Hitting properties of parabolic s.p.d.e.'s with reflection

We study the hitting properties of the solutions \(u\) of a class of parabolic stochastic partial differential equations with singular drifts that prevent \(u\) from becoming negative. The drifts can be a reflecting term or a nonlinearity \(cu^{-3}\), with \(c>0\). We prove that almost surely, fo...

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Bibliographic Details
Published in:arXiv.org 2006-09
Main Authors: Dalang, Robert C, Mueller, C, Zambotti, L
Format: Article
Language:English
Subjects:
Partial differential equations
Online Access:Get full text
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Summary:We study the hitting properties of the solutions \(u\) of a class of parabolic stochastic partial differential equations with singular drifts that prevent \(u\) from becoming negative. The drifts can be a reflecting term or a nonlinearity \(cu^{-3}\), with \(c>0\). We prove that almost surely, for all time \(t>0\), the solution \(u_t\) hits the level 0 only at a finite number of space points, which depends explicitly on \(c\). In particular, this number of hits never exceeds 4 and if \(c>15/8\), then level 0 is not hit.
ISSN:2331-8422
DOI:10.48550/arxiv.0410414