The Ornstein$-$Uhlenbeck process on $\mathscr P_2$ with a volatility operator

Abstract

We analyze a diffusion ${(μ_t)}_{t\geq 0}$ on the $2$-Wasserstein space $\mathscr P_2$ over $\mathbb R^d$ for which \begin{equation*} |μ_t|_2^2-|μ_0|_2^2-2ct+2\int_0 ^t|μ_s|_2^2\,d s,\qquad t\geq 0, \end{equation*} is a martingale, where the constant $c\in(0,\infty)$ equals the trace of a volatility operator on a Hilbert space and $|μ_t|_2:=(\int_{\mathbb R^d}x^T xμ_t(d x ))^{1/2}$. The invariant measure of ${(μ_t)}_{t\geq 0}$ is a Gaussian on $\mathscr P_2$, as introduced by P. Ren and F.-Y. Wang. Moreover, the Dirichlet form and its generator are given explicitly on a dense subspace of $L^2$.