Florian Völlering

Random walks in dynamic random environments: A transference principle

We study a general class of random walks driven by a uniquely ergodic Markovian environment. Under a coupling condition on the environment we obtain strong ergodicity properties for the environment as seen from the position of the walker, that is, the environment process. We can transfer the rate of mixing in time of the environment to the rate of mixing of the environment process with a loss of at most polynomial order. Therefore the method is applicable to environments with sufficiently fast polynomial mixing. We obtain unique ergodicity of the environment process. Moreover, the unique invariant measure of the environment process depends continuously on the jump rates of the walker. As a consequence we obtain the law of large numbers and a central limit theorem with nondegenerate variance for the position of the walk.

Symmetric exclusion as a random environment: hydrodynamic limits

We consider a one-dimensional continuous time random walk with transition rates depending on an underlying autonomous simple symmetric exclusion process starting out of equilibrium. This model represents an example of a random walk in a slowly nonuniform mixing dynamic random environment. Under a proper space-time rescaling in which the exclusion is speeded up compared to the random walk, we prove a hydrodynamic limit theorem for the exclusion as seen by this walk and we derive an ODE describing the macroscopic evolution of the walk. The main difficulty is the proof of a r lemma for the exclusion as seen from the walk without explicit knowledge of its invariant measures. We further discuss how to obtain similar results for several variants of this model.

Explicit LDP for a slowed RW driven by a symmetric exclusion process

We consider a random walk (RW) driven by a simple symmetric exclusion process (SSE). Rescaling the RW and the SSE in such a way that a joint hydrodynamic limit theorem holds we prove a joint path large deviation principle. The corresponding large deviation rate function can be split into two components, the rate function of the SSE and the one of the RW given the path of the SSE. These components have different structures (Gaussian and Poissonian, respectively) and to overcome this difficulty we make use of the theory of Orlicz spaces. In particular, the component of the rate function corresponding to the RW is explicit.

A new look at duality for the symbiotic branching model

The symbiotic branching model is a spatial population model describing the dynamics of two interacting types that can only branch if both types are present. A classical result for the underlying stochastic partial differential equation identifies moments of the solution via a duality to a system of Brownian motions with dynamically changing colors. In this paper, we revisit this duality and give it a new interpretation. This new approach allows us to extend the duality to the limit as the branching rate

Talagrand’s inequality for interacting particle systems satisfying a log-Sobolev inequality

\gamma

Absolute continuity and weak uniform mixing of random walk in dynamic random environment

is sent to infinity. This limit is particularly interesting since it captures the large scale behaviour of the system. As an application of the duality, we can explicitly identify the

A variance inequality for Glauber dynamics applicable to high and low temperature regimes

\gamma = \infty

Entrance laws for annihilating Brownian motions

limit when the driving noises are perfectly negatively correlated. The limit is a system of annihilating Brownian motions with a drift that depends on the initial imbalance between the types.

The Algebraic Approach to Duality: An Introduction

Talagrands inequality for independent Bernoulli random variables is extended to many interacting particle systems (IPS). The main assumption is that the IPS satisfies a log-Sobolev inequality. In this context it is also shown that a slightly stronger version of Talagrands inequality is equivalent to a log-Sobolev inequality. Additionally we also look at a common application, the relation between the probability of increasing events and the influences on that event by changing a single spin.