The Ornstein$-$Uhlenbeck process on ${\mathcal{P}}_2$ with a volatility operator

Abstract

We analyze a diffusion ${(\hat{I}{¼}_t)}_{t\ge 0}$ on the $2$-Wasserstein space ${\mathcal{P}}_2$ 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 $|\hat{I}{¼}_t{|}_2:=({\int }_{{\mathbb{R}}^d}{x}^{T}x\hat{I}{¼}_t(dx){)}^{1/2}$. The invariant measure of ${(\hat{I}{¼}_t)}_{t\ge 0}$ is a Gaussian on ${\mathcal{P}}_2$, as introduced by P. Ren and F.-Y. Wang. Moreover, the Dirichlet form and its generator are given explicitly on a dense subspace of $L^2$.