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Person# Robert Dalang

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Stochastic differential equation

A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs have m

Stochastic process

In probability theory and related fields, a stochastic (stəˈkæstɪk) or random process is a mathematical object usually defined as a sequence of random variables, where the index of the sequence has

Dimension

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a d

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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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We consider a system of d non-linear stochastic fractional heat equations in spatial dimension 1 driven by multiplicative d-dimensional space-time white noise. We establish a sharp Gaussian-type upper bound on the two-point probability density function of (u(s, y), u(t, x)). From this result, we deduce optimal lower bounds on hitting probabilities of the process {u(t, x) : (t, x) is an element of [0, infinity[xR} in the non-Gaussian case, in terms of Newtonian capacity, which is as sharp as that in the Gaussian case. This also improves the result in Dalang et al. (2009) for systems of classical stochastic heat equations. We also establish upper bounds on hitting probabilities of the solution in terms of Hausdorff measure. (C) 2020 Elsevier B.V. All rights reserved.

We study vector-valued solutions u(t, x) is an element of R-d to systems of nonlinear stochastic heat equations with multiplicative noise, partial derivative/partial derivative t u(t, x) = partial derivative(2)/partial derivative x(2) u(t, x) + sigma (u(t, x)(W) over dot (t, x). Here, t >= 0, x is an element of R and (W) over dot (t, x) is an R-d-valued space-time white noise. We say that a point z is an element of R-d is polar if P{u(t, x) = z for some t > 0 and x is an element of R} = 0. We show that, in the critical dimension d = 6, almost all points in R-d are polar.