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Stationary Ornstein–Uhlenbeck score and probability flow

Example

Assume AC. Fix $γ > 0$ , $θ > 0$ , and $T > 0$ . Let $(Ω, F, (F_{t})_{t \geq 0}, P)$ be a complete filtered probability space with a right-continuous filtration. Let $W$ be an adapted standard one-dimensional Brownian motion such that, for every $0 \leq s < t$ , $W_{t} - W_{s}$ is independent of $F_{s}$ and has law $N (0, t - s)$ . Let $X_{0}$ be a finite $F_{0}$ -measurable random variable with law $N (0, θ)$ , independent of the Brownian process. Define

$X_{t} = e^{- γ t} X_{0} + \sqrt{2 γ θ} \int_{0}^{t} e^{- γ (t - r)} d W_{r}, 0 \leq t \leq T .$

This process solves $d X_{t} = - γ X_{t} d t + \sqrt{2 γ θ} d W_{t}$ . Every marginal has density $p (x) = (2 π θ)^{- 1 / 2} e^{- x^{2} / (2 θ)}$ and score $s (x) = - x / θ$ . Its forward Fokker–Planck equation has zero right-hand side, and the probability-flow vector field $v (x) = - γ x - \frac{1}{2} (2 γ θ) s (x)$ is identically zero. Thus the probability-flow ODE is stationary and the exact finite-time terminal law is $P_{T} = N (0, θ)$ .

Example references

Särkkä and Solin, Section 5.2, Theorem 5.4 and Example 5.7; Section 6.1

Raginsky, Section 2.3 and Chapter 2, Problem 8

Song et al., Section 4.3 and Appendix D.1

Facts & Assumptions

Given: AC, $W$ , $X_{0}$ , and $X_{t}$ as in the Example.

[A1]

The Axiom of Choice is assumed by the Brownian, Itô, and normal-law interfaces used in the verification. The Axiom of Choice

[F1]

Under the stated filtration and increment conditions, the displayed process has a continuous Brownian Itô representation with pathwise integrable drift and a localized Itô integral. Continuous Brownian Ito processes

[F2]

For a one-dimensional Brownian Itô process and $f \in C^{1, 2}$ , $d f (t, X_{t}) = (\partial_{t} f + b \partial_{x} f + \frac{1}{2} σ^{2} \partial_{x}^{2} f) (t, X_{t}) d t + σ_{t} \partial_{x} f (t, X_{t}) d W_{t}$ . One-dimensional Ito formula

[F3]

A deterministic square-integrable integrand $h$ has centered normal Itô integral with variance $\int_{0}^{t} h (r)^{2} d r$ . Deterministic Ito integrals are Gaussian

[F4]

The deterministic Itô integral is an $L^{2}$ limit of elementary sums of Brownian increments. These sums are measurable with respect to the Brownian process; $L^{2}$ convergence implies convergence in probability, which has an almost-surely convergent subsequence. Defining its limit on the measurable convergence event gives a Brownian-process-measurable version. Measurable functions of independent random elements remain independent, and replacing a random variable by an almost-surely equal version preserves its joint law with $X_{0}$ . Independent random elements Ito integral for square-integrable predictable processes Measurable coordinatewise functions preserve independence An almost-surely convergent subsequence from convergence in probability

[F5]

The characteristic function of $N (m, q)$ is $u \mapsto \exp (i m u - q u^{2} / 2)$ , including $q = 0$ . Independent variables have the product characteristic function, and equal characteristic functions determine equal Borel probability laws. Thus independent scalar normal variables have a normal sum with the summed means and variances. Characteristic function of a normal law Characteristic functions under affine maps and independent sums Uniqueness of a law from its characteristic function

[F6]

$N (0, θ)$ for $θ > 0$ has density $(2 π θ)^{- 1 / 2} e^{- x^{2} / (2 θ)}$ . A positive-variance normal law has its location-scale density

[F7]

At positive density points, the marginal score is $\partial_{x} \log p (x)$ . Marginal score function

[F8]

For positive finite normalized densities $p_{t}$ of Borel probability laws $P_{t}$ satisfying the forward equation, with $a = σ^{2}$ in the scalar case and all displayed derivatives continuous (one-sided in time at endpoints), assume the probability-flow field has the global jointly $C^{1}$ flow of diffeomorphisms with both time-derivative identities specified in the cited theorem. Starting that flow with law $P_{0}$ gives law $P_{t}$ at every time. Probability-flow ODE has the same one-time marginals

Verification

1.1A1F1F2given

Put $M_{t} = \int_{0}^{t} e^{γ r} d W_{r}$ , so $X_{t} = e^{- γ t} (X_{0} + \sqrt{2 γ θ} M_{t})$ . Apply [F2] to $f (t, x) = e^{- γ t} x$ and the continuous Itô process $X_{0} + \sqrt{2 γ θ} M_{t}$ . Its drift is zero and its diffusion coefficient is $\sqrt{2 γ θ} e^{γ t}$ ; the product rule gives $d X_{t} = - γ X_{t} d t + \sqrt{2 γ θ} d W_{t}$ , as asserted.

1.2A1F3F4F5F6givenalgebra

For fixed $t$ , [F3] gives $M_{t} \sim N (0, \int_{0}^{t} e^{2 γ r} d r)$ . Equivalently the stochastic term in the Example is centered normal with variance $2 γ θ \int_{0}^{t} e^{- 2 γ (t - r)} d r = θ (1 - e^{- 2 γ t})$ . Its deterministic-integrand Itô approximants are measurable functions of Brownian increments, and [F4] gives a Brownian-process-measurable version of their limit; that version is independent of $X_{0}$ , and almost-sure equality preserves the joint law. The first term is $N (0, θ e^{- 2 γ t})$ , and [F5] gives $X_{t} \sim N (0, θ e^{- 2 γ t} + θ (1 - e^{- 2 γ t})) = N (0, θ)$ . By [F6], its density is the one displayed in the Example.

2.1F6F7step 1.2algebra

Differentiating the density in [F6] gives $\partial_{x} p = - (x / θ) p$ and $\partial_{x x} p = (x^{2} / θ^{2} - 1 / θ) p$ . With $b (x) = - γ x$ and $a = 2 γ θ$ , the Fokker–Planck right-hand side is $- \partial_{x} (b p) + \frac{1}{2} \partial_{x x} (a p) = γ \partial_{x} (x p) + γ θ \partial_{x x} p = 0$ by substitution of these derivatives. Thus $p_{t} = p$ is a positive smooth solution of the forward equation and [F7] gives $s (x) = - x / θ$ .

3.1F7F8step 1.2step 2.1algebra∎

The diffusion covariance $a = σ^{2} = 2 γ θ$ is constant, so $\partial_{x} a = 0$ . Substituting $b = - γ x$ , $a = 2 γ θ$ , and $s = - x / θ$ into the probability-flow velocity gives $v (x) = - γ x - \frac{1}{2} (2 γ θ) (- x / θ) = 0$ . Its global flow is $Φ_{t, s} (x) = x$ , jointly smooth and a diffeomorphism satisfying the flow composition and both time-derivative identities since $v = 0$ . Step 1.2 identifies $p_{t} = p$ as a normalized probability density; $p$ , $b$ , and $a$ are smooth, and step 2.1 verifies the forward equation. Thus every hypothesis of [F8] holds, so the deterministic flow started with law $N (0, θ)$ preserves that law at each time, including $T$ .

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Sources

Simo Särkkä and Arno Solin, Applied Stochastic Differential Equations, Section 5.2, Theorem 5.4 and Example 5.7; Section 6.1 (standard reference, not scraped)
Maxim Raginsky, Stochastic Differential Equations: A Systems-Theoretic Approach, Section 2.3 and Chapter 2, Problem 8 (standard reference, not scraped)
Yang Song et al., Score-Based Generative Modeling through Stochastic Differential Equations, Section 4.3 and Appendix D.1 (standard reference, not scraped)