Stochastic analysis and partial differential equations: theory and numerics

Abstract

This thesis is concerned with problems at the interface of stochastic analysis and partial differential equations (PDEs). In particular, we focus on two different classes of problem: the transport of measures under the flow of nonlinear PDEs and the global well-posedness of singular stochastic PDEs. First we present a work, joint with J. Forlano, studying the transport of Gaussian measures on periodic functions under the flow of the fractional nonlinear Schrödinger equation on the 1-dimensional torus. In particular, we prove that Gaussian measures are quasi-invariant under this flow. We present another work, joint with P. Sosoe and T. Xiao, demonstrating that, in the 3-dimensional setting, Gaussian measures on periodic functions are quasi-invariant under the flow of nonlinear wave equations with polynomial nonlinearity. With regard to singular stochastic PDEs, under certain conditions on the coefficients, we prove the global well-posedness of the renormalised stochastic complex Ginzburg-Landau equation on the 2-dimensional torus. To conclude this thesis, from a numerical perspective we study the transport of Gaussian measures, under the flow of nonlinear PDEs, by way of Monte-Carlo simulation. More specifically, for several PDEs, we generate a large number of solutions, with initial data sampled from a Gaussian measure, and then examine statistical properties of the ensemble of solutions. These simulations illustrate the equations and problems studied in this thesis and give insight into conjectures beyond current theoretical techniques.