Mosco convergence of gradient forms with non-convex interaction potential
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| Format: | Preprint |
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2021
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| author | Grothaus, Martin Wittmann, Simon |
| author_facet | Grothaus, Martin Wittmann, Simon |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2105_05140 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Mosco convergence of gradient forms with non-convex interaction potential Grothaus, Martin Wittmann, Simon Probability 60J46, 47D07, 82M10 (Primary) 60B12 (Secondary) 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. |
| title | Mosco convergence of gradient forms with non-convex interaction potential |
| topic | Probability 60J46, 47D07, 82M10 (Primary) 60B12 (Secondary) |
| url | https://arxiv.org/abs/2105.05140 |