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January 2004 Occupation densities for SPDEs with reflection
Lorenzo Zambotti
Ann. Probab. 32(1A): 191-215 (January 2004). DOI: 10.1214/aop/1078415833

Abstract

We consider the solution $(u,\eta)$ of the white-noise driven stochastic partial differential equation with reflection on the space interval $[0,1]$ introduced by Nualart and Pardoux, where $\eta$ is a reflecting measure on $[0,\infty)\times(0,1)$ which forces the continuous function u, defined on $[0,\infty)\times[0,1]$, to remain nonnegative and $\eta$ has support in the set of zeros of u. First, we prove that at any fixed time $t>0$, the measure $\eta([0,t]\times d\theta)$ is absolutely continuous w.r.t. the Lebesgue measure $d\theta$ on $(0,1)$. We characterize the density as a family of additive functionals of u, and we interpret it as a renormalized local time at $0$ of $(u(t,\theta))_{t\geq 0}$. Finally, we study the behavior of $\eta$ at the boundary of $[0,1]$. The main technical novelty is a projection principle from the Dirichlet space of a Gaussian process, vector-valued solution of a linear SPDE, to the Dirichlet space of the process u.

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Lorenzo Zambotti. "Occupation densities for SPDEs with reflection." Ann. Probab. 32 (1A) 191 - 215, January 2004. https://doi.org/10.1214/aop/1078415833

Information

Published: January 2004
First available in Project Euclid: 4 March 2004

zbMATH: 1121.60069
MathSciNet: MR2040780
Digital Object Identifier: 10.1214/aop/1078415833

Subjects:
Primary: 60H15, 60J55
Secondary: 60J55

Rights: Copyright © 2004 Institute of Mathematical Statistics

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Vol.32 • No. 1A • January 2004
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