Brownian Motion With Partial Resetting Conditioned to Stay Positive

We consider Brownian motion with partial resetting, which has recently attracted a lot of attention in physics as well as the mathematics literature. We analyze the speed of convergence of this process towards stationarity as well as its quasistationary behavior. In particular, we prove the existence of a Yaglom limit and hence of a minimal quasistationary distribution. We use these results to study our main topic, namely the process conditioned on staying positive using methods which are well adapted to this specific process. It turns out that this process can be described explicitly as a three‐dimensional Bessel process with partial resetting with the same parameter but a modified resetting rate. This can be interpreted as an effect due to entropic repulsion.