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Leapfrog is reversible and volume-preserving

Statement

Assume Countable Choice. Use the notation of One-step leapfrog map as force and drift shears on $R^{d} \times R^{d}$ , equipped with full Euclidean Lebesgue measure $λ_{2 d}$ . Every kick $K_{a}$ and drift $D_{b}$ preserves that measure. Consequently $Φ_{ϵ}^{L}$ and $T_{ϵ, L} = R \circ Φ_{ϵ}^{L}$ preserve phase volume, and $T_{ϵ, L}^{2} = I .$ For $U \in C^{2}$ , the classical Jacobian matrices of the kick and drift are block triangular with determinant one; under only $U \in C^{1}$ , the volume conclusion means measure preservation and does not presume a classical Jacobian exists.

Facts & Assumptions

Given: The inputs and maps in One-step leapfrog map as force and drift shears, finite $L \geq 1$ , and Countable Choice for the stated Lebesgue-measure supplier.

[A1]

Countable Choice supplies the measure construction used by the cited reflection result. The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F1]

For $U \in C^{1}$ , the kick and drift have the formulas in One-step leapfrog map as force and drift shears; $\nabla U$ is continuous, and their inverses are $K_{- a}$ and $D_{- b}$ .

[F2]

Translating a Lebesgue-measurable set preserves its Lebesgue measure. Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation

[F3]

Sections of product-measurable sets are measurable; a continuous map pulls Borel sets back to Borel sets. Every section of a product-measurable function is measurable, A continuous map has Borel preimages of Borel sets

[F4]

Under Countable Choice, Lebesgue measure is sigma-finite: the increasing finite boxes $(- n, n]^{d}$ cover $R^{d}$ , and Borel sets are Lebesgue measurable. A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included, Assuming countable choice, every Borel subset of $R^{n}$ is Lebesgue measurable

[F5]

The product of sigma-finite Lebesgue measures exists with the rectangle formula, and Tonelli applies to nonnegative product-measurable functions. For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique, Tonelli's theorem for nonnegative measurable functions on a sigma-finite product

[F6]

Reflection $p \mapsto - p$ preserves Lebesgue measure on $R^{d}$ under Countable Choice. For a nonzero real $c$ , dilation by $c$ multiplies Lebesgue outer measure by $∣ c ∣^{n}$ , and reflection in the origin preserves it

[F7]

The determinant of a triangular matrix is the product of its diagonal entries. The determinant of a triangular matrix is the product of its diagonal entries

[F8]

Under Countable Choice, the Borel sigma-algebra of $R^{2 d}$ is the product of the Borel sigma-algebras of the two factors, and the product of their Borel-restricted Lebesgue measures agrees with full Euclidean measure $λ_{2 d}$ . The Borel product of R^m and R^n is the Borel sigma-algebra of R^{m+n}, On Borel subsets of R^{m+n}, the product lambda_m times lambda_n agrees with lambda_{m+n}

[F9]

Under Countable Choice, every Lebesgue-measurable subset of $R^{2 d}$ differs from a Borel set by a subset of a Borel null set, and $λ_{2 d}$ is the completion of its Borel restriction. $L (R^{n})$ is exactly the completion of the restriction of $λ_{n}$ to the Borel sets

Proof

technique · slice the shears and use the symmetric composition

1.1A1F1F2F3F4F5F8construct

Fix $a \in R$ and a Borel set $E \subseteq R^{d} \times R^{d}$ . Continuity of $K_{a}$ makes $K_{a}^{- 1} (E)$ Borel, and [F3, F8] make these sets product-measurable with measurable sections $E_{q}$ . For fixed $q$ , the momentum section of $K_{a}^{- 1} (E)$ is $E_{q} + a \nabla U (q)$ , a translate of $E_{q}$ , so [F2] gives equal $λ_{d}$ -measures. Tonelli [F5] on the product measure from [F4] gives equal product measures, and [F8] identifies them with full Euclidean measures: $λ_{2 d} (K_{a}^{- 1} (E)) = λ_{2 d} (E)$ .

1.2A1F1F2F3F4F5F6F8algebra

For each fixed $p$ , continuity of $D_{b}$ from [F1] and [F3, F8] make the position section of $D_{b}^{- 1} (E)$ measurable; it is $E^{p} - b M^{- 1} p$ , also a translate. The same [F2] and Tonelli calculation, with [F8], gives $λ_{2 d} (D_{b}^{- 1} (E)) = λ_{2 d} (E)$ . The map $R$ preserves full phase volume because it reflects each momentum section, so [A1, F5, F6] apply. Thus every kick, drift, and flip preserves phase volume on Borel sets.

2.1F1F3F9step 1.1step 1.2

Let $E \subseteq R^{2 d}$ be Lebesgue measurable. By [F9], choose Borel sets $A, Z$ and $N \subseteq Z$ with $E = A \cup N$ and $λ_{2 d} (Z) = 0$ . For any kick, drift, or flip $S$ , [F1] and [F3] make $S^{- 1} (A)$ and $S^{- 1} (Z)$ Borel; steps 1.1 and 1.2 give $λ_{2 d} (S^{- 1} (A)) = λ_{2 d} (A)$ and $λ_{2 d} (S^{- 1} (Z)) = 0$ . Thus $S^{- 1} (E)$ differs from $S^{- 1} (A)$ by a subset of the null Borel set $S^{- 1} (Z)$ , so [F9] gives its measurability and $λ_{2 d} (S^{- 1} (E)) = λ_{2 d} (E)$ . This proves preservation for every Lebesgue-measurable set.

3.1F1step 2.1algebra

Since $Φ_{ϵ} = K_{ϵ / 2} \circ D_{ϵ} \circ K_{ϵ / 2}$ , its finite powers and their composition with $R$ preserve volume by step 2.1.

4.1F1step 3.1algebra

Direct substitution gives $R K_{a} R = K_{- a}$ and $R D_{b} R = D_{- b}$ . Hence $R Φ_{ϵ} R = K_{- ϵ / 2} \circ D_{- ϵ} \circ K_{- ϵ / 2} = Φ_{ϵ}^{- 1} .$ For $T_{ϵ, L} = R Φ_{ϵ}^{L}$ , this yields $T_{ϵ, L}^{2} = R Φ_{ϵ}^{L} R Φ_{ϵ}^{L} = Φ_{ϵ}^{- L} Φ_{ϵ}^{L} = I .$ This proves reversibility by momentum flip and the involution claim.

5.1F1F7step 2.1step 4.1algebra∎

If $U \in C^{2}$ , differentiation gives $D K_{a} = (\begin{array}{cc} I & 0 \\ - a \nabla^{2} U (q) & I \end{array}), D D_{b} = (\begin{array}{cc} I & b M^{- 1} \\ 0 & I \end{array});$ by [F7] both determinants are one, and $∣ \det D R ∣ = 1$ . Under merely $C^{1}$ regularity these derivative matrices need not exist, but steps 1.1–4.1 already prove measure preservation without them.

Remarks

The statement “unit Jacobian” is classical only when the Hessian exists. For the design's base $C^{1}$ potential, the slice-translation proof is the volume-preservation argument.

Depends on

Used by

The Metropolized Hamiltonian chain preserves its target Theorem

Dependency tree · two levels

83 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Radford M. Neal, MCMC Using Hamiltonian Dynamics, Handbook of Markov Chain Monte Carlo, Chapter 5 (standard reference, not scraped)
Michael Betancourt, A Conceptual Introduction to Hamiltonian Monte Carlo (standard reference, not scraped)