Derivation of the Chapman-Kolmogorov type equation from a stochastic hybrid system

TL;DR

This paper explores how to derive a Chapman-Kolmogorov type PDE from a stochastic hybrid system, bridging stochastic and PDE modeling approaches in biology to enhance understanding and analysis.

Contribution

It provides a method to connect stochastic hybrid systems with PDE models, facilitating better analysis and parameterization in biological modeling.

Findings

Established a derivation method linking stochastic hybrid systems to PDEs.

Highlighted the close connection between stochastic and PDE models.

Facilitated potential integration of biological phenomena into PDE frameworks.

Abstract

Both stochastic and PDE modeling approaches have been used and compared in various context in biology. Typically, stochastic models are easier to parameterize, can be used to integrate underlying biological phenomena, but hard to analyze mathematically and usually computational expensive. PDE models are amenable to mathematical analysis and computational methods are well-developed, but biological processes are usually lumped together in them and parameters harder to obtain. Therefore it is important to find connections of different approaches. In this note, I give some investigation of methods to derive a Chapman-Kolmogorov type PDE model from a stochastic hybrid system, highlighting the close connection between these two.