On mild and weak solutions for stochastic heat equations with piecewise-constant conductivity

TL;DR

This paper studies stochastic heat equations with piecewise-constant conductivity, introducing a weak solution concept and proving its equivalence to the mild solution, advancing understanding of such equations with discontinuous coefficients.

Contribution

It defines a weak solution for stochastic heat equations with discontinuous coefficients and proves its equivalence to the existing mild solution, enhancing theoretical understanding.

Findings

Weak solution concept introduced for stochastic heat equations with piecewise-constant conductivity.

Equivalence between weak and mild solutions established.

Provides a foundation for analyzing stochastic PDEs with discontinuous coefficients.

Abstract

We investigate a stochastic partial differential equation with second order elliptic operator in divergence form, having a piecewise constant diffusion coefficient, and driven by a space-time white noise. We introduce a notion of weak solution of this equation and prove its equivalence to the already known notion of mild solution.