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, for all time t >...

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Bibliographic Details
Published in:The Annals of probability 2006-07, Vol.34 (4), p.1423-1450
Main Authors: Dalang, Robert C., Mueller, C., Zambotti, L.
Format: Article
Language:English
Subjects:
60H15
60J45
Boundary conditions
Brownian motion
Coordinate systems
Exact sciences and technology
Heat equation
Mathematical analysis
Mathematical models
Mathematical monotonicity
Mathematical theorems
Mathematics
Partial differential equations
Probability and statistics
Probability theory and stochastic processes
Random variables
Rectangles
reflecting nonlinearity
Sciences and techniques of general use
singular coefficients
Stochastic analysis
Stochastic models
stochastic obstacle problem
Stochastic partial differential equations
Studies
White noise
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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:0091-1798
2168-894X
DOI:10.1214/009117905000000792