Non-equilibrium Dynamics for a Widom–Rowlinson Type Model with Mutations

Kondratiev²

2018

J Stat Phys

Self Cite

We study a spatial birth-and-death process on the phase space of locally finite configurations Γ`ˆΓ´over R d . Dynamics is described by an non-equilibrium evolution of states obtained from the Fokker-Planck equation and associated with the Markov operator L`pγ´q`1 ε L´, ε ą 0. Here L´describes the environment process on Γ´and L`pγ´q describes the system process on Γ`, where γ´indicates that the corresponding birth-and-death rates depend on another locally finite configuration γ´P Γ´. We prove that, for a certain class of birth-and-death rates, the corresponding Fokker-Planck equation is well-posed, i.e. there exists a unique evolution of states µ ε t on Γ`ˆΓ´. Moreover, we give a sufficient condition such that the environment is ergodic with exponential rate. Let µ inv be the invariant measure for the environment process on Γ´. In the main part of this work we establish the stochastic averaging principle, i.e. we prove that the marginal of µ ε t onto Γ`converges weakly to an evolution of states on Γ`associated with the averaged Markov birth-and-death operator L " ş Γ´L`p γ´qdµ inv pγ´q.

“…In this section we closely follow the arguments in [FK16b] (see also [Fri17]). Since the necessary computations are very similar to the latter works, we give only the main steps of proof.…”

Section: Two Interacting Bdlp-modelsmentioning

confidence: 96%

Section: Preliminariesmentioning

confidence: 99%

Section: The Environmentmentioning

confidence: 99%

Section: ´|mentioning

confidence: 99%

See 2 more Smart Citations

Stochastic Averaging Principle for Spatial Birth-and-Death Evolutions in the Continuum

Kondratiev²

2018

J Stat Phys

Self Cite

“…It is worth noting that the weak formulation of the Fokker-Planck equation (1.3) does not require that µ t also provides a classical solution to (1.4). An extension of this uniqueness statement (without requiring that k µt is a classical solution to (1.4)) was given in [FK16,Fri17,FK18a]. The latter result was essentially based on [WZ02,WZ06].…”

Section: Introduction 1motivationmentioning

confidence: 99%

Linear evolution equations in scales of Banach spaces

Journal of Functional Analysis

2019

Self Cite

A variant of the abstract Cauchy-Kovalevskaya theorem is considered. We prove existence and uniqueness of classical solutions to the nonlinear, non-autonomous initial value problemin a scale of Banach spaces. Here A(t) is the generator of an evolution system acting in a scale of Banach spaces and B(u, t) obeys an Ovcyannikov-type bound. Continuous dependence of the solution with respect to A(t), B(u, t) and x is proved. The results are applied to the Kimura-Maruyama equation for the mutation-selection balance model. This yields a new insight in the construction and uniqueness question for nonlinear Fokker-Planck equations related with interacting particle systems in the continuum.

Nonlinear perturbations of evolution systems in scales of Banach spaces

Kutoviy

2020

Nonlinearity

Self Cite

A variant of the abstract Cauchy-Kovalevskaya theorem is considered. We prove existence and uniqueness of classical solutions to the nonlinear, nonautonomous initial value problem du(t) dt = A(t)u(t) + B(u(t), t), u(0) = x in a scale of Banach spaces. Here A(t) is the generator of an evolution system acting in a scale of Banach spaces and B(u, t) obeys an Ovcyannikov-type bound. Continuous dependence of the solution with respect to A(t), B(u, t) and x is proved. The results are applied to the Kimura-Maruyama equation for the mutation-selection balance model. This yields a new insight in the construction and uniqueness question for nonlinear Fokker-Planck equations related with interacting particle systems in the continuum.