# Time fractional equations and probabilistic representation

## TL;DR

This paper investigates fractional-time parabolic equations, establishing existence, uniqueness, and probabilistic representations via inverse subordinators, and relates occupation measures of time-changed Markov processes to original processes.

## Contribution

It provides new results on the existence, uniqueness, and probabilistic representation of solutions for fractional-time parabolic equations, including explicit relations for occupation measures.

## Key findings

- Established existence and uniqueness of solutions.
- Derived probabilistic representations using inverse subordinators.
- Connected occupation measures of time-changed and original Markov processes.

## Abstract

In this paper, we study the existence and uniqueness of solutions for general fractional-time parabolic equations of mixture type, and their probabilistic representations in terms of the corresponding inverse subordinators with or without drifts. An explicit relation between occupation measure for Markov processes time-changed by inverse subordinator in open sets and that of the original Markov process in the open set is also given.