Explicit contraction rates for a class of degenerate and infinite-dimensional diffusions

Abstract

Given a separable and real Hilbert space $\mathbb{H}$ and a trace-class, symmetric and non-negative operator $\mathcal{G}:\mathbb{H}\rightarrow\mathbb{H}$, we examine the equation \begin{align*} dX_t = -X_t\, dt + b(X_t) \, dt + \sqrt{2} \, dW_t, \qquad X_0=x\in\mathbb{H}, \end{align*} where $(W_t)$ is a $\mathcal{G}$-Wiener process on $\mathbb{H}$ and $b:\mathbb{H}\rightarrow\mathbb{H}$ is Lipschitz. We assume there is a splitting of $\mathbb{H}$ into a finite-dimensional space $\mathbb{H}^l$ and its orthogonal complement $\mathbb{H}^h$ such that $\mathcal{G}$ is strictly positive definite on $\mathbb{H}^l$ and the non-linearity $b$ admits a contraction property on $\mathbb{H}^h$. Assuming a geometric drift condition, we derive a Kantorovich ($L^1$ Wasserstein) contraction with an explicit rate for the corresponding Markov kernels. The estimates for the rate are based on the eigenvalues of $\mathcal{G}$ on the space $\mathbb{H}^l$, a Lipschitz bound on $b$ and a geometric drift condition. The results are derived using coupling methods.