Sticky-Reflected Stochastic Heat Equation Driven by Colored Noise

We prove the existence of a sticky-reflected solution to the heat equation on the spatial 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 now the solution obeys the usual stochastic heat equation except points where it reaches zero. At zero the solution has no noise and a drift pushes it to stay positive. The proof is based on a new approach that can also be applied to other types of SPDEs with discontinuous coefficients.