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January 2003 Integration by parts on $\bolds{\delta}$-Bessel bridges, $\bolds{\delta>3}$, and related SPDEs
Lorenzo Zambotti
Ann. Probab. 31(1): 323-348 (January 2003). DOI: 10.1214/aop/1046294313

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 $\delta>3$.

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Lorenzo Zambotti. "Integration by parts on $\bolds{\delta}$-Bessel bridges, $\bolds{\delta>3}$, and related SPDEs." Ann. Probab. 31 (1) 323 - 348, January 2003. https://doi.org/10.1214/aop/1046294313

Information

Published: January 2003
First available in Project Euclid: 26 February 2003

zbMATH: 1019.60062
MathSciNet: MR1959795
Digital Object Identifier: 10.1214/aop/1046294313

Subjects:
Primary: 60H07 , 60H15
Secondary: 31C25 , 37L40

Keywords: Bessel bridges , integration by parts formulae , Invariant measures , Stochastic partial differential equations

Rights: Copyright © 2003 Institute of Mathematical Statistics

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Vol.31 • No. 1 • January 2003
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