# On the fundamental solution of heat and stochastic heat equations

## TL;DR

This paper establishes bounds for the fundamental solution of divergence form heat equations with irregular time coefficients and explores the existence of solutions for related stochastic PDEs under natural noise assumptions.

## Contribution

It provides new upper bounds for the fundamental solution with minimal regularity assumptions and demonstrates the existence of mild solutions for stochastic heat equations with anticipating stochastic integration.

## Key findings

- Spatial derivatives of fundamental solutions follow Aronson type bounds.
- Existence of mild solutions for stochastic heat equations with natural noise assumptions.
- Bounds depend only on ellipticity and coefficient derivatives' norms.

## Abstract

We consider the generic divergence form second order parabolic equation with coefficients that are regular in the spatial variables and just measurable in time. We show that the spatial derivatives of its fundamental solution admit upper bounds that agree with the Aronson type estimate and only depend on the ellipticity constants of the equation and the L $\infty$ norm of the spatial derivatives of its coefficients. We also study the corresponding stochastic partial differential equations and prove that under natural assumptions on the noise the equation admits a mild solution, given by anticipating stochastic integration.