Using Malliavin calculus to solve a chemical diffusion master equation

TL;DR

This paper introduces a novel approach using Malliavin calculus to solve an infinite system of coupled Fokker-Planck equations modeling chemical diffusion, linking probabilistic and PDE methods for better analysis.

Contribution

It reformulates a complex infinite-dimensional system into a single evolution equation using Malliavin derivatives, enabling new analytical insights.

Findings

Reformulation of the infinite system as a single evolution equation.

Connection between finite-dimensional projections and Ornstein-Uhlenbeck PDEs.

Application of Gaussian analysis tools to chemical reaction modeling.

Abstract

We propose a novel method to solve a chemical diffusion master equation of birth and death type. This is an infinite system of Fokker-Planck equations where the different components are coupled by reaction dynamics similar in form to a chemical master equation. This system was proposed in [3] for modelling the probabilistic evolution of chemical reaction kinetics associated with spatial diffusion of individual particles. Using some basic tools and ideas from infinite dimensional Gaussian analysis we are able to reformulate the aforementioned infinite system of Fokker-Planck equations as a single evolution equation solved by a generalized stochastic process and written in terms of Malliavin derivatives and differential second quantization operators. Via this alternative representation we link certain finite dimensional projections of the solution of the original problem to the solution of a single partial differential equations of Ornstein-Uhlenbeck type containing as many variables as the dimension of the aforementioned projection space.