Fractional diffusion limit of a linear kinetic transport equation in a bounded domain
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Speakers: Pedro Aceves- Sanchez, Department of Mathematics, University of Arizona
Title: Fractional diffusion limit of a linear kinetic transport equation in a bounded domain
Abstract: In recent years, the study of evolution equations with a fractional Laplacian has received much attention due to the fact that they have been successfully applied to the modeling of a wide variety of phenomena ranging from biology to physics to finance. The stochastic process behind fractional operators is associated with an $\alpha$-stable process in all space, in contrast to the Laplacian operator, which is associated with a Brownian stochastic process.
In addition, evolution equations involving fractional Laplacians offer new interesting, and very challenging mathematical problems. There are several equivalent definitions of the fractional Laplacian in the entire domain. In a bounded domain, however, there are several possibilities depending on the stochastic process under consideration.
In this talk, we will present results on the rigorous transition from a velocity-jumping stochastic process in a bounded domain to a macroscopic evolution equation with a fractional Laplace operator. More precisely, we will consider the long-time/small mean-free path asymptotic behavior of the solutions of a rescaled linear kinetic transport equation in a smooth bounded domain.