Interacting Particle Systems and Stochastic PDEs

The heat equation is a mathematical equation used to describe the flow of heat through a purely conducting substance. This equation and many variations of it have been studied for around two hundred years, and the mathematical theory is now well developed. Over the last few decades mathematicians and physicists have been interested in the more realistic stochastic heat equation which considers the effect of having impurities randomly scattered in the medium. Certain transformations of this equation are also used to describe many other physical phenomena, for example motion of a turbulent fluid, deposition of snow and bacterial growth.

Although many advances have been made in our understanding of the stochastic heat equation, much more remains to be discovered. One of the goals of this proposal is to describe features of this equation for a wide class of impurities in the medium. The proposal also looks into ways of approximating this equation so that one can simulate the equation on a computer, which in turn will give us further intuition on its behaviour. We shall also explore connections of this equation to other systems exhibiting randomness.

Stochastic partial differential equations (SPDEs), of which the stochastic heat equation is the most well studied example, are increasingly being used to model several physical phenomena like flow of heat through a random medium, motion of a strand of DNA, internal structure of the Sun, motion of a turbulent fluid, bacterial growth, deposition of snow etc. These equations are therefore of great interest to researchers in the physical sciences.

The primary beneficiaries of this research are mathematicians, particularly probabilists, and physicists. Some parts of this proposal, especially the part dealing with approximations of SPDEs, will be of immediate interest to researchers interested in simulations of these equations on the computer.

The papers coming from this proposal shall be put on arXiv so that all researchers have access to it. I shall speak about the research in seminars and international conferences to increase the visibility of the work.

I plan to organize a 3 day workshop in 2017 to bring together researchers from the U.K. and abroad working in the areas related to this proposal. I will also look to provide support for young researchers to attend this workshop.