Generalized Dean-Kawasaki equations for interacting Brownian particles in partially absorbing media
The Dean-Kawasaki (DK) equation for the empirical measure of an interacting system of overdamped Brownian particles is a stochastic analyst’s worst nightmare! On the other hand, taking expectations of the DK equation with respect to the Gaussian noise processes under a mean field ansatz provides a direct route to the corresponding McKean-Vlasov (MV) equation. The latter provides a framework for investigating the existence of multiple stationary solutions and their associated phase transitions in the thermodynamic limit. In this talk, we use an encounter-based method in order to generalize the derivation of the DK equation to the case of interacting Brownian particles in a domain with a partially absorbing boundary. Encounter-based methods provide a general probabilistic framework for modelling surface absorption, whereby an absorption event occurs when the boundary local time exceeds a random threshold. If the probability distribution of the latter is an exponential function, then one recovers the Markovian example of absorption at a constant rate, whereas a non-exponential distribution signifies non-Markovian absorption. We assume that each particle is independently absorbed when its individual boundary local time exceeds a random threshold. The empirical measure is defined in terms of the set of particle positions, particle local times, and local time thresholds. Averaging the resulting DK equation with respect to the Gaussian noise processes and the local time thresholds yields an MV equation in a domain with a partially absorbing boundary. We illustrate the theory for Brownian motion on the half-line.
The talk will be followed by refreshments in the common room at 4pm.