It is known that the transition probabilities of a classical Brownian motion satisfy the associated forward Kolmogorov equation, which is a diffusion equation involving a first-order time derivative. In many applications, however, Kolmogorov type equations with fractional order time derivatives are employed to model "sub-diffusions," in which particles spread more slowly than the classical Brownian motion predicts. Generally speaking, stochastic processes describing such sub-diffusions are obtained by applying a time-change to classical processes such as Brownian motion and Levy processes, where the simplest time-change to be considered is the generalized inverse, or equivalently, the first hitting time process, of a stable subordinator.
In this talk, I will first introduce properties of the time-change, followed by discussions on derivations of some important classes of fractional order PDEs. A lot of interesting papers on time-changed processes have been published, some of which will be presented along with related open problems during the talk. ( Slides)
Speaker: Kei Kobayashi, Department of Mathematics, Tufts University