A variational approach to Ornstein-Uhlenbeck evolution equations and obstacle problems

Abstract

We develop a variational formulation of the Ornstein-Uhlenbeck evolution equation based on a Gaussian divergence structure, in which the drift term is absorbed into the first variation of a weighted Dirichlet energy rather than treated as a lower-order perturbation. For the linear equation, the distributional, energy weak, and variational notions of solution coincide and determine the same unique solution. The same framework yields a variational formulation of the associated parabolic obstacle problem, for which we prove existence and then uniqueness under a standard additional regularity assumption on the obstacle. The same mechanism applies to nonlinear Gaussian gradient flows associated with convex integrands of standard $p$-growth. Finally, we extend the Gaussian-divergence formulation to the force-free kinetic Fokker-Planck equation with periodic position variables, establishing the same equivalence for the linear equation.