Fix $x_{0} \in R^{d}$ and set $μ_{n} = δ_{x_{0}} K^{n}$ . The difference $μ_{1} - μ_{0}$ has finite $w$ norm, and step 5.1 gives $∥ μ_{n + 1} - μ_{n} ∥_{w} \leq r^{n} ∥ μ_{1} - μ_{0} ∥_{w}$ . The geometric tail sum makes $(μ_{n})$ Cauchy, so step 1.1 gives a signed-measure limit $π$ with finite $w$ norm. Norm convergence implies $π (A) = \lim_{n} μ_{n} (A) \geq 0$ for every Borel $A$ and $π (R^{d}) = 1$ . Thus $π$ is a probability and $π V < \infty$ . Step 5.1 gives $∥ μ_{n} K - π K ∥_{w} \leq r ∥ μ_{n} - π ∥_{w} \to 0$ ; since $μ_{n} K = μ_{n + 1} \to π$ , it follows that $π K = π$ . Finally drift iteration gives $μ_{n} V \leq a^{n} V (x_{0}) + b (1 - a^{n}) / (1 - a)$ . Because $β V \leq w$ , norm convergence gives $μ_{n} V \to π V$ , whence $π V \leq b / (1 - a)$ .

7.1F4F6F7step 5.1step 6.1algebra∎

Apply step 5.1 to $δ_{x} - π$ . Its $V$ norm is at most $V (x) + π V$ , giving the pointwise asserted bound. If $λ$ is any invariant probability, without a moment assumption, then for each Borel $A$ the pointwise convergence $K^{n} (x, A) \to π (A)$ and $0 \leq K^{n} (x, A) \leq 1$ give $λ (A) = \int K^{n} (x, A) λ (d x) \to π (A)$ by dominated convergence. Thus $λ = π$ . Together with steps 5.1 and 6.1 this proves all assertions.

Remarks

This is a weighted signed-measure contraction and an invariant-law construction. Minorization is needed only on one sufficiently large sublevel. For a family of kernels its constants depend on the drift and minorization constants; no uniformity over an additional discretization parameter is asserted.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

The ULA Gaussian proposal is a probability kernel with a positive density

Statement

Assume AC and the hypotheses of Unadjusted Langevin algorithm (ULA). For $γ > 0$ , its proposal family $Q_{γ}$ is a probability kernel on $R^{d}$ and has the everywhere positive finite jointly continuous Lebesgue density $q_{γ} (y ∣ x) = (4 π γ)^{- d / 2} \exp ⁣ (- \frac{∥ y - x + γ \nabla U (x) ∥^{2}}{4 γ}) .$

Facts & Assumptions

Given: AC, a finite $d \geq 1$ , $U \in C^{1} (R^{d})$ , $γ > 0$ , and the proposal in Unadjusted Langevin algorithm (ULA).

[A1]

AC holds as required by the algorithm and multivariate normal law. The Axiom of Choice Unadjusted Langevin algorithm (ULA)

[F1]

For positive-definite covariance, $N_{d} (m, Σ)$ has the displayed Gaussian Lebesgue density. A positive-definite multivariate normal law has its Gaussian density

[F2]

$N_{d} (m, Σ)$ is a probability law when $Σ$ is symmetric positive semidefinite; its coordinates and projections follow the multivariate-normal convention. Multivariate normal law, including singular covariance

[F3]

The definition of a probability kernel requires each section to be a probability measure and each evaluation on a Borel set to be measurable in the starting point. Measure kernel and probability kernel

[F4]

Integration of a nonnegative product-measurable function against a uniformly sigma-finite kernel is measurable in the starting point. Measurability of integration against a kernel

[F5]

Lebesgue measure on $R^{d}$ is sigma-finite in the sense of Finite, sigma-finite, and semifinite measures, with finite mass on bounded boxes. Lebesgue measurable sets, the family $L (R^{n})$ , and the restricted set function $λ_{n}$ Finite, sigma-finite, and semifinite measures Lebesgue measure is sigma-finite, and every metrically bounded subset of $R^{n}$ has finite outer measure

[F6]

AC implies countable choice, which is the hypothesis of the continuous-map measurability result. AC supplies countable selections and prescribed serial paths The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F7]

A continuous Euclidean map is Borel measurable in the sense of Borel measurable and Lebesgue measurable functions on $R^{n}$ , under countable choice. Continuous functions on Euclidean spaces are Borel measurable Borel measurable and Lebesgue measurable functions on $R^{n}$

[F8]

The Borel sigma-algebra The Borel sigma-algebra of a topological space of $R^{2 d}$ is the product of the two Borel sigma-algebras on $R^{d}$ . The Borel product of R^m and R^n is the Borel sigma-algebra of R^{m+n}

Proof

Given: The hypotheses and conventions above.

1.1A1F1F2given

The covariance $2 γ I_{d}$ is symmetric positive definite. By [F2], the transition law in Unadjusted Langevin algorithm (ULA) is a probability measure for each $x$ . Applying [F1] with $m = x - γ \nabla U (x)$ gives the displayed density; its normalizing factor is finite and positive because $γ > 0$ .

2.1F3F4F5F6F7F8step 1.1∎

The map $(x, y) \mapsto x - y + γ \nabla U (x)$ is continuous, so the displayed $q_{γ} (y ∣ x)$ is Borel on $R^{2 d}$ by [F7], hence product-measurable by [F8]. For a Borel set $A$ , the constant kernel $K (x, B) = λ_{d} (B)$ is uniformly sigma-finite: $[- n, n]^{d}$ exhausts $R^{d}$ and has finite Lebesgue measure by [F5]. Applying [F4] to $f (x, y) = 1_{A} (y) q_{γ} (y ∣ x)$ gives measurability of $x \mapsto \int_{A} q_{γ} (y ∣ x) d y = Q_{γ} (x, A)$ . Together with step 1.1 this is exactly [F3], so $Q_{γ}$ is a probability kernel.

TheoremStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

The Gaussian ULA invariant law has variance $2 / (2 - γ)$

Statement

Assume AC, let $U (x) = x^{2} / 2$ on $R$ , and let $π = N (0, 1)$ . For every $0 < γ < 2$ , the one-dimensional ULA kernel of Unadjusted Langevin algorithm (ULA) has the invariant probability $π_{γ} = N ⁣ (0, \frac{2}{2 - γ}) .$ Every invariant probability with finite second moment has variance $2 / (2 - γ)$ . Thus its invariant variance is not the target variance $1$ . No uniqueness claim is made for invariant laws without a finite second moment.

Facts & Assumptions

Given: AC, the standard Gaussian target, and the ULA update with $U (x) = x^{2} / 2$ .

[A1]

AC is assumed in the standard-normal construction and characteristic-function uniqueness. The Axiom of Choice

[F1]

For this potential, $\nabla U (x) = x$ and the ULA update is $X^{'} = (1 - γ) X + \sqrt{2 γ} Z$ , where $Z \sim N (0, 1)$ is independent of the current state. Unadjusted Langevin algorithm (ULA) The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case

[F2]

The normal-law convention is Standard normal and normal laws; the characteristic function of $N (m, v)$ is $\exp (i m t - v t^{2} / 2)$ , and its mean and variance are $m$ and $v$ . Characteristic function of a normal law Moments, variance, and covariance on a probability space

[F3]

Affine transforms and finite independent sums multiply the corresponding characteristic functions. Characteristic functions under affine maps and independent sums

[F4]

Equal characteristic functions determine equal Borel probability laws. Uniqueness of a law from its characteristic function

[F5]

For independent square-integrable variables, expectations factor and finite linear combinations commute with expectation. Expectations factor over finite products of independent random variables The Lebesgue integral is linear on $L^{1} (μ)$

[F6]

A square-integrable variable is integrable by Cauchy–Schwarz on a probability space. Cauchy-Schwarz for random variables

Proof

Given: AC, $0 < γ < 2$ , and the ULA transition in [F1].

1.1A1F1F2F3F4algebra

Set $v_{γ} = 2 / (2 - γ) > 0$ and let $X \sim N (0, v_{γ})$ be independent of $Z \sim N (0, 1)$ . By [F2]–[F3], the updated state has characteristic function $φ_{X^{'}} (t) = \exp ⁣ (- \frac{(1 - γ)^{2} v_{γ} t^{2}}{2}) \exp (- γ t^{2}) = \exp ⁣ (- \frac{((1 - γ)^{2} v_{γ} + 2 γ) t^{2}}{2}) .$ Direct algebra gives $(1 - γ)^{2} v_{γ} + 2 γ = v_{γ}$ , so [F2] and [F4] imply $X^{'} \sim N (0, v_{γ})$ . This proves that $π_{γ}$ is invariant.

2.1F1F2F5F6algebra∎

Let $X$ have any invariant law with finite second moment, independently of the fresh standard normal $Z$ . By [F6], $X$ is integrable, and [F5] gives $Var (X^{'}) = (1 - γ)^{2} Var (X) + 2 γ,$ because $Var (Z) = 1$ and the cross-covariance of the independent variables is zero. Invariance gives $Var (X^{'}) = Var (X)$ , so $Var (X) = \frac{2 γ}{1 - (1 - γ)^{2}} = \frac{2}{2 - γ} .$ Since $γ > 0$ , this exceeds $1$ .

LemmaStatement: Literature-sourcedProof: AI-adaptedOpen item page →

ULA's Gaussian kernel satisfies an exponential drift bound

Statement

Assume AC. Let $d \geq 1$ , $U \in C^{2} (R^{d})$ , and suppose $\nabla U$ is $L$ -Lipschitz for $L > 0$ . Let $x_{⋆}$ be a global minimizer with $U (x_{⋆}) = 0$ . Suppose that for $ρ > 0$ , $α \in (1, 2]$ , and $M_{ρ} \geq 0$ , $⟨ \nabla U (x), x - x_{⋆} ⟩ \geq ρ ∥ x - x_{⋆} ∥^{α} if ∥ x - x_{⋆} ∥ \geq M_{ρ} .$ For $0 < γ < L^{- 1}$ , define $K_{γ} : = \max {M_{ρ}, {(\frac{2 d [- \log (1 - L γ)] / γ + 4}{ρ^{2}})}^{1 / (2 (α - 1))}}, V (x) : = e^{U (x) / 2},$ and let $P_{γ}$ be the ULA proposal kernel. Then $P_{γ} V (x) : = \int V (y) P_{γ} (x, d y) \leq e^{- γ} V (x) + e^{L K_{γ}^{2} / 4} (1 - L γ)^{- d / 2} 1_{{∥ x - x_{⋆} ∥ \leq K_{γ}}} .$

Facts & Assumptions

Given: AC; $d, L, U, x_{⋆}, ρ, α, M_{ρ}, γ$ as in the Statement; and the ULA update.

[A1]

$U$ is twice continuously differentiable, has $L$ -Lipschitz gradient, has the stated global minimizer and radial drift, and is normalized by $U (x_{⋆}) = 0$ . The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case

[F1]

At a differentiable local minimum in Euclidean space, the gradient is zero. Fermat's theorem: an interior differentiable local extremum has zero gradient

[F2]

The derivative of a composition with a differentiable path is given by the total-derivative chain rule, and a continuous derivative integrates to its endpoint difference. The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ Newton–Leibniz needs only continuity on $[a, b]$ , differentiability on $(a, b)$ , and a Riemann-integrable extension of the interior derivative

[F3]

$∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ . The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation

[F4]

The standard $d$ -dimensional normal has density $(2 π)^{- d / 2} e^{- ∥ z ∥^{2} / 2}$ , and the ULA proposal has the displayed Gaussian density. The Axiom of Choice Multivariate normal law, including singular covariance A positive-definite multivariate normal law has its Gaussian density The ULA Gaussian proposal is a probability kernel with a positive density Unadjusted Langevin algorithm (ULA) Lebesgue measurable sets, the family $L (R^{n})$ , and the restricted set function $λ_{n}$

[F5]

The Borel function $V = e^{U / 2}$ is measurable under countable choice; AC implies countable choice. Continuous functions on Euclidean spaces are Borel measurable AC supplies countable selections and prescribed serial paths The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F6]

A nonnegative integral is unchanged by the stated $C^{1}$ change of variables, and $\int_{R} e^{- z^{2}} d z = \sqrt{π}$ . A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions The Gaussian integral $\int_{- \infty}^{\infty} e^{- x^{2}} d x = \sqrt{π}$

[F7]

Finite-dimensional Lebesgue measure is the finite product of one-dimensional Lebesgue measures on Borel sets, and Tonelli factors nonnegative product-measurable integrals. On Borel subsets of R^{m+n}, the product lambda_m times lambda_n agrees with lambda_{m+n} Tonelli's theorem for nonnegative measurable functions on a sigma-finite product

[F8]

Under [A1], $U (x) \geq c ∥ x - x_{⋆} ∥^{α} - a_{α}$ , where $c = ρ / (α + 1) > 0$ and $a_{α}$ is finite. Radial gradient drift forces potential coercivity

[F9]

Under countable choice, $\int_{R} e^{- ∣ z ∣} d z < \infty$ . The real-line exponential tail is Lebesgue integrable

Proof

Given: The hypotheses in the Statement.

1.1A1F1F2F3

By [F1], $\nabla U (x_{⋆}) = 0$ . For $h = y - x$ , apply [F2] to $t \mapsto U (x + t h)$ and subtract $⟨ \nabla U (x), h ⟩$ from the endpoint difference. The remaining integral is $\int_{0}^{1} ⟨ \nabla U (x + t h) - \nabla U (x), h ⟩ d t \leq \int_{0}^{1} L t ∥ h ∥^{2} d t = \frac{L}{2} ∥ h ∥^{2},$ by [A1] and [F3]. Thus $U (y) \leq U (x) + ⟨ \nabla U (x), y - x ⟩ + L ∥ y - x ∥^{2} / 2$ for all $x, y$ .

1.2A1F3algebra

Put $r = ∥ x - x_{⋆} ∥$ . If $r \geq M_{ρ}$ and $r > 0$ , the radial condition and [F3] give $ρ r^{α} \leq ⟨ \nabla U (x), x - x_{⋆} ⟩ \leq ∥ \nabla U (x) ∥ r,$ so $∥ \nabla U (x) ∥ \geq ρ r^{α - 1}$ .

1.3F4F6F7

For $c < 1 / 2$ and $β \in R$ , completing the square and applying [F6] gives $\frac{1}{\sqrt{2 π}} \int_{R} e^{- (1 / 2 - c) z^{2} + β z} d z = (1 - 2 c)^{- 1 / 2} e^{β^{2} / (2 (1 - 2 c))} .$ For $b \in R^{d}$ , apply this with $β = b_{i}$ for each $i < d$ . The product density in [F4], Lebesgue product identification, and Tonelli's theorem [F7] therefore give, for $Z \sim N_{d} (0, I_{d})$ , $E e^{c ∥ Z ∥^{2} + ⟨ b, Z ⟩} = (1 - 2 c)^{- d / 2} \exp ⁣ (\frac{∥ b ∥^{2}}{2 (1 - 2 c)}) .$

1.4A1F3F5F6F7F8F9step 1.1algebra

Verify the target normalizer before using the proposal-kernel conclusion of [F4]. Put $r = ∥ x - x_{⋆} ∥$ , $c = ρ / (α + 1)$ , and $k = c / \sqrt{d} > 0$ . Since $r^{α} \geq r - 1$ for $r \geq 0$ and $\sum_{i < d} ∣ x_{i} - x_{⋆, i} ∣ \leq \sqrt{d} r$ by [F3], [F8] gives $e^{- U (x)} \leq e^{a_{α} + c} e^{- c r} \leq e^{a_{α} + c} \prod_{i < d} e^{- k ∣ x_{i} - x_{⋆, i} ∣} .$ Translation and scaling [F6], followed by [F7] and [F9], show that the right-hand side has finite integral. Step 1.1 also gives $U (x) \leq L r^{2} / 2$ , so [F6]–[F7] imply $\int_{R^{d}} e^{- U (x)} d x \geq \int_{R^{d}} e^{- L ∥ x - x_{⋆} ∥^{2} / 2} d x = (2 π / L)^{d / 2} > 0.$ Continuity supplies Borel measurability as in [F5]. Thus all target hypotheses of the ULA algorithm and its proposal-kernel lemma hold.

2.1A1F4F5step 1.1step 1.3step 1.4algebra

Write $g = \nabla U (x)$ and $Y = x - γ g + \sqrt{2 γ} Z$ , the ULA update. Step 1.1 and [F4] imply $\frac{E V (Y)}{V (x)} \leq E \exp ⁣ (\frac{1}{2} ⟨ g, - γ g + \sqrt{2 γ} Z ⟩ + \frac{L}{4} ∥ - γ g + \sqrt{2 γ} Z ∥^{2}) .$ In step 1.3 take $c = L γ / 2$ and $b = \sqrt{2 γ} (1 - L γ) g / 2$ . Since $1 - 2 c = 1 - L γ > 0$ , expanding the exponent and collecting its $∥ g ∥^{2}$ terms gives $P_{γ} V (x) \leq V (x) (1 - L γ)^{- d / 2} \exp ⁣ (- \frac{γ ∥ g ∥^{2}}{4}) .$

3.1A1step 1.2step 2.1algebra

If $r \geq K_{γ}$ , then step 1.2 and the definition of $K_{γ}$ give $\frac{d}{2} [- \log (1 - L γ)] - \frac{γ ∥ g ∥^{2}}{4} \leq \frac{d}{2} [- \log (1 - L γ)] - \frac{γ ρ^{2} K_{γ}^{2 (α - 1)}}{4} \leq - γ .$ Step 2.1 therefore yields $P_{γ} V (x) \leq e^{- γ} V (x)$ on this region.

4.1A1F1step 1.1step 2.1step 3.1algebra∎

If $r \leq K_{γ}$ , use step 1.1 with starting point $x_{⋆}$ and [F1] to obtain $U (x) \leq L r^{2} / 2$ , so $V (x) \leq e^{L K_{γ}^{2} / 4}$ . Bounding the exponential factor in step 2.1 by one gives $P_{γ} V (x) \leq e^{L K_{γ}^{2} / 4} (1 - L γ)^{- d / 2} .$ Together with step 3.1, this proves the asserted drift inequality on both regions.

AlgorithmAlgorithm: Literature-sourcedProof: Not applicablejudge pass (gpt-6-luna)Open item page →

Metropolis-adjusted Langevin algorithm (MALA)

Definition

Assume AC (The Axiom of Choice) and the target-potential hypotheses of Unadjusted Langevin algorithm (ULA). For a step size $γ > 0$ , use the ULA Gaussian proposal $Q_{γ} (x, d y) = q_{γ} (y ∣ x) d y, q_{γ} (y ∣ x) = (4 π γ)^{- d / 2} \exp ⁣ (- \frac{∥ y - x + γ \nabla U (x) ∥^{2}}{4 γ}),$ whose kernel properties are in The ULA Gaussian proposal is a probability kernel with a positive density. The target density is $p_{U} (x) = Z_{U}^{- 1} e^{- U (x)}$ . At state $x$ , propose $y$ from $Q_{γ} (x, \cdot)$ , set $α_{γ} (x, y) : = \min {1, \frac{p_{U} (y) q_{γ} (x ∣ y)}{p_{U} (x) q_{γ} (y ∣ x)}},$ and use the Metropolis–Hastings transition kernel $P_{γ} (x, A) = \int_{A} α_{γ} (x, y) Q_{γ} (x, d y) + (1 - \int α_{γ} (x, z) Q_{γ} (x, d z)) 1_{A} (x) .$ Thus every rejected proposal remains at the current state. The target and proposal densities are strictly positive, so the displayed ratio has no zero-denominator case. The ratio is the common-density accepted-flow version from The common-density Metropolis–Hastings ratio; this procedure is the general Metropolis–Hastings algorithm Metropolis–Hastings kernel specialized to the ULA proposal. The Euclidean Borel space is standard Borel in the sense of Standard Borel spaces, by Finite-dimensional Euclidean space with its Borel sigma-algebra is standard Borel, as required by that general Metropolis–Hastings procedure.

LemmaStatement: AI-adaptedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Finite-horizon Brownian change of drift and relative entropy

Statement

Assume AC. Let $T > 0$ , let $W$ be a standard $d$ -dimensional Brownian motion on a complete usual filtered probability space, with increments after $s$ independent of $F_{s}$ , and let $θ$ be predictable with $A_{T} : = \int_{0}^{T} ∣ θ_{s} ∣^{2} d s < \infty$ almost surely. Set $M_{t} = \int_{0}^{t} θ_{s} \cdot d W_{s}, A_{t} = \int_{0}^{t} ∣ θ_{s} ∣^{2} d s, Z_{t} = \exp (M_{t} - A_{t} / 2) .$ If $Z$ is a true martingale on $[0, T]$ , then $Q = Z_{T} P$ is equivalent to $P$ , its restriction to $F_{0}$ equals that of $P$ , and $B_{t} = W_{t} - \int_{0}^{t} θ_{s} d s$ is standard Brownian motion under $Q$ relative to the same filtration. If $A_{T}$ is bounded by a deterministic constant, $Z$ is a true martingale. If additionally $E_{P} A_{T} < \infty$ , then $D (P ∥ Q) = \frac{1}{2} E_{P} A_{T} .$

Here is a sufficient condition that permits unbounded $A_{T}$ . Fix a globally Lipschitz vector field $b : R^{d} \to R^{d}$ , a finite deterministic grid $0 = t_{0} < \dots < t_{N} = T$ , and $η (t) = t_{k}$ for $t_{k} \leq t < t_{k + 1}$ . Let each of the two path drifts $a, c$ be one of $0$ , $b (X_{t})$ , or $b (X_{η (t)})$ . Suppose $X_{t} = X_{0} + \int_{0}^{t} a_{s} (X) d s + \sqrt{2} W_{t}, E ∣ X_{0} ∣^{2} < \infty,$ where $X$ is adapted with continuous paths and the equation holds on one probability-one event for all $t \in [0, T]$ , while $X_{0}$ is $F_{0}$ -measurable. For $θ_{t} = (c_{t} (X) - a_{t} (X)) / \sqrt{2}$ , the exponential $Z$ above is a true martingale, and under $Q$ the process $X$ has the unique path law with initial law $L_{P} (X_{0})$ and drift $c$ and dispersion $\sqrt{2} I_{d}$ . Moreover $E_{P} A_{T} < \infty$ , so the entropy identity applies. Values of the piecewise coefficients at grid points do not affect any integral.

Finally, for any probability measures $R ≪ S$ and measurable statistic $F$ , relative entropy decreases under pushforward: $D (F_{#} R ∥ F_{#} S) \leq D (R ∥ S) .$ All equivalence and entropy assertions concern this finite horizon.

Facts & Assumptions

Given: AC and the filtered Brownian data, integrand, horizon and, for the sufficient condition, the vector field, grid, initial value and solution specified in the Statement.

[F1]

Localized predictable Brownian integrals exist; stopping commutes with integration; finite-energy integrals are continuous square-integrable mean-zero martingales and obey their isometry. The multidimensional Itô formula applies to Brownian Itô processes with locally integrable drift and locally square-integrable diffusion. In particular it applies to exponentials of $(M, A)$ and products of two such processes. Localized Ito integral Stopping an Ito integral The Ito integral process has a continuous martingale version Ito isometry and linearity in predictable L2 Multidimensional Ito formula for Brownian-driven processes

[F2]

A continuous vector local martingale starting at zero with covariation matrix $t I_{d}$ is Brownian relative to its filtration; its increments are independent of the earlier sigma-algebra. Vector Levy characterization

[F3]

Conditional expectations are characterized by event integrals and preserve order; the tower property holds. Conditional expectations of a fixed integrable terminal variable are UI; UI and convergence in probability imply $L^{1}$ convergence. Bounded optional sampling applies to the finite discrete-time martingale obtained by sampling a continuous martingale on a deterministic grid; the continuous-time extension is proved in step 1.2 below. Conditional expectation as an ae class Basic algebra and order properties of conditional expectation Tower property of conditional expectation Uniform integrability of conditional expectations of one variable Uniform integrability plus convergence in probability implies $L^{1}$ convergence Optional sampling for bounded stopping times

[F4]

Each Brownian coordinate satisfies $E \sup_{t \leq T} ∣ W_{t}^{i} ∣^{2} \leq 4 T$ . The globally Lipschitz diffusion has a unique strong solution with finite second path moment, constructed by Picard iteration. Doob maximal bound for the Ito integral Global strong well-posedness for Lipschitz Itô SDEs

[F5]

Continuous adapted processes are predictable; interval step processes with coefficients measurable at the left endpoint have predictable versions. Adapted continuous processes are progressively measurable Progressively measurable and predictable processes

[F6]

Fatou, dominated convergence, monotone convergence, Tonelli and Cauchy–Schwarz apply to the displayed integrable or nonnegative quantities. Radon–Nikodym densities are determined by their event integrals and change measure in integrals. Relative entropy is directional, and $x \log x \geq - 1 / e$ for $x \geq 0$ , with $0 \log 0 = 0$ . Fatou's lemma Dominated convergence Monotone convergence for the integral Tonelli's theorem for nonnegative measurable functions on a sigma-finite product Cauchy-Schwarz for random variables A sigma-finite signed measure that is absolutely continuous with respect to a sigma-finite positive measure has a unique almost-everywhere density Likelihood-ratio change of measure Kullback–Leibler divergence

[F7]

The Borel sigma-algebra of continuous path space is generated by coordinate evaluations. The finite-horizon version follows by restriction and constant extension, and the vector version by finite products of scalar path spaces. Thus independence on finite coordinate cylinders extends to independence on the full Borel path space. Borel sigma-algebra of continuous path space is generated by coordinates

[A1]

AC is inherited from the Brownian Itô, conditional-expectation, Levy, and SDE suppliers; the integer localizers and deterministic meshes below require no additional choice principle. The Axiom of Choice

Proof

1.1F1F3F5F6given

By [F1], applied to $f (m, a) = e^{m - a / 2}$ , $d Z_{t} = Z_{t} θ_{t} \cdot d W_{t}$ . Stopping at integer levels of $A$ , $∣ M ∣$ and $Z$ makes the integral finite-energy, so $Z$ is a positive continuous local martingale. All these stops tend beyond every time strictly less than $T$ , and stabilize at $T$ , because the paths and energy are bounded on $[0, T]$ . A nonnegative local martingale $N$ is a supermartingale: Fatou first gives integrability, and conditional Fatou gives $E [N_{t} ∣ F_{s}] \leq {lim inf}_{j} E [N_{t \land ρ_{j}} ∣ F_{s}] = {lim inf}_{j} N_{s \land ρ_{j}} = N_{s}$ for a localizing sequence. The conditional Fatou assertion follows by taking the increasing infima of the tails, using conditional order and monotone convergence; conditional monotone convergence follows by ordinary monotone convergence in the defining event integrals. Thus $E Z_{t} \leq 1$ , and the same assertion applies to the stopped exponential for every scalar multiple of $θ$ .

1.2F3F6

We record the continuous stopping consequence needed below. Let $V$ be a continuous true martingale on $[0, T]$ . For $ρ \leq T$ , round $ρ$ upward to stopping times $ρ_{j}$ on finite grids whose meshes tend to zero. Discrete bounded optional sampling in [F3] gives $V_{ρ_{j}} = E [V_{T} ∣ F_{ρ_{j}}]$ . This family is UI and converges almost surely to $V_{ρ}$ , hence in $L^{1}$ . If $A \in F_{ρ}$ , then $A \in F_{ρ_{j}}$ , as is checked by the defining stopped-event tests, so passage in those tests gives $V_{ρ} = E [V_{T} ∣ F_{ρ}]$ . In particular the family of all $V_{ρ}$ , $ρ \leq T$ , is UI. Stopping preserves true martingality: for fixed $s < t$ , choose the grids on $[0, t]$ to contain $s$ ; the finite-grid identity $E [1_{A} V_{t \land ρ_{j}}] = E [1_{A} V_{s \land ρ_{j}}]$ for $A \in F_{s}$ follows by telescoping grid increments with the left-endpoint measurable stopping indicators. Continuity and the preceding UI argument pass it to $V^{ρ}$ . Thus both continuous stopping facts used below have been derived from the discrete supplier.

1.3F3F6algebra

To prove data processing, write $r = d R / d S$ , and let $u$ be the density of $F_{#} R$ relative to $F_{#} S$ ; absolute continuity follows immediately by taking inverse images of null sets. Testing events in $σ (F)$ gives $u \circ F = E_{S} [r ∣ σ (F)]$ . If $D (R ∥ S) = \infty$ the desired inequality is immediate. Otherwise $ϕ (r)$ is integrable, where $ϕ (x) = x \log x$ on $x > 0$ and $ϕ (0) = 0$ , since its negative part is bounded by $1 / e$ . For positive rational $a$ , the tangent line $ℓ_{a} (x) = (1 + \log a) x - a$ lies below $ϕ (x)$ on $x \geq 0$ , by differentiating their difference, and $ϕ (x) = \sup_{a \in Q, a > 0} ℓ_{a} (x)$ , including $x = 0$ by $a ↓ 0$ . Conditional order gives $ℓ_{a} (E_{S} [r ∣ σ (F)]) \leq E_{S} [ϕ (r) ∣ σ (F)]$ . Intersect these countably many full events, take the supremum and then expectations. This yields $D (F_{#} R ∥ F_{#} S) \leq D (R ∥ S)$ , with left integrability following from the lower bound $- 1 / e$ and the displayed conditional upper bound.

2.1F1F3step 1.1algebra

If $A_{T} \leq n$ , then at every stopping time $ρ \leq T$ , $Z_{ρ}^{2} = \exp (2 M_{ρ} - 2 A_{ρ}) \exp (A_{ρ})$ , whence $E Z_{ρ}^{2} \leq e^{n}$ by step 1.1 applied to the stopped integrand $2 θ 1_{[0, ρ]}$ . Apply this bound to localizing stops for $Z$ . Their terminal values are UI, since their tail integrals are at most $e^{n} / R$ above level $R$ . They converge almost surely, and hence in $L^{1}$ , to $Z_{T}$ . The same holds at every earlier time, so passage in martingale event tests proves true martingality. In particular $Z_{t} = E [Z_{T} ∣ F_{t}]$ .

2.2F1F3F5F6step 1.1

Suppose $Z$ is true and put $Q = Z_{T} P$ . Since $0 < Z_{T} < \infty$ almost surely, these probabilities are equivalent. For $A \in F_{t}$ , $Q (A) = E_{P} [Z_{T} 1_{A}] = E_{P} [Z_{t} 1_{A}]$ by [F3], so $Q ∣ F_{t} = Z_{t} P ∣ F_{t}$ ; at $t = 0$ this is $Q ∣ F_{0} = P ∣ F_{0}$ . The measures have identical null sets, so completeness and the usual filtration conditions persist. Let $B = W - \int θ d s$ , whose drift has finite total variation by pathwise Cauchy–Schwarz. For a coordinate $i$ , let $σ_{r}$ be the first time $∣ B^{i} ∣$ reaches $r$ , capped at $T$ . These are stopping times because continuous adapted path suprema are measurable by their rational-time suprema. Product Itô gives $d (Z_{t} B_{t \land σ_{r}}^{i}) = Z_{t} 1_{[0, σ_{r}]} (t) d W_{t}^{i} + B_{t \land σ_{r}}^{i} Z_{t} θ_{t} \cdot d W_{t} .$ The finite-variation terms cancel against $d [Z, W^{i}]_{t} = Z_{t} θ_{t}^{i} d t$ . The right side is locally square integrable, so the product is a $P$ -local martingale.

3.1F3F6step 1.2step 2.2

For every $ρ \leq T$ , $∣ Z_{ρ} B_{ρ \land σ_{r}}^{i} ∣ \leq r Z_{ρ}$ . By step 1.2, $Z_{ρ} = E [Z_{T} ∣ F_{ρ}]$ , and these random variables form a UI family. Hence the localizing stopped products in step 2.2 are UI. Passing their event tests in $L^{1}$ makes $Z B^{i, σ_{r}}$ a true $P$ -martingale. For $A \in F_{s}$ , change of measure at times $s, t$ now gives $E_{Q} [1_{A} B_{t \land σ_{r}}^{i}] = E_{P} [1_{A} Z_{t} B_{t \land σ_{r}}^{i}] = E_{P} [1_{A} Z_{s} B_{s \land σ_{r}}^{i}] = E_{Q} [1_{A} B_{s \land σ_{r}}^{i}]$ . Thus $B^{i}$ is a $Q$ -local martingale. Its localizers increase to the horizon because continuous paths are bounded on each compact interval.

3.2F1F6step 2.2

If $E_{P} A_{T} < \infty$ , [F1] gives $E_{P} M_{T} = 0$ and $E_{P} M_{T}^{2} = E_{P} A_{T}$ . Since $d Q / d P = Z_{T} > 0$ , $d P / d Q = 1 / Z_{T}$ and $- \log Z_{T} = - M_{T} + A_{T} / 2$ is $P$ -integrable. The definition of relative entropy therefore gives $D (P ∥ Q) = E_{P} [- \log Z_{T}] = E_{P} A_{T} / 2$ . This establishes the asserted identity whenever $Z$ is true; it does not yet presume true martingality in the unbounded-energy sufficient condition.

4.1F1F2F6step 3.1

Subtracting a continuous finite-variation path preserves quadratic covariations: its squared-increment sum is at most its largest increment times its total variation, and its mixed-increment sum with $W^{i}$ is at most the largest Brownian increment times that variation. Both errors tend to zero pathwise along every vanishing-mesh partition. The Brownian covariation limits also hold in $Q$ -probability: $P (E_{n}) \to 0$ implies $Q (E_{n}) = E_{P} [Z_{T} 1_{E_{n}}] \to 0$ by integrability of $Z_{T}$ . Consequently $[B^{i}, B^{j}]_{t} = δ_{i j} t$ under $Q$ , and [F2] proves that $B$ is $Q$ -Brownian. The cited proof applies on $[0, T]$ because it uses only finite pairs of times. Equivalently, if the Brownian space is defined for all times, set $θ = 0$ after $T$ and keep $Z$ constant; the argument applies on every horizon and uses [F2] literally. A space defined only up to $T$ can first be extended by an independent Brownian continuation.

5.1F4F5step 2.1step 4.1given

Consider that sufficient condition. The three candidate drifts are predictable by [F5] and obey $∣ v_{t} (X) ∣ \leq K (1 + \sup_{u \leq t} ∣ X_{u} ∣)$ for some common $K$ . The base Brownian and Euler equations are explicit, and the diffusion equation is covered by [F4]. Their solutions have finite second path moment: for Euler this follows immediately from its finite recursion and the Brownian maximal bound. Pathwise boundedness gives $A_{T} < \infty$ . Set $τ_{n} = \inf {t : A_{t} \geq n} \land T$ and $Z_{t}^{n} = \exp (M_{t \land τ_{n}} - A_{t \land τ_{n}} / 2)$ . By step 2.1 it is a true martingale. Under $Q_{n} = Z_{T}^{n} P$ , step 4.1 gives Brownian motion $B^{n} = W - \int_{0}^{\cdot \land τ_{n}} θ_{s} d s$ , and substitution in the additive equation gives $d X_{t} = (1_{[0, τ_{n}]} (t) c_{t} (X) + 1_{(τ_{n}, T]} (t) a_{t} (X)) d t + \sqrt{2} d B_{t}^{n} .$ The initial law is unchanged and the switched drift obeys the same $K$ bound.

6.1F4F6step 5.1algebra

Uniformly in $n$ , $E_{Q_{n}} \sup_{t \leq T} ∣ X_{t} ∣^{2} \leq C_{T} : = (1 + 3 E ∣ X_{0} ∣^{2} + 24 d T) e^{6 K^{2} T^{2}} - 1.$ To establish finiteness without assuming it, first stop $X$ at its first exit from a ball of radius $R$ . If $∣ X_{0} ∣ \geq R$ the stop is zero and the stopped path is $X_{0}$ , which retains its integrable second moment. Before stopping, the drift bound uses the corresponding stopped path supremum. Cauchy–Schwarz, $∣ x + y + z ∣^{2} \leq 3 (∣ x ∣^{2} + ∣ y ∣^{2} + ∣ z ∣^{2})$ , and [F4] therefore bound its expected squared supremum $q_{n, R} (t)$ by $3 E ∣ X_{0} ∣^{2} + 24 d T + 6 K^{2} T \int_{0}^{t} (1 + q_{n, R} (s)) d s$ . These quantities are finite for fixed $R$ . Put $H (t) = 1 + 3 E ∣ X_{0} ∣^{2} + 24 d T + 6 K^{2} T \int_{0}^{t} (1 + q_{n, R} (s)) d s$ ; then $1 + q_{n, R} \leq H$ and $H^{'} \leq 6 K^{2} T H$ almost everywhere, so the integrating factor gives $q_{n, R} (T) \leq C_{T}$ . The ball stops tend to $T$ on every bounded continuous path; Fatou proves the displayed bound.

7.1F1F5F6step 4.1step 5.1step 6.1

For the integrands used here, elementary left-endpoint sums on deterministic partitions refining the Euler grid give the identity $M_{T \land τ_{n}} = \int_{0}^{T} 1_{[0, τ_{n}]} (s) θ_{s} \cdot d B_{s}^{n} + A_{T \land τ_{n}} .$ Here is a direct justification across the measure change. All path coefficients are continuous on each of finitely many open mesh intervals; multiplying by the stopping indicator adds at most one discontinuity. Stop further on bounded path and coefficient levels. Their left-endpoint staircases converge for almost every time on each path, are bounded, and hence converge in predictable $L^{2}$ under both equivalent measures by dominated convergence. Both Brownian integrals converge by their respective isometries, while the drift integral converges by Cauchy–Schwarz. For each elementary sum the identity follows by substituting $d W = d B^{n} + 1_{[0, τ_{n}]} θ d s$ . Equivalence identifies the probability limits; removing the extra increasing stops proves the displayed identity. The $Q_{n}$ -Brownian integral has mean zero since its energy is at most $n$ . As $∣ θ_{t} ∣^{2} \leq 4 K^{2} (1 + \sup_{u \leq t} ∣ X_{u} ∣^{2})$ , step 6.1 yields $E_{P} [Z_{T}^{n} \log Z_{T}^{n}] = \frac{1}{2} E_{Q_{n}} A_{T \land τ_{n}} \leq 2 K^{2} T (1 + C_{T}) .$

8.1F3F6step 1.1step 3.2step 5.1step 7.1

Since $z \log z \geq - 1 / e$ , step 7.1 gives, for $R > 1$ , $E_{P} [Z_{T}^{n} 1_{{Z_{T}^{n} > R}}] \leq (2 K^{2} T (1 + C_{T}) + 1 / e) / \log R$ . Thus $(Z_{T}^{n})$ is UI. On every path with $A_{T} < \infty$ , $τ_{n} = T$ for all sufficiently large $n$ , so $Z_{T}^{n} \to Z_{T}$ almost surely and in $L^{1}$ . In particular $E Z_{T} = 1$ . Step 1.1 gives a supermartingale with initial mean one and terminal mean one, so every intermediate mean equals one; the nonnegative conditional supermartingale deficit has mean zero, proving true martingality of $Z$ . Apply step 4.1 without stops to get the target equation. The base finite second path moment in step 5.1 and the coefficient bound imply $E_{P} A_{T} < \infty$ , so step 3.2 applies.

9.1F2F4F6F7step 4.1step 8.1

The target path law is uniquely determined, as follows. For target zero it is $X_{0} + \sqrt{2} B$ ; for Euler target it is the finite mesh recursion, a Borel function of $(X_{0}, B)$ . For target $b (X_{t})$ , use the Picard recursion $X^{m + 1} = X_{0} + \int_{0}^{\cdot} b (X_{s}^{m}) d s + \sqrt{2} B$ starting with $X_{t}^{0} = X_{0}$ . Its iterates are Borel functions on $R^{d} \times C ([0, T], R^{d})$ : integration of a continuous path is a uniform limit of its deterministic Riemann sums. For every fixed input $(x, w)$ , put $C = T ∣ b (x) ∣ + \sqrt{2} \sup ∣ w ∣$ . The first iterate difference has supremum at most $C$ , and induction in the Lipschitz integral inequality bounds the $m$ th difference up to time $t$ by $C L^{m - 1} t^{m - 1} / (m - 1)!$ . Their uniformly summable majorant proves uniform convergence for every continuous input. The limit is consequently a Borel function of that input. The deterministic Lipschitz integral inequality makes any solution with the same input agree with this limit: if $h (t)$ is the path supremum of their difference, $h (t) \leq L \int_{0}^{t} h (s) d s$ , and the integrating factor forces $h = 0$ . Under the changed measure, $B$ has Wiener law and is independent of $X_{0}$ , because its future increments are independent of $F_{0}$ ; independence of the entire path follows first on finite-time cylinder events and then on their generated Borel sigma-algebra by [F7]. Thus $(X_{0}, B)$ has the same product law on every such space, and its Borel solution map proves the asserted uniqueness in law.

10.1A1step 2.1step 4.1step 3.2step 8.1step 9.1step 1.3∎

If $θ = 0$ , then $Z = 1$ , $Q = P$ and the entropy is zero. A zero mesh interval is omitted; the stated grid has strictly positive lengths, and no lower bound on those lengths is needed. At time zero all integrals vanish and the initial law is preserved. The excluded horizon $T = 0$ has the same trivial conclusions. AC has exactly the inherited uses in [A1].

Source notes

Van der Vaart, Theorem 6.9 and Theorem 6.14 with their proofs, supply the exponential/product route and its source provenance. The bounded-energy argument above replaces the source's general Novikov/time-change argument. The entropy localization is supplied explicitly for the required linear-growth path drifts. No general Girsanov, Novikov, semimartingale measure-invariance, or weak-uniqueness theorem is taken as an unproved premise.

TheoremStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

MALA preserves its target probability exactly

Statement

Assume AC and the hypotheses of Metropolis-adjusted Langevin algorithm (MALA), with $d \geq 1$ , $U \in C^{1} (R^{d})$ , $0 < Z_{U} = \int e^{- U (x)} d x < \infty$ , and $γ > 0$ . The MALA transition kernel $P_{γ}$ is reversible with respect to $π_{U} (d x) = Z_{U}^{- 1} e^{- U (x)} d x,$ and therefore $π_{U}$ is invariant. This conclusion asserts neither irreducibility nor uniqueness, convergence, a rate, nor a central limit theorem.

Facts & Assumptions

Given: AC; the target and proposal specified by Metropolis-adjusted Langevin algorithm (MALA).

[A1]

AC supplies the accepted-flow Radon–Nikodym version in the general Metropolis–Hastings construction. The Axiom of Choice Metropolis–Hastings kernel

[F1]

MALA uses the ULA Gaussian proposal and explicit common-density acceptance ratio, with rejection mass placed at $x$ . Metropolis-adjusted Langevin algorithm (MALA)

[F2]

The proposal $Q_{γ}$ is a probability kernel with finite, positive, jointly measurable density $q_{γ} (y ∣ x)$ . The ULA Gaussian proposal is a probability kernel with a positive density

[F3]

The target density $p_{U} (x) = Z_{U}^{- 1} e^{- U (x)}$ is finite, positive, Borel measurable, and integrates to one, hence defines a probability measure in the sense of Probability measures and probability spaces. Unadjusted Langevin algorithm (ULA) Probability measures and probability spaces

[F4]

For a target density $p$ and jointly measurable proposal density $q$ against a common sigma-finite measure, the pointwise minimum ratio is a version of the accepted-flow Radon–Nikodym derivative. The common-density Metropolis–Hastings ratio

[F5]

A Metropolis–Hastings kernel built from a target probability and proposal probability kernel is reversible and leaves the target invariant. Metropolis–Hastings flow is reversible and invariant

[F6]

Lebesgue measure on $R^{d}$ is sigma-finite in the sense of Finite, sigma-finite, and semifinite measures. Lebesgue measurable sets, the family $L (R^{n})$ , and the restricted set function $λ_{n}$ Finite, sigma-finite, and semifinite measures Lebesgue measure is sigma-finite, and every metrically bounded subset of $R^{n}$ has finite outer measure

[F7]

The Euclidean Borel space, with Borel sigma-algebra The Borel sigma-algebra of a topological space, is standard Borel in the sense of Standard Borel spaces, as required by the general Metropolis–Hastings suppliers. Finite-dimensional Euclidean space with its Borel sigma-algebra is standard Borel Standard Borel spaces The Borel sigma-algebra of a topological space

Proof

Given: The hypotheses in the Statement.

1.1A1F1F2F3F4F6F7

By [F3], $π_{U}$ is a probability with positive finite density $p_{U}$ . By [F2], $Q_{γ}$ is a probability kernel with jointly measurable finite density $q_{γ}$ relative to the sigma-finite Lebesgue measure [F6]. The state-space hypothesis [F7] is also met. Thus the common-density ratio result [F4] applies at every pair $(x, y)$ with $p_{U} (x) q_{γ} (y ∣ x) > 0$ , which holds because both factors are strictly positive. It identifies the MALA acceptance function in [F1] with a version of the accepted-flow derivative; the chosen pointwise version is measurable and belongs to $[0, 1]$ .

2.1F1F5step 1.1∎

Step 1.1 and [F1] show that $P_{γ}$ is precisely the Metropolis–Hastings transition built from target $π_{U}$ , proposal $Q_{γ}$ , and an accepted-flow derivative version, including its rejected self-transition. Applying [F5] gives reversibility and invariance of $π_{U}$ . No ergodicity or convergence premise is among the hypotheses, so no such conclusion follows here.

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Pathwise relative-entropy error of the Langevin Euler interpolation

Statement

Assume AC. Let $d \geq 1$ , let $U \in C^{1} (R^{d})$ have globally $L$ -Lipschitz gradient $g = \nabla U$ , and let $0 = t_{0} < \dots < t_{N} = T$ , $γ_{k + 1} = t_{k + 1} - t_{k}$ , $η (t) = t_{k}$ on $[t_{k}, t_{k + 1})$ . On a complete usual filtered Brownian space, start with an $F_{0}$ -measurable $X_{0}$ satisfying $E ∣ X_{0} ∣^{2} < \infty$ and interpolate the Euler recursion by $X_{t} = X_{t_{k}} - (t - t_{k}) g (X_{t_{k}}) + \sqrt{2} (W_{t} - W_{t_{k}}), t_{k} \leq t \leq t_{k + 1} .$ Let $E$ be its law on $C ([0, T], R^{d})$ and let $D$ be the path law of $d Y_{t} = - g (Y_{t}) d t + \sqrt{2} d B_{t}$ with the same initial law. Then $E$ and $D$ are equivalent and $D (E ∥ D) = \frac{1}{4} E \int_{0}^{T} ∣ g (X_{t}) - g (X_{η (t)}) ∣^{2} d t \leq \frac{L^{2}}{4} \sum_{k = 0}^{N - 1} {\frac{γ_{k + 1}^{3}}{3} E ∣ g (X_{t_{k}}) ∣^{2} + d γ_{k + 1}^{2}} .$ In particular the same upper bound holds for $D (L (X_{T}) ∥ L (Y_{T}))$ . The expectations in the bound are under the Euler law; its gradient moments are finite under the stated hypotheses. No convexity or stationarity is required, and the direction of the relative entropy is Euler relative to diffusion.

Facts & Assumptions

Given: AC, the potential, Lipschitz constant, finite mesh, square-integrable initial value, Brownian filtration, interpolation and path laws above.

[F1]

The finite-horizon change-of-drift lemma applies to drift choices $b (X_{t})$ and $b (X_{η (t)})$ for globally Lipschitz $b$ , with the same square-integrable initial law. Its exponential is a true density, identifies the target path law, and satisfies the base-measure entropy identity and measurable-statistic data processing. Finite-horizon Brownian change of drift and relative entropy

[F2]

The Langevin diffusion is strongly well-posed and has finite second path moment. Each Brownian coordinate has squared maximal expectation at most $4 T$ ; finite-energy vector Brownian integrals obey their isometry. Global strong well-posedness for Lipschitz Itô SDEs Doob maximal bound for the Ito integral Ito isometry and linearity in predictable L2

[F3]

Brownian increments after $t_{k}$ are independent of $F_{t_{k}}$ , centered and have covariance $(t - t_{k}) I_{d}$ . $d$ -dimensional Brownian motion

[F4]

Relative entropy is directional and uses the Radon–Nikodym density; change of measure follows by integrating that density. Dominated convergence, Tonelli and Cauchy–Schwarz justify the integrable operations below. Kullback–Leibler divergence Likelihood-ratio change of measure Dominated convergence Tonelli's theorem for nonnegative measurable functions on a sigma-finite product Cauchy-Schwarz for random variables

[A1]

AC is inherited from the Brownian, Itô, SDE, change-of-measure and relative-entropy suppliers. The Axiom of Choice

Proof

1.1F1F2F3given

The finite Euler recursion and $∣ g (x) ∣ \leq ∣ g (0) ∣ + L ∣ x ∣$ give $E ∣ X_{t_{k}} ∣^{2} < \infty$ by induction, using the finite Brownian second moments. Its interpolation has $E \sup_{t \leq T} ∣ X_{t} ∣^{2} < \infty$ : on each of the finitely many intervals its supremum is bounded by $∣ X_{t_{k}} ∣ + γ_{k + 1} ∣ g (X_{t_{k}}) ∣ + \sqrt{2} \sup_{t_{k} \leq t \leq t_{k + 1}} ∣ W_{t} - W_{t_{k}} ∣$ , whose square is integrable by [F2]. Thus every stated Euler gradient moment is finite. The diffusion law exists by [F2]. Apply [F1] on the Euler probability space with $b = - g$ , $a_{t} = - g (X_{η (t)})$ , $c_{t} = - g (X_{t})$ and $θ_{t} = \frac{g (X_{η (t)}) - g (X_{t})}{\sqrt{2}}, Z_{T} = \exp (\int_{0}^{T} θ_{t} \cdot d W_{t} - \frac{1}{2} \int_{0}^{T} ∣ θ_{t} ∣^{2} d t) .$ It is a true positive density, and under $Q = Z_{T} P$ , $X$ has diffusion path law $D$ .

2.1F1F3F4step 1.1algebra

For $h = t - t_{k}$ , the increment representation and [F3] imply $E ∣ X_{t} - X_{t_{k}} ∣^{2} = h^{2} E ∣ g (X_{t_{k}}) ∣^{2} + 2 d h .$ The mixed term vanishes because $g (X_{t_{k}})$ is $F_{t_{k}}$ -measurable and square integrable, while $W_{t} - W_{t_{k}}$ is centered and independent of that sigma-algebra. Lipschitz continuity and Tonelli now give $E \int_{0}^{T} ∣ θ_{t} ∣^{2} d t \leq \frac{L^{2}}{2} \sum_{k = 0}^{N - 1} {\frac{γ_{k + 1}^{3}}{3} E ∣ g (X_{t_{k}}) ∣^{2} + d γ_{k + 1}^{2}} < \infty .$ Indeed integrate $h^{2}$ and $2 d h$ on $[0, γ_{k + 1}]$ . The entropy identity of [F1] yields $D (P ∥ Q) = \frac{1}{2} E \int ∣ θ_{t} ∣^{2} d t$ .

3.1F2F4step 1.1step 2.1construct

The exact path-law identity, rather than just its upper bound, follows because $Z_{T}$ is a measurable function of the Euler path. In fact $W_{t} = \frac{X_{t} - X_{0} + \int_{0}^{t} g (X_{η (s)}) d s}{\sqrt{2}}$ is a Borel functional of that continuous path. On deterministic refining partitions containing all mesh points, the left-endpoint step approximations to $θ$ converge at almost every time on every continuous path: on an open Euler interval $θ$ is continuous and at its left endpoint it equals zero. Their squared errors are bounded by a fixed constant times $1 + \sup_{t \leq T} ∣ X_{t} ∣^{2}$ , integrable by step 1.1. Dominated convergence and the Itô isometry therefore make the corresponding stochastic sums converge to $\int θ d W$ in $L^{2} (P)$ . Each sum is a Borel path functional, since both $θ$ and the recovered Brownian increments are such functionals. Choose a subsequence whose squared $L^{2}$ errors are at most $2^{- 3 j}$ ; Markov's inequality bounds the probability of error exceeding $2^{- j}$ by $2^{- j}$ , so the union bound on the tails shows almost sure convergence. Define its Borel limit arbitrarily on the Borel nonconvergence set. Also $\int ∣ θ ∣^{2} d t$ is a Borel path functional, by ordinary interval Riemann approximation. Hence $Z_{T} = z (X)$ almost surely for some positive Borel function $z$ (reset its value to one where the constructed exponential is not finite and positive).

4.1F2F4step 2.1step 3.1algebra

For every Borel path set $C$ , $D (C) = Q (X \in C) = E_{P} [Z_{T} 1_{{X \in C}}] = \int_{C} z d E$ . Since $z > 0$ almost everywhere, the two path laws are equivalent and $d D / d E = z$ . The logarithm is integrable under $E$ because its pullback equals $\int θ d W - \frac{1}{2} \int ∣ θ ∣^{2} d t$ , with the first term square integrable and the second integrable by step 2.1. Thus $D (E ∥ D) = E_{P} [- \log Z_{T}] = \frac{1}{2} E \int_{0}^{T} ∣ θ_{t} ∣^{2} d t = \frac{1}{4} E \int_{0}^{T} ∣ g (X_{t}) - g (X_{η (t)}) ∣^{2} d t .$ Combining with step 2.1 proves the asserted upper bound.

5.1A1F1step 4.1∎

Terminal evaluation is a measurable statistic on continuous path space. Data processing in [F1] therefore gives $D (L (X_{T}) ∥ L (Y_{T})) \leq D (E ∥ D)$ . If $L = 0$ , $g$ is constant, Euler interpolation already solves the diffusion and both entropies and the displayed bound are zero. At $t = 0$ the path laws have the same initial law; the trivial zero-horizon case likewise has entropy zero. All positive step lengths, including a single interval, were covered without a minimum-step restriction. AC has exactly the inherited uses stated in [A1].

Source notes

Durmus–Moulines v3, equations (12)–(15), motivates the interpolation bound. Its Euler expectations give the directional entropy $D (E ∥ D)$ used here. The Brownian exponential in step 1.1 fixes the density sign without relying on a path-coordinate sign convention in a source formula. The change-of-measure and entropy arguments are supplied by the proved local supplier, rather than by an external Girsanov citation.

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Langevin diffusion has a reversible density and compact minorization

Statement

Assume AC. Let $d \geq 1$ , $U \in C^{2} (R^{d})$ , $\nabla U$ be $L$ -Lipschitz, and let $x_{⋆}$ be a global minimizer with $U (x_{⋆}) = 0$ . The solutions of $d X_{t} = - \nabla U (X_{t}) d t + \sqrt{2} d B_{t}, X_{0} = x,$ define a Borel Markov semigroup $P_{t}$ . For every $t > 0$ it has a positive jointly Borel density $p_{t} (x, y)$ relative to Lebesgue measure, and $e^{- U (x)} p_{t} (x, y) = e^{- U (y)} p_{t} (y, x) .$ If $Z = \int e^{- U} d x \in (0, \infty)$ , then $Z^{- 1} e^{- U (x)} d x$ is invariant. For every $R > 0$ and $t > 0$ , there are $ϵ > 0$ and a probability $ν$ such that $P_{t} (x, A) \geq ϵ ν (A)$ for every Borel $A$ and $∥ x - x_{⋆} ∥ \leq R$ .

Facts & Assumptions

Given: AC and the potential, minimizer, and Brownian equation in the Statement.

[A1]

AC is inherited through Brownian, Gaussian, Ito and change-of-measure constructions. The Axiom of Choice

[F1]

Globally Lipschitz coefficients with linear growth have a unique strong solution from each square-integrable initial state on every finite horizon. Global strong well-posedness for Lipschitz Itô SDEs

[F2]

A true stochastic exponential changes a Brownian drift by its integrand. Finite-horizon Brownian change of drift and relative entropy

[F3]

Multidimensional Ito's formula applies to $U (x + \sqrt{2} B_{t})$ and to its exponential compositions. Multidimensional Ito formula for Brownian-driven processes

[F4]

Brownian finite-dimensional laws are Gaussian with covariance $\min (s, u) I_{d}$ , independent future increments, and continuous paths; uncorrelated jointly Gaussian finite blocks are independent. $d$ -dimensional Brownian motion Uncorrelated jointly Gaussian blocks are independent

[F5]

Continuous-path Borel sets are generated by rational evaluations; equality on a generating pi-system extends by the pi-lambda theorem. The same assertion on a finite interval and in finitely many coordinates follows by restriction, constant extension, and finite products. Borel sigma-algebra of continuous path space is generated by coordinates Dynkin's pi-lambda theorem

[F6]

The Gaussian increment $\sqrt{2} B_{t}$ has density $h_{t} (z) = (4 π t)^{- d / 2} e^{- ∥ z ∥^{2} / (4 t)}$ . Each coordinate Brownian path obeys $E \sup_{s \leq t} ∣ B_{s}^{i} ∣^{2} \leq 4 t$ . A positive-definite multivariate normal law has its Gaussian density Doob maximal bound for the Ito integral

[F7]

Tonelli interchanges nonnegative integrals, and dominated convergence applies under a fixed integrable majorant. Tonelli's theorem for nonnegative measurable functions on a sigma-finite product Dominated convergence

[F8]

A differentiable global minimizer has zero gradient. Fermat's theorem: an interior differentiable local extremum has zero gradient

Proof

Given: The hypotheses in the Statement.

1.1A1F1F4F8construct

Put $g = \nabla U$ . By [F8], $g (x_{⋆}) = 0$ , so $∥ g (x) ∥ \leq L ∥ x - x_{⋆} ∥$ ; the equation meets [F1]. Its additive-noise integral equation is $X_{s} = x - \int_{0}^{s} g (X_{u}) d u + \sqrt{2} B_{s}$ . For each continuous input path the deterministic Picard iteration converges uniformly on finite intervals: successive differences are bounded by a fixed finite constant times $L^{n} s^{n} / n!$ , by repeated integration. The limit is the unique solution; subtraction and the same iterative integral bound give $\sup_{s \leq t} ∥ X_{s}^{x, w} - X_{s}^{z, v} ∥ \leq e^{L t} (∥ x - z ∥ + \sqrt{2} \sup_{s \leq t} ∥ w_{s} - v_{s} ∥)$ . Thus the solution is a continuous function of its initial point and driving path. Its flow composition identity follows from uniqueness of the integral equation. Independent Brownian future increments [F4] now give the Markov and semigroup identities; integration of its Borel solution map against the Brownian law gives a Borel transition kernel. Solutions on integer horizons agree by uniqueness and hence define all $t \geq 0$ .

1.2F4F5F6construct

Define $β_{s} = \sqrt{2} (B_{s} - s B_{t} / t)$ for $0 \leq s \leq t$ . For any finite list of times its coordinates together with $B_{t}$ are jointly Gaussian and $Cov (B_{s} - s B_{t} / t, B_{t}) = 0$ . By [F4] that finite bridge block is independent of $B_{t}$ . Apply [F5] to rational-time cylinders to obtain independence of the entire bridge path and $B_{t}$ . Thus Brownian paths disintegrate explicitly as $Y_{s} = x + (s / t) (y - x) + β_{s}$ when their endpoint has value $y$ , with endpoint density $h_{t} (y - x)$ and bridge law independent of $y$ . This identity holds first for cylinder functions by independence and the Gaussian endpoint density, then for all bounded Borel path functions by [F5], and for nonnegative ones by monotone approximation. The bridge has covariance $2 (\min (s, u) - s u / t) I_{d}$ , which is unchanged when $(s, u)$ is replaced by $(t - s, t - u)$ ; its mean is zero. Gaussian finite-dimensional laws and [F5] therefore also give equality in law of $β_{s}$ and $β_{t - s}$ .

2.1F2F3F7step 1.1algebra

Under Brownian law put $Y_{s} = x + \sqrt{2} B_{s}$ . The stochastic exponential with integrand $- g (Y_{s}) / \sqrt{2}$ is, by [F3], $Z_{s} = \exp ⁣ (\frac{U (x) - U (Y_{s})}{2} + \int_{0}^{s} {\frac{Δ U (Y_{u})}{2} - \frac{∥ g (Y_{u}) ∥^{2}}{4}} d u) .$ Lipschitz continuity gives $∣ \partial_{i i} U ∣ \leq L$ by taking coordinate difference quotients. Since $U \geq 0$ , for $s \leq t$ this positive local martingale is bounded above by $e^{U (x) / 2 + t d L / 2}$ . Stop at increasing energy/local-martingale levels. Dominated convergence [F7] in the stopped martingale identities makes $Z$ a true martingale of mean one. By [F2], under $Z_{t}$ times Brownian law the process $B_{s} + \int_{0}^{s} g (Y_{u}) / \sqrt{2} d u$ is Brownian, so $Y$ solves the Langevin equation. The explicit unique solution map of step 1.1 identifies its law with $P_{t} (x, \cdot)$ .

3.1F5F7step 2.1step 1.2algebra

Put $H (z) = Δ U (z) / 2 - ∥ g (z) ∥^{2} / 4$ and $b_{s}^{x, y} = x + (s / t) (y - x) + β_{s}$ . Combining steps 2.1 and 1.2 gives the density $p_{t} (x, y) = h_{t} (y - x) e^{(U (x) - U (y)) / 2} E \exp ⁣ (\int_{0}^{t} H (b_{s}^{x, y}) d s) .$ The integral is a continuous function of $(x, y, β)$ in the uniform path topology, since $H$ is continuous and uniformly continuous on each compact set containing the convergent paths. Its exponential is strictly positive and bounded above by $e^{t d L / 2}$ . Consequently its expectation is positive, finite, and continuous in $(x, y)$ by dominated convergence; the density is jointly Borel. Its integral in $y$ is one by step 2.1. Bridge reversal from step 1.2 interchanges $x, y$ without changing the integral of $H$ , and $h_{t} (y - x) = h_{t} (x - y)$ . Multiplication by $e^{- U (x)}$ proves the asserted symmetry. If $Z < \infty$ , Tonelli and the mass-one identity give $\int Z^{- 1} e^{- U (x)} p_{t} (x, y) d x = Z^{- 1} e^{- U (y)} \int p_{t} (y, x) d x = Z^{- 1} e^{- U (y)},$ proving invariance.

4.1F6F7step 1.1step 3.1algebra∎

By [F6], $E \sup_{s \leq t} ∥ B_{s} ∥^{2} \leq 4 d t$ , and $\sup ∥ β_{s} ∥^{2} \leq 8 \sup ∥ B_{s} ∥^{2}$ . With $a_{0} = \sqrt{64 d t}$ , Markov's inequality (integrate $1_{{Y > a_{0}^{2}}} \leq Y / a_{0}^{2}$ for $Y = \sup ∥ β ∥^{2}$ ) gives $\Pr (\sup ∥ β ∥ \leq a_{0}) \geq 1 / 2$ . If both endpoints lie in the ball $D = {∥ z - x_{⋆} ∥ \leq R}$ , on this event the bridge lies in the radius $R + a_{0}$ ball. Therefore $H (b_{s}^{x, y}) \geq - d L / 2 - L^{2} (R + a_{0})^{2} / 4$ and the bridge expectation in step 3.1 is at least one half times the exponential of $- t [d L / 2 + L^{2} (R + a_{0})^{2} / 4]$ . The heat factor is at least $(4 π t)^{- d / 2} e^{- R^{2} / t}$ on $D^{2}$ , and $e^{(U (x) - U (y)) / 2} \geq e^{- \sup_{D} U / 2} > 0$ . Thus a constant $c > 0$ bounds $p_{t}$ below on $D^{2}$ . Let $ν$ be normalized Lebesgue measure on $D$ and $ϵ = \min {c Leb (D), 1 / 2} > 0$ . Integrating the density over $A \cap D$ proves the stated minorization. The radius is positive, so the normalizing volume is finite and nonzero.

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ULA invariant laws have uniform exponential moments on the needed step ranges

Statement

Assume AC and the hypotheses of ULA's Gaussian kernel satisfies an exponential drift bound. Let $R_{γ}$ be the ULA kernel, $V = e^{U / 2}$ , $W = e^{U / 4}$ , and $γ_{0} = (2 L)^{- 1}$ . For every $0 < γ < L^{- 1}$ there is a unique invariant probability $π_{γ}$ , and $π_{γ} V < \infty$ . There are finite constants $M_{V}, M_{W}, M_{g}$ , independent of $γ$ , such that $\sup_{0 < γ \leq γ_{0}} π_{γ} V \leq M_{V}, \sup_{0 < γ < L^{- 1}} π_{γ} W \leq M_{W}, ∥ \nabla U (x) ∥^{2} + ∥ x - x_{⋆} ∥^{2} \leq M_{g} V (x) .$

Facts & Assumptions

Given: AC, the stated potential and radial constants, and $0 < γ < L^{- 1}$ .

[A1]

AC is inherited through the Gaussian and weighted Hahn/Jordan interfaces. The Axiom of Choice

[F1]

The ULA kernel has a positive continuous Gaussian density, with proposal $x - γ \nabla U (x) + \sqrt{2 γ} Z$ . Unadjusted Langevin algorithm (ULA) The ULA Gaussian proposal is a probability kernel with a positive density

[F2]

The fixed-step Gaussian kernel satisfies $R_{γ} V \leq e^{- γ} V + b_{γ} 1_{D_{γ}}$ for a finite constant and a compact ball. ULA's Gaussian kernel satisfies an exponential drift bound

[F3]

Radial drift gives $U (x) \geq c ∥ x - x_{⋆} ∥^{α} - C$ for some $c > 0$ ; hence V-sublevel sets are contained in compact balls. Radial gradient drift forces potential coercivity

[F4]

Global geometric drift and minorization on a sufficiently large weight-sublevel set give invariant existence, a finite weight moment, and uniqueness among all invariant probabilities. Global drift and one-step minorization give weighted contraction and a unique invariant law

[F5]

An exponential with positive exponent dominates every fixed nonnegative integer power at infinity. The exponential dominates every fixed nonnegative integer power at $+ \infty$

[F6]

Fermat gives a zero gradient at the minimizer. The chain rule and the fundamental theorem integrate the derivative along a segment, and Euclidean Cauchy–Schwarz bounds its inner products. Fermat's theorem: an interior differentiable local extremum has zero gradient The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ Newton–Leibniz needs only continuity on $[a, b]$ , differentiability on $(a, b)$ , and a Riemann-integrable extension of the interior derivative Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation

[F7]

Standard Gaussian density, the scalar Gaussian integral, change of variables, Lebesgue product identification and Tonelli evaluate nonnegative finite-dimensional Gaussian integrals. A positive-definite multivariate normal law has its Gaussian density The Gaussian integral $\int_{- \infty}^{\infty} e^{- x^{2}} d x = \sqrt{π}$ A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions On Borel subsets of R^{m+n}, the product lambda_m times lambda_n agrees with lambda_{m+n} Tonelli's theorem for nonnegative measurable functions on a sigma-finite product

Proof

Given: The hypotheses in the Statement.

1.1A1F1F2F3

Fix $γ$ . Dropping the indicator in [F2] gives a global drift $R_{γ} V \leq e^{- γ} V + b_{γ}$ with finite $b_{γ}$ . Each sufficiently large V-sublevel is contained in a compact ball by [F3]. On the product of that starting ball and any fixed positive-radius target ball, the continuous positive density [F1] has a positive minimum: explicitly its Gaussian exponent is bounded because both balls and the continuous gradient are bounded. Restriction to the target ball and normalization of its positive finite volume give a common minorization. These are precisely the drift and minorization hypotheses for the weight V.

1.2F6givenalgebra

By [F6], $g (x_{⋆}) = 0$ , and therefore Lipschitz continuity gives $∥ g (x) ∥ \leq L ∥ x - x_{⋆} ∥$ . If $r = ∥ x - x_{⋆} ∥$ is positive and at least the radial radius, Cauchy–Schwarz and the radial hypothesis give $∥ g (x) ∥ r \geq ⟨ g (x), x - x_{⋆} ⟩ \geq ρ r^{α}$ , so $∥ g (x) ∥ \geq ρ r^{α - 1}$ . For $h = y - x$ , integrate the derivative along $x + t h$ and subtract $⟨ g (x), h ⟩$ to obtain $U (y) - U (x) - ⟨ g (x), h ⟩ = \int_{0}^{1} ⟨ g (x + t h) - g (x), h ⟩ d t \leq \frac{L}{2} ∥ h ∥^{2} .$

1.3F7algebra

Completing the square, [F7] gives for $c < 1 / 2$ and scalar v the identity $(2 π)^{- 1 / 2} \int_{R} e^{- (1 / 2 - c) z^{2} + v z} d z = (1 - 2 c)^{- 1 / 2} e^{v^{2} / (2 (1 - 2 c))} .$ Finite multiplication and Tonelli therefore give $E e^{c ∥ Z ∥^{2} + ⟨ v, Z ⟩} = (1 - 2 c)^{- d / 2} e^{∥ v ∥^{2} / (2 (1 - 2 c))}$ for $Z \sim N_{d} (0, I_{d})$ .

2.1F4step 1.1

Apply [F4] to the fixed-step kernel and weight V verified in step 1.1. It supplies a unique invariant probability $π_{γ}$ with finite V moment.

2.2F1step 1.2step 1.3algebra

Put $g = \nabla U (x)$ and $H_{s} = e^{s U}$ for $0 < s < 1$ with $2 s L γ < 1$ . Apply step 1.2 to $Y = x - γ g + \sqrt{2 γ} Z$ , multiply by s, and exponentiate. Step 1.3, with quadratic coefficient $s L γ$ and linear coefficient $s \sqrt{2 γ} (1 - L γ) g$ , gives $\frac{R_{γ} H_{s} (x)}{H_{s} (x)} \leq (1 - 2 s L γ)^{- d / 2} \exp ⁣ (- \frac{s γ (1 - s - L γ / 2)}{1 - 2 s L γ} ∥ g ∥^{2}) .$ The coefficient follows by combining $s γ (- 1 + L γ / 2)$ with $s^{2} γ (1 - L γ)^{2} / (1 - 2 s L γ)$ over the common denominator. All Gaussian integrals are finite under $2 s L γ < 1$ .

3.1step 2.1step 1.2step 2.2algebra

For $s = 1 / 2$ , step 2.2 reduces to $R_{γ} V \leq V \exp ⁣ (\frac{d}{2} [- \log (1 - L γ)] - \frac{γ}{4} ∥ g ∥^{2}) .$ If $γ \leq γ_{0}$ , then $- \log (1 - L γ) \leq 2 L γ$ , since $- \log (1 - z) = \int_{0}^{z} (1 - u)^{- 1} d u \leq 2 z$ for $z \leq 1 / 2$ . Choose a radius R at least the radial radius and large enough that $ρ^{2} R^{2 (α - 1)} / 4 \geq d L + 1$ . Outside this ball step 1.2 makes the exponent at most $- γ$ . Inside it let $M = \sup_{∥ x - x_{⋆} ∥ \leq R} V (x) < \infty$ . Since $e^{a γ} - e^{- γ} = \int_{- γ}^{a γ} e^{u} d u \leq (a + 1) γ e^{a γ_{0}}$ for $a = d L$ , the preceding bound gives $R_{γ} V \leq e^{- γ} V + b γ, b = M (d L + 1) e^{d L γ_{0}},$ uniformly for small steps. Integrate against the finite-V invariant law from step 2.1 to obtain $π_{γ} V \leq b γ / (1 - e^{- γ}) \leq b e^{γ_{0}}$ , using $1 - e^{- γ} = \int_{0}^{γ} e^{- u} d u \geq γ e^{- γ_{0}}$ . This proves the first uniform bound.

4.1step 2.1step 1.2step 2.2step 3.1algebra

For $s = 1 / 4$ and $0 < γ < L^{- 1}$ , the logarithm of the prefactor in step 2.2 is at most $d L γ / 2$ , by the same integral estimate with $z = L γ / 2 < 1 / 2$ . Moreover $s (1 - s - L γ / 2) / (1 - 2 s L γ) \geq 1 / 16$ , because the numerator is at least $1 / 16$ and the denominator is at most one. Choose R' at least the radial radius such that $ρ^{2} (R^{'})^{2 (α - 1)} / 16 \geq d L / 2 + 1$ . Outside this ball $R_{γ} W \leq e^{- γ} W$ ; inside use $M^{'} = \sup_{∥ x - x_{⋆} ∥ \leq R^{'}} W (x)$ and the exponential-difference estimate from step 3.1 with $a = d L / 2$ and upper horizon $L^{- 1}$ . Thus $R_{γ} W \leq e^{- γ} W + b^{'} γ, b^{'} = M^{'} (d L / 2 + 1) e^{d / 2},$ for the whole step range. Since $W \leq V$ , its invariant moment is finite by step 2.1. Integrating and using $1 - e^{- γ} \geq γ e^{- 1 / L}$ gives $π_{γ} W \leq b^{'} e^{1 / L}$ , uniformly for every allowed step.

5.1F3F5step 1.2step 3.1step 4.1algebra∎

Put $r = ∥ x - x_{⋆} ∥$ . For $r \geq 1$ , [F3] gives $V (x) \geq e^{- C / 2} e^{c r^{α} / 2} \geq e^{- C / 2} e^{c r / 2}$ because $α > 1$ . By [F5], $r^{2} / e^{c r / 2}$ is bounded on $[1, \infty)$ ; on $[0, 1]$ use $r^{2} \leq 1 \leq V$ . Thus $r^{2} \leq A V$ for some finite A. The Lipschitz bound in step 1.2 gives $∥ \nabla U (x) ∥^{2} \leq L^{2} r^{2}$ . Taking $M_{g} = (1 + L^{2}) A$ , $M_{V} = b e^{γ_{0}}$ and $M_{W} = b^{'} e^{1 / L}$ proves all asserted estimates. All constants depend only on the fixed potential and its displayed parameters.

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Radial drift gives weighted geometric contraction of Langevin diffusion

Statement

Assume AC and the smoothness, minimizer, normalization and radial hypotheses of Exponential Lyapunov drift for the formal Langevin generator. Suppose $Z = \int e^{- U (x)} d x \in (0, \infty)$ and put $π (d x) = Z^{- 1} e^{- U (x)} d x$ . For $0 < s < 1$ , write $H_{s} = e^{s U}$ and let $a_{s} = s d L$ and $b_{s} = β_{s}$ be the constants in that lemma. The Langevin semigroup of Langevin diffusion has a reversible density and compact minorization satisfies $P_{t} H_{s} (x) \leq e^{- a_{s} t} H_{s} (x) + \frac{b_{s}}{a_{s}} (1 - e^{- a_{s} t}), π H_{s} < \infty .$ There are $C_{s} < \infty$ and $v_{s} > 0$ , depending on the potential and $s$ , such that for any probability measures $μ, ν$ with finite $H_{s}$ moments and any $t \geq 0$ , $∥ μ P_{t} - ν P_{t} ∥_{H_{s}} \leq C_{s} e^{- v_{s} t} ∥ μ - ν ∥_{H_{s}} .$ The norm is Weighted total variation for probability measures.

Facts & Assumptions

Given: AC, $U$ , its constants, the semigroup, and $s \in (0, 1)$ in the Statement.

[A1]

AC is inherited through the stochastic and Hahn/Jordan interfaces. The Axiom of Choice

[F1]

The Langevin equation has a Borel Markov semigroup, invariant target $π$ , and one-step minorization on every compact ball for every fixed positive time. Langevin diffusion has a reversible density and compact minorization

[F2]

For $H_{s} = e^{s U}$ the formal generator satisfies $L_{U} H_{s} \leq - a_{s} H_{s} + b_{s} 1_{{∥ x - x_{⋆} ∥ \leq K_{s}}}$ , with $a_{s} > 0$ and finite $b_{s}$ . Exponential Lyapunov drift for the formal Langevin generator

[F3]

Radial drift gives $U (x) \geq c ∥ x - x_{⋆} ∥^{α} - C$ for $c > 0$ . Consequently every sublevel set of $H_{s}$ is contained in a finite-radius ball. Radial gradient drift forces potential coercivity

[F4]

Ito's formula applies to the solution; after stopping in a compact ball its stochastic integral is square-integrable and has mean zero. Multidimensional Ito formula for Brownian-driven processes Stopping an Ito integral Ito isometry and linearity in predictable L2

[F5]

Fatou passes nonnegative stopped estimates to the unstopped limit; Tonelli integrates the resulting nonnegative kernel bounds over initial laws. Fatou's lemma Tonelli's theorem for nonnegative measurable functions on a sigma-finite product

[F6]

Global geometric drift and a common minorization on one sufficiently large weight-sublevel set give a unique invariant probability, finite weight moment, and geometric contraction in that weighted norm. Uniqueness holds among all invariant probabilities. Global drift and one-step minorization give weighted contraction and a unique invariant law

Proof

Given: The hypotheses in the Statement.

1.1A1F1F2F4F5algebra

Fix a deterministic starting point $x$ and horizon $t$ . Stop the continuous solution at its first exit $τ_{n}$ from the radius- $n$ ball about $x_{⋆}$ , taking $n > ∥ x - x_{⋆} ∥$ . Apply [F4] to $e^{a_{s} u} H_{s} (X_{u})$ up to $t \land τ_{n}$ . Its drift is at most $b_{s} e^{a_{s} u}$ by [F2]. Its stochastic integrand is bounded on this stopped finite cylinder, so its expectation is zero. Hence $E_{x} [e^{a_{s} (t \land τ_{n})} H_{s} (X_{t \land τ_{n}})] \leq H_{s} (x) + \frac{b_{s}}{a_{s}} (e^{a_{s} t} - 1) .$ The globally defined continuous paths in [F1] are bounded on $[0, t]$ , so $τ_{n} > t$ eventually on each such path. Fatou [F5] and division by $e^{a_{s} t}$ give the claimed semigroup moment bound.

2.1F1F3F6step 1.1

In particular $P_{1} H_{s} \leq a H_{s} + b$ with $a = e^{- a_{s}} \in (0, 1)$ and $b = (b_{s} / a_{s}) (1 - e^{- a_{s}}) < \infty$ . Any sufficiently large level set ${H_{s} \leq R}$ is contained in a ball by [F3], and inherits the common minorization for $P_{1}$ from [F1]. The weighted drift-minorization result [F6] therefore gives an invariant law $ξ_{s}$ with finite $H_{s}$ moment and constants $M_{s} < \infty$ , $r_{s} \in (0, 1)$ such that $∥ μ P_{n} - ν P_{n} ∥_{H_{s}} \leq M_{s} r_{s}^{n} ∥ μ - ν ∥_{H_{s}} (n \in N) .$ Since [F1] already makes $π$ invariant and [F6] gives uniqueness among every invariant probability, $ξ_{s} = π$ and $π H_{s} < \infty$ .

3.1F5F6step 1.1step 2.1algebra∎

For $0 \leq u < 1$ , step 1.1 gives $P_{u} H_{s} \leq H_{s} + b_{s} / a_{s} \leq (1 + b_{s} / a_{s}) H_{s}$ , since $H_{s} \geq 1$ . For every Borel $f$ with $∣ f ∣ \leq H_{s}$ , therefore $∣ P_{u} f ∣ \leq (1 + b_{s} / a_{s}) H_{s}$ . Taking the test-function supremum shows $∥ μ P_{u} - ν P_{u} ∥_{H_{s}} \leq (1 + b_{s} / a_{s}) ∥ μ - ν ∥_{H_{s}} .$ All integrals are finite by step 1.1 and [F5]. Write $t = n + u$ with $n = ⌊ t ⌋$ and apply this estimate after the integer contraction in step 2.1. With $v_{s} = - \log r_{s} > 0$ and $C_{s} = M_{s} (1 + b_{s} / a_{s}) r_{s}^{- 1}$ , the result follows from $r_{s}^{n} \leq r_{s}^{- 1} e^{- v_{s} t}$ . This includes $t = 0$ and all stated finite-moment initial laws.

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ULA's invariant-law bias under smoothness and radial drift

Statement

Assume AC (The Axiom of Choice) and the hypotheses of Unadjusted Langevin algorithm (ULA). In addition, let $U \in C^{2} (R^{d})$ , let $\nabla U$ be $L$ -Lipschitz for $L > 0$ , and suppose there are a global minimizer $x_{⋆}$ , $ρ > 0$ , $α \in (1, 2]$ , and $M_{ρ} \geq 0$ such that $⟨ \nabla U (x), x - x_{⋆} ⟩ \geq ρ ∥ x - x_{⋆} ∥^{α} whenever ∥ x - x_{⋆} ∥ \geq M_{ρ} .$ Assume $0 < Z_{U} = \int e^{- U (x)} d x < \infty$ and normalize $U (x_{⋆}) = 0$ ; subtracting a constant leaves the target and ULA kernel unchanged and only rescales the weight by a constant. Put $π_{U} (d x) = Z_{U}^{- 1} e^{- U (x)} d x$ and $V (x) = e^{U (x) / 2}$ . For every $0 < γ < L^{- 1}$ , the ULA kernel has a unique invariant probability $π_{γ}$ , both $π_{U}$ and $π_{γ}$ have finite $V^{1 / 2}$ moments, and $∥ π_{U} - π_{γ} ∥_{V^{1 / 2}} \leq C γ^{1 / 2},$ where the finite source-dependent constant $C$ is independent of $γ$ . The norm is the weighted total-variation norm of Weighted total variation for probability measures.

Facts & Assumptions

Given: AC and the potential, target, weights and step size in the Statement.

[A1]

AC is inherited through the Brownian/Gaussian, Ito, change-of-measure and Hahn/Jordan interfaces. The Axiom of Choice

[F1]

The ULA kernel $R_{γ}$ is that of Unadjusted Langevin algorithm (ULA). It has a unique invariant $π_{γ}$ with finite $V$ moment for every $0 < γ < L^{- 1}$ . With $γ_{0} = (2 L)^{- 1}$ , constants $M_{V}, M_{W}, M_{g} < \infty$ independent of the step satisfy $π_{γ} V \leq M_{V}$ for $γ \leq γ_{0}$ , $π_{γ} W \leq M_{W}$ for all allowed steps, and $∥ \nabla U (x) ∥^{2} + ∥ x - x_{⋆} ∥^{2} \leq M_{g} V (x)$ , where $W = V^{1 / 2}$ . ULA invariant laws have uniform exponential moments on the needed step ranges

[F2]

The Langevin semigroup leaves $π_{U}$ invariant, has $π_{U} V < \infty$ , satisfies $P_{t} V \leq V + B_{V}$ for a fixed finite $B_{V}$ , and contracts zero-mass differences in the W norm: $∥ μ P_{t} - ν P_{t} ∥_{W} \leq C_{W} e^{- v_{W} t} ∥ μ - ν ∥_{W}$ , with $C_{W} < \infty$ and $v_{W} > 0$ . Radial drift gives weighted geometric contraction of Langevin diffusion

[F3]

From any finite-second-moment initial probability, the endpoint relative entropy of an m-step Euler interpolation against the Langevin endpoint satisfies $D (Law (X_{m}) ∥ Law (Y_{m γ})) \leq (L^{2} / 4) \sum_{k = 0}^{m - 1} (γ^{3} E ∥ \nabla U (X_{k}) ∥^{2} / 3 + d γ^{2})$ . Pathwise relative-entropy error of the Langevin Euler interpolation

[F4]

For probabilities with finite V moments, weighted Pinsker with $W^{2} = V$ gives $∥ μ - ν ∥_{W}^{2} \leq 2 (μ V + ν V) D (μ ∥ ν)$ . A weighted total-variation bound from relative entropy

[F5]

Weighted total variation is the supremum over Borel tests $∣ f ∣ \leq W$ and obeys the triangle inequality; it is at most $μ W + ν W$ for two probabilities with finite W moments. Weighted total variation for probability measures

Proof

Given: The hypotheses in the Statement, with $W = e^{U / 4}$ and $γ_{0} = (2 L)^{- 1}$ .

1.1A1F1F2algebrachoose

By [F1], for each allowed step the unique invariant probability $π_{γ}$ exists and has finite V moment, hence finite W moment since $1 \leq W \leq V$ . By [F2], $π_{U} V < \infty$ , so its W moment is finite too. Choose a fixed $T_{0} > 0$ so that $C_{W} e^{- v_{W} T_{0}} \leq 1 / 2$ , possible because $v_{W} > 0$ . All these constants depend only on the potential and its hypotheses.

2.1A1F1F3step 1.1algebraconstruct

First suppose $0 < γ \leq γ_{0}$ . Put $m = ⌈ T_{0} / γ ⌉$ and $T = m γ$ , so $T_{0} \leq T < T_{0} + γ \leq T_{0} + γ_{0}$ . Start Euler with law $μ = π_{γ}$ , independently of its Brownian increments; every grid-time law is $μ$ by invariance. Its second moment is finite by [F1], since $∥ x ∥^{2} \leq 2 ∥ x - x_{⋆} ∥^{2} + 2 ∥ x_{⋆} ∥^{2}$ . Moreover $E ∥ \nabla U (X_{k}) ∥^{2} \leq M_{g} M_{V}$ at every grid time. Apply [F3] to the diffusion with the same initial law. Euler's endpoint law is $μ$ and diffusion's is $μ P_{T}$ , giving $D (μ ∥ μ P_{T}) \leq \frac{L^{2}}{4} m (\frac{γ^{3}}{3} M_{g} M_{V} + d γ^{2}) \leq K_{D} γ, K_{D} = \frac{L^{2}}{4} (T_{0} + γ_{0}) (d + \frac{γ_{0}}{3} M_{g} M_{V}) .$ The initial Brownian/Gaussian construction uses only the inherited AC assumption.

3.1F1F2F4F5step 1.1step 2.1algebra

By [F1], $μ V \leq M_{V}$ ; by [F2], $(μ P_{T}) V \leq M_{V} + B_{V}$ . Weighted Pinsker [F4] and step 2.1 therefore imply $∥ μ - μ P_{T} ∥_{W} \leq K_{E} \sqrt{γ}, K_{E} = \sqrt{2 (2 M_{V} + B_{V}) K_{D}} .$ Since $π_{U} P_{T} = π_{U}$ , the contraction [F2], the choice of $T_{0}$ , and $T \geq T_{0}$ give $∥ μ - π_{U} ∥_{W} \leq K_{E} \sqrt{γ} + ∥ μ P_{T} - π_{U} P_{T} ∥_{W} \leq K_{E} \sqrt{γ} + \frac{1}{2} ∥ μ - π_{U} ∥_{W} .$ The last norm is finite by step 1.1, so subtraction yields $∥ π_{γ} - π_{U} ∥_{W} \leq 2 K_{E} \sqrt{γ}$ throughout the small-step range.

4.1F1F5step 1.1step 3.1algebra∎

If $γ_{0} < γ < L^{- 1}$ , [F1] and [F5] give $∥ π_{γ} - π_{U} ∥_{W} \leq M_{W} + π_{U} W \leq (M_{W} + π_{U} W) γ_{0}^{- 1 / 2} \sqrt{γ}$ . Together with step 3.1, this proves the stated bound with $C = \max {2 K_{E}, (M_{W} + π_{U} W) γ_{0}^{- 1 / 2}} < \infty,$ independent of $γ$ over the entire specified open interval. Step 1.1 supplies both finite weight moments and uniqueness/existence for every such step.

5 · Examples, counterexamples and false statements

None yet.

Langevin algorithms: ULA bias and MALA exactness

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

Weighted total variation for probability measures

Definition

Radial gradient drift forces potential coercivity

Statement

Facts & Assumptions

Proof

Exponential Lyapunov drift for the formal Langevin generator

Statement

Facts & Assumptions

Proof

Unadjusted Langevin algorithm (ULA)

Definition

Remarks

A weighted total-variation bound from relative entropy

Statement

Facts & Assumptions

Proof

Global drift and one-step minorization give weighted contraction and a unique invariant law

Statement

Facts & Assumptions

Proof

Remarks

The ULA Gaussian proposal is a probability kernel with a positive density

Statement

Facts & Assumptions

Proof

The Gaussian ULA invariant law has variance 2/(2−γ)2/(2−γ)

Statement

Facts & Assumptions

Proof

ULA's Gaussian kernel satisfies an exponential drift bound

Statement

Facts & Assumptions

Proof

Metropolis-adjusted Langevin algorithm (MALA)

Definition

Finite-horizon Brownian change of drift and relative entropy

Statement

Facts & Assumptions

Proof

Source notes

MALA preserves its target probability exactly

Statement

Facts & Assumptions

Proof

Pathwise relative-entropy error of the Langevin Euler interpolation

Statement

Facts & Assumptions

Proof

Source notes

Langevin diffusion has a reversible density and compact minorization

Statement

Facts & Assumptions

Proof

ULA invariant laws have uniform exponential moments on the needed step ranges

Statement

Facts & Assumptions

Proof

Radial drift gives weighted geometric contraction of Langevin diffusion

Statement

Facts & Assumptions

Proof

ULA's invariant-law bias under smoothness and radial drift

Statement

Facts & Assumptions

Proof

5 · Examples, counterexamples and false statements

Sources

The Gaussian ULA invariant law has variance $2 / (2 - γ)$