Nonlinear Integrals, Diffusion in Random Environments and Stochastic Partial Differential Equations

Abstract

In this dissertation, we investigate various problems in the analysis of stochastic (partial) differential equations. A part of the dissertation introduces several notions of nonlinear integrations. Some differential equations associated with nonlinear integrations are investigated. Examples include transport differential equations in space-time random fields and parabolic equations with potentials of the type $\partial_t W$, where $W$ is continuous in time variable and smooth in the spatial variables. Another part of the dissertation studies nonlinear stochastic convolution equations driven by a multiplicative Gaussian noise which is white in time and which has the covariance of a fractional Brownian motion with Hurst parameter $H\in(1/4,1/2)$ in the spatial variable. The other part of the dissertation gives rigorous meaning to the Brox differential equation $X(t)=\cB(t)-\frac12\int_0^t \dot{W}(X(s))ds$ where $\cB$ and $W$ are independent Brownian motions. Furthermore, it is shown that the Brox differential equation has a unique strong solution which is a time-changed spatial transformation of a Brownian motion. Along the way, some appropriate tools are developed in order to solve these problems. In particular, we establish a multiparameter version of Garsia-Rodemich-Rumsey inequality which allows one to control rectangular increments in any dimensions of multivariate functions, definitions and compact criteria for some new functions spaces are developed. The methodologies employed form a combination of stochastic analysis, Malliavin calculus and functional analytic tools. Several parts of the dissertation are joint work of the author with Yaozhong Hu, Jingyu Huang, David Nualart, Leonid Mytnik and Samy Tindel.