Abstract
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 $(u,\nu)$ where $u$ is a predictable continuous process which takes values in a proper Sobolev space and $\nu$ 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.
Highlights
In this paper we study the following SPDE with obstacle: dut (x) =∂i (ai,j (t, x)∂j ut(x)) dt +∂igi (t, x, ut(x), ∇ut (x))dt f (t, x, ut (x), ∇ut(x))dt +∞+ hj(t, x, ut(x), ∇ut(x))dBtj + ν(t, dx), j=1
Given an obstacle S : Ω × [0, T ] × O → R, we study the obstacle problem for the SPDE (1), i.e. we want to find a solution of (1) which satisfies "u ≥ S" where the obstacle S is regular in some sense and controlled by the solution of a SPDE
In [11], still in the homogeneous case, we have obtained a maximum principle for local solutions
Summary
In this paper we study the following SPDE with obstacle (in short OSPDE): dut (x). L. In [11], still in the homogeneous case, we have obtained a maximum principle for local solutions In these papers we have assumed that a does not depend on time and so many proofs are based on the notion of semigroup associated to the second order operator and on the regularizing property of the semigroup. The main results of this paper are first an existence and uniqueness Theorem for the solution with null Dirichlet condition and a maximum principle for local solutions. · ∞,∞;t is the uniform norm on [0, t] × O
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