Thesis work
My thesis work was in the field of probabilistic potential theory. I was primarily interested in processes associated to integro-differential operators with kernels possessing critically low singularities, and in questions such as the presence of Harnack inequalities, continuity of harmonic functions and well-posedness of martingale problems for such operators.
These operators are connected to well known processes which I considered in my research. These include a large class of subordinate Brownian motions, in particular the geometric stable processes.
Probabilistic potential theory lies in the intersection of many broad fields such as partial differential equations and stochastic processes. For example, any Feller process possesses an infinitesimal generator which gives rise to a class of Dirichlet problems. Solutions to these Dirichlet problems can be expressed and studied using functionals of the process, and probabilistic estimates may be used to study their regularity. Similarly, there are many connections between potential theory and stochastic processes. Thus, the field is filled with numerous problems and utilizes techniques from various areas. I dealt with three of these problems
Formulating an "elliptic" Harnack inequality for geometric stable processes over the whole of R^d, d>= 1.
Proving that such processes have "infinite conformal walk dimension", in that even after time-changing the process and changing the underlying metric, such processes cannot satisfy a "parabolic" Harnack inequality. The above two questions accomplished the aim of wedging the gap between the elliptic and parabolic Harnack inequalities, and opening up further questions on the relation between them.
Proving the well-posedness of martingale problems associated to low-regularity integro-differential operators. Here's a very, very rough idea : imagine a Markov process on R^d which is pure-jump, but the way it jumps out of a point (that is, it's random decision on where to go next from here) depends upon the point it's jumping out of. Let's call this probability measure the "jump kernel" at each point. We prove that if someone tells me that a process should have a jump kernel "similar" to a geometric stable process at each point, and if close-by points have similar jump kernels, then there is a unique process that can be constructed according to this person's requirements. Essentially, one can stitch together a process that jumps out of different points in different ways, and this jump is very heavy/fat tailed.
Present research work
However, subordinate Brownian motions go beyond pure mathematical applications. Random patterns that are associated to such objects appear in multiple places, including extreme value theory (where the rare events of a well-behaved stochastic process can be tilted to look like functionals of a subordinated BM which has applications in climate and finance), fractal geometry (where running a jump process on a fractal, or any nice geometric object, is capable of exposing rarer facets of its geometry quicker than a diffusion : this is used, for instance, to sample from irregularly organized data like image spaces and text spaces) and more.
My present research work lies in understanding some of these applications and their real-life implications. For instance, we have tried to understand, recently, whether exponential tilting of random variables and vectors can be performed using diffusion. It can be argued that for exponential tilting, using a diffusion that isn't driven by a Gaussian noise but rather a fat-tailed or Levy noise could allow us to perform rare-event sampling better. We have some results in this direction. Similarly, we are also trying to understand the implications of discretizing Levy process-driven differential equations, and how much this solution differs from the original continuous solution.