Journal
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
Volume 35, Issue 11, Pages 5185-5202Publisher
AMER INST MATHEMATICAL SCIENCES
DOI: 10.3934/dcds.2015.35.5185
Keywords
Parabolic potential; regular measure; stochastic partial differential equations; non-homogeneous second order operator; obstacle problem; penalization method; Ito's formula; comparison theorem; space-time white noise
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We prove the existence and uniqueness of solution of the obstacle problem for quasilinear Stochastic PDEs with non-homogeneous second order operator. Our method is based on analytical technics coming from the parabolic potential theory. The solution is expressed as a pair (u, v) where u is a predictable continuous process which takes values in a proper Sobolev space and v is a random regular measure satisfying minimal Skohorod condition. Moreover, we establish a maximum principle for local solutions of such class of stochastic PDEs. The proofs are based on a version of ItO's formula and estimates for the positive part of a local solution which is non-positive on the lateral boundary.
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