Solutions to the Linear Stochastic Heat and Wave Equation

Abstract

The solutions to certain stochastic partial differential equations with linear Gaussian noise constitute interesting examples of self-similar processes. In this chapter we analyze these classes of self-similar processes. We focus on the solution to the linear heat and wave equation driven by a Gaussian noise which behaves as a Brownian motion or fractional Brownian motion with respect to the time variable and is white or colored with respect to the space variable. We consider various aspects of these self-similar processes. In particular we present the conditions for the existence of the solution, the sharp regularity of their trajectories, we study the law of the solution to the linear heat equation and its connection with the bifractional Brownian motion.