Subordinate Brownian Motions and Their Applications

A Levy process is simply a process with stationary and independent increments. Brownian motion is an example of a Levy process. A non-Brownian Levy process has discontinuous trajectories and usually has heavy tails. Levy processes form a very rich family and they are widely used in various fields. However, general Levy processes are not very tractable.

An increasing Levy process S_t with S_0=0 is called a subordinator. A subordinate Brownian motion can be obtained from a Brownian motion by replacing its time parameter by an independent subordinator. More precisely, if B_t is a Brownian motion and S_t is a subordinator independent of B, then the process X_t=B_{S_t} is called a subordinate Brownian motion. The subordinator S can be thought of as "operational time" or "physical time". Subordinate Brownian motions form a very large subclass of Levy processes and they have been applied in various fields. Compared to general Levy processes, subordinate Brownian motions are much more tractable.

In recent years, a lot of progress has been made in the study of subordinate Brownian motion. In this talk, I will give a survey of some of these recent results and mention some areas where subordinate Brownian motions have been applied to.

Speaker Bio: Renming Song is a Professor in the Department of Mathematics at the University of Illinois. His research interests include stochastic analysis, Markov processes, potential theory, and Dirichlet forms. He previously taught at the University of Michigan and received his Ph.D. from the University of Florida. He has published over 90 research papers and 3 books, and is an editor of the Journal of the Korean Mathematical Society. He has also edited two collections: Selected Works of Donald L. Burkholder and A Collection of Articles in Honor of Don Burkholder.

Presentation Slides: PDF

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