# Heat Kernel Estimates for Fractional Heat Equation

## TL;DR

This paper investigates the long-term behavior of solutions to fractional heat equations, focusing on their fundamental solutions and the effects of different types of fractional derivatives on their asymptotic properties.

## Contribution

It provides new insights into the asymptotic behavior of fundamental solutions for fractional evolution equations with various fractional derivatives.

## Key findings

- Cesaro means of solutions exhibit specific long-time decay properties.
- Stable subordinators influence the fractional derivatives considered.
- Results extend to distributed order derivatives and general convolution operators.

## Abstract

We study the long-time behavior of the Cesaro means of fundamental solutions for fractional evolution equations corresponding to random time changes in the Brownian motion and other Markov processes. We consider both stable subordinators leading to equations with the Caputo-Djrbashian fractional derivative and more general cases corresponding to differential-convolution operators, in particular, distributed order derivatives.