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

We analyze a diffusion ${({\mu }_t)}_{t\ge 0}$ 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 $c\in (0,\mathrm{\infty })$ equals the trace of a volatility operator on a Hilbert space and $\mathrm{\mid }{\mu }_t{\mathrm{\mid }}_2:=({\int }_{{\mathbb{R}}^d}{x}^{T}x{\mu }_t(dx){)}^{1\mathrm{/}2}$. The invariant measure of ${({\mu }_t)}_{t\ge 0}$ 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 $L^2$.