We study ordinary differential equations (ODEs) with vector fields given by general Schwartz distributions, and we show that if we perturb such an equation by adding an “infinitely regularizing” path, then it has a unique solution and it induces an infinitely smooth flow of diffeomorphisms. We also introduce a criterion under which the sample paths of a Gaussian process are infinitely regularizing

We aim to obtain a homogenization theorem for a passive tracer interacting with a fractional, possibly non-Gaussian, noise. To do so, we analyze limit theorems for normalized functionals of Hermite–Volterra processes and extend existing results to cover power series with fast decaying coefficients. We obtain either convergence to a Wiener process, in the short-range dependent case, or to a Hermite

We consider a basic one-dimensional model which allows to obtain a diversity of diffusive regimes whose speed depends on the moments of a per-site trapping time. This models a discrete subordinated random walk, closely related to the continuous time random walks widely studied in the literature. The model we consider lends itself to a detailed elementary treatment, based on the study of recurrence

In this paper, we are interested in proving the existence of a weak martingale solution of the stochastic smectic-A liquid crystal system driven by a pure jump noise in both 2D and 3D bounded domains. We prove the existence of a global weak martingale solution under some non-Lipschitz assumptions on the coefficients. The construction of the solution is based on a Faedo–Galerkin approximation, a compactness

This work is devoted to the study of a stochastic logistic growth model with and without the Allee effect. Such a model describes the evolution of a population under environmental stochastic fluctuations and is in the form of a stochastic differential equation driven by multiplicative Gaussian noise. With the help of the associated Fokker–Planck equation, we analyze the population extinction probability

Let ℬ⊆ℕ∖{1} be a primitive set, ℱℬ:=ℤ∖⋃b∈ℬbℤ, η:=1ℱℬ∈{0,1}ℤ, and denote by Xη the orbit closure of η under the shift. We complement results on heredity of Xη from [Dymek et al., ℬ-free sets and dynamics, Trans. Amer. Math. Soc. 370 (2018) 5425–5489] in two directions: In the proximal case we prove that a certain subshift Xφ⊇Xη, which coincides with Xη when ℬ is taut, is always hereditary. (In particular

In this paper, we investigate the multifractal decomposition of the limit set of a finitely generated, free Fuchsian group with respect to the mean cusp-winding number. We completely determine its multifractal spectrum by means of a certain free energy function and show that the Hausdorff dimension of sets consisting of limit points with the same scaling exponent coincides with the Legendre transform

Given a non-conformal repeller Λ of a C1 map f, we give a variational principle of a dimensional upper bound of the non-conformal repellers. If f is C1+α then we prove the joint continuity of topological pressure for sub-additive singular-valued potentials on maps and parameters.

In this paper, we study fluctuations of the volume of a stable sausage defined via a d-dimensional rotationally invariant α-stable process. As the main results, we establish a functional central limit theorem (in the case when d/α>3/2) with a standard one-dimensional Brownian motion in the limit, and Khintchine’s and Chung’s laws of the iterated logarithm (in the case when d/α>9/5).

In this work, we are interested in characterizing typical (generic) dimensional properties of invariant measures associated with the full-shift system, T, in a product space whose alphabet is a perfect Polish metric space (thus, uncountable). More specifically, we show that the set of invariant measures with upper Hausdorff dimension equal to zero and lower packing dimension equal to infinity is a

We consider random perturbations of a topologically transitive local diffeomorphism of a Riemannian manifold. We show that if an absolutely continuous ergodic stationary measures is expanding (all Lyapunov exponents positive), then there is a random Gibbs–Markov–Young structure which can be used to lift that measure. We also prove that if the original map admits a finite number of expanding invariant

In this paper, we investigate a generalization of Brownian motion, called sticky skew Brownian motion, which has two interesting characteristics: stickiness and skewness. This kind of processes spends a lot more time at its sticky points so that the time they spend at the sticky points has positive Lebesgue measure. By using time change, we obtain an SDE for the sticky skew Brownian motion. Then, we

In this paper, we provide a concrete construction of Furstenberg entropy values of τ-boundaries of the group ℤ[1p1,…,1pl]⋊{p1n1⋯plnl:ni∈ℤ} by choosing an appropriate random walk τ. We show that the boundary entropy spectrum can be realized as the subsum-set for any given finite sequence of positive numbers.

We study a general class of fully coupled backward–forward stochastic differential equations of mean-field type (MF-BFSDE). We derive existence and uniqueness results for such a system under weak monotonicity assumptions and without the non-degeneracy condition on the forward equation. This is achieved by suggesting an implicit approximation scheme that is shown to converge to the solution of the system

A set P⊂ℕ is called predictive if for any zero entropy finite-valued stationary process (Xi)i∈ℤ, X0 is measurable with respect to (X−i)i∈P. We know that ℕ is a predictive set. In this paper, we give sufficient conditions and necessary ones for a set to be predictive. We also discuss linear predictivity, predictivity among Gaussian processes and relate these to Riesz sets which arise in harmonic analysis

We prove uniqueness of equilibrium states for subshifts corresponding to intermediate beta transformations with β>2 having the property that the orbit of 0 is bounded away from 1.

We present a notion of almost periodicity which can be applied to random dynamical systems as well as almost periodic stochastic differential equations in Hilbert spaces (abstract stochastic partial differential equations). This concept allows for improvements of known results of almost periodicity in distribution, for general random processes and for solutions to stochastic differential equations

The Wong–Zakai approximations given by a stationary process and attractors for stochastic degenerate parabolic equations are considered in this paper. We first establish the existence and uniqueness of tempered pullback attractors for the Wong–Zakai approximations of stochastic degenerate parabolic equations. We then prove that the attractors of Wong–Zakai approximations converge to the attractor of

In this paper, we study the joint Laplace transform of the sticky Brownian motion on an interval, its occupation time at zero and its integrated process. The perturbation approach of Li and Zhou [The joint Laplace transforms for diffusion occupation times, Adv. Appl. Probab.45 (2013) 1049–1067] is adopted to convert the problem into the computation of three Laplace transforms, which is essentially

Recently, Hao and Li [Fully coupled forward-backward SDEs involving the value function. Nonlocal Hamilton–Jacobi–Bellman equations, ESAIM: Control Optim, Calc. Var.22(2016) 519–538] studied a new kind of forward-backward stochastic differential equations (FBSDEs), namely the fully coupled FBSDEs involving the value function in the case where the diffusion coefficient σ in forward stochastic differential

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

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 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).

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

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

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.

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 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

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

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

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