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

We study the nonlinear multiplicative stochastic heat equation on Dirichlet spaces with white in time noise under weak disorder. We show that positive invariant random fields with uniformly bounded second moments are in one-to-one correspondence with bounded positive harmonic functions. The proof combines a remote past pullback construction with a uniqueness argument that applies a contraction inspired by chaos expansion. As a consequence, the space of invariant measures inherits geometric structure from the Martin Boundary. We further establish a small-noise Gaussian fluctuation result within each harmonic sector and show that the long-time behavior of solutions is completely determined by the Martin boundary data of the initial condition. These results reveal a direct connection between stochastic PDE dynamics and boundary theory in potential analysis.

💡 Research Summary

The paper investigates the nonlinear multiplicative stochastic heat equation (SHE) on a general Dirichlet space (M,d,µ) driven by space‑time white noise. The equation is written as
∂ₜu(t,x)=Lu(t,x)+β f(u(t,x)) · Ẇ(t,x), t>0, x∈M,
where L is the self‑adjoint generator of the heat semigroup, β>0 is an inverse‑temperature parameter, f is a Lipschitz function with f(0)=0, and Ẇ is a centered Gaussian noise with covariance kernel R. The authors impose a key integrability condition (Assumption 1.1) on R, namely that the double integral of the product of two heat kernels against |R| is uniformly bounded by a constant Λ. This condition is verified in Euclidean spaces of dimension d≥3, in negatively curved manifolds (e.g., hyperbolic space), and on regular trees, among other settings.

The first main result (Theorem 1.2) shows that if (β L_f)² Λ<1 (where L_f is the Lipschitz constant of f), the system lies in the weak‑disorder regime: the second moment of the solution remains uniformly bounded for all time and space. Moreover, under the same condition there is a bijective correspondence between positive invariant random fields Z(x) (i.e., stationary in law under the SHE dynamics) with uniformly bounded second moments and bounded positive harmonic functions h on M. In particular, E