RTG 2131 - Workshop 2018

Markov processes on metric measure spaces and Gaussian fields




Diffusions on metric measure trees

Siva Athreya (Indian Statistical Institute, Bangalore)

In this talk we will give an overview of the work on diffusions on complete, locally compact tree-like metric spaces $(T,r)$ on their ``natural scale'' with boundedly finite speed measure $\nu$. We will provide a natural Dirichlet form construction of them, an invariance principle and path properties they satisfy on specific examples.

Random walks on Galton-Watson trees

Adam Bowditch (University of Warwick)

In a wide range of models of random walks in random environments, the asymptotic behaviour of the walk is driven by a trapping mechanism arising due to adverse regions in the environment. Random walks on Galton-Watson trees conditioned to survive are a natural example as dead-ends in the environment form traps which slow the walk. We discuss how differences in the geometric structure of the tree influence the limiting behaviour. In particular, we focus on questions concerning the existence of a limiting speed, the scale of the fluctuations and the issues concerning functional results. These questions link closely to open problems for random walks on supercritical percolation clusters which exhibit similar phenomena.


Percolation with long-range correlations via isomorphism theorems

Alexander Drewitz (University of Collogne)

We illustrate how two percolation models with long-range correlations, vacant set percolation for random interlacements as well as level set percolation for the Gaussian free field, can be understood more profoundly by relating them through the use of isomorphism theorems. As an application, this can be used to show that the critical parameter for level set percolation in $Z^d$, $d\ge 3,$ is positive in any dimension larger than or equal to three, which had been conjectured by Bricmont, Lebowitz and Maes in 1987. If time admits, we will consider applications to other graphs also. This talk is based on joint works with A. Prévost (Köln) and P.-F. Rodriguez (Los Angeles).

On permanental processes

Nathalie Eisenbaum (University Paris 5)

We will introduce the permanental processes. The most known permanental processes are the squared Gaussian processes (i.e.  $(G^2(x), x \in E)$ for $(G(x), x \in E)$ centered Gaussian process). We will show how they are connected to the local times of Markov processes and how these connections can be exploited.

Stein's method for functional approximations

Mikolaj Kasprzak (Oxford University)

We extend the ideas of the famous paper of Andrew Barbour from 1990 and use Stein's method to measure distances between distributions of stochastic processes. We consider scaled $p$-dimensional sums of random variables under various dependence schemes and their distance from a $p$-dimensional (correlated) Brownian Motion. Furthermore, we make the first step towards estimating the speed of convergence of a class of scaled Markov Chains to diffusion processes, as described in the Stroock-Varadhan theory of diffusion approximation. In addition, we show how the celebrated method of exchangeable pairs can be adapted to the functional setting. Our findings are applied in a number of examples, including the asymptotic behaviour of processes used in population genetics, a combinatorial central limit theorem and convergence properties of scaled U-statistics.

Pathwise contraction of Brownian motion and lower Ricci curvature bounds

Eva Kopfer (University of Bonn)

We explain the connection between pathwise contraction of Brownian motions and lower Ricci curvature bounds on manifolds as well as on metric measure spaces with lower Ricci curvature in the sense of Lott-Sturm-Villani. We will see that this connection has been generalized to time-dependent Brownian motions on manifolds evolving as a super-Ricci flow. We will define time-dependent Brownian motions on metric measure spaces $(X,d_t,m_t)_{t\in I}$. If $(X,d_t,m_t)_{t\in I}$ evolve as a super-Ricci flow in the sense of Sturm we construct couplings of Brownian motions and obtain pathwise contraction of their trajectories.

More on the extreme and large-value landscape of the discrete Gaussian free field and friends.

Oren Louidor (Technion, Haifa)

I will discuss some new results concerning extreme and large values of the 2D discrete Gaussian free field and related processes. These include finer structural properties of its extremal landscape, scaling limits for its high (but not extreme) level sets and the asymptotic growth of the infinite volume pinned DGFF. Based on joint work (some in progress) with M. Biskup, A. Cortines, L. Hartung and D. Yeo.

Convergence of Diffusion Processes on RCD spaces

Kohei Suzuki (University of Bonn)

The motivation of this talk is to study the weak convergence of diffusion processes in terms of a geometric convergence of the underlying spaces. We first show that the weak convergence of Brownian motions is equivalent to the measured Gromov-Hausdorff convergence of the underlying metric measure spaces under the synthetic lower Ricci curvature bound. Secondly, we discuss the weak convergence of non-symmetric diffusion processes.