Degree

Doctor of Philosophy (PhD)

Department

Mathematics

Document Type

Dissertation

Abstract

The Boltzmann equation describes the time evolution of the density function in position-velocity space for a classical particle subjected to possible collisions by other particles in a diluted gas that expands in vacuum for a given initial distribution. While many authors have studied the probabilistic interpretation of the spatially homogeneous Boltzmann equation, there is a dearth of articles on the stochastic framework of the full (that is, spatially inhomogeneous) Boltzmann equation. In this thesis, we examine a stochastic process, developed by S. Albevario, B. Ruediger, and P. Sundar, whose law is a weak solution to a mollified Boltzmann equation. This process is aptly named the Boltzmann-Enskog process, and it is driven by a Poisson random measure whose compensator includes the law of the Boltzmann-Enskog process. We establish the existence of a probability density for the velocity component of Boltzmann-Enskog processes using a functional-analytic criterion due to A. Debussche and M. Romito. This work was inspired by a similar treatment of the spatially homogeneous Boltzmann equation, by N. Fournier. In a later work by M. Friesen, B. Ruediger, and P. Sundar, they establish the uniqueness of solutions under sufficient moment estimates. Assuming these moment estimates are satisfied, we establish the convergence in distribution of a sequence of Boltzmann-Enskog processes under the assumption of varying collision kernels using the convergence of martingale problems. To show this, we establish tightness of this sequence and use a criterion due to A.G. Bhatt and R.L. Karandikar.

Date

3-27-2025

Committee Chair

Sundar, Padmanabhan

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