# Stochastic models for fully coupled systems of nonlinear parabolic equations

## TL;DR

This paper develops a probabilistic framework for representing solutions to complex coupled nonlinear parabolic PDE systems modeling interacting populations, enabling new stochastic analysis tools.

## Contribution

It introduces a novel probabilistic representation for fully coupled nonlinear parabolic systems using stochastic equations and Markov processes.

## Key findings

- Probabilistic representation of coupled PDE systems established.
- Stochastic equations derived for nonlinear Markov processes.
- Framework applicable to models of spatial population segregation.

## Abstract

We construct a probabilistic representation of a system of fully coupled parabolic equations arising as a model describing spatial segregation of interacting population species. We derive a closed system of stochastic equations such that its solution allows to obtain a probabilistic representation of a weak solution of the Cauchy problem for the PDE system. The corresponded stochastic system is presented in the form of a system of stochastic equations describing nonlinear Markov processes and their multiplicative functionals.