Exploring families of energy-dissipation landscapes via tilting -- three types of EDP convergence

Abstract

We introduce two new concepts of convergence of gradient systems $(\mathbf{Q},{\mathcal{E}}_{\epsilon },{\mathcal{R}}_{\epsilon })$ to a limiting gradient system $(\mathbf{Q},{\mathcal{E}}_0,{\mathcal{R}}_0)$. These new concepts are called `EDP convergence with tilting' and `contact--EDP convergence with tilting'. Both are based on the Energy-Dissipation-Principle (EDP) formulation of solutions of gradient systems, and can be seen as refinements of the Gamma-convergence for gradient flows first introduced by Sandier and Serfaty. The two new concepts are constructed in order to avoid the `unnatural' limiting gradient structures that sometimes arise as limits in EDP-convergence. EDP-convergence with tilting is a strengthening of EDP-convergence by requiring EDP-convergence for a full family of `tilted' copies of $(\mathbf{Q},{\mathcal{E}}_{\epsilon },{\mathcal{R}}_{\epsilon })$. It avoids unnatural limiting gradient structures, but many interesting systems are non-convergent according to this concept. Contact--EDP convergence with tilting is a relaxation of EDP convergence with tilting, and still avoids unnatural limits but applies to a broader class of sequences $(\mathbf{Q},{\mathcal{E}}_{\epsilon },{\mathcal{R}}_{\epsilon })$. In this paper we define these concepts, study their properties, and connect them with classical EDP convergence. We illustrate the different concepts on a number of test problems.

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• Cosh gradient systems and tilting
• Homogenization of Wasserstein gradient flows
• Evolutionary $\Gamma$-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions
• The Energy-Dissipation Principle for stochastic parabolic equations
• Large Deviations of Irreversible Processes