Regularity of Stochastic Burgers’-Type Equations

Abstract

In classical partial differential equations (PDEs), it is well known that the solution to Burgers' equation in one spatial dimension with positive viscosity can be solved by the so called Hopf-Cole transformation, which linearizes the PDE. In particular, this converts Burgers' equation to the linear heat equation, which can be solved explicitly. On the other hand, the Feynman-Kac formula is a tool that can be used to solve the heat equation probabilistically. An interesting and perhaps surprising result which we prove is that one can still make sense of these approaches to Burgers' equation in the presence of space-time white noise, which is very rough. After proving that a suitable Feynman-Kac representation solves stochastic Burgers' equation under a Hopf-Cole transformation, we study some regularity properties of this solution. In particular, we prove moment estimates and Holder continuity, which can be thought of as how ``big'' the solution gets in time and space, and how ``rough'' this solution can be. From this, we then obtain sub-exponential moments and bounds on the tails of the probability distribution for the solution. Prior to this work, no results about any kinds of moment estimates or tails of distributions for stochastic Burgers'-type equations had been established. Furthermore, only one publication on Burgers' equation contains a discussion of Holder regularity. Given the solution to a stochastic partial differential equation (SPDE), it is natural to ask whether this stochastic process has a well-behaved probability law. For example, does the solution have a smooth probability density function or just an absolutely continuous one? Using some powerful tools from Malliavin calculus, we answer this question for stochastic Burgers' equation with our Hopf-Cole solution. Finally, we study regularity of the probability law of the solution to a more general class of semilinear SPDEs which contain Burgers' equation as an example. These results take a less tangible approach since there is no explicit representation for solutions to these equations. However, as we will see, there are some clever techniques and interesting results that can be used to establish such properties. For example, we prove a comparison theorem for this class of SPDEs which, interestingly enough, will be instrumental in obtaining regularity of the probability density function of the solution at fixed points in time and space. The projects in this thesis are joint work of the author and David Nualart.