Mean field limit with proliferation

Abstract

An interacting particle system with long range interaction is considered. Particles, in addition to the interaction, proliferate with a rate depending on the empirical measure. We prove convergence of the empirical measure to the solution of a parabolic equation with non-local nonlinear transport term and proliferation term of logistic type. T a,N 1 := lim t↑T a,N 1 X a,N t , we impose T a,N 0 = T (a,−),N 1 and X a,N T a,N 0 = X (a,−),N T (a,−),N 1 . Denote by A N the set of all labels a and 2010 Mathematics Subject Classification. Primary: 60K35, 35K57; Secondary: 60F17, 35K58.

Uniform convergence of proliferating particles to the FKPP equation

“…But a direct link between the particle system and the limit PDE (the so called hydrodynamic limit problem) leads to a non-linearity of the form ∆f (u) where f (u) is not necessarily u 2 (see [27], [25]). For a proof of the mean field result of convergence of S N 0 ,N t to u N 0 as N → ∞, see for instance [6], [10]. The issue of uniform convergence of θ N * S N t to u does not arise and weak convergence of the measures S N 0 ,N t is sufficient.…”

“…Traces of this approach can be found in [15] and [6], but have been used for other purposes. In the work [10] it is remarked that uniform convergence can be obtained as a by-product of energy inequalities and Sobolev convergence, under the assumption β < d d+2 , but only in dimension d = 1, where the condition is more restrictive than β < 1/2.…”

“…t to u N 0 as N → ∞, see for instance[6],[10]. The issue of uniform convergence of θ N * S N t to u does not arise and weak convergence of the measures S N 0 ,N t is sufficient.…”

Epidemic Models as Scaling Limits of Individual Dynamics

“…We also mention that applications of Probability Theory to compartmental models involve furthermore Large Deviation Principles for stochastic models, [27,19,28]. Finally, besides topic related literature, we would like to mention some works whose techniques inspired the present one, [10,11,14,1,12,13].…”

The KPP equation as a scaling limit of locally interacting Brownian particles

“…Of course, here we are assuming that particles are relatively homogeneously distributed in space at all times down to the microscopic scale (which is not always proven). For the system with generator (1), the intermediate scaling regime with β ∈ (0, 1/2) has been studied and its scaling limit to F-KPP equation established in [8], with earlier results [17] for a shorter range of β, and so is the mean-field case whose limit is a different kind of equation [2,7]; our aim here is to study the local regime, subject to a modification of the rate (2).…”