Fully Nonlinear Stochastic Partial Differential Equations

This thesis is concerned with the theory and applications of certain fully nonlinear stochastic partial differential equations. First, we present several new results regarding the well-posedness of the equations. Among these are proofs of the comparison principle for equations with nontrivial spatial dependence. We also prove some new path-stability estimates, and we give a very general proof of existence using Perron's method, which characterizes the unique solution as the maximal sub-solution. We also discuss a general framework for approximating solutions numerically. A variety of convergent approximation schemes are considered, including finite difference schemes and Trotter-Kato splitting formulas, and the results are general enough to allow for many more examples. For first-order equations, we derive explicit error estimates. Finally, we introduce a family of homogenization problems that arise from scaling limits of fully nonlinear equations with highly oscillatory spatio-temporal dependence. We prove, under suitable assumptions on the nonlinearities and the random dependence, that the limiting behavior is governed by a spatially homogenous, stochastic Hamilton-Jacobi equation.