Markov Jump Dynamics with Additive Intensities in Continuum: State Evolution and Mesoscopic Scaling

Abstract

We investigate stochastic (conservative) non-equilibrium jump dynamics of interacting particles in continuum. The corresponding evolutions of correlation functions are constructed. The mesoscopic scaling (Vlasov scaling) of the dynamics is studied and the corresponding kinetic equations for the particle densities are derived. Keywords Interacting particle system • Jump dynamics • Non-equilibrium evolution • Vlasov scaling • Kinetic equation 1 Introduction Usually, an infinite group of identical particles with different locations in R d can be characterized by locally finite subset of R d , the points of which are

Stochastic models of tumour development and related mesoscopic equations

“…The rigorous derivation of the kinetic description was already done in [4]. Scaling the potential as b → εb and rescaling, we see that only the last terms in L and L ∆ will be multiplied by ε.…”

Linear evolution equations in scales of Banach spaces

“…It is necessary and sufficient that k t is positive definite in the sense of Lennard, see [KK02] and the references therein. Such a positivity property was studied for particular models in [FKK13a,BKK15], while a general result based on semigroup methods was obtained in [FK16]. As a consequence, using the theory of semigroups one is able to obtain existence of solutions to (1.3), while uniqueness holds among all solutions µ t for which the associated sequence of correlation functions k µt is a classical solution to (1.4).…”