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Publication# Optimal lower bounds on hitting probabilities for stochastic heat equations in spatial dimension k >= 1

Abstract

We establish a sharp estimate on the negative moments of the smallest eigenvalue of the Malliavin matrix gamma z of Z := (u(s, y), u(t , x) - u(s, y)), where u is the solution to a system of d non-linear stochastic heat equations in spatial dimension k >= 1. We also obtain the optimal exponents for the L-p-modulus of continuity of the increments of the solution and of its Malliavin derivatives. These lead to optimal lower bounds on hitting probabilities of the process {u(t,x) : (t, x) is an element of [0, infinity[xR(k)} in the non-Gaussian case in terms of Newtonian capacity, and improve a result in Dalang, Khoshnevisan and Nualart [Stoch PDE: Anal Comp 1 (2013) 94-151].

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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.
We first study the properties of the probability density functions of the solution to non-linear systems of stochastic fractional heat equations driven by multiplicative space-time white noise. Using the techniques of Malliavin calculus, we prove that the one-point probability density function of the solution is infinitely differentiable, uniformly bounded and positive everywhere. Moreover, a Gaussian-type upper bound on the two-point probability density function is obtained by a detailed analysis of the small eigenvalues of the Malliavin matrix. We establish an optimal lower bound on hitting probabilities for the (non-Gaussian) solution, which is as sharp as that for the Gaussian solution to a system of linear equations.
We develop a new method to study the upper bound on hitting probabilities, from the perspective of probability density functions. For the solution to the linear stochastic heat equation, we prove that the random vector, which consists of the solution and the supremum of a linear increment of the solution over a time segment, has an infinitely differentiable probability density function. We derive a formula for this density and establish a Gaussian-type upper bound. The smoothness property and Gaussian-type upper bound for the density of the supremum of the solution over a space-time rectangle touching the $t = 0$ axis are also studied. Furthermore, we extend these results to the solutions of systems of linear stochastic fractional heat equations.
For a system of linear stochastic heat equations with Dirichlet boundary conditions, we present a sufficient condition for certain sets to be hit with probability one.