Sticky couplings of multidimensional diffusions with different drifts

TL;DR

This paper introduces a new coupling method for multidimensional diffusions with different drifts, using a sticky boundary process to control their distance and provide explicit bounds on their convergence probability.

Contribution

It develops a novel sticky coupling approach for multidimensional Itô processes with different drifts, including explicit bounds and stability analysis.

Findings

Explicit bounds for the probability of the processes meeting.

Construction of the coupling as a weak limit of Markovian couplings.

Long-time stability results for the coupling.

Abstract

We present a novel approach of coupling two multidimensional and non-degenerate It\^o processes $(X_t)$ and $(Y_t)$ which follow dynamics with different drifts. Our coupling is sticky in the sense that there is a stochastic process $(r_t)$, which solves a one-dimensional stochastic differential equation with a sticky boundary behavior at zero, such that almost surely $|X_t-Y_t|\leq r_t$ for all $t\geq 0$. The coupling is constructed as a weak limit of Markovian couplings. We provide explicit, non-asymptotic and long-time stable bounds for the probability of the event $\{X_t=Y_t\}$.