We establish Freidlin–Wentzell results for a nonlinear ordinary differential equation starting close to the stable state 0, say, subject to a perturbation by a stochastic integral which is driven by an 𝜀-small and (1/𝜀)-accelerated Lévy process with exponentially light jumps. For this purpose, we derive a large deviations principle for the stochastically perturbed system using the weak convergence

This paper studies typical dynamics and fluctuations for a slow–fast dynamical system perturbed by a small fractional Brownian noise. Based on an ergodic theorem with explicit rates of convergence, which may be of independent interest, we characterize the asymptotic dynamics of the slow component to two orders (i.e. the typical dynamics and the fluctuations). The limiting distribution of the fluctuations

In this paper, we focus on the so-called identification problem for a BSDE driven by a continuous local martingale and a possibly non-quasi-left-continuous random measure. Supposing that a solution (Y,Z,U) of a BSDE is such that Yt=v(t,Xt), where X is an underlying process and v is a deterministic function, solving the identification problem consists in determining Z and U in terms of v. We study the

The objective of this paper is to prove the existence of a global solution to a certain stochastic partial differential equation subject to the L2-norm being constrained. The corresponding evolution equation can be seen as the projection of the unconstrained problem onto the tangent space of the unit sphere ℳ in a Hilbert space L2.

This paper is devoted to an heuristic discussion of the merging mechanism between two clusters of point vortices, supported by some numerical simulations. A concept of renormalized Onsager function is introduced, elaboration of the solutions of the mean field equation. It is used to understand the shape of the single cluster observed as a result of the merging process. Potential implications for the

We prove uniform synchronization by noise with rates for the stochastic quantization equation in dimensions two and three. The proof relies on a combination of coming down from infinity estimates and the framework of order-preserving Markov semigroups derived in [O. Butkovsky and M. Scheutzow, Couplings via comparison principle and exponential ergodicity of SPDEs in the hypoelliptic setting, preprint

We consider the Mean Field limit of Gibbsian ensembles of 2-dimensional (2D) point vortices on the torus. It is a classical result that in such limit correlations functions converge to 1, that is, point vortices decorrelate: We compute the rate at which this convergence takes place by means of Gaussian integration techniques, inspired by the correspondence between the 2D Coulomb gas and the Sine-Gordon

We consider a discrete model in which particles are characterized by two quantities X and Y; both quantities evolve in time according to stochastic dynamics and the equation that governs the evolution of Y is also influenced by mean-field interaction between the particles. We allow for discontinuous coefficients and random initial condition and, under suitable assumptions, we prove that in the limit

We consider a class of generalized stochastic porous media equations with multiplicative Lipschitz continuous noise. These equations can be related to physical models exhibiting self-organized criticality. We show that these SPDEs have unique SVI solutions which depend continuously on the initial value. In order to formulate this notion of solution and to prove uniqueness in the case of a slowly growing

We prove uniform synchronization by noise with rates for the stochastic quantization equation in dimensions two and three. The proof relies on a combination of coming down from infinity estimates and the framework of order-preserving Markov semigroups derived in [O. Butkovsky and M. Scheutzow, Couplings via comparison principle and exponential ergodicity of SPDEs in the hypoelliptic setting, preprint

We show the existence of generalized compensation functions for a particular type of one-block factor maps π:X→Y between countable subshifts X and Y. For factor maps between compact spaces, continuous compensation functions were studied by Walters in relation to the theory of relative pressure. Applying the thermodynamic formalism for sequences on countable subshifts, we generalize some existing results

For the Ornstein–Uhlenbeck process in stationary and explosive cases, this paper studies Cramér-type moderate deviations for the log-likelihood ratio process. As an application, we give the negative regions of drift testing problem, and also obtain the decay rates of the error probabilities. The main methods of this paper consist of mod-ϕ convergence approach, deviation inequalities for multiple Wiener–Itô

Recently, the Cox–Ingersoll–Ross (CIR) model with Markov switching has been discussed extensively. However, the covariance function and the kth moment for this model are still open. In this paper, we consider some characterizations for the CIR model with Markov switching. First, the conditional moment generating functions for CIR model with Markov switching are given. Then, explicit expressions for

We obtain asymptotic expansions for the large deviation principle (LDP) for continuous time stochastic processes with weakly-dependent increments. As a key example, we show that additive functionals of solutions of stochastic differential equations (SDEs) satisfying Hörmander condition on a d-dimensional compact manifold admit these asymptotic expansions of all orders.

A Freidlin–Wentzell type large deviation principle is established for stochastic partial differential equations with slow and fast time-scales, where the slow component is a one-dimensional stochastic Burgers equation with small noise and the fast component is a stochastic reaction-diffusion equation. Our approach is via the weak convergence criterion developed in [A. Budhiraja and P. Dupuis, A variational

We study a non-conservation second-order stochastic partial differential equation (SPDE) driven by multi-parameter anisotropic fractional Lévy noise (AFLN) and under different initial and/or boundary conditions. It includes the time-dependent linear heat equation and quasi-linear heat equation under the fractional noise as special cases. Unique existence and expressions of solution to the equation

We consider a class of planar dispersing billiards with a cusp at a point of vanishing curvature. Convergence to a stable law and to the corresponding Lévy process in the ℳ1 and ℳ2 Skorohod topologies has been studied in recent work. Here, we show that certain sufficient conditions for ℳ2-convergence are also necessary.

We extend the notions of joint and generalized spectral radii to cocycles acting on Banach spaces and obtain a version of Berger–Wang’s formula when restricted to the space of cocycles taking values in the space of compact operators. Moreover, we observe that the previous quantities depends continuously on the underlying cocycle.

Let ℱ be a C2 random partially hyperbolic dynamical system. For the unstable foliation, the corresponding unstable metric entropy, unstable topological entropy and unstable pressure via the dynamics of ℱ on the unstable foliation are introduced and investigated. A version of Shannon–McMillan–Breiman Theorem for unstable metric entropy is given, and a variational principle for unstable pressure (and

We present a numerical method to calculate resonances of Schottky surfaces based on Selberg theory, transfer operator techniques and Lagrange–Chebyshev approximation. This method is an alternative to the method based on periodic orbit expansion used previously in this context.

In this paper, we formulate a stochastic logistic fish growth model driven by both white noise and non-Gaussian noise. We focus our study on the mean time to extinction, escape probability to measure the noise-induced extinction probability and the Fokker–Planck equation for fish population X(t). In the Gaussian case, these quantities satisfy local partial differential equations while in the non-Gaussian

This work described here is part of a project to document the life and work of Kurt Heegner (1893–1965) whose papers have been deposited in the Handschriftenabteilung of the SUB Göttingen. The question with which this paper is concerned is the rationality of the variety which describes the connection of the volume of a tetrahedron with its side-lengths (Cayley–Menger formula).

In this work, we prove a version of Hörmander’s theorem for a stochastic evolution equation driven by a trace-class fractional Brownian motion with Hurst exponent 12

This paper studies the (random) uniform attractor for a class of non-autonomous stochastic evolution equations driven by a time-periodic forcing and multiplicative fractional noise with Hurst parameter bigger than 1/2. We first establish the existence and uniqueness results for the solution to the considered equation and show that the solution generates a jointly continuous non-autonomous random dynamical

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

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.

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 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

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.

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

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

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

Random attractors and their higher-order regularity properties are studied for stochastic reaction–diffusion equations on time-varying domains. Some new a priori estimates for the difference of solutions near the initial time and the continuous dependence in initial data in H01 are proved. Then attraction of the random attractors in the higher integrability space L2+δ for any δ∈[0,∞) and the regular

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

A G-process is briefly a process ([A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, Vol. 182 (Springer, 2013)], [C. M. Dafermos, An invariance principle for compact processes, J. Differential Equations9 (1971) 239–252], [P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys

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