Stationary measures for stochastic differential equations with jumps

In the paper, stationary measures of stochastic differential equations with jumps are considered. Under some general conditions, existence of stationary measures is proved through Markov measures and Lyapunov functions. Moreover, for two special cases, stationary measures are given by solutions of Fokker–Planck equations and long time limits for the distributions of system states.     

Stability of infinite systems of stochastic equations

A study is presented of processes with local interaction and infinite systems of Itô equations, which are small perturbations of independent processes at each point of a lattice with noncompact set of values. A survey of results and a complete exposition are devoted to the technique of cluster expansions, which uses estimates of the Lyapunov function type. This technique permits total control over the temporal evolution of the system and, in particular, enables one to prove that the temporal evolution is exponentially convergent.Translated from Itogi Nauki i Tekhniki, Seriya Teoriya Veroyatnostei, Matematicheskaya Statistika, Teoreticheskaya Kibernetika, Vol. 27, pp. 79–128, 1990.     

Ergodicity of infinite systems of stochastic equations

Translated from Matematicheskie Zametki, Vol. 45, No. 4, pp. 78–88, April, 1989.     

Dynamical behavior for stochastic lattice systems

Random attractors describe the long term behavior of the random dynamical systems. This paper is devoted to a general first order stochastic lattice dynamical systems (SLDS) with some dissipative nonlinearity. We prove the asymptotic compactness of the random dynamical system and obtain the random attractor, which is a compact random invariant set with tempered bound.     

Infinite systems of stochastic differential equations and some lattice models on compact Riemannian manifolds

Stochastic dynamics associated with Gibbs measures on an infinite product of compact Riemannian manifolds is constructed. The probabilistic representations for the corresponding Feller semigroups are obtained. The uniqueness of the dynamics is proved.     

Stationary and stationarizable regimes in normal stochastic differential systems

Narrow-sense stationary regimes are considered for multi-dimensional non-linear systems described by Ito stochastic differential equations with Wiener processes. The conditions for the existence of stationary and stationarizable one-dimensional distributions are derived. Exact expressions are obtained for stationary distributions in some mechanical systems.     

Random attractors for second-order stochastic lattice dynamical systems

In this paper, the asymptotic behavior of second-order stochastic lattice dynamical systems is considered. We firstly show the existence of an absorbing set. Then an estimate on tails of the solutions is derived when the time is large enough, which ensures the asymptotic compactness of the random dynamical system. Finally, the existence of the random attractor is provided.     

Networks of interacting stochastic fluid models with infinite and finite buffers

Queueing Systems - Stochastic fluid models have been widely used to model the level of a resource that changes over time, where the rate of variation depends on the state of some continuous-time...     

Stationary probability distributions of some lotka-volterra types of stochastic predation systems

This paper provides exact solutions to the stationary probability distributions in some stochastic predation systems. These are derived by solving the Fokker-Planck equations for:

(i) a generalized stochastic Lotka-Volterra predator-prey system, and

(ii) a generalised stochastic Lotka-Volterra food chain.

In all these systems the growth dynamics of all levels of species are subject to stochastic shocks. Since stationary probability distributions provide the most comprehensive characterization of a stochastic system in a steady state, system stability can be analysed accordingly     

Stationary Stokes and Navier-Stokes systems in an infinite open channel. I

Institute of Mathematics and Cybernetics, Academy of Sciences of the Lithuanian SSR. Leningrad Division of the Mathematical Institute, Academy of Sciences of the USSR. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 29, No. 1, pp. 90–108, January–March, 1989.     

Stationary stokes and Navier-Stokes systems in an infinite open channel. II

Institute of Mathematics and Cybernetics, Academy of Sciences of the Lithuanian SSR. Leningrad Section of the Mathematical Institute, Academy of Sciences of the USSR. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 29, No. 2, pp. 347–367, April–June, 1989.     

Non reversible stationary measures for infinite interacting particle systems

Summary Two results concerning the local conditional distributions of a stationary measure for a spin flip process with strictly positive and continuous rates are obtained: 1) The local conditional distributions and the rates of the reversed process determine each other. 2) Either all shift invariant stationary measures are Gibbs with the same potential or no shift invariant stationary Gibbs measure exist.Research supported by the Japan Society for the Promotion of Science     

Asymptotic Log-Harnack inequality and applications for stochastic systems of infinite memory

The asymptotic log-Harnack inequality is established for several kinds of models on stochastic differential systems with infinite memory: non-degenerate SDEs, neutral SDEs, semi-linear SPDEs, and stochastic Hamiltonian systems. As applications, the following properties are derived for the associated segment Markov semigroups: asymptotic heat kernel estimate, uniqueness of the invariant probability measure, asymptotic gradient estimate (hence, asymptotically strong Feller property), as well as asymptotic irreducibility.     

Attractors for stochastic lattice dynamical systems with a multiplicative noise

In this paper, we consider a stochastic lattice differential equation with diffusive nearest neighbor interaction, a dissipative nonlinear reaction term, and multiplicative white noise at each node. We prove the existence of a compact global random attractor which, pulled back, attracts tempered random bounded sets.      

Random attractors for stochastic lattice dynamical systems in weighted spaces

In this paper, we first provide some sufficient conditions for the existence of global compact random attractors for general random dynamical systems in weighted space (p?1) of infinite sequences. Then we consider the existence of global compact random attractors in weighted space for stochastic lattice dynamical systems with random coupled coefficients and multiplicative/additive white noises. Our results recover many existing ones on the existence of global random attractors for stochastic lattice dynamical systems with multiplicative/additive white noises in regular l² space of infinite sequences.     

The doubling property of conformal measures of infinite iterated function systems

We provide sufficient conditions for the conformal measures induced by regular conformal infinite iterated function systems to satisfy the doubling property. We apply these conditions to iterated function systems derived from the continued fraction algorithm—continued fractions with restricted entries. For these systems our conditions are expressed in terms of the asymptotic density properties of the allowed entries. As examples, we give some relatively large classes of sets of continued fractions with restricted entries for which the corresponding conformal measures have the doubling property. Similarly, we give some other classes for which the conformal measure does not have the doubling property.     

Natural gradient algorithm for stochastic distribution systems with output feedback

In this paper, we use natural gradient algorithm to control the shape of the conditional output probability density function for the stochastic distribution systems from the viewpoint of information geometry. The considered system here is of multi-input and single output with an output feedback and a stochastic noise. Based on the assumption that the probability density function of the stochastic noise is known, we obtain the conditional output probability density function whose shape is only determined by the control input vector under the condition that the output feedback is known at any sample time. The set of all the conditional output probability density functions forms a statistical manifold (M), and the control input vector and the output feedback are considered as the coordinate system. The Kullback divergence acts as the distance between the conditional output probability density function and the target probability density function. Thus, an iterative formula for the control input vector is proposed in the sense of information geometry. Meanwhile, we consider the convergence of the presented algorithm. At last, an illustrative example is utilized to demonstrate the effectiveness of the algorithm.