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Abstract

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.

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