Date of Award
1-1-2025
Document Type
Dissertation
Degree Name
Ph.D. in Mathematics
First Advisor
Jeremy Clark
Second Advisor
Jeremy Clark
Third Advisor
Kevin Beach
School
University of Mississippi
Relational Format
dissertation/thesis
Abstract
We construct a family of time-inhomogeneous two-dimensional diffusion processes, defined over a finite time interval [0, T]. These diffusions have transition density functions that are given by the integral kernels of the semigroup corresponding to the two-dimensional Schrödinger operator with a point potential at the origin. Although, in a few ways, our processes of interest are closely related to two-dimensional Brownian motion, they have a singular drift pointing in the direction of the origin that is strong enough to enable the possibility of visiting there with positive probability. Our main focus is on characterizing a local time process at the origin for these diffusions and on studying the law of its process inverse. Before introducing our model, we provide an overview of some preliminary topics related to Brownian motion, including the Wiener measure, the transience versus recurrence behavior determined by the dimension of Brownian motion, and Brownian local time. Additionally, we review some key concepts in stochastic calculus that are useful for understanding the material discussed in the later chapters.
Recommended Citation
Mian, Barkat, "On Planar Brownian Motion Singularly Tilted Through A Point Potential" (2025). Electronic Theses and Dissertations. 3332.
https://egrove.olemiss.edu/etd/3332