We consider the following stochastic partial differential equation on , where we consider to be the circle with end points identified,
is 2-parameter d-dimensional vector valued white noise and σ is function from to space of symmetric matrices which is Lipschitz in u. We assume that σ is uniformly elliptic and that g is uniformly bounded. Assuming that , we prove small ball probabilities for the solution u. We also prove a support theorem for solutions, when is not necessarily zero.
The first author was supported in part by MATRICS and CPDA grants. The second author was supported in part by a CPDA grant. The third author was supported in part by a Simons grant.
A significant part of this work was done during a Research in Pairs visit by the authors to Centre International De Recontres Mathématiques (CIRM) in July 2019. We thank the centre for the wonderful environment and hospitality.
"Small ball probabilities and a support theorem for the stochastic heat equation." Ann. Probab. 49 (5) 2548 - 2572, September 2021. https://doi.org/10.1214/21-AOP1515