Mosco convergence of gradient forms with non-convex potentials
A talk in the Bielefeld Stochastic Afternoon series by
Simon Wittmann
| Abstract: | The work presented in this talk provides a new approach to address Mosco convergence of gradient-type Dirichlet forms, $\mathcal E^N$ on $L^2(E,\mu_N)$ for $N\in\mathbb N$,
in the framework of converging Hilbert spaces by K.~Kuwae and T.~Shioya.
The basic assumption is the weak measure convergence of the family of
${(\mu_N)}_{N}$ on the state space $E$ - either the Euclidean space or a locally convex topological vecor space.
Apart from that, the conditions on ${(\mu_N)}_{N}$ try to impose as little restrictions as possible.
The problem has fully been solved if the family ${(\mu_N)}_{N}$
contain only log-concave measures, due to L. Ambrosio, G. Savaré and L. Zambotti.
Although a complete understanding of the non-log-concave case is out of reach, some conditions under which it can be handled
are known thanks to the results of A.~Kolesnikov.
This talk shows an alternative approach and complements those results.
The techniques of our proofs are genuinely new, as they incorporate methods of Finite Elements, familiar from numerical analysis.
Via the well-known relation between Mosco convergence and weak convergence of marginals of Markov processes,
we reach the central application, a statement on the scaling limit in the context of dynamical stochastic interface models.
We generalize a convergence result from, where skew reflection in the corresponding SPDE leads to a non-log-concave, non-continuous
density perturbing the Gaussian reference measure of the asymptotic Dirichlet form.
The model we consider provides more versatility in the class of approximating reference measures $\mu_N$. Within the CRC this talk is associated to the project(s): A5, B1 |