[Paper Review] Stochastic Partial Differential Equations on Evolving Surfaces and Evolving Riemannian Manifolds

Charles M. Elliott, Martin Hairer+1 more|arXiv (Cornell University)|Aug 29, 2012

Nonlinear Partial Differential Equations15 references3 citations

TL;DR

This paper formulates and establishes existence and uniqueness for stochastic partial differential equations (SPDEs) on evolving Riemannian manifolds using the variational approach, extending SPDE theory to moving surfaces and manifolds with random metrics. It provides a rigorous framework for SPDEs on compact, oriented, evolving hypersurfaces and general evolving Riemannian manifolds, including nonlinear operators like the p-Laplace-Beltrami operator, with applications to stochastic heat equations under general geometric evolution.

ABSTRACT

We formulate stochastic partial differential equations on Riemannian manifolds, moving surfaces, general evolving Riemannian manifolds (with appropriate assumptions) and Riemannian manifolds with random metrics, in the variational setting of the analysis to stochastic partial differential equations. Considering mainly linear stochastic partial differential equations, we establish various existence and uniqueness theorems.

Motivation & Objective

• To develop a variational framework for SPDEs on Riemannian manifolds, addressing the lack of such theory in the mathematical literature.
• To extend SPDE theory to moving surfaces and evolving Riemannian manifolds, where the metric evolves in time or is random.
• To establish existence and uniqueness results for linear and nonlinear SPDEs, including stochastic heat and p-Laplace-Beltrami equations, on geometrically evolving domains.
• To enable the use of natural noise structures on evolving manifolds by reinterpreting evolution as a time-dependent metric rather than a family of surfaces.
• To lay the foundation for future research on SPDEs with manifold-valued solutions and long-time behavior under topological and metric evolution.

Proposed method

• Adopt the variational approach to SPDEs, following Prévôt and Röckner (2007), to define solutions in Hilbert spaces of functions on evolving manifolds.
• Define Sobolev spaces and prove the Poincaré inequality on compact Riemannian manifolds to ensure coercivity and compactness for existence theorems.
• Formulate SPDEs on evolving hypersurfaces using a geometric conservation law, leading to the stochastic heat equation with normal velocity evolution.
• Reinterpret evolving manifolds as a single manifold with a time-dependent metric, enabling a more natural choice of noise and equivalence to surface-based formulations under technical assumptions.
• Use the Laplace-Beltrami and p-Laplace-Beltrami operators in local coordinates via the metric tensor $ g_{ij} $, with the stochastic term driven by a Hilbert-Schmidt operator acting on a Wiener process.
• Address circularity in solution-dependent spaces by considering isotropic metric evolution and seeking alternative solution spaces or notions like viscosity solutions.

Experimental results

Research questions

• RQ1 How can the variational approach to SPDEs be extended to Riemannian manifolds, particularly when the manifold itself evolves in time?
• RQ2 What conditions ensure existence and uniqueness of solutions to SPDEs on moving hypersurfaces evolving with normal velocity?
• RQ3 How does the formulation of SPDEs change when the evolution is described by a time-dependent metric rather than a family of surfaces?
• RQ4 Can the theory be extended to SPDEs with nonlinearities in the differential operator, such as the p-Laplace-Beltrami operator, on evolving domains?
• RQ5 What are the implications for long-time behavior and topological changes in the manifold when the metric or surface evolves stochastically?

Key findings

• The paper establishes existence and uniqueness for linear SPDEs on compact, oriented, evolving Riemannian manifolds under standard variational assumptions (H1–H4).
• For the stochastic heat equation on a moving surface, existence and uniqueness are proven under the assumption that points evolve with normal velocity only and the surface remains compact and without boundary.
• The nonlinear stochastic heat equation with the p-Laplace-Beltrami operator is shown to have a unique solution on a general moving surface, with nonlinearity only in the diffusion term and not in the derivatives.
• Under technical conditions, the formulation of SPDEs on evolving manifolds via a time-dependent metric is shown to be equivalent to the surface-based formulation, enabling a more natural noise structure.
• The paper identifies a circular dependency in defining solution spaces when the metric depends on the solution, suggesting the need for alternative solution concepts such as viscosity solutions.
• The framework opens pathways for future research on SPDEs with manifold-valued solutions and long-time behavior under stochastic metric and topological evolution.