A nonlinear stochastic heat equation: Hölder continuity and smoothness of the density of the solution |
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Authors: | Yaozhong Hu David Nualart Jian Song |
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Institution: | 1. Department of Mathematics, University of Kansas, Lawrence, KS, 66045, United States;2. Department of Mathematics, Rutgers University, Hill Center - Busch Campus, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, United States |
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Abstract: | 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 nonhomogeneous Gaussian noise: first, an explicit expression for the Malliavin derivatives of the solutions is obtained. 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 Hölder continuity of the solutions. |
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Keywords: | Fractional noise Stochastic heat equations Feynman&ndash Kac formula Exponential integrability Absolute continuity Hö lder continuity Chaos expansion |
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