Underdamped Langevin equation and canonical Gibbs density

Assume the Axiom of Choice. Let $d\ge 1$ be finite, let $M\in {\mathbb{R}}^{d\times d}$ be symmetric positive definite (that is, $x^{T}Mx>0$ for every nonzero $x$), and let $\beta ,\gamma >0$. Then $M$ is invertible, and $M^{1\mathrm{/}2}$ denotes its unique symmetric positive square root. Let $W_t$ be standard $d$-dimensional Brownian motion relative to a complete usual filtration, and let $V:{\mathbb{R}}^d\to \mathbb{R}$ be differentiable. The underdamped Langevin equation is the It^o SDE on phase space ${\mathbb{R}}_q^d\times {\mathbb{R}}_p^d$ $d{Q}_t=M^{-1}{P}_{t}dt,d{P}_t=-\mathrm{\nabla }V(Q_t)dt-\gamma P_{t}dt+\sqrt{2\gamma {\beta }^{-1}}{M}^{1\mathrm{/}2}d{W}_t\mathrm{.}$ Its Hamiltonian is $H(q,p)=\frac{1}{2}{p}^T{M}^{-1}p+V(q)$. If $Z={\int }_{{\mathbb{R}}^{2d}}{e}^{-\beta H(q,p)}dqdp$ is finite and positive, the canonical Gibbs probability is the probability measure with density $\rho (q,p)=Z^{-1}{e}^{-\beta H(q,p)}$ with respect to $2d$-dimensional Lebesgue measure. This definition alone asserts no stationarity or ergodicity.

For the finite-time order results below, $D^{j}V$ denotes the $j$th derivative as a multilinear map. The notation $C_b^k$ means that a function and each of its derivatives through order $k$ are continuous and bounded. The notation $C_P^k$ means that a function and each of its derivatives through order $k$ are continuous and bounded by $C(1+\mathrm{\mid }z{\mathrm{\mid }}^r)$ for some finite constants $C,r$ that may depend on the derivative. These growth conventions are used only where stated.

The noise covariance is $2\gamma {\beta }^{-1}M$, so the Maxwell momentum covariance in the canonical density is ${\beta }^{-1}M$. The friction and covariance convention is fixed by the displayed equation. $\beta$ is inverse temperature and $\gamma$ is the scalar friction rate.