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We prove the null controllability of forward and backward linear stochastic parabolic equations with first and zero order coupling terms, and with bounded coefficients. The null controllability results rely on novel Carleman estimates for the backward and forward adjoint equations, established through the application of the duality technique. In doing so, we resolve an open question (see Remark 2.2 in [Tang and Zhang in SIAM J Control Optim 48:2191–2216, 2009]). Moreover, we provide more accurate estimates for the null-control costs. [ABSTRACT FROM AUTHOR]

Loosely speaking, the Navier–Stokes- α model and the Navier–Stokes equations differ by a spatial filtration parametrized by a scale denoted α . Starting from a strong two-dimensional solution to the Navier–Stokes- α model driven by a multiplicative noise, we demonstrate that it generates a strong solution to the stochastic Navier–Stokes equations under the condition α → 0 . The initially introduced probability space and the Wiener process are maintained throughout the investigation, thanks to a local monotonicity property that abolishes the use of Skorokhod’s theorem. High spatial regularity a priori estimates for the fluid velocity vector field are carried out within periodic boundary conditions. [ABSTRACT FROM AUTHOR]

This work is concerned with the modified stochastic Swift–Hohenberg equation in a 3D thin domain. Although the diffusion motion of molecules is irregular with the interference of the film-fluid fluctuation, the invariant measure on the trajectory space reveals delicate transition of the dynamical behavior when the interior forces change. We therefore prove that the invariant measure of the system converges weakly to the unique counterpart of the stochastic Swift–Hohenberg equation in a 2D bounded domain with a concrete convergence rate, as the modified parameter and the thickness of the thin domain tend to zero. Furthermore, we address that the smooth density of the limit invariant measure fulfills a Fokker–Planck equation. [ABSTRACT FROM AUTHOR]

In this work, we consider a system of fast and slow time-scale stochastic partial differential equations with reflection, where the slow component is the one-dimensional stochastic Burgers equation, the fast component is the stochastic reaction-diffusion equation, and both the fast and slow components have two reflecting walls. The well-posedness of this system is established. Our approach is based on the penalized method by giving the delicate estimation of the penalized terms, which do not resort to splitting the reflected system into stochastic system without reflection and deterministic system with reflection. Then by means of penalized method and combining the classical Khasminskii’s time discretization, we prove the averaging principle for a class of reflected stochastic partial differential equations. In particular, due to the existence and uniqueness of invariant measure for fast component with frozen slow component, the ergodicity for frozen equations are given for different initial function spaces, which plays an important role. [ABSTRACT FROM AUTHOR]

Considering irreducibility is fundamental for studying the ergodicity of stochastic dynamical systems. In this paper, we establish the irreducibility of stochastic complex Ginzburg-Laudau equations driven by pure jump noise. Our results are dimension free and the conditions placed on the driving noises are very mild. A crucial role is played by criteria developed by the authors of this paper and T. Zhang for the irreducibility of stochastic equations driven by pure jump noise. As an application, we obtain the ergodicity of stochastic complex Ginzburg-Laudau equations. We remark that our ergodicity result covers the weakly dissipative case with pure jump degenerate noise. [ABSTRACT FROM AUTHOR]

In this paper, we study the approximate optimal control problems for a class of fractional partial stochastic differential inclusions driven by Rosenblatt process and non-instantaneous impulses in a Hilbert space. Firstly, we prove an existence result of mild solutions for the control systems by using stochastic analysis, the fractional calculus, the measure of noncompactness, properties of sectorial operators and fixed point theorems. Secondly, we derive the existence conditions of approximate solutions to optimal control problems governed by fractional impulsive partial stochastic differential control systems with the help of minimizing sequence method. Finally, an example is given for the illustration of the obtained theoretical results. [ABSTRACT FROM AUTHOR]

This paper is mainly concerned with the asymptotic dynamics of non-autonomous stochastic 3D globally modified Navier–Stokes equations driven by nonlinear noise. Based on the well-posedness of such equations, we first show the existence and uniqueness of weak pullback mean random attractors. Then we investigate the existence of (periodic) invariant measures, the zero-noise limit of periodic invariant measures and their limit as the modification parameter N → N 0 ∈ (0 , + ∞) . Furthermore, under weaker conditions, we obtain the existence of invariant measures as well as their limiting behaviors when the external term is independent of time. Finally, by using weak convergence method, we establish the large deviation principle for the solution processes. [ABSTRACT FROM AUTHOR]

This paper deals with the limiting behavior of random attractors of stochastic wave equations with supercritical drift driven by linear multiplicative white noise defined on unbounded domains. We first establish the uniform Strichartz estimates of the solutions with respect to noise intensity, and then prove the convergence of the solutions of the stochastic equations with respect to initial data as well as noise intensity. To overcome the non-compactness of Sobolev embeddings on unbounded domains, we first utilize the uniform tail-ends estimates to truncate the solutions in a bounded domain and then employ a spectral decomposition to establish the pre-compactness of the collection of all random attractors. We finally prove the upper semicontinuity of random attractor as noise intensity approaches zero. [ABSTRACT FROM AUTHOR]

This paper is devoted to proving the strong averaging principle for slow–fast stochastic partial differential equations with locally monotone coefficients, where the slow component is a stochastic partial differential equations with locally monotone coefficients and the fast component is a stochastic partial differential equations with strongly monotone coefficients. The result is applicable to a large class of examples, such as the stochastic porous medium equation, the stochastic p-Laplace equation, the stochastic Burgers type equation and the stochastic 2D Navier–Stokes equation, which are the nonlinear stochastic partial differential equations. The main techniques are based on time discretization and the variational approach to stochastic partial differential equations. [ABSTRACT FROM AUTHOR]

This paper deals with a nonlinear filtering problem in which a multi-dimensional signal process is additively affected by a process ν whose components have paths of bounded variation. The presence of the process ν prevents from directly applying classical results and novel estimates need to be derived. By making use of the so-called reference probability measure approach, we derive the Zakai equation satisfied by the unnormalized filtering process, and then we deduce the corresponding Kushner–Stratonovich equation. Under the condition that the jump times of the process ν do not accumulate over the considered time horizon, we show that the unnormalized filtering process is the unique solution to the Zakai equation, in the class of measure-valued processes having a square-integrable density. Our analysis paves the way to the study of stochastic control problems where a decision maker can exert singular controls in order to adjust the dynamics of an unobservable Itô-process. [ABSTRACT FROM AUTHOR]

This paper is devoted to investigating the Freidlin–Wentzell's large deviation principle for a class of McKean–Vlasov quasilinear SPDEs perturbed by small multiplicative noise. We adopt the variational framework and the modified weak convergence criteria to prove the Laplace principle for McKean–Vlasov type SPDEs, which is equivalent to the large deviation principle. Moreover, we do not assume any compactness condition of embedding in the Gelfand triple to handle both the cases of bounded and unbounded domains in applications. The main results can be applied to various McKean–Vlasov type SPDEs such as distribution dependent stochastic porous media type equations and stochastic p-Laplace type equations. [ABSTRACT FROM AUTHOR]

In the paper, an averaging principle problem of stochastic McKean-Vlasov equations with slow and fast time-scale is considered. Firstly, existence and uniqueness of the strong solutions of stochastic McKean-Vlasov equations with two time-scale is proved by using the Picard iteration. Secondly, we show that there exists an exponential convergence to the invariant measure for solutions of the fast equation of stochastic McKean-Vlasov equations with slow and fast time-scale. Finally, strong averaging principle for two-time-scale stochastic McKean-Vlasov equations is investigated. [ABSTRACT FROM AUTHOR]

We consider the existence and first order conditions of optimality for a stochastic optimal control problem inspired by the celebrated FitzHugh–Nagumo model, with nonlinear diffusion term, perturbed by a linear multiplicative Brownian-type noise. The main novelty of the present paper relies on the application of the rescaling method which allows us to reduce the original problem to a random optimal one. [ABSTRACT FROM AUTHOR]

We study the Boussinesq system in a two-dimensional domain. In case only the velocity equation is forced by a white-in-time multiplicative noise, we prove the global existence of a martingale solution in the following two cases: positive horizontal viscous diffusion but zero vertical viscous diffusion and zero thermal diffusion; positive horizontal thermal diffusion but zero viscous diffusion and zero vertical thermal diffusion. Finally, in case both velocity and temperature equations are forced by a white-in-time multiplicative noise, and the velocity equation has zero viscous diffusion while the temperature equation has full thermal diffusion, the global existence of a martingale solution was shown in Yamazaki (Stoch Anal Appl, 34:404–426, 2016); in this manuscript we additionally prove significantly higher regularity. [ABSTRACT FROM AUTHOR]

This paper investigates the time optimal control for a class of non-instantaneous impulsive Clarke subdifferential type stochastic evolution inclusions in Hilbert spaces. We focus first on the existence of mild solutions for these systems by using the measure of non-compactness and a fixed-point theorem wth the properties of Clarke subdifferential. Then, the existence of time optimal control of governed by stochastic control systems is also obtained. We do not assume that the evolution operator and the values of multi-valued map are compact. Finally, an example is given to illustrate the effectiveness of the results. [ABSTRACT FROM AUTHOR]

Well-posedness is proved for the stochastic viscous Cahn–Hilliard equation with homogeneous Neumann boundary conditions and Wiener multiplicative noise. The double-well potential is allowed to have any growth at infinity (in particular, also super-polynomial) provided that it is everywhere defined on the real line. A vanishing viscosity argument is carried out and the convergence of the solutions to the ones of the pure Cahn–Hilliard equation is shown. Some refined regularity results are also deduced for both the viscous and the non-viscous case. [ABSTRACT FROM AUTHOR]

This paper addresses approximate controllability for a class of control systems represented by second order stochastic differential equations driven by Teugels martingales associated with a Lévy process. The main technique is the fundamental solution theory constructed through Laplace transformation. By using the so-called resolvent condition and cosine family of linear operators, stochastic analysis, and the technique of stochastic control theory, a new set of sufficient conditions for the approximate controllability of the considered second order stochastic differential system are formulated and proved. Due to the fundamental solution theory, the nonlinear terms are only required to be partly uniformly bounded. As an illustration of the applications of the obtained results, an example is also provided in the end. [ABSTRACT FROM AUTHOR]

Stochastic partial differential equations of second order with two unknown parameters are studied. Based on ergodicity, two suitable families of minimum contrast estimators are introduced. Strong consistency and asymptotic normality of estimators are proved. The results are applied to hyperbolic equations perturbed by Brownian noise. [ABSTRACT FROM AUTHOR]

We introduce the Hilbert-valued fractional Ornstein–Uhlenbeck of the second kind as the mild solution of a stochastic evolution equation with fractional-type Gaussian noise. We study the stationarity and the ergodicity for this infinite-dimensional process. Finally, via Malliavin calculus, we also analyze the least squares estimator of the drift parameter of the fractional Ornstein–Uhlenbeck of the second kind. [ABSTRACT FROM AUTHOR]

We are concerned with a nonlinear nonautonomous model represented by an equation describing the dynamics of an age-structured population diffusing in a space habitat O, governed by local Lipschitz vital factors and by a stochastic behavior of the demographic rates possibly representing emigration, immigration and fortuitous mortality. The model is completed by a random initial condition, a flux type boundary conditions on ∂ O with a random jump in the population density and a nonlocal nonlinear boundary condition given at age zero. The stochastic influence is expressed by a linear multiplicative Gaussian noise perturbation in the equation. The main result proves that the stochastic model is well-posed, the solution being in the class of path-wise continuous functions and satisfying some particular regularities with respect to the age and space. The approach is based on a rescaling transformation of the stochastic equation into a random deterministic time dependent hyperbolic-parabolic equation with local Lipschitz nonlinearities. The existence and uniqueness of a strong solution to the random deterministic equation is proved by combined semigroup, variational and approximation techniques. The information given by these results is transported back via the rescaling transformation towards the stochastic equation and enables the proof of its well-posedness. [ABSTRACT FROM AUTHOR]

We present a version of the stochastic maximum principle (SMP) for ergodic control problems. In particular we give necessary (and sufficient) conditions for optimality for controlled dissipative systems in finite dimensions. The strategy we employ is mainly built on duality techniques. We are able to construct a dual process for all positive times via the analysis of a suitable class of perturbed linearized forward equations. We show that such a process is the unique bounded solution to a backward SDE on infinite horizon from which we can write a version of the SMP. [ABSTRACT FROM AUTHOR]

In this paper, controlled linear stochastic evolution equations driven by square integrable Lévy processes are studied in the Hilbert space setting. The control operator may be unbounded which makes the results obtained in the abstract setting applicable to parabolic SPDEs with boundary or point control. The first part contains some preliminary technical results, notably a version of Itô formula which is applicable to weak/mild solutions of controlled equations. In the second part, the ergodic control problem is solved: The feedback form of the optimal control and the formula for the optimal cost are found. As examples, various parabolic type controlled SPDEs are studied. [ABSTRACT FROM AUTHOR]

In this work we first present the existence, uniqueness and regularity of the strong solution of the tidal dynamics model perturbed by Lévy noise. Monotonicity arguments have been exploited in the proofs. We then formulate a martingale problem of Stroock and Varadhan associated to an initial value control problem and establish existence of optimal controls. [ABSTRACT FROM AUTHOR]

The theory of micropolar fluids emphasizes the micro-structure of fluids by coupling the Navier-Stokes equations with micro-rotational velocity, and is widely viewed to be well fit, better than the Navier-Stokes equations, to describe fluids consisting of bar-like elements such as liquid crystals made up of dumbbell molecules or animal blood. Following the work of Weinan et al. (Commun Math Phys 224:83-106, 2001), we prove the existence of a unique stationary measure for the stochastic micropolar fluid system with periodic boundary condition, forced by only the determining modes of the noise and therefore a type of finite-dimensionality of micropolar fluid flow. The novelty of the manuscript is a series of energy estimates that is reminiscent from analysis in the deterministic case. [ABSTRACT FROM AUTHOR]

In the present paper we study the convergence of the solution of the two dimensional (2-D) stochastic Leray- $$\alpha $$ model to the solution of the 2-D stochastic Navier-Stokes equations. We are mainly interested in the rate, as $$\alpha \rightarrow 0$$ , of the following error function where $$\mathbf {u}^\alpha $$ and $$\mathbf {u}$$ are the solution of stochastic Leray- $$\alpha $$ model and the stochastic Navier-Stokes equations, respectively. We show that when properly localized the error function $$\varepsilon _\alpha $$ converges in mean square as $$\alpha \rightarrow 0$$ and the convergence is of order $$O(\alpha )$$ . We also prove that $$\varepsilon _\alpha $$ converges in probability to zero with order at most $$O(\alpha )$$ . [ABSTRACT FROM AUTHOR]

We discuss $$L_p$$ -estimates for finite difference schemes approximating parabolic, possibly degenerate, SPDEs, with initial conditions from $$W^m_p$$ and free terms taking values in $$W^m_p.$$ Consequences of these estimates include an asymptotic expansion of the error, allowing the acceleration of the approximation by Richardson's method. [ABSTRACT FROM AUTHOR]

We study a filtering problem for non-Markovian SDE's where the drift vector fields commute with diffusion vector fields. The evolution of the conditioned mean value will be decribed using a backward parabolic equation with parameters. [ABSTRACT FROM AUTHOR]

We consider the mean-variance hedging problem for pricing bond options using the yield curve as the observation. The model considered contains infinite-dimensional noise sources with the stochastically- varying risk premium. Hence our model is incomplete. We consider mean-variance hedging under the real world measure and obtain an explicit form of the optimal hedging strategy. [ABSTRACT FROM AUTHOR]

In this paper we consider Ito's stochastic differential equation in Hilbert spaces. A strong solution is generated by the difference approximation. A regularity result is obtained for solutions to a class of parabolic stochastic partial differential equations. Hyperbolic stochastic evolution equations are also discussed. [ABSTRACT FROM AUTHOR]

We introduce a class of distribution-valued stochastic processes that arise in the study of pulse reflection from random media and we analyze their asymptotic properties when they are scaled in a natural way. [ABSTRACT FROM AUTHOR]