# Markovian Integral Equations

## TL;DR

This paper studies multidimensional Markovian integral equations linked to time-inhomogeneous Markov processes, establishing key properties like uniqueness, stability, and existence, and connecting solutions to path-dependent PDEs.

## Contribution

It introduces new conditions for solution continuity, provides a multidimensional Feynman-Kac formula, and proves global existence and uniqueness in one dimension.

## Key findings

- Established uniqueness, stability, and existence of solutions.
- Provided a multidimensional Feynman-Kac representation.
- Proved global existence and uniqueness in one dimension.

## Abstract

We analyze multidimensional Markovian integral equations that are formulated with a time-inhomogeneous progressive Markov process that has Borel measurable transition probabilities. In the case of a path-dependent diffusion process, the solutions to these integral equations lead to the concept of mild solutions to semilinear parabolic path-dependent partial differential equations (PPDEs). Our goal is to establish uniqueness, stability, existence, and non-extendibility of solutions among a certain class of maps. By requiring the Feller property of the Markov process, we give weak conditions under which solutions become continuous. Moreover, we provide a multidimensional Feynman-Kac formula and a one-dimensional global existence- and uniqueness result.