Pathwise Geometry of Parabolic Flows

This paper explores the pathwise geometric properties of solutions to parabolic flows, particularly those driven by stochastic noise. Traditional studies of parabolic partial differential equations often focus on properties of expectations or ensemble averages. However, in many physical and mathematical contexts, the behavior of individual sample paths is crucial. We develop a framework for analyzing the geometric features of these paths, such as their regularity, curvature, and topological evolution, within the context of stochastic differential geometry. By employing tools from rough path theory, Malliavin calculus, and geometric measure theory, we establish conditions for pathwise smoothness and investigate how noise influences the intrinsic geometry of the evolving manifolds or interfaces. Our findings contribute to a deeper understanding of the interplay between stochastic perturbations and geometric evolution, with implications for fields ranging from materials science to general relativity.