Small ball probabilities and a support theorem for the stochastic heat equation

Article Type

Research Article

Publication Title

Annals of Probability

Abstract

We consider the following stochastic partial differential equation on t > 0, x e [0, J ], J≥ 1, where we consider [0, J ] to be the circle with end points identified, (Formula presented) W (t, x) is 2-parameter d-dimensional vector valued white noise and σ is function from ℝ+ × ℝ×ℝd to space of symmetric d × d matrices which is Lipschitz in u. We assume that σ is uniformly elliptic and that g is uniformly bounded. Assuming that u (0, x) ≡ 0, we prove small ball probabilities for the solution u. We also prove a support theorem for solutions, when u (0, x) is not necessarily zero.

First Page

2548

Last Page

2572

DOI

10.1214/21-AOP1515

Publication Date

9-1-2021

Comments

Open Access, Green

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