Heat Kernels, Stochastic Processes and Functional Inequalities

The general topic of the 2013 workshop Heat kernels, stochastic processes and functional inequalities was the study of linear and non-linear diffusions in geometric environments: finite and infinite-dimensional manifolds, metric spaces, fractals and graphs, including random environments. The workshop brought together leading researchers from analysis, probability and geometry and provided a unique opportunity for interaction of established and young scientists from these areas. Unifying themes were heat kernel analysis, mass transport problems and related functional inequalities such as Poincar´e, Sobolev, logarithmic Sobolev, Bakry-Emery, Otto-Villani and Talagrand inequalities. These concepts were at the heart of Perelman’s proof of Poincar´e’s conjecture, as well as of the development of the Otto calculus, and the synthetic Ricci bounds of Lott-Sturm-Villani. The workshop provided participants with an opportunity to discuss how these techniques can be used to approach problems in optimal transport for non-local operators, subelliptic operators in finite and infinite dimensions, analysis on singular spaces, as well as random walks in random media

Homogenization of Random Media: Random Walks, Diffusions and Stochastic Interface Models

This thesis concerns homogenization results, in particular scaling limits and heat kernel estimates, for random processes moving in random environments and for stochastic interface models. The first chapter will survey recent research and introduce three models of interest: the random conductance model, the Ginzburg-Landau ∇φ model, and the symmetric diffusion process in a random medium. In the second chapter we present some novel research on the random conductance model; a random walk on an infinite lattice, usually taken to be Ζ^d with nearest neighbour edges, whose law is determined by random weights on the edges. In the setting of degenerate, ergodic weights and general speed measure, we present a quenched local limit theorem for this model. This states that for almost every instance of the random environment, the heat kernel, once suitably rescaled, converges to that of Brownian motion with a deterministic, non-degenerate covariance matrix. The quenched local limit theorem is proven under ergodicity and moment conditions on the environment. Under stronger, non-optimal moment conditions, we also prove annealed local limit theorems for the static RCM with general speed measure and for the dynamic RCM. The dynamic model allows for the random weights, or conductances, to vary with time. Our focus turns to the Ginzburg-Landau gradient model in the subsequent chapter. This is a model for a stochastic interface separating two distinct thermodynamic phases, using an infinite system of coupled stochastic differential equations (SDE). Our main assumption is that the potential in the SDE system is strictly convex with second derivative uniformly bounded below. The aforementioned annealed local limit theorem for the dynamic RCM is applied via a coupling relation to prove a scaling limit result for the space-time covariances in the Ginzburg-Landau model. We also show that the associated Gibbs distribution scales to a Gaussian free field. In the final chapter, we study a symmetric diffusion process in divergence form in a stationary and ergodic random environment. This is a continuum analogue of the random conductance model and similar analytical techniques are applicable here. The coefficients are assumed to be degenerate and unbounded but satisfy a moment condition. We derive upper off-diagonal estimates on the heat kernel of this process for general speed measure. Lower off-diagonal estimates are also proven for a natural choice of speed measure under an additional decorrelation assumption on the environment. Finally, using these estimates, a scaling limit for the Green’s function is derived

On the definition and the properties of the principal eigenvalue of some nonlocal operators

In this article we study some spectral properties of the linear operator $\mathcal{L}\_{\Omega}+a$ defined on the space $C(\bar\Omega)$ by :$$\mathcal{L}\_{\Omega}[\varphi] +a\varphi:=\int\_{\Omega}K(x,y)\varphi(y)\,dy+a(x)\varphi(x)$$ where $\Omega\subset \mathbb{R}^N$ is a domain, possibly unbounded, $a$ is a continuous bounded function and $K$ is a continuous, non negative kernel satisfying an integrability condition. We focus our analysis on the properties of the generalised principal eigenvalue $\lambda\_p(\mathcal{L}\_{\Omega}+a)$ defined by \lambda\_p(\mathcal{L}\_{\Omega}+a):= \sup\{\lambda \in \mathbb{R} \,|\, \exists \varphi \in C(\bar \Omega), \varphi\textgreater{}0, \textit{such that}\, \mathcal{L}\_{\Omega}[\varphi] +a\varphi +\lambda\varphi \le 0 \, \text{in}\;\Omega\}. We establish some new properties of this generalised principal eigenvalue $\lambda\_p$. Namely, we prove the equivalence of different definitions of the principal eigenvalue. We also study the behaviour of $\lambda\_p(\mathcal{L}\_{\Omega}+a)$ with respect to some scaling of $K$. For kernels $K$ of the type, $K(x,y)=J(x-y)$ with $J$ a compactly supported probability density, we also establish some asymptotic properties of $\lambda\_{p} \left(\mathcal{L}\_{\sigma,m,\Omega} -\frac{1}{\sigma^m}+a\right)$ where $\mathcal{L}\_{\sigma,m,\Omega}$ is defined by $\displaystyle{\mathcal{L}\_{\sigma,m,\Omega}[\varphi]:=\frac{1}{\sigma^{2+N}}\int\_{\Omega}J\left(\frac{x-y}{\sigma}\right)\varphi(y)\, dy}$. In particular, we prove that $$\lim\_{\sigma\to 0}\lambda\_p\left(\mathcal{L}\_{\sigma,2,\Omega}-\frac{1}{\sigma^{2}}+a\right)=\lambda\_1\left(\frac{D\_2(J)}{2N}\Delta +a\right),$$where $D\_2(J):=\int\_{\mathbb{R}^N}J(z)|z|^2\,dz$ and $\lambda\_1$ denotes the Dirichlet principal eigenvalue of the elliptic operator. In addition, we obtain some convergence results for the corresponding eigenfunction $\varphi\_{p,\sigma}$

Invariance principle for the random conductance model in a degenerate . . .

We study a continuous time random walk, X, on Z d in an environment of random conductances taking values in (0, ∞). We assume that the law of the conductances is ergodic with respect to space shifts. We prove a quenched invariance principle for X under some moment conditions on the environment. The key result on the sublinearity of the corrector is obtained by Moser's iteration scheme

Dirichlet Form Theory and its Applications

Theory of Dirichlet forms is one of the main achievements in modern probability theory. It provides a powerful connection between probabilistic and analytic potential theory. It is also an effective machinery for studying various stochastic models, especially those with non-smooth data, on fractal-like spaces or spaces of infinite dimensions. The Dirichlet form theory has numerous interactions with other areas of mathematics and sciences. This workshop brought together top experts in Dirichlet form theory and related fields as well as promising young researchers, with the common theme of developing new foundational methods and their applications to specific areas of probability. It provided a unique opportunity for the interaction between the established scholars and young researchers

Large Scale Stochastic Dynamics

The goal of this workshop was to explore the recent advances in the mathematical understanding of the macroscopic properties which emerge on large space-time scales from interacting microscopic particle systems. There were 55 participants, including postdocs and graduate students, working in diverse intertwining areas of probability and statistical mechanics. During the meeting, 29 talks of 45 minutes were scheduled and an evening session was organised with 10 more short talks of 10 minutes, mostly by younger participants. These talks addressed the following topics : randomness emerging from deterministic dynamics, hydrodynamic limits, interface growth models and slow convergence to equilibrium in kinetically constrained dynamics