Martin Boundary and the Nonlinear Multiplicative Stochastic Heat Equation in Weak Disorder

We study invariant random fields of nonlinear multiplicative stochastic heat equations in the weak disorder regime. Under a natural second-moment condition, we show that positive invariant fields are in one-to-one correspondence with bounded positive harmonic functions of the underlying space. This implies that the space of invariant fields inherits the structure of the Martin boundary. We also show that whenever the deterministic heat flow converges to a bounded harmonic function, the stochastic evolution converges to the corresponding invariant field. The results apply to many settings with nontrivial Martin boundary, such as negatively curved manifolds and trees.