Continuum Random Walks in Random Environments

Random walks in random environment are a class of well-studied models of Statistical Mechanics, aiming to describe diffusion through an inhomogeneous material. The mathematical description consists of a Markov process X_t (describing individual particles moving through the environment) whose transition probabilities are themselves random (describing the inhomogeneity of the environment). Typical questions that have been addressed in the literature include:
- is the Markov process recurrent or transient,
- if it is transient, is its leading order behaviour ballistic (i.e. linear motion) or not,
- what are the fluctuations around the leading order behaviour.

Almost all of the literature so far has been implemented on the level of discrete lattice models, i.e. both the
process and the environment are defined on a discrete graph (mostly Z^d). A continuum setting is in principle very
natural, however, due to technical difficulties in even defining these models, it has hardly been considered.

The understanding of how to treat such continuum models from Statistical Mechanics has improved dramatically over the last years, key contributions being Hairer's work on Regularity Structures and Gubinelli's work on Paracontrolled distributions. Based on these ideas it is now possible to give a mathematically rigorous meaning to most random walk in random environment models in continuum, and the aim of this project is to revisit the classical questions listed above in this new technical framework.

More specifically, a stochastic differential equation with a random drift term which is itself the solution of a linear stochastic partial differential equation will be considered. This model is a natural continuum analogue for the random walk in dynamic random environment considered recently by Kious et al. The first aims are

- develop a rigorous well-posedness theory for this SDE. This should be a relatively simple modification of recent work by Delarue and Diel.

- with this solution theory at hand, consider the simplest parameter regime where the SDE has an additional drift term. In this regime establish rigorously the ballistic motion of the particles as well as the CLT around this motion.