K-th order Hydrodynamic limits

In this thesis, we study stochastic duality under hydrodynamic scaling in the context of interacting particles on a grid. The approach is inspired and motivated by the relation between duality and local equilibria. We identify duality relations in terms of the expectation of the density field for which the hydrodynamic limit is recovered. This...

The Hydrodynamic Limit of The Random Waiting Time Model and The Fractional Kinetics Process

In this thesis, the hydrodynamic limit (HDL) for two trapping models is studied, the Random Waiting Time Model (RWTM) and the fractional kinetics process (FKP), on a discrete lattice Zd . The RWTM is studied for dimension d ≥ 1 and E[wi ],∞where wi denotes the waiting time at position i . On the other hand, the FKP is studied for d ≥ 3 and a...

Advances in Stochastic Duality for Interacting Particle Systems: from many to few

Interacting particle systems (IPS) is a subfield of probability theory that provided a fruitful framework in which several questions of physical interests have been answered with mathematical rigor. An interacting particle system is a stochastic system consisting of a very large number of particles interacting with each other. The class of IPS...

Hydrodynamic Limit for the Symmetric Inclusion Process with Slowly Varying Inhomogeneities

In this thesis we study an interacting particle system: the Symmetric Inclusion Process with slowly varying inhomogeneities (SIP(α)). In the SIP(α) particles display random walk like behaviour subjected to an attractive type of interaction whilst evolving in an inhomogeneous environment. We set out to prove its hydrodynamic limit. The main tool...

The Symmetric Exclusion Process and the Gausian Free Field on compact Riemannian manifolds

In this thesis we study the Symmetric Exclusion Process (SEP) and the Discrete Gaussian Free Field (DGFF) on compact Riemannian manifolds. In particular, we obtain the hydrodynamic limit and the equilibrium fluctuations of SEP and we show that the DGFF converges to its continuous counterpart. To define these discrete models, we construct grids...

Ergodic theory and hydrodynamic limit for run-and-tumble particle processes

In this thesis we will study the ergodic measures and the hydrodynamic limit of independent run-and-tumble particle processes, i.e., an interacting particle system for particles with an internal energy source, which makes them move in a preferred direction that changes at random times. We start by providing some basic concepts and theory of...

Hydrodynamics for the partial exclusion process in random environment

In this paper, we introduce a random environment for the exclusion process in Z^d obtained by assigning a maximal occupancy to each site. This maximal occupancy is allowed to randomly vary among sites, and partial exclusion occurs. Under the assumption of ergodicity under translation and uniform ellipticity of the environment, we...

Symmetric simple exclusion process in dynamic environment: Hydrodynamics

We consider the symmetric simple exclusion process in Z^d with quenched bounded dynamic random conductances and prove its hydrodynamic limit in path space. The main tool is the connection, due to the self-duality of the process, between the invariance principle for single particles starting from all points and the macroscopic...

Hydrodynamic Limit of the Symmetric Exclusion Process on a Compact Riemannian Manifold

We consider the symmetric exclusion process on suitable random grids that approximate a compact Riemannian manifold. We prove that a class of random walks on these random grids converge to Brownian motion on the manifold. We then consider the empirical density field of the symmetric exclusion process and prove that it converges to the...

Invariance principle and hydrodynamic limits on Riemannian manifolds

In this report we study Markov processes on compact and connected Riemannian manifolds. We define a random walk on such manifolds and give a direct proof of the invariance principle. This principle says that under some conditions on the jumping distributions (i.e. the distributions of single steps), the random walk converges to Brownian motion...

Hydrodynamic Limit for the Symmetric Inclusion Process

This thesis deals with the proof of the hydrodynamic limit for the symmetric inclusion process (SIP). After introduction of our process and some basic concepts on the theory of Markov processes we proved self-duality for the SIP. Once equipped with SIP self-duality, and together with an assumption on the initial configuration, we were able to...