We consider a ϕ-mixing shift T on a sequence space Ω and study the number 𝒩N of returns {TqN(n)ω∈Ana} at times qN(n) to a cylinder Ana constructed by a sequence a∈Ω where n runs either until a fixed integer N or until a time τN of the first return {TqN(n)ω∈Amb} to another cylinder Amb constructed by b∈Ω. Here, qN(n) are certain functions of n taking on nonnegative integer values when n runs from 0

We study a mean field approximation for the 2D Euler vorticity equation driven by a transport noise. We prove that the Euler equations can be approximated by interacting point vortices driven by a regularized Biot–Savart kernel and the same common noise. The approximation happens by sending the number of particles N to infinity and the regularization 𝜖 in the Biot–Savart kernel to 0, as a suitable

An interacting particle system made of diffusion processes with local interaction is considered and the macroscopic limit to a nonlinear PDE is investigated. Few rigorous results exists on this problem and in particular the explicit form of the nonlinearity is not known. This paper reviews this subject, some of the main ideas to get the limit nonlinear PDE and provides both heuristic and numerical

In this paper, we consider the maximum likelihood estimation for the symmetric α-stable Ornstein–Uhlenbeck (SαS-OU) processes based on discrete observations. Since the closed-form expression of maximum likelihood function is hard to obtain in the Lévy case, we choose a mixture of Cauchy and Gaussian distribution to approximate the probability density function (PDF) of the SαS distribution. By means

The asymptotic behavior of stochastic modified quasi-geostrophic equations with damping driven by colored noise is analyzed. In fact, the existence of random attractors is established in W2α−,p(ℝ2). In particular, we prove also the existence of a global compact attractor for autonomous quasi-geostrophic equations with damping in W2α−,p(ℝ2). Here, we do not add any modifying factor on the nonlinear

For an annealed type random dynamical system arising from non-uniformly expanding maps which admits uniformly contractive branches, we establish the existence of an absolutely continuous σ-finite invariant measure. We also show when the invariant measure is infinite.

We prove existence and uniqueness of martingale solutions to a (slightly) hyper-viscous stochastic Navier–Stokes equation in 2d with initial conditions absolutely continuous with respect to the Gibbs measure associated to the energy, getting the results both in the torus and in the whole space setting.

This paper aims to investigate the controllability for impulsive neutral stochastic delay partial differential equations (PDEs) driven by fractional Brownian motion (fBm) with Hurst index H∈(12,1) and Lévy noise in Hilbert spaces. By using a fixed point approach without imposing a severe compactness condition on the semigroup, a new set of sufficient conditions is derived. The results in this paper

Quantum Bernoulli noises (QBN) are the family of annihilation and creation operators acting on Bernoulli functionals, which satisfy a canonical anti-commutation relation in equal-time. In this paper, we aim to investigate quantum Feller semigroups in terms of QBN. We first investigate local structure of the algebra generated by identity operator and QBN. We then use our new results obtained here to

This paper focuses on systems of stochastic partial differential equations with impulse effects. We establish an averaging principle such that the solution to the complex original nonlinear impulsive stochastic evolution equations can be approximated by that to the more simplified averaged stochastic evolution equations without impulses. By adopting stochastic analysis theory, semigroup approach and

We establish weak well-posedness for critical symmetric stable driven SDEs in ℝd with additive noise Z, d≥1. Namely, we study the case where the stable index of the driving process Z is α=1 which exactly corresponds to the order of the drift term having the coefficient b which is continuous and bounded. In particular, we cover the cylindrical case when Zt=(Zt1,…,Ztd) and Z1,…,Zd are independent one-dimensional

We consider a Piecewise Deterministic Markov Process given by random switching between finitely many vector fields vanishing at 0. It has been shown recently that the behavior of this process is mainly determined by the signs of Lyapunov exponents. However, results have only been given when all these exponents have the same sign. In this paper, we consider the degenerate case where the process leaves

Based on our deterministic models for cholera epidemics, we propose a stochastic model for cholera epidemics to incorporate environmental fluctuations which is a nonlinear system of Itô stochastic differential equations. We conduct an asymptotical analysis of dynamical behaviors for the model. The basic stochastic reproduction value ℛs is defined in terms of the basic reproduction number R0 for the

In this paper, we generalize the recently introduced concept of fair measure [M. Misiurewicz and A. Rodrigues, Counting preimages, Ergod. Theor. Dyn. Syst.38 (2018) 1837–1856]. We study fair measures for Markov and mixing interval maps with countably many branches. We investigate them in terms of the recurrence properties of some underlying countable Markov shifts, both from the stochastic viewpoint

We consider the sequence of spatial quadratic variations of the solution to the stochastic heat equation with space-time white noise. This sequence satisfies a Central Limit Theorem. By using Malliavin calculus, we refine this result by proving the convergence of the sequence of densities and by finding the second-order term in the asymptotic expansion of the densities. In particular, our proofs are

Several variants of the classic Fibonacci inflation tiling are considered in an illustrative fashion, in one and in two dimensions, with an eye on changes or robustness of diffraction and dynamical spectra. In one dimension, we consider extension mechanisms of deterministic and of stochastic nature, while we look at direct product variations in a planar extension. For the pure point part, we systematically

In this paper, we shall prove the homeomorphism property of solutions for multidimensional stochastic differential equations (SDEs) driven by G-Brownian motion with non-Lipschitz coefficients with respect to the initial values.

In this paper, we consider a stochastic Cahn–Hilliard Navier–Stokes system in a bounded domain of ℝd,d=2,3. The system models the evolution of an incompressible isothermal mixture of binary fluids under the influence of stochastic external forces. We prove the existence of a global weak martingale solution. The proof is based on the splitting-up method as well as some compactness method.

In this paper, we introduce the conceptions of multi-transitivity, Δ-transitivity and Δ-mixing property for free semigroup actions and give some equivalent conditions for a free semigroup action to be multi-transitive, multi-transitive with respect to vectors and strongly multi-transitive, respectively. For instance, we prove that a free semigroup action is multi-transitive or multi-transitive with

We consider on the 2D torus the modified Surface Quasi-Geostrophic (mSQG) equation with L2-initial data and perturbed by multiplicative transport noise. Under a suitable scaling of the noises, we show that the solutions converge weakly to the unique solution of the dissipative mSQG equation.

Our aim in this paper is to establish some strong stability properties of solutions of mean-field stochastic differential equations. These latter are stochastic differential equations where the coefficients depend not only on the state of the unknown process but also on its probability distribution. The results are obtained assuming that the pathwise uniqueness property holds and using Skorokhod’s

This paper is concerned with the relationship between backward stochastic Volterra integral equations (BSVIEs, for short) and a kind of non-local quasilinear (and possibly degenerate) parabolic equations. As a natural extension of BSVIEs, the extended BSVIEs (EBSVIEs, for short) are introduced and investigated. Under some mild conditions, the well-posedness of EBSVIEs is established and some regularity

We consider a linearized dynamical system modeling the flow rate of water along the rivers and hillslopes of an arbitrary watershed. The system is perturbed by a random rainfall in the form of a compound Poisson process. The model describes the evolution, at daily time scales, of an interconnected network of linear reservoirs and takes into account the differences in flow celerity between hillslopes

We introduce a fractional stochastic heat equation with second-order elliptic operator in divergence form, having a piecewise constant diffusion coefficient, and driven by an infinite-dimensional fractional Brownian motion. We characterize the fundamental solution of its deterministic part, and prove the existence and the uniqueness of its solution.

We link a general method for modeling random phenomena using systems of stochastic differential equations (SDEs) to the class of affine SDEs. This general construction emphasizes the central role of the Duffie–Kan system [Duffie and Kan, A yield-factor model of interest rates, Math. Finance6 (1996) 379–406] as a model for first-order approximations of a wide class of nonlinear systems perturbed by

Let D be a bounded Lipschitz domain of ℝd. We consider the complement value problem (Δ+aαΔα/2+b⋅∇+c)u+f=0inD,u=gonDc. Under mild conditions, we show that there exists a unique bounded continuous weak solution. Moreover, we give an explicit probabilistic representation of the solution. The theory of semi-Dirichlet forms and heat kernel estimates play an important role in our approach.

We study a Markov birth-and-death process on a space of locally finite configurations, which describes an ecological model with a density-dependent fecundity regulation mechanism. We establish existence and uniqueness of this process and analyze its properties. In particular, we show global time-space boundedness of the population density and, using a constructed Foster–Lyapunov-type function, we study

In this paper, we use the techniques of fractional calculus to study the existence of a unique solution to semilinear fractional differential equation driven by a γ-Hölder continuous function 𝜃 with γ∈(23,1). Here, the initial condition is a function that may not be defined at zero and the involved integral with respect to 𝜃 is the extension of the Young integral [An inequality of the Hölder type

We first give an existence criterion for a random uniform exponential attractor for a jointly continuous non-autonomous random dynamical system defined on the product space of N-weighted spaces of infinite sequences. Then, based on this criterion, we prove the existence of random uniform exponential attractors for stochastic lattice systems and stochastic FitzHugh–Nagumo lattice systems that are both

A stochastic approach to the (generic) mean-field limit in Bose–Einstein Condensation is described and the convergence of the ground-state energy as well as of its components are established. For the one-particle process on the path space, a total variation convergence result is proved. A strong form of Kac’s chaos on path-space for the k-particles probability measures is derived from the previous

A generalized dynamically evolving random network and a game model taking place on the evolving network are presented. We show that there exists a high-dimensional critical curved surface of the parameters related the probabilities of adding or removing vertices or edges such that the evolving network may exhibit three kinds of degree distributions as the time goes to infinity when the parameters belong

In this paper, we investigate the time asymptotic behavior of solutions to fractional in time evolution equations which appear as results of random time changes in Markov processes. We consider inverse subordinators as random times and use the subordination principle for the solutions to forward Kolmogorov equations. The classes of subordinators for which asymptotic analysis may be realized are described

Let (X,d) be a locally compact separable ultrametric space. Given a measure m on X and a function C(B) defined on the set of all non-singleton balls B of X, we consider the hierarchical Laplacian L=LC. The operator L acts in ℒ2(X,m), is essentially self-adjoint and has a purely point spectrum. Choosing a sequence {𝜀(B)} of i.i.d. random variables, we consider the perturbed function C(B,ω) and the

We consider exponential large deviations estimates for unbounded observables on uniformly expanding dynamical systems. We show that uniform expansion does not imply the existence of a rate function for unbounded observables no matter the tail behavior of the cumulative distribution function. We give examples of unbounded observables with exponential decay of autocorrelations, exponential decay under

We consider a class of Markov processes with resettings, where at random times, the Markov processes are restarted from a predetermined point or a region. These processes are frequently applied in physics, chemistry, biology, economics, and in population dynamics. In this paper, we establish the local large deviation principle (LLDP) for the Wiener processes with random resettings, where the resettings

This paper studies the convergence of the stochastic algorithm of the modified Robbins–Monro form for a Markov random field (MRF), in which some of the nodes are clamped to be observed variables while the others are hidden ones. Based on the theory of stochastic approximation, we propose proper assumptions to guarantee the Hölder regularity of both the update function and the solution of the Poisson

We study the thermodynamic formalism for particular types of sub-additive sequences on a class of subshifts over countable alphabets. The subshifts we consider include factors of irreducible countable Markov shifts under certain conditions, which we call irreducible countable sofic shifts. We show the variational principle for topological pressure for some sub-additive sequences with tempered variation

We show regularization effect of nonlinear gradient noise to the solution of 1D stochastic parabolic equation. We demonstrate convergence to a martingale (independent upon space variable) when we rescale noise at the extremum points of the process.

We consider the exit problem for a one-dimensional system with random switching near an unstable equilibrium point of the averaged drift. In the infinite switching rate limit, we show that the exit time satisfies a limit theorem with a logarithmic deterministic term and a random correction converging in distribution. Thus, this setting is in the universality class of the unstable equilibrium exit under

Existence of expansivity for group action φ:G×X→X depends on algebraic properties of G and the topology of X. We give an expansive action of a solvable group on S1 while there is no expansive action of a solvable group on a dendrite X. We prove that a continuous action φ:G×X→X on a compact metric space X is expansive if and only if there exists an open cover 𝒰 such that for any other open cover 𝒱

For an Ornstein–Uhlenbeck process driven by fractional Brownian motion with Hurst index H∈[12,34], we show the Berry–Esséen bound of the least squares estimator of the drift parameter based on the continuous-time observation. We use an approach based on Malliavin calculus given by Kim and Park [Optimal Berry–Esséen bound for statistical estimations and its application to SPDE, J. Multivariate Anal

We study dynamical systems arising as time-dependent compositions of Pomeau-Manneville-type intermittent maps. We establish central limit theorems for appropriately scaled and centered Birkhoff-like partial sums, with estimates on the rate of convergence. For maps chosen from a certain parameter range, but without additional assumptions on how the maps vary with time, we obtain a self-norming CLT provided

We present an adaptation of Stein’s method of normal approximation to the study of both discrete- and continuous-time dynamical systems. We obtain new correlation-decay conditions on dynamical systems for a multivariate central limit theorem augmented by a rate of convergence. We then present a scheme for checking these conditions in actual examples. The principal contribution of our paper is the method

Any Borel probability measure supported on a Cantor set included in [0,1] and of zero Lebesgue measure on the real line possesses a discrete inverse measure. We study the validity of the multifractal formalism for the inverse measures of random weak Gibbs measures. The study requires, in particular, to develop in this context of random dynamics a suitable version of the results known for heterogeneous

Linear filtering problem for infinite-dimensional Gaussian processes is studied, the observation process being finite-dimensional. Integral equations for the filter and for covariance of the error are derived. General results are applied to linear SPDEs driven by Gauss–Volterra process observed at finitely many points of the domain.

We investigate the asymptotic behavior of a class of non-autonomous stochastic FitzHugh–Nagumo systems driven by additive white noise on unbounded thin domains. For this aim, we first show the existence and uniqueness of random attractors for the considered equations and their limit equations. Then, we establish the upper semicontinuity of these attractors when the thin domains collapse into a lower-dimensional

In this paper, we prove a central limit theorem and a moderate deviation principle for a perturbed stochastic Cahn–Hilliard equation defined on [0,T]×[0,π]d with d∈{1,2,3}. This equation is driven by a space-time white noise. The weak convergence approach plays an important role.

A new modification of the minimum-contrast estimator (the weighted MCE) of drift parameter in a linear stochastic evolution equation with additive fractional noise is introduced in the setting of the spectral approach (Fourier coordinates of the solution are observed). The reweighing technique, which utilizes the self-similarity property, achieves strong consistency and asymptotic normality of the

In this paper, we prove a moderate deviation principle for the multivalued stochastic differential equations whose proof are based on recently well-developed weak convergence approach. As an application, we obtain the moderate deviation principle for reflected Brownian motion.

We consider a parameter estimation problem for one-dimensional stochastic heat equations, when data is sampled discretely in time or spatial component. We prove that, the real valued parameter next to the Laplacian (the drift), and the constant parameter in front of the noise (the volatility) can be consistently estimated under somewhat surprisingly minimal information. Namely, it is enough to observe