Parabolic Frequency on Gaussian Spaces and Unique Continuation

Abstract

We establish an almost-monotonicity formula for a parabolic frequency on Gaussian spaces for solutions of the Ornstein-Uhlenbeck heat equation with lower-order terms: $$\partial_t u = L_\gamma u + b(x,t) \cdot \nabla u + c(x,t)u, $$ where $L_\gamma = \Delta - x \cdot \nabla$ is the Ornstein-Uhlenbeck operator. In contrast to classical results that require $b$ and $c$ to be bounded, we only assume that $b$ is bounded and $c$ satisfies a linear growth condition, while the solution $u$ is allowed to have at most exponential quadratic growth. The key innovation is a weighted $L^2$ framework that uses the backward Mehler kernel as a weight, which naturally encodes the underlying measure and compensates for the unbounded coefficients. From the frequency monotonicity, we derive the strong unique continuation principle. This extends Poon's seminal results and complements recent geometric generalizations by Colding and Minicozzi in the context of Gaussian measure spaces. We further apply our framework to establish unique continuation for equations with potentials exhibiting quadratic growth or certain singularities.