Sticky-Reflected Stochastic Heat Equation Driven by Colored Noise

Abstract

We prove the existence of a sticky-reflected solution to the heat equation on the space interval [0, 1] driven by colored noise. The process can be interpreted as an infinite-dimensional analog of the sticky-reflected Brownian motion on the real line but, in this case, the solution obeys the ordinary stochastic heat equation, except the points where it reaches zero. The solution has no noise at zero and a drift pushes it to stay positive. The proof is based on a new approach that can also be applied to some other types of SPDEs with discontinuous coefficients.