Integration by parts on $\bolds{\delta}$-Bessel bridges, $\bolds{\delta>3}$, and related SPDEs

Abstract

We study a white-noise driven semilinear partial differential equation on the spatial interval $[0,1]$ with Dirichlet boundary condition and with a singular drift of the form $c u^{-3}$, $c>0$. We prove existence and uniqueness of a non-negative continuous adapted solution $u$ on $[0,\infty)\times[0,1]$ for every nonnegative continuous initial datum $x$, satisfying $x(0)=x(1)=0$. We prove that the law $\pi_\delta$ of the Bessel bridge on $[0,1]$ of dimension $\delta>3$ is the unique invariant probability measure of the process $x\mapsto u$, with $c=(\delta-1)(\delta-3)/8$ and, if $\delta\in{\mathbb N}$, that $u$ is the radial part in the sense of Dirichlet forms of the ${\mathbb R}^\delta$-valued solution of a linear stochastic heat equation. An explicit integration by parts formula w.r.t. $\pi_\delta$ is given for all $\delta>3$.