Particle approximation of the Wasserstein diffusion

Abstract

We construct a system of interacting two-sided Bessel processes on the unit interval and show that the associated empirical measure process converges to the Wasserstein diffusion (von Renesse and Sturm (2009) [25]), assuming that Markov uniqueness holds for the generating Wasserstein Dirichlet form. The proof is based on the variational convergence of an associated sequence of Dirichlet forms in the generalized Mosco sense of Kuwae and Shioya (2003) [19].

Emergence of particle clusters in a one-dimensional model: connection to condensation processes

“…This choice is simple from a theoretical point of view and is consistent with [13], but the model is difficult from a numerical perspective because of the large values of ∂E/∂x i that appear when particles approach one another. In the absence of any interparticle forces (E = 0), the x Figure 1.…”

“…As well as offering a new twist on condensation, this work is also motivated by connections between this model system and a recent mathematical study [13] where it was found that the dynamics of the particle density in a similar model should be described by a stochastic partial differential equation (PDE) with a stochastic term that does not vanish in the hydrodynamic limit. Usually, one expects to recover (almost surely) deterministic behaviour in the hydrodynamic limit: for example, the deterministic diffusion equation describes the spreading of a large number of random walkers.…”

“…Usually, one expects to recover (almost surely) deterministic behaviour in the hydrodynamic limit: for example, the deterministic diffusion equation describes the spreading of a large number of random walkers. Moreoever, the stochastic PDE found in [13] is closely related to the Dean equation [14], which describes (in this case) the diffusive motion of a finite number of non-interacting particles. This result offers the possibility that the clusters that form in our model might themselves act as free particles that diffuse through the system.…”

Singular perturbations to semilinear stochastic heat equations

“…Note that the function α → C α is bounded on any closed interval not including 1 2 or 3 2 , but it diverges at these two values. Similarly, the functionC α diverges as it approaches 1 2 from below.…”

“…However, starting with the seminal works of Mosco [25,26], a number of authors have investigated sufficient conditions for the resolvent convergence of Dirichlet forms, see for example [2,3,21,24,28]. While our setting does not formally seem to be covered by these works, it 'morally' falls into the same category.…”