Stochastic analysis

Our research focus here employs probabilistic and analytic methods to explore complex random systems. At its core, the methods developed and utilized include a wide array of tools from stochastic analysis, regularity structures, paracontrolled distributions and point processes. For instance, stochastic analysis provides a framework for studying systems influenced by randomness, allowing us to model and predict the evolution of these systems over time. Regularity structures and paracontrolled distributions, relatively recent developments in the field, offer a powerful way to handle singularities and irregularities that often arise in singular stochastic partial differential equations (SPDEs), such as the Kardar-Parisi-Zhang (KPZ) equation as well as the dynamic $\Phi^4$ models appearing in the realm of quantum field theory.

In the domain of Kardar-Parisi-Zhang (KPZ) universality class, a unique perspective emerges from stochastic PDEs which are supercritical within the framework of regularity structures and paracontrolled distributions. An example of such a stochastic PDE is  given by the KPZ equation in supercritical dimensions which is directly linked to a random directed polymers. Our research builds on new developments between the latter and the theory of Gaussian multiplicative chaos on path spaces.

Point processes, which are random collections of points in a given space, serve as another fundamental aspect of our research.  The objectives here, on one hand, is to construct geometries or metrics on the space of point processes to understand how evolve and interact, both spatially and temporally. On the other hand, our incentive is to develop and blend tools from point processes, stochastic analysis and large deviations to understand effective/ large scale behavior of Gibbs measures  defined by singular interactions of Brownian motions. A pertinent example here is the construction of the Polaron measure and analyzing its effective mass.