Large scale asymptotics for Markov processes in the analytic framework of Mosco-Kuwae-Shioya

  • In this thesis, a new concept to prove Mosco convergence of gradient-type Dirichlet forms within the \(L^2\)-framework of K.~Kuwae and T.~Shioya for varying reference measures is developed. The goal is, to impose as little additional conditions as possible on the sequence of reference measure \({(\mu_N)}_{N\in \mathbb N}\), apart from weak convergence of measures. Our approach combines the method of Finite Elements from numerical analysis with the topic of Mosco convergence. We tackle the problem first on a finite-dimensional substructure of the \(L^2\)-framework, which is induced by finitely many basis functions on the state space \(\mathbb R^d\). These are shifted and rescaled versions of the archetype tent function \(\chi^{(d)}\). For \(d=1\) the archetype tent function is given by \[\chi^{(1)}(x):=\big((-x+1)\land(x+1)\big)\lor 0,\quad x\in\mathbb R.\] For \(d\geq 2\) we define a natural generalization of \(\chi^{(1)}\) as \[\chi^{(d)}(x):=\Big(\min_{i,j\in\{1,\dots,d\}}\big(\big\{1+x_i-x_j,1+x_i,1-x_i\big\}\big)\Big)_+,\quad x\in\mathbb R^d.\] Our strategy to obtain Mosco convergence of \(\mathcal E^N(u,v)=\int_{\mathbb R^d}\langle\nabla u,\nabla v\rangle_\text{euc}d\mu_N\) towards \(\mathcal E(u,v)=\int_{\mathbb R^d}\langle\nabla u,\nabla v\rangle_\text{euc}d\mu\) for \(N\to\infty\) involves as a preliminary step to restrict those bilinear forms to arguments \(u,v\) from the vector space spanned by the finite family \(\{\chi^{(d)}(\frac{\,\cdot\,}{r}-\alpha)\) \(|\alpha\in Z\}\) for a finite index set \(Z\subset\mathbb Z^d\) and a scaling parameter \(r\in(0,\infty)\). In a diagonal procedure, we consider a zero-sequence of scaling parameters and a sequence of index sets exhausting \(\mathbb Z^d\). The original problem of Mosco convergence, \(\mathcal E^N\) towards \(\mathcal E\) w.r.t.~arguments \(u,v\) form the respective minimal closed form domains extending the pre-domain \(C_b^1(\mathbb R^d)\), can be solved by such a diagonal procedure if we ask for some additional conditions on the Radon-Nikodym derivatives \(\rho_N(x)=\frac{d\mu_N(x)}{d x}\), \(N\in\mathbb N\). The essential requirement reads \[\frac{1}{(2r)^d}\int_{[-r,r]^d}|\rho_N(x)- \rho_N(x+y)|d y \quad \overset{r\to 0}{\longrightarrow} \quad 0 \quad \text{in } L^1(d x),\, \text{uniformly in } N\in\mathbb N.\] As an intermediate step towards a setting with an infinite-dimensional state space, we let $E$ be a Suslin space and analyse the Mosco convergence of \(\mathcal E^N(u,v)=\int_E\int_{\mathbb R^d}\langle\nabla_x u(z,x),\nabla_x v(z,x)\rangle_\text{euc}d\mu_N(z,x)\) with reference measure \(\mu_N\) on \(E\times\mathbb R^d\) for \(N\in\mathbb N\). The form \(\mathcal E^N\) can be seen as a superposition of gradient-type forms on \(\mathbb R^d\). Subsequently, we derive an abstract result on Mosco convergence for classical gradient-type Dirichlet forms \(\mathcal E^N(u,v)=\int_E\langle \nabla u,\nabla v\rangle_Hd\mu_N\) with reference measure \(\mu_N\) on a Suslin space $E$ and a tangential Hilbert space \(H\subseteq E\). The preceding analysis of superposed gradient-type forms can be used on the component forms \(\mathcal E^{N}_k\), which provide the decomposition \(\mathcal E^{N}=\sum_k\mathcal E^{N}_k\). The index of the component \(k\) runs over a suitable orthonormal basis of admissible elements in \(H\). For the asymptotic form \(\mathcal E\) and its component forms \(\mathcal E^k\), we have to assume \(D(\mathcal E)=\bigcap_kD(\mathcal E^k)\) regarding their domains, which is equivalent to the Markov uniqueness of \(\mathcal E\). The abstract results are tested on an example from statistical mechanics. Under a scaling limit, tightness of the family of laws for a microscopic dynamical stochastic interface model over \((0,1)^d\) is shown and its asymptotic Dirichlet form identified. The considered model is based on a sequence of weakly converging Gaussian measures \({(\mu_N)}_{N\in\mathbb N}\) on \(L^2((0,1)^d)\), which are perturbed by a class of physically relevant non-log-concave densities.

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Author:Simon Wittmann
Advisor:Martin Grothaus
Document Type:Doctoral Thesis
Language of publication:English
Publication Date:2023/03/07
Year of Publication:2023
Publishing Institute:Rheinland-Pfälzische Technische Universität Kaiserslautern-Landau
Granting Institute:Rheinland-Pfälzische Technische Universität Kaiserslautern-Landau
Acceptance Date of the Thesis:2023/11/04
Date of the Publication (Server):2023/03/07
Tag:Coxeter-Freudenthal-Kuhn triangulation; Mosco convergence; infinite-dimensional analysis
Number of page:IV, 117
Faculties / Organisational entities:Kaiserslautern - Fachbereich Mathematik
DDC-Cassification:5 Naturwissenschaften und Mathematik / 510 Mathematik
MSC-Classification (mathematics):60-XX PROBABILITY THEORY AND STOCHASTIC PROCESSES (For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX)
Licence (German):Creative Commons 4.0 - Namensnennung, nicht kommerziell, keine Bearbeitung (CC BY-NC-ND 4.0)