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This thesis contains three manuscripts addressing the application of stochastic processes to the analysis and solution of partial differential equations (PDEs) in mathematical physics.
In the first manuscript, one dimensional diffusion and Burgers equation are considered. The Fourier transform of the solution to each PDE is represented as the expected value of a multiplicative functional on a branching stochastic process. Monte Carlo simulation schemes are then
developed to perform accurate numerical calculations of the solution.
The second manuscript considers an advection-diffusion PDE in a cylinder where the diffusion coefficient and flow velocity are constant in the direction of fluid flow, but are arbitrarily non-smooth in the transversal direction. The stochastic process associated with the PDE is constructed as a diffusion process with the appropriate infinitesimal generator. The properties of ergodic Markov processes are then used to obtain a homogenization result for the solution of the PDE.
The third manuscript studies the stochastic process associated with the one-dimensional diffusion equation in the case where the
diffusion coefficient is piecewise constant with a countable set of discontinuities. The resulting process generalizes skew Brownian motion to the case of countable many interfaces. Finally, some applications to advection-diffusion phenomena in two-dimensional layered media are outlined. |
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