Coupling for Ornstein--Uhlenbeck processes with jumps

TL;DR

This paper studies the coupling properties and regularity of solutions to linear Ornstein-Uhlenbeck processes driven by Lévy noise with jumps, establishing bounds on transition probabilities and conditions for strong Feller properties.

Contribution

It provides new coupling bounds and regularity results for Ornstein-Uhlenbeck processes with jumps under specific Lévy measure conditions.

Findings

Total variation distance decays as 1/√t for large t

Processes admit successful couplings under given conditions

Harnack inequality and ultracontractivity are established

Abstract

Consider the linear stochastic differential equation (SDE) on $\mathbb{R}^n$: \[\mathrm {d}{X}_t=AX_t\,\mathrm{d}t+B\,\mathrm{d}L_t,\] where $A$ is a real $n\times n$ matrix, $B$ is a real $n\times d$ real matrix and $L_t$ is a L\'{e}vy process with L\'{e}vy measure $\nu$ on $\mathbb{R}^d$. Assume that $\nu(\mathrm {d}{z})\ge \rho_0(z)\,\mathrm{d}z$ for some $\rho_0\ge 0$. If $A\le 0,\operatorname {Rank}(B)=n$ and $\int_{\{|z-z_0|\le\varepsilon\}}\rho_0(z)^{-1}\,\mathrm{d}z<\infty$ holds for some $z_0\in \mathbb{R}^d$ and some $\varepsilon>0$, then the associated Markov transition probability $P_t(x,\mathrm {d}{y})$ satisfies \[\|P_t(x,\cdot)-P_t(y,\cdot)\|_{\mathrm{var}}\le \frac{C(1+|x-y|)}{\sqrt{t}}, x,y\in \mathbb{R}^d,t>0,\] for some constant $C>0$, which is sharp for large $t$ and implies that the process has successful couplings. The Harnack inequality, ultracontractivity and the strong Feller property are also investigated for the (conditional) transition semigroup.