SMALL BALL PROBABILITIES AND A SUPPORT THEOREM FOR THE STOCHASTIC HEAT EQUATION
成果类型：
Article
署名作者：
Athreya, Siva; Joseph, Mathew; Mueller, Carl
署名单位：
Indian Statistical Institute; Indian Statistical Institute Kolkata; University of Rochester
刊物名称：
ANNALS OF PROBABILITY
ISSN/ISSBN：
0091-1798
DOI：
10.1214/21-AOP1515
发表日期：
2021
页码：
2548-2572
关键词：
摘要：
We consider the following stochastic partial differential equation on t >= 0, x is an element of [0, J], J >= 1, where we consider [0, J] to be the circle with end points identified, partial derivative(t)u(t, x) = 1/2 partial derivative(2)(x) u(t, x) + g(t, x, u) + sigma(t, x, u) (W) over dot (t, x), (W) over dot (t, x) is 2-parameter d-dimensional vector valued white noise and sigma is function from R+ x R x R-d to space of symmetric d x d matrices which is Lipschitz in u. We assume that sigma 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.