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

Authors

Siva Athreya, Indian Statistical Institute, Kolkata
Mathew Joseph, Indian Statistical Institute, Kolkata
Carl Mueller, University of Rochester

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

Recommended Citation

Athreya, Siva; Joseph, Mathew; and Mueller, Carl, "Small ball probabilities and a support theorem for the stochastic heat equation" (2021). Journal Articles. 1806.
https://digitalcommons.isical.ac.in/journal-articles/1806