turkmath.org

There is a strong connection between Analysis and Probability Theory. The classical results of Analysis can be obtained by using tools of Martingale Theory, and Brownian motion as the underlying process. Brownian motion is a very specific Levy process which embraces many nice properties where these properties allows one to reproduce probabilistic alternatives of classical tools. On the other hand, it is possible to study more general Levy processes with some cost. In this talk, we will discuss the question that to what extent we can generalise this process provided that the connection with Analysis is not lost. We will consider some results on bounded linear operators on $L^p(\mathbb{R}^d)$ when the underlying process is taken to be the symmetric stable process which shares some fundamental properties of Brownian motion. We will argue the tools obtained by means of this more general process, and define a new class of harmonic functions to work with. We will also provide a Harnack's Inequality for this new class of harmonic functions.