RIT on Levy Processes and the Fractional Laplacian Archives for Academic Year 2021


Heuristic derivations of fractional-order problems from random walks with long jumps

When: Wed, September 8, 2021 - 12:00pm
Where: https://umd.zoom.us/j/92886976341?pwd=Sm9Mampobm8rUDdnNndFNkVhVFlHQT09
Speaker: Michael Rozowski (UMD-CP) -
Abstract: We will present heuristic derivations of two problems: 1) the heat equation on $\mathbb{R}^d$ with the fractional Laplacian and 2) the Dirichlet problem for the fractional Laplacian. Our starting point will be random walks with long jumps. It follows the first chapter in the text Nonlocal Diffusion and Applications by Claudia Bucur and Enrico Valdinoci.

During this talk, we will highlight questions that the RIT aims to answer by the end of the semester. Such questions include: on what spaces of functions is the fractional Laplacian well-defined, in what spaces we should look for solutions to fractional-order problems, and what are "reasonable" boundary conditions to impose for problems featuring the fractional Laplacian? It is our perspective that the starting point for answering the last question should be stochastic processes, which motivates our study of Lévy processes for a good chunk of the semester.

A brisk review of necessary probability theory

When: Wed, September 15, 2021 - 12:00pm
Where: https://umd.zoom.us/j/92886976341?pwd=Sm9Mampobm8rUDdnNndFNkVhVFlHQT09
Speaker: Michael Rozowski (UMD-CP) -
Abstract: We'll review the probability theory necessary for our development of Lévy processes. We will focus on material that is particular to probability but not necessarily measure theory. Such topics include the Doob-Dynkin lemma, conditional expectation, the Chebyshev-Markov inequality, independence, convergence in distribution, and characteristic functions. The material follows Section 1.1 of Lévy Processes and Stochastic Calculus, 2ed, by David Applebaum.

The conclusion of the probability review and an intro to infinitely divisible laws

When: Wed, September 22, 2021 - 12:00pm
Where: https://umd.zoom.us/j/92886976341?pwd=Sm9Mampobm8rUDdnNndFNkVhVFlHQT09
Speaker: Stavros Papathanasiou (UMD-CP) -
Abstract: We will finish our presentation on the necessary background on probability theory that is perhaps not included in a course on measure theory. This includes material on independence, convergence in law, and characteristic functions. Then, we will start the material on infinitely divisible distributions. The reading material can be found in Sections 1.1 and 1.2 of David Applebaum's text, Lévy Processes and Stochastic Calculus, 2ed.

An Introduction to Infinitely Divisible Laws and the Lévy-Khintchine Formula

When: Wed, September 29, 2021 - 12:00pm
Where: https://umd.zoom.us/j/92886976341?pwd=Sm9Mampobm8rUDdnNndFNkVhVFlHQT09
Speaker: Michael Rozowski (UMD-CP) -
Abstract: We will finish discussing the results on characteristic functions and then introduce the class of infinitely divisible laws. Infinitely divisible laws are essential to our study of Lévy processes since the value of a Lévy process at any fixed time is a random vector whose distribution is in this class. After presenting some examples of such laws, we turn to a characterization of them in terms of their characteristic function, which is the Lévy-Khintchine formula. This material can be found in Section 1.2 of David Applebaum's text Lévy Processes and Stochastic Calculus, 2ed.

The Lévy-Khintchine Formula and an Intro to Stable Laws

When: Wed, October 6, 2021 - 12:00pm
Where: https://umd.zoom.us/j/92886976341?pwd=Sm9Mampobm8rUDdnNndFNkVhVFlHQT09
Speaker: Michael Rozowski (UMD-CP) -
Abstract: We will present the Lévy-Khintchine formula, which characterizes the characteristic function of an infinitely divisible law on $\mathbb{R}^d$. We present one half of the proof of this characterization, namely that the function provided by the formula is a characteristic function of an infinitely divisible law. Towards this end, we present an argument that is doomed to fail, but in pursuing this strategy a solution to the argument's deficiency is revealed. Moreover, this solution points to a generic feature of infinitely divisible laws: they are weak limits of compound Poisson laws. Then we describe stable laws, which are a subset of the infinitely divisible laws, and introduce the rotationally invariant stable law's relationship with the fractional Laplacian.

An informal review of probability theory and infinitely divisible distributions

When: Wed, October 13, 2021 - 12:00pm
Where: https://umd.zoom.us/j/92886976341?pwd=Sm9Mampobm8rUDdnNndFNkVhVFlHQT09
Speaker: Leonid Koralov (UMD-College Park) - https://www.math.umd.edu/~koralov/
Abstract: We will re-visit some of the probability concepts presented so-far, avoiding technicalities and focusing on their significance in the context of examples. The aim is to reinforce what has been discussed up until this point informally, answer questions, and discuss a few key points so we may have a common language and point of departure before advancing into deeper material.

Informal review of characteristic functions and infinite divisibility

When: Wed, October 20, 2021 - 12:00pm
Where: https://umd.zoom.us/j/92886976341?pwd=Sm9Mampobm8rUDdnNndFNkVhVFlHQT09
Speaker: Leonid Koralov (UMD-College Park) - https://www.math.umd.edu/~koralov/
Abstract: We will continue the informal review. This includes characteristic functions and infinite divisibility and perhaps the Lévy-Khintchine formula.

An Introduction to Stable Laws

When: Wed, October 27, 2021 - 12:00pm
Where: https://umd.zoom.us/j/92886976341?pwd=Sm9Mampobm8rUDdnNndFNkVhVFlHQT09
Speaker: Michael Rozowski (UMD-College Park) -
Abstract: Stable laws are a subset of the infinitely divisible laws that we've been studying. They can be viewed with two equivalent lenses. First, they are a class containing the possible limiting distributions from suitably normalized sums of independent random variables. Thus, the stable laws appear in generalizations of the central limit theorem. Secondly, they may be viewed as distributions which are ``nearly invariant'' under convolution, up to a translation and scaling. After examining some properties, we'll look at the isotropic $\alpha$-stable law, which is closely related to the fractional Laplacian.