Interacting Particle Systems and Stochastic PDEs

Lead Research Organisation: University of Sheffield
Department Name: Probability and Statistics

Abstract

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.

Planned Impact

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.

Publications

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Joseph M (2018) An invariance principle for the stochastic heat equation in Stochastics and Partial Differential Equations: Analysis and Computations

 
Description Stochastic Partial differential equations are mathematical equations which are used to describe various physical phenomena, for example the flow of heat, turbulent fluid motion, growth of interfaces etc. The simulation of these equations (for example, on a computer) is not an easy task due to their highly irregular behavior. My collaborators and I have come up with approximation schemes for a class of these equations which would be useful for researchers interested in modelling such phenomena.
Exploitation Route We consider approximation schemes for a special class of stochastic partial differential equations. One could investigate whether these schemes could be applicable for other models.
Sectors Education,Other

URL https://arxiv.org/abs/1611.06829
 
Description LMS conference grant Scheme 1
Amount £4,500 (GBP)
Funding ID 11672 
Organisation London Mathematical Society 
Sector Learned Society
Country United Kingdom
Start 06/2017 
End 07/2017
 
Description Mathematics and Statistics Research Centre Funding
Amount £1,300 (GBP)
Organisation University of Sheffield 
Sector Academic/University
Country United Kingdom
Start 06/2017 
End 07/2017
 
Description Paper with Foondun and Li 
Organisation University of Strathclyde
Department Department of Electronic & Electrical Engineering
Country Unknown 
Sector Academic/University 
PI Contribution The collaboration with Mohammud Foondun and Shiu-Tang Li was on a paper about the approximation of stochastic partial differential equations. It was a truly collaborative effort in which we were all involved in formulating and proving the main results in the paper.
Collaborator Contribution It was a joint effort, see above.
Impact A paper "An approximation result for a class of stochastic heat equations with colored noise" has been submitted to the Annals of Applied Probability. The EPSRC grant was used to cover a trip to visit Mohammud Foondun in the final stages of the preparation of the article.
Start Year 2014
 
Description Paper with Foondun and Li 
Organisation University of Utah
Department Department of Mathematics
Country United States 
Sector Academic/University 
PI Contribution The collaboration with Mohammud Foondun and Shiu-Tang Li was on a paper about the approximation of stochastic partial differential equations. It was a truly collaborative effort in which we were all involved in formulating and proving the main results in the paper.
Collaborator Contribution It was a joint effort, see above.
Impact A paper "An approximation result for a class of stochastic heat equations with colored noise" has been submitted to the Annals of Applied Probability. The EPSRC grant was used to cover a trip to visit Mohammud Foondun in the final stages of the preparation of the article.
Start Year 2014