### RIT on Levy Processes and the Fractional Laplacian Archives for Fall 2021 to Spring 2022

#### 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.

#### Stable Laws - II

When: Wed, November 3, 2021 - 12:00pm
Where: Zoom Meeting ID: 928 8697 6341
Speaker: Michael Rozowski (UMD-College Park) -
Abstract: We will continue the discussion on stable laws. First, we present some physical motivation arising from anomalous diffusion. The second piece of motivation is related to the distribution of "hitting locations" of Brownian motion. This material is taken from chapters 11 and 12 of Uchaikin and Zolotarev's book Chance and Stability: Stable Distributions and their Applications. Then, we'll start a proof characterizing stable distributions on $\mathbb{R}$ as those arising from a generalized central limit theorem. This follows Shiryaev's presentation in his text Probability-1.

#### Stable Distributions III: Equivalence of self similarity and a generalized central limit theorem

When: Wed, November 10, 2021 - 12:00pm
Where: Zoom Meeting ID: 928 8697 6341
Speaker: Michael Rozowski (UMD-College Park) -
Abstract: We first demonstrate that all stable distributions are infinitely divisible. Then, we will show that the self similarity property they possess, which we take as the definition of a stable law, is equivalent to the distribution arising from a central limit theorem wherein the hypothesis of finite second moments is dropped. The proof will be completed for distributions on $\mathbb{R}$, and then some comments will be make about adjusting the proof to work on $\mathbf{R}^d$. This material is largely from the third edition of Albert Shiryaev's book Probability-I.

#### Stable Laws IV - The Range of the Index of Stability

When: Wed, November 17, 2021 - 12:00pm
Where: Zoom Meeting ID - 928 8697 6341
Speaker: Michael Rozowski (UMD-College Park) -
Abstract: We'll continue the discussion on stable laws. We will say a few words about adapting the proof of the generalized central limit theorem for random variables on $\mathbb{R}^d$. This essentially boils down to adapting the proof of the convergence of types lemma. Without proof, we mention that it is possible to characterize the form of the norming constants $a_n$ in the definition of stable laws. In particular, if $X$ is a stable random variable, then for each $n \in \mathbb{N}$ there exists $(X_k)_{k=1}^n$ i.i.d. with the same distribution as $X$ and constants $\sigma > 0$, $\alpha > 0$, and $b_n \in \mathbb{R}^d$ such that $X_1 + \cdots + X_n =_d \sigma n^{1/\alpha} X + b_n$. The constant $\alpha$ is called the index of stability, and its value is central to many of a stable law's properties. For example, it describes the tail behavior of the distribution and constrains the moments that exist. We will prove that the index of stability is constrained to $0 < \alpha \le 2$ by adapting an argument made in Chapter 3, Section 13 of Ken-iti Sato's book Lévy Processes and Infinitely Divisible Distributions. This is relevant for our study of the fractional Laplacian because it indicates what powers of the fractional Laplacian may be obtained through this probabilistic approach.

#### Stable Laws V - Range on Index of Stability

When: Wed, November 24, 2021 - 12:00pm
Where: Zoom Meeting ID: 928 8697 6341
Speaker: Michael Rozowski (UMD-College Park) -
Abstract: We will continue the march towards showing the index of stability $\alpha$ of a stable distribution satisfies $0 < \alpha \le 2$. The strategy is to show that, for any stable random variable, the norming constants in the definition of stability take the form $a_n = n^{1/\alpha}$. For this, we follow John Nolan's approach in his book Univariate Stable Distributions. Since any stable law is infinitely divisible, the Lévy-Khintchine formula specifies the form of a stable law's characteristic function. Then, using the uniqueness of the characteristic triplet in the Lévy-Khintchine formula and the definition of stability, we can argue that the index $\alpha$ must satisfy the proposed bounds, $0 < \alpha \le 2$. For this, we follow Ken-iti Sato's book Lévy Processes and Infinitely Divisible Distributions.

#### Introduction to Lévy Processes

When: Wed, December 8, 2021 - 12:00pm
Where: Zoom: 928 8697 6341
Speaker: Michael Rozowski (UMD, College Park) -
Abstract: We will introduce Lévy processes, some fundamental examples, and some essential distributional properties before we move on to the semigroups they induce. This material follows the beginning of Chapter 2 from David Applebaum's text Lévy Processes and Stochastic Calculus.

#### Feller semigroups, the infinitesimal generator of a Lévy process, and the fractional heat equation

When: Wed, December 15, 2021 - 10:00am
Where: https://umd.zoom.us/j/92886976341?pwd=Sm9Mampobm8rUDdnNndFNkVhVFlHQT09
Speaker: Michael Rozowski (UMD, College Park) -
Abstract: In our last meeting of the semester, we will discuss the semigroup of operators on the bounded Borel functions that is induced by a Markov transition function defined from a Lévy process. Such a semigroup enjoys a number of nice regularity properties that make it a Feller semigroup. We'll take a bird's eye view for a bit to discuss some results regarding the infinitesimal generator of a Feller semigroup, such as an abstract ODE it provides and whose solution is described by the action of the semigroup. We then return to earth and provide a characterization of the infinitesimal generator for a semigroup induced by Lévy process. By specializing further to an $\alpha$-stable rotationally invariant Lévy process, we obtain the fractional Laplacian. By combining this with the abstract ODE, we obtain a probabilistic solution to the fractional heat equation on $\mathbb{R}^d$.

#### Equivalent forms of the fractional Laplacian and an introduction to fractional Sobolev spaces

When: Wed, February 2, 2022 - 11:00am
Where: https://umd.zoom.us/j/93208511028
Speaker: Michael Rozowski (UMD, College Park) -
Abstract: Last semester we ended with two equivalent forms of the fractional Laplacian: 1) the infinitesimal generator of the semigroup induced by the $2s$-stable isotropic Lévy process ($0 \lt s \lt 1$), and 2) as a Fourier multiplier with symbol $\vert\xi\vert^{2s}$. We examine two other forms of this operator in terms of integrated difference quotients in the physical variables. We also introduce fractional Sobolev spaces as those which are natural domains for these operators, and then we show their equivalence.

#### A survey of fractional Laplacians on bounded domains

When: Wed, February 9, 2022 - 11:00am
Where: https://umd.zoom.us/j/93208511028
Speaker: Michael Rozowski (UMD, College Park) -
Abstract: We turn our attention to the possibilities for fractional Laplacians on bounded domains in $\mathbb{R}^d$, including the spectral, censored, and integral fractional Laplacians. Comparisons of their action on functions common to their domains are made. We describe the stochastic processes for which each operator is the infinitesimal generator.

#### A discussion on a fractional Poisson problem

When: Wed, February 16, 2022 - 11:00am
Where: https://umd.zoom.us/j/93208511028
Speaker: Michael Rozowski (University of Maryland, College Park) -
Abstract: We will prove existence, uniqueness, and stability in $\tilde{H}^s(\Omega)$ of weak solutions to the fractional Poisson problem $(-\Delta)^s u = f$ on a bounded domain $\Omega \subset \bR^d$ with exterior Dirichlet conditions. An ingredient needed will be a Poincar\'e-type inequality, which we'll prove along the way. We then show weak solutions satisfy a maximum principle, which will imply a comparison principle.

#### Survey of regularity for solutions of the fractional Poisson equation

When: Wed, March 9, 2022 - 11:00am
Where: https://umd.zoom.us/j/93208511028
Speaker: Michael Rozowski (University of Maryland, College Park) -

#### Sobolev and Besov Regularity for the fractional Poisson problem

When: Wed, March 16, 2022 - 11:00am
Where: https://umd.zoom.us/j/93208511028?pwd=S3VyNWd4ZjV6engyTm9Yc2x6TEZ6dz09
Speaker: Michael Rozowski and Ricardo H. Nochetto (University of Maryland, College Park) -
Abstract: We will discuss Sobolev regularity of solutions to
\label{eq:fractionalPoisson}
\begin{aligned}
\left\{
(-\Delta)^s u &amp;= f, &amp;&amp; \text{on } \Omega, \\
u &amp;= 0, &amp;&amp; \text{on } \mathbb{R}^d \setminus \Omega,
\end{aligned}
\right.

where $\Omega \subset \mathbb{R}^d$ is a bounded domain. Surprisingly, the possible regularity will mimic the case when $f$ is a constant and $\Omega$ is the open unit ball.

Then Prof. Nochetto will introduce Besov regularity for solutions to \eqref{eq:fractionalPoisson}.

#### Besov regularity of solutions to the fractional Poisson problem

When: Wed, March 23, 2022 - 11:00am
Where: https://umd.zoom.us/meeting/93208511028?occurrence=1648047600000
Speaker: Ricardo H. Nochetto (University of Maryland, College Park) - https://www.math.umd.edu/~rhn/

#### Sobolev and Besov Regularity Results in Fractional Laplacian

When: Wed, April 27, 2022 - 11:00am
Where: https://umd.zoom.us/j/93208511028?pwd=S3VyNWd4ZjV6engyTm9Yc2x6TEZ6dz09
Speaker: Yaqi Wu (UMD, College Park) -
Abstract: Before moving to numerical methods for integral fractional Laplacian, we will finish the discussion of regularity results in this talk. I will review several Sobolev regularity results based on Holder and Besov regularity. Later on, I will need these Sobolev regularity results for error estimates in the following 2-3 weeks. In the presentation, some intuitions of weighted Sobolev space and Besov space will be covered. I will also give a brief idea of the proof of weighted Sobolev regularity result in the case $$1/2 &lt; s &lt; 1$$. If time permits, the whole proof will be given. For the case $$0 &lt; x &lt; 1/2$$ and $$s = 1/2$$, the strategy is similar.

#### A priori error estimates of finite element scheme with quasi-uniform meshes for fractional Laplacian problem

When: Wed, May 4, 2022 - 11:00am
Where: Zoom Meeting ID: 932 0851 1028
Speaker: Yaqi Wu (University of Maryland, College Park) -
Abstract: In this talk, we will introduce the finite element discretization of fractional Laplacian problem with homogeneous Dirichlet boundary condition, and briefly mention several difficulties in the implementation. After that, we will focus on the error estimates (in energy norm and $$L^2$$-norm) on quasi-uniform meshes. Full proof of error estimates in energy norm will be given by combining regularity results and local interpolation estimates. If time permits, we will show the proof of $$L^2$$-error estimates with a duality argument.

#### Finite element approximation of integral fractional Laplacian with graded meshes: error analysis and mesh construction

When: Wed, May 11, 2022 - 11:00am
Where: Zoom Meeting ID: 932 0851 1028
Speaker: Yaqi Wu (UMD, College Park) -
Abstract: In this talk, we will present error analysis of finite element scheme with graded meshes. Based on the behavior of solution near the boundary, we have weighted Sobolev regularity with weight depending on some power of distance to the boundary. A natural consideration is to apply graded meshes to capture it. We will talk about the derivation of grading structure and give error estimates in energy norm and $$L^2$$ norm. For simplicity, we only focus on dimension 2 case. Such result can be extended to higher dimensions.