We show the heat kernel type variance decay $t^{-\frac{d}{2}}$, up to a logarithmic correction, for the semigroup of an infinite particle system on $\mathbb{R}^d$, where every particle evolves following a divergence-form operator with diffusivity coefficient that depends on the local configuration of particles. The proof relies on the strategy from $\mathbb{Z}^d$ zero range model, and generalizes the localization estimate to the continuum configuration space introduced by S. Albeverio, Y.G. Kondratiev and M. Röckner.
We study the evolution of genealogies of a population of individuals, whose type frequencies result in an interacting Fleming-Viot process on $\Z$. We construct and analyze the genealogical structure of the population in this genealogy-valued Fleming-Viot process as a marked metric measure space, with each individual carrying its spatial location as a mark. We then show that its time evolution converges to that of the genealogy of a continuum-sites stepping stone model on $\R$, if space and time are scaled diffusively. We construct the genealogies of the continuum-sites stepping stone model as…
A conservative Feller evolution on continuous bounded functions is constructed from a weakly continuous, time-inhomogeneous transition function describing a pure jump process on a locally compact Polish space. The transition function is assumed to satisfy a Foster-Lyapunov type condition. The results are applied to interacting particle systems in continuum, in particular to general birth-and-death processes (including jumps). Particular examples such as the BDLP and Dieckmann-Law model are considered in the end.
We consider an interacting particle system modeled as a system of $N$ stochastic differential equations driven by Brownian motions. We prove that the (mollified) empirical process converges, uniformly in time and space variables, to the solution of the two-dimensional Navier-Stokes equation written in vorticity form. The proofs follow a semigroup approach.
We consider additive functionals of Markov processes in continuous time with general (metric) state spaces. We derive concentration bounds for their exponential moments and moments of finite order. Applications include diffusions, interacting particle systems and random walks. The method is based on coupling estimates and not spectral theory, hence reversibility is not needed. We bound the exponential moments(or the moments of finite order) in terms of a so-called coupled function difference, which in turn is estimated using the generalized coupling time. Along the way we prove a general relat…
We introduce two partially overlapping classes of pathwise dualities between interacting particle systems that are based on commutative monoids (semigroups with a neutral element) and semirings, respectively. For interacting particle systems whose local state space has two elements, this approach yields a unified treatment of the well-known additive and cancellative dualities. For local state spaces with three or more elements, we discover several new dualities.
For the weakly interacting one-dimensional multi-particle Anderson model in the continuum space of configurations, we prove the spectral exponential and the strong dynamical localization. The results require the interaction amplitude to be sufficiently small. The general strategy uses the multi-scale analysis bounds. Actually, we show that the multi-scale analysis bounds of the single particle model remain stable when passing to multi-particle systems, provided that the inter-particle interaction is sufficiently small. The common probability distribution of the i.i.d. random external potential…
We consider a stochastic interacting particle system in a bounded domain with reflecting boundary, including creation of new particles on the boundary prescribed by a given source term. We show that such particle system approximates 2d Navier-Stokes equations in vorticity form and impermeable boundary, the creation of particles modeling vorticity creation at the boundary. Kernel smoothing, more specifically smoothing by means of the Neumann heat semigroup on the space domain, allows to establish uniform convergence of regularized empirical measures to (weak solutions of) Navier-Stokes equation…
Analogues of stepping--stone models are considered where the site--space is continuous, the migration process is a general Markov process, and the type--space is infinite. Such processes were defined in previous work of the second author by specifying a Feller transition semigroup in terms of expectations of suitable functionals for systems of coalescing Markov processes. An alternative representation is obtained here in terms of a limit of interacting particle systems. It is shown that, under a mild condition on the migration process, the continuum--sites stepping--stone process has continuou…
We consider Markov processes that alternate continuous motions and jumps in a general locally compact polish space. Starting from a mechanistic construction, a first contribution of this article is to provide conditions on the dynamics so that the associated transition kernel forms a Feller semigroup, and to deduce the corresponding infinitesimal generator. In a second contribution, we investigate the ergodic properties in the special case where the jumps consist of births and deaths, a situation observed in several applications including epidemiology, ecology and microbiology. Based on a coup…
We show that some classes of birth-and-death processes in continuum (Glauber dynamics) may be derived as a scaling limit of a dynamics of interacting hopping particles (Kawasaki dynamics)
Quantum stochastic cocycles provide a basic model for time-homogeneous Markovian evolutions in a quantum setting, and a direct counterpart in continuous time to quantum random walks, in both the Schrodinger and Heisenberg pictures. This paper is a sequel to one in which correspondences were established between classes of quantum stochastic cocycle on an operator space or C*-algebra, and classes of Schur-action `global' semigroup on associated matrix spaces over the operator space. In this paper we investigate the stochastic generation of cocycles via the generation of their corresponding globa…
An infinite system of point particles placed in $\mathds{R}^d$ is studied. Its constituents perform random jumps with mutual repulsion described by a translation-invariant jump kernel and interaction potential, respectively. The pure states of the system are locally finite subsets of $\mathds{R}^d$, which can also be interpreted as locally finite Radon measures. The set of all such measures $Γ$ is equipped with the vague topology and the corresponding Borel $σ$-field. For a special class $\mathcal{P}_{\rm exp}$ of (sub-Poissonian) probability measures on $Γ$, we prove the existence of a unique…
We discuss the interrelation between phase transitions in interacting lattice or continuum models, and the existence of infinite clusters in suitable random-graph models. In particular, we describe a random-geometric approach to the phase transition in the continuum Ising model of two species of particles with soft or hard interspecies repulsion. We comment also on the related area-interaction process and on perfect simulation.
The approximation of the Feynman-Kac semigroups by systems of interacting particles is a very active research field, with applications in many different areas. In this paper, we study the parallelization of such approximations. The total population of particles is divided into sub-populations, referred to as \emph{islands}. The particles within each island follow the usual selection / mutation dynamics. We show that the evolution of each island is also driven by a Feynman-Kac semigroup, whose transition and potential can be explicitly related to ones of the original problem. Therefore, the sam…
We give a general existence result for interacting particle systems with local interactions and bounded jump rates but noncompact state space at each site. We allow for jump events at a site that affect the state of its neighbours. We give a law of large numbers and functional central limit theorem for additive set functions taken over an increasing family of subcubes of $Z^d$. We discuss application to marked spatial point processes with births, deaths and jumps of particles, in particular examples such as continuum and lattice ballistic deposition and a sequential model for random loose sphe…
We consider a class of discrete $q$-state spin models defined in terms of a translation-invariant quasilocal specification with discrete clock-rotation invariance which have extremal Gibbs measures $μ'_φ$ labeled by the uncountably many values of $φ$ in the one-dimensional sphere (introduced by van Enter, Opoku, Külske [J. Phys. A 44 (2011) 475002, 11]). In the present paper we construct an associated Markov jump process with quasilocal rates whose semigroup $(S_t)_{t\geq0}$ acts by a continuous rotation $S_t(μ'_φ)=μ'_{φ+t}$. As a consequence our construction provides examples of interacting p…
We prove the Wegner bounds for the one-dimensional interacting multi-particle Anderson models in the continuum. The results apply to singular probability distribution functions such as the Bernoulli's measures. The proofs need the amplitude of the inter-particle interaction potential to be sufficiently weak. As a consequence, the results imply the Anderson localization via the multi-scale analysis.
This article is concerned with the design and analysis of discrete time Feynman-Kac particle integration models with geometric interacting jump processes. We analyze two general types of model, corresponding to whether the reference process is in continuous or discrete time. For the former, we consider discrete generation particle models defined by arbitrarily fine time mesh approximations of the Feynman-Kac models with continuous time path integrals. For the latter, we assume that the discrete process is observed at integer times and we design new approximation models with geometric interacti…
Motivated by discrete directed polymers in one space and one time dimension, we construct a continuum directed random polymer that is modeled by a continuous path interacting with a space-time white noise. The strength of the interaction is determined by an inverse temperature parameter beta, and for a given beta and realization of the noise the path evolves in a Markovian way. The transition probabilities are determined by solutions to the one-dimensional stochastic heat equation. We show that for all beta > 0 and for almost all realizations of the white noise the path measure has the same Ho…
We develop a general technique, based on a Bochner-type identity, to estimate spectral gaps of a class of Markov operator. We apply this technique to various interacting particle systems. In particular, we give a simple and short proof of the diffusive scaling of the spectral gap of the Kawasaki model at high temperature. Similar results are derived for Kawasaki-type dynamics in the lattice without exclusion, and in the continuum. New estimates for Glauber-type dynamics are also obtained.
In the context of interacting particle systems, we study the influence of the action of the semigroup on the concentration property of Lipschitz functions. As an application, this gives a new approach to estimate the relaxation speed to equilibrium of interacting particle systems. We illustrate our approach in a variety of examples for which we obtain several new results with short and non-technical proofs. These examples include the symmetric and asymmetric exclusion process and high-temperature spin-flip dynamics ("Glauber dynamics"). We also give a new proof of the Poincaré inequality, base…
