We study the problem of identification of a proper state-space for the stochastic dynamics of free particles in continuum, with their possible birth and death. In this dynamics, the motion of each separate particle is described by a fixed Markov process $M$ on a Riemannian manifold $X$. The main problem arising here is a possible collapse of the system, in the sense that, though the initial configuration of particles is locally finite, there could exist a compact set in $X$ such that, with probability one, infinitely many particles will arrive at this set at some time $t>0$. We assume that $X$ has infinite volume and, for each $α\ge1$, we consider the set $Θ_α$ of all infinite configurations in $X$ for which the number of particles in a compact set is bounded by a constant times the $α$-th power of the volume of the set. We find quite general conditions on the process $M$ which guarantee that the corresponding infinite particle process can start at each configuration from $Θ_α$, will never leave $Θ_α$, and has cadlag (or, even, continuous) sample paths in the vague topology. We consider the following examples of applications of our results: Brownian motion on the configuration space, free Glauber dynamics on the configuration space (or a birth-and-death process in $X$), and free Kawasaki dynamics on the configuration space. We also show that if $X=\mathbb R^d$, then for a wide class of starting distributions, the (non-equilibrium) free Glauber dynamics is a scaling limit of (non-equilibrium) free Kawasaki dynamics.
Let $Γ$ denote the space of all locally finite subsets (configurations) in $\mathbb R^d$. A stochastic dynamics of binary jumps in continuum is a Markov process on $Γ$ in which pairs of particles simultaneously hop over $\mathbb R^d$. We discuss a non-equilibrium dynamics of binary jumps. We prove the existence of an evolution of correlation functions on a finite time interval. We also show that a Vlasov-type mesoscopic scaling for such a dynamics leads to a generalized Boltzmann non-linear equation for the particle density.
Let $Γ$ denote the space of all locally finite subsets (configurations) in $R^d$. A stochastic dynamics of binary jumps in continuum is a Markov process on $Γ$ in which pairs of particles simultaneously hop over $R^d$. In this paper, we study an equilibrium dynamics of binary jumps for which a Poisson measure is a symmetrizing (and hence invariant) measure. The existence and uniqueness of the corresponding stochastic dynamics are shown. We next prove the main result of this paper: a big class of dynamics of binary jumps converge, in a diffusive scaling limit, to a dynamics of interacting Brown…
We deal with two following classes of equilibrium stochastic dynamics of infinite particle systems in continuum: hopping particles (also called Kawasaki dynamics), i.e., a dynamics where each particle randomly hops over the space, and birth-and-death process in continuum (or Glauber dynamics), i.e., a dynamics where there is no motion of particles, but rather particles die, or are born at random. We prove that a wide class of Glauber dynamics can be derived as a scaling limit of Kawasaki dynamics. More precisely, we prove the convergence of respective generators on a set of cylinder functions,…
In this paper we investigate Nash equilibrium payoffs for two-player nonzero-sum stochastic differential games whose cost functionals are defined by a system of coupled backward stochastic differential equations. We obtain an existence theorem and a characterization theorem for Nash equilibrium payoffs. For this end the problem is described equivalently by a stochastic differential game with jumps. But, however, unlike the work by Buckdahn, Hu and Li [9], here the important tool of a dynamic programming principle for stopping times has to be developed. Moreover, we prove that the lower and upp…
The convergence to the stationary regime is studied for Stochastic Differential Equations driven by an additive Gaussian noise and evolving in a semi-contractive environment, i.e. when the drift is only contractive out of a compact set but does not have repulsive regions. In this setting, we develop a synchronous coupling strategy to obtain sub-exponential bounds on the rate of convergence to equilibrium in Wasserstein distance. Then by a coalescent coupling close to terminal time, we derive a similar bound in total variation distance.
We model spatially expanding populations by means of two spatial $Λ$-Fleming Viot processes (or SLFVs) with selection: the k-parent SLFV and the $\infty$-parent SLFV. In order to do so, we fill empty areas with type 0 ''ghost'' individuals with a strong selective disadvantage against ''real'' type 1 individuals, quantified by a parameter k. The reproduction of ghost individuals is interpreted as local extinction events due to stochasticity in reproduction. When k $\rightarrow$ +$\infty$, the limiting process, corresponding to the $\infty$-parent SLFV, is reminiscent of stochastic growth models…
In this paper we study the equilibrium energy fluctuation field of a one-dimensional reversible non gradient model. We prove that the limit fluctuation process is governed by a generalized Ornstein- Uhlenbeck process, which covariances are given in terms of the diffusion coefficient. Adapting the non gradient method introduced by Varadhan, we are able to derive the diffusion coefficient. The fact that the conserved quantity (energy) is not a linear functional of the coordinates of the system, introduces new difficulties of geometric nature when applying the nongradient method.
We consider a stochastic model, describing the growth of two competing infections on $\mathbb{R}^d$. The growth takes place by way of spherical outbursts in the infected region, an outburst in the type 1 (2) infected region causing all previously uninfected points within a stochastic distance from the outburst location to be type 1 (2) infected. The main result is that, if the infection types have the same intensity, then there is a strictly positive probability that both infection types grow unboundedly.
We prove phase transition for the non-symmetric continuum Potts model with background interaction, by generalizing the methods introduced in the symmetric case by Georgii and Häggström. The proof relies on a Fortuin-Kasteleyn representation, percolation and stochastic domination arguments.
We show that every two-player stochastic game with finite state and action sets and bounded, Borel-measurable, and shift-invariant payoffs, admits an $\ep$-equilibrium for all $\varepsilon>0$.
The main objective of the paper is to study the long-time behavior of general discrete dynamics driven by an ergodic stationary Gaussian noise. In our main result, we prove existence and uniqueness of the invariant distribution and exhibit some upper-bounds on the rate of convergence to equilibrium in terms of the asymptotic behavior of the covariance function of the Gaussian noise (or equivalently to its moving average representation). Then, we apply our general results to fractional dynamics (including the Euler Scheme associated to fractional driven Stochastic Differential Equations). Whent…
In this paper, we investigate Nash equilibrium payoffs for nonzero-sum stochastic differential games with reflection. We obtain an existence theorem and a characterization theorem of Nash equilibrium payoffs for nonzero-sum stochastic differential games with nonlinear cost functionals defined by doubly controlled reflected backward stochastic differential equations.
In this paper, we study Nash equilibrium payoffs for nonzero-sum stochastic differential games via the theory of backward stochastic differential equations. We obtain an existence theorem and a characterization theorem of Nash equilibrium payoffs for nonzero-sum stochastic differential games with nonlinear cost functionals defined with the help of a doubly controlled backward stochastic differential equation. Our results extend former ones by Buckdahn, Cardaliaguet and Rainer (2004) and are based on a backward stochastic differential equation approach.
This paper is related to nonzero-sum stochastic differential games in the Markovian framework. We show existence of a Nash equilibrium point for the game when the drift is no longer bounded and only satisfies a linear growth condition. The main tool is the notion of backward stochastic differential equations which, in our case, are multidimensional with continuous coefficient and stochastic linear growth.
We study the limiting behavior of the two-dimensional singular stochastic stochastic cubic nonlinear complex Ginzburg-Landau with Gibbs measure initial data. We show that in the appropriate small viscosity and small noise regimes, the limiting dynamics is given by the deterministic cubic nonlinear Schrödinger equation at Gibbs equilibrium. In order to obtain this convergence, our approach combines both heat and Schrödinger analysis, within the framework of the Fourier restriction norm method of Bourgain (1993).
In this paper, we continue our study on a general time-inconsistent stochastic linear--quadratic (LQ) control problem originally formulated in [6]. We derive a necessary and sufficient condition for equilibrium controls via a flow of forward--backward stochastic differential equations. When the state is one dimensional and the coefficients in the problem are all deterministic, we prove that the explicit equilibrium control constructed in \cite{HJZ} is indeed unique. Our proof is based on the derived equivalent condition for equilibria as well as a stochastic version of the Lebesgue differentia…
A Kawasaki dynamics in continuum is a dynamics of an infinite system of interacting particles in $\mathbb R^d$ which randomly hop over the space. In this paper, we deal with an equilibrium Kawasaki dynamics which has a Gibbs measure $μ$ as invariant measure. We study a diffusive limit of such a dynamics, derived through a scaling of both the jump rate and time. Under weak assumptions on the potential of pair interaction, $ϕ$, (in particular, admitting a singularity of $ϕ$ at zero), we prove that, on a set of smooth local functions, the generator of the scaled dynamics converges to the generato…
An optimal control problem is considered for a stochastic differential equation with the cost functional determined by a backward stochastic Volterra integral equation (BSVIE, for short). This kind of cost functional can cover the general discounting (including exponential and non-exponential) situation with a recursive feature. It is known that such a problem is time-inconsistent in general. Therefore, instead of finding a global optimal control, we look for a time-consistent locally near optimal equilibrium strategy. With the idea of multi-person differential games, a family of approximate e…
The existence of a (partial) market equilibrium price is proved in a complete, continuous time finite-agent market setting. The economic agents act as price takers in a fully competitive setting and maximize exponential utility from terminal wealth. As the number $N$ of economic agents goes to infinity, the BSDE system of $N$ equations characterizing the equilibrium asset price dynamics decouples. Due to the system's symmetry, the influence of the mean field of the agents, conditionally on the common noise, becomes deterministic.
In this paper we study macroscopic thermodynamic properties of a stochastic microscopic heat conduction model that is reduced from deterministic problems. Our goal is to numerically check how the `low energy site effect' inherited from the deterministic model would affect the macroscopic thermodynamic properties such as the thermal conductivity and the local thermodynamic equilibrium. After a series of numerical computations, our conclusion is that neither the thermal conductivity nor the existence of local thermodynamic equilibrium is qualitatively changed by this effect.
We extend the duality of Kipnis Marchioro and Presutti to inhomogeneous lattice gas systems where either the components have different degrees of freedom or the rate of interaction depends on the spatial location. Then the dual process is applied to prove local equilibrium in the hydrodynamic limit for some inhomogeneous high dimensional systems andin the nonequilibrium steady state for one dimensional systems with arbitrary inhomogeneity.
The distribution $μ_{cl}$ of a Poisson cluster process in $X=\mathbb{R}^{d}$ (with i.i.d. clusters) is studied via an auxiliary Poisson measure on the space of configurations in $\mathfrak{X}=\sqcup_{n} X^n$, with intensity measure defined as a convolution of the background intensity of cluster centres and the probability distribution of a generic cluster. We show that the measure $μ_{cl}$ is quasi-invariant with respect to the group of compactly supported diffeomorphisms of $X$ and prove an integration-by-parts formula for $μ_{cl}$. The corresponding equilibrium stochastic dynamics is then co…
We provide simple and constructive proofs of Harris-type theorems on the existence and uniqueness of an equilibrium and the speed of equilibration of discrete-time and continuous-time stochastic semigroups. Our results apply both to cases where the relaxation speed is exponential (also called geometric) and to those with no spectral gap, with non-exponential speeds (also called subgeometric). We give constructive estimates in the subgeometric case and discrete-time statements which seem both to be new. The method of proof also differs from previous works, based on semigroup and interpolation a…
A Markov evolution of a system of point particles in $\mathbb{R}^d$ is described at micro-and mesoscopic levels. The particles reproduce themselves at distant points (dispersal) and die, independently and under the influence of each other (competition). The microscopic description is based on an infinite chain of equations for correlation functions, similar to the BBGKY hierarchy used in the Hamiltonian dynamics of continuum particle systems. The mesoscopic description is based on a Vlasov-type kinetic equation for the particle's density obtained from the mentioned chain via a scaling procedur…
Non-equilibrium steady states are subject to intense investigations but still poorly understood. For instance, the derivation of Fourier law in Hamiltonian systems is a problem that still poses several obstacles. In order to investigate non-equilibrium systems, stochastic models of energy-exchange have been introduced and they have been used to identify universal properties of non-equilibrium. In these notes, after a brief review of the problem of anomalous transport in 1-dimensional Hamiltonian systems, some boundary-driven interacting random systems are considered and the "duality approach" …
We present a review of recent work on the statistical mechanics of non equilibrium processes based on the analysis of large deviations properties of microscopic systems. Stochastic lattice gases are non trivial models of such phenomena and can be studied rigorously providing a source of challenging mathematical problems. In this way, some principles of wide validity have been obtained leading to interesting physical consequences.
We study the approach to equilibrium, described by a Gibbs measure, for a system on a $d$-dimensional torus evolving according to a stochastic nonlinear Schrödinger equation (SNLS) with a high frequency truncation. We prove exponential approach to the truncated Gibbs measure both for the focusing and defocusing cases when the dynamics is constrained via suitable boundary conditions to regions of the Fourier space where the Hamiltonian is convex. Our method is based on establishing a spectral gap for the non self-adjoint Fokker-Planck operator governing the time evolution of the measure, which …
