A support theorem for parabolic stochastic PDEs with nondegenerate Hölder diffusion coefficients
Yi Han
TL;DR
The article addresses support and small-ball properties for parabolic SPDEs with nondegenerate Hölder diffusion in the solution variable by combining a Girsanov-based reduction with sharp Gaussian-type estimates for the stochastic convolution. It derives explicit two-sided small-ball probabilities and shows that, for Hölder exponent $\alpha\in(0,1]$, the solution law has full support on Wiener space, despite the lack of Lipschitz continuity in $\sigma$. The core method includes a truncation argument, a Gaussian comparison, and a refined time-space scaling to manage Hölder regularity, producing upper and lower bounds whose exponents depend on $\alpha$ and the Hölder constant. This extends support theorem results beyond Lipschitz settings and highlights how regularization by noise interacts with parabolic scaling to yield probabilistic control on fine scales. The findings have implications for understanding robustness of SPDEs under rough coefficients and for quantifying small-ball behavior in nonlinear stochastic heat equations.
Abstract
In this paper we work with parabolic SPDEs of the form $$ \partial_t u(t,x)=\partial_x^2 u(t,x)+g(t,x,u)+σ(t,x,u)\dot{W}(t,x) $$ with Neumann boundary conditions, where $x\in[0,1]$, $\dot{W}(t,x)$ is the space-time white noise on $(t,x)\in[0,\infty)\times [0,1]$, $g$ is uniformly bounded, and the solution $u\in\mathbb{R}$ is real valued. The diffusion coefficient $σ$ is assumed to be uniformly elliptic but only Hölder continuous in $u$. Previously, support theorems for SPDEs have only been established assuming that $σ$ is Lipschitz continuous in $u$. We obtain new support theorems and small ball probabilities in this $σ$ Hölder continuous case via the recently established sharp two sided estimates of stochastic integrals.