Finite Speed of Propagation and Waiting Times for the Stochastic Porous Medium Equation: A Unifying Approach

In this paper, we develop an energy method to study finite speed of propagation and waiting time phenomena for the stochastic porous-media equation with linear multiplicative noise in up to three spatial dimensions. Based on a novel iteration technique and on stochastic counterparts of weighted integral estimates used in the deterministic setting, we formulate a sufficient criterion on the growth of initial data which locally guarantees a waiting time phenomenon to occur almost surely. Up to a logarithmic factor, this criterion coincides with the optimal criterion known from the deterministic setting. Our technique can be modified to prove finite speed of propagation as well.