The Annals of Probability

Occupation densities for SPDEs with reflection

Lorenzo Zambotti

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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.

Article information

Source
Ann. Probab., Volume 32, Number 1A (2004), 191-215.

Dates
First available in Project Euclid: 4 March 2004

Permanent link to this document
https://projecteuclid.org/euclid.aop/1078415833

Digital Object Identifier
doi:10.1214/aop/1078415833

Mathematical Reviews number (MathSciNet)
MR2040780

Zentralblatt MATH identifier
1121.60069

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60] 60J55: Local time and additive functionals
Secondary: 60J55: Local time and additive functionals

Keywords
Stochastic partial differential equations with reflection local times and additive functionals

Citation

Zambotti, Lorenzo. Occupation densities for SPDEs with reflection. Ann. Probab. 32 (2004), no. 1A, 191--215. doi:10.1214/aop/1078415833. https://projecteuclid.org/euclid.aop/1078415833


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