#### Title

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

#### 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

## Comments

Open Access, Green