# Space-time coupled evolution equations and their stochastic solutions

## TL;DR

This paper introduces a class of space-time coupled evolution equations derived from subordination of the heat operator, extending non-Markovian process models with broad applications, and provides theoretical results on their solutions.

## Contribution

It develops a general framework for CEEs with initial conditions, spatial operators, and forcing terms, including existence, uniqueness, and stochastic representations of solutions.

## Key findings

- Established existence and uniqueness of solutions.
- Derived stochastic representations for solutions.
- Extended models of non-Markovian processes with broader applicability.

## Abstract

We consider a class of space-time coupled evolution equations (CEEs), obtained by a subordination of the heat operator. Our CEEs reformulate and extend known governing equations of non-Markovian processes arising as scaling limits of continuous time random walks, with widespread applications. In particular we allow for initial conditions imposed on the past, general spatial operators on Euclidean domains and a forcing term. We prove existence, uniqueness and stochastic representation for solutions.