A nonlinear stochastic heat equation: Hölder continuity and smoothness of the density of the solution

Details A nonlinear stochastic heat equation: Hölder continuity and smoothness of the density of the solution A nonlinear stochastic heat equation: Hölder continuity and smoothness of the density of the solution

2011-10-21

arxiv.org/abs/1110.4855v1

Mathematics

Probability

Scientific paper

In this paper, we establish a version of the Feynman-Kac formula for multidimensional stochastic heat equation driven by a general semimartingale. This Feynman-Kac formula is then applied to study some nonlinear stochastic heat equations driven by nonhomogenous Gaussian noise: First, it is obtained an explicit expression for the Malliavin derivatives of the solutions. Based on the representation we obtain the smooth property of the density of the law of the solution. On the other hand, we also obtain the H\"older continuity of the solutions.

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