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.
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BibTeX
@misc{ipocoana2026variational,
title = {{A variational approach to Ornstein-Uhlenbeck evolution equations and obstacle problems}},
author = {Erica Ipocoana and Annalaura Rebucci},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21003},
eprint = {2609.21003},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21003},
}
RIS
TY - UNPB
TI - A variational approach to Ornstein-Uhlenbeck evolution equations and obstacle problems
AU - Erica Ipocoana
AU - Annalaura Rebucci
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.21003
UR - https://arxiv.org/abs/2609.21003
AB - 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.
ER -