Stochastic parabolic Anderson model with time-homogeneous generalized potential: Mild formulation of solution

Hyun-Jung Kim

doi:10.3934/cpaa.2019038

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2019, Volume 18, Issue 2: 795-807. Doi: 10.3934/cpaa.2019038

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Stochastic parabolic Anderson model with time-homogeneous generalized potential: Mild formulation of solution

Hyun-Jung Kim

Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA

Received: January 31, 2018

Revised: May 31, 2018

Published: October 2018

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Abstract

A mild formulation for stochastic parabolic Anderson model with time-homogeneous Gaussian potential suggests a way of defining a solution to obtain its optimal regularity. Two different interpretations in the equation or in the mild formulation are possible with usual pathwise product and the Wick product: the usual pathwise interpretation is mainly discussed. We emphasize that a modified version of parabolic Schauder estimates is a key idea for the existence and uniqueness of a mild solution. In particular, the mild formulation is crucial to investigate a relation between the equations with usual pathwise product and the Wick product.

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Mathematics Subject Classification: Primary: 35R60; Secondary: 35K20, 35C15, 58C30.

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