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Publication# On the density of the supremum of the solution to the linear stochastic heat equation

Abstract

We study the regularity of the probability density function of the supremum of the solution to the linear stochastic heat equation. Using a general criterion for the smoothness of densities for locally nondegenerate random variables, we establish the smoothness of the joint density of the random vector whose components are the solution and the supremum of an increment in time of the solution over an interval (at a fixed spatial position), and the smoothness of the density of the supremum of the solution over a space-time rectangle that touches thet=0 axis. Applying the properties of the divergence operator, we establish a Gaussian-type upper bound on these two densities respectively, which presents a close connection with the Holder-continuity properties of the solution.

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In this thesis, we study systems of linear and/or non-linear stochastic heat equations and fractional heat equations in spatial dimension $1$ driven by space-time white noise. The main topic is the study of hitting probabilities for the solutions to these systems.
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