The Fundamental Solution for the 1+1-dimensional Stochastic Heat Equation

Abstract

I will explain some regularity properties of the fundamental solution to the stochastic heat equation with multiplicative noise, in one spatial dimension. This fundamental solution is a random analogue of the heat kernel. Just as the heat kernel describes Brownian motion, this fundamental solution describes a random process interacting with a random environment called the continuum directed random polymer. I will discuss how some properties of the random polymer can be used to show properties of the fundamental solution.