Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Mosco convergence of gradient forms with non-convex interaction potential

Martin Grothaus, Simon Wittmann

Abstract

This article provides a new approach to address Mosco convergence of gradient-type Dirichlet forms, $\mathcal E^N$ on $L^2(E,μ_N)$ for $N\in\mathbb N$, in the framework of converging Hilbert spaces by K.~Kuwae and T.~Shioya. The basic assumption is weak measure convergence of the family ${(μ_N)}_{N}$ on the state space $E$ - either a separable Hilbert space or a locally convex topological vector space. Apart from that, the conditions on ${(μ_N)}_{N}$ try to impose as little restrictions as possible. The problem has fully been solved if the family ${(μ_N)}_{N}$ contain only log-concave measures, due to L.~Ambrosio, G.~Savaré and L.~Zambotti, 2009. However for a large class of convergence problems the assumption of log-concavity fails. The article suggests a way to overcome this hindrance, as it presents a new approach. Combining the theory of Dirichlet forms with methods from numerical analysis we find abstract criteria for Mosco convergence of standard gradient forms with varying reference measures. These include cases in which the measures are not log-concave. To demonstrate the accessibility of our abstract theory we discuss a first application, generalizing an approximation result by S.~K.~Bounebache and L.~Zambotti, 2014.

Mosco convergence of gradient forms with non-convex interaction potential

Abstract

This article provides a new approach to address Mosco convergence of gradient-type Dirichlet forms, on for , in the framework of converging Hilbert spaces by K.~Kuwae and T.~Shioya. The basic assumption is weak measure convergence of the family on the state space - either a separable Hilbert space or a locally convex topological vector space. Apart from that, the conditions on try to impose as little restrictions as possible. The problem has fully been solved if the family contain only log-concave measures, due to L.~Ambrosio, G.~Savaré and L.~Zambotti, 2009. However for a large class of convergence problems the assumption of log-concavity fails. The article suggests a way to overcome this hindrance, as it presents a new approach. Combining the theory of Dirichlet forms with methods from numerical analysis we find abstract criteria for Mosco convergence of standard gradient forms with varying reference measures. These include cases in which the measures are not log-concave. To demonstrate the accessibility of our abstract theory we discuss a first application, generalizing an approximation result by S.~K.~Bounebache and L.~Zambotti, 2014.

Paper Structure

This paper contains 13 sections, 13 theorems, 286 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Mosco convergence of gradient forms
  3. Interface models with skew reflection
  4. Outline
  5. Finite Elements
  6. Triangulation and tent functions
  7. L2 and energy estimates
  8. Preliminaries on Mosco convergence and main results
  9. Basic terminology and the theorem of Mosco-Kuwae-Shioya
  10. Convergence of superposed standard gradient forms
  11. Application to infinite-dimensional problems and a first example
  12. Mosco convergence of standard gradient forms on Fréchet spaces
  13. A Gaussian measure and orthogonal projections: An example with a non-convex perturbing potential

Key Result

Theorem 2.1

Let $r\in(0,\infty)$. The space $\mathbb R^d$ admits a partition $\{D_T|T\in \mathscr T_r\}$ and a partition of unity ${(\chi_r^{\alpha})}_{\alpha\in r\mathbb Z^d}$ with the following properties.

Figures (2)

  • Figure 1: The Coxeter-Freudenthal-Kuhn triangulation of the unit cube for $d=3$.
  • Figure 2: The primal tent function $\chi_1^0$ for $d=2$ with its hexagonal support.

Theorems & Definitions (31)

  • Theorem 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Remark 3.3
  • Theorem 3.4
  • ...and 21 more