The Annals of Probability

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

Lorenzo Zambotti

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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 $\delta>3$.

Article information

Ann. Probab., Volume 31, Number 1 (2003), 323-348.

First available in Project Euclid: 26 February 2003

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Zentralblatt MATH identifier

Primary: 60H15: Stochastic partial differential equations [See also 35R60] 60H07: Stochastic calculus of variations and the Malliavin calculus
Secondary: 37L40: Invariant measures 31C25: Dirichlet spaces

Stochastic partial differential equations Bessel bridges invariant measures integration by parts formulae


Zambotti, Lorenzo. Integration by parts on $\bolds{\delta}$-Bessel bridges, $\bolds{\delta>3}$, and related SPDEs. Ann. Probab. 31 (2003), no. 1, 323--348. doi:10.1214/aop/1046294313.

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