The Asymmetric Brownian Energy Process

Abstract

The focus of this thesis is the hydrodynamic limit of the Brownian Energy Process (BEP) and the
Asymmetric Brownian Energy Process (ABEP) in infinite volume. The thesis starts by introducing some general theory about Markov processes, after which the topic of Markov duality is introduced. A number of relevant interacting particle systems (IPS) will be introduced, and we will show how the BEP and the ABEP can be related to the simpler Symmetric Inclusion Process (SIP) through Markov duality. Using these tools, the first main result is proven, which states that the hydrodynamic limit of the BEP is a weak solution to the heat equation. As a consequence of this, in the second main result we use the relation between the BEP and the ABEP to prove that the hydrodynamic limit of the ABEP is a weak solution to the viscous Burgers’ equation. We then attempt to show a similar result for a newly developed IPS, the Dynamic ABEP. Finally, we prove propagation of chaos for the BEP and the ABEP, where for the latter we argue that this is only possible in a finite volume. As a part of the proof of this we argue that the SIP in infinite volume is very ’similar’ to Independent Random Walkers (IRW) when we look at a long enough time-scale, where we give a rough sketch of a proof that significantly improves upon a quantification of this ’similarity’ established in the literature.