In this note, we consider the parabolic Anderson model on , driven by a Gaussian noise which is fractional in time with index and fractional in space with index such that . Under a general condition on the initial data, we prove the existence and uniqueness of the mild solution and obtain its exponential upper bounds in time for all p-th moments with .
Research supported by a grant from the Natural Sciences and Engineering Research Council of Canada (R. Balan) and partially by the Travel Support for Mathematicians from Simons Foundation (L. Chen).
The authors thank the anonymous referee for the careful reading and valuable comments.
"Parabolic Anderson model with rough noise in space and rough initial conditions." Electron. Commun. Probab. 27 1 - 12, 2022. https://doi.org/10.1214/22-ECP506