## 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.