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In this paper, we present a stochastic augmented Lagrangian approach on (possibly infinite-dimensional) Riemannian manifolds to solve stochastic optimization problems with a finite number of deterministic constraints. We investigate the convergence of the method, which is based on a stochastic approximation approach with random stopping combined with an iterative procedure for updating Lagrange multipliers. The algorithm is applied to a multi-shape optimization problem with geometric constraints and demonstrated numerically. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the three-dimensional globally modified Navier–Stokes equations driven by additive white noise on some unbounded domains 풪. By using the Ornstein–Uhlenbeck process, we transfer the original equation to a random dynamical system, and then we prove the existence of pullback attractors for the random dynamical system equations under suitable conditions. Due to the unboundedness of the domains, we get the asymptotic compactness of the solutions by Ball's idea of energy equations. The periodicity of the attractors is also proved when the deterministic non-autonomous external terms are periodic in time. [ABSTRACT FROM AUTHOR]

We consider a stochastic nonlinear defocusing Schrödinger equation with zero-order linear damping, where the stochastic forcing term is given by a combination of a linear multiplicative noise in the Stratonovich form and a nonlinear noise in the Itô form. We work at the same time on compact Riemannian manifolds without boundary and on relatively compact smooth domains with either the Dirichlet or the Neumann boundary conditions, always in dimension two. We construct a martingale solution using a modified Faedo–Galerkin's method, following Brzeźniak et al (2019 Probab. Theory Relat. Fields 174 1273–338). Then by means of the Strichartz estimates deduced from Blair et al (2008 Proc. Am. Math. Soc. 136 247–56) but modified for our stochastic setting we show the pathwise uniqueness of solutions. Finally, we prove the existence of an invariant measure by means of a version of the Krylov–Bogoliubov method, which involves the weak topology, as proposed by Maslowski and Seidler (1999 Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 10 69–78). This is the first result of this type for stochastic nonlinear Schrödinger equation (NLS) on compact Riemannian manifolds without boundary and on relatively compact smooth domains even for an additive noise. Some remarks on the uniqueness in a particular case are provided as well. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence result for a mild solution of a fractional stochastic evolution inclusion involving the Caputo derivative in the Hilbert space driven by a fractional Brownian motion with the Hurst parameter H > 1 2 . The results are obtained by using fractional calculation, operator semigroups and the fixed point theorem for multivalued mappings. [ABSTRACT FROM AUTHOR]

We consider a system of forward backward stochastic differential equations (FBSDEs) with a time-delayed generator driven by Lévy-type noise. We establish a non-linear Feynman–Kac representation formula associating the solution given by the FBSDEs system to the solution of a path dependent nonlinear Kolmogorov equation with both delay and jumps. Obtained results are then applied to study a generalization of the so-called large investor problem, where the stock price evolves according to a jump-diffusion dynamic. [ABSTRACT FROM AUTHOR]

We consider 3D stochastic globally modified Navier–Stokes equations in bounded domains with homogeneous Dirichlet boundary conditions and infinite dimensional Wiener process. We study stability properties of stationary solutions. We also show that one can stabilize an unstable stationary solution by using a multiplicative Itô noise of sufficient intensity or a linear internal feedback controller with support large enough. [ABSTRACT FROM AUTHOR]

This paper concerns a class of fractional impulsive neutral functional differential equations with an infinite delay driven by the Rosenblatt process. A set of sufficient conditions are established for the existence of new mild solutions using fixed point theory. Finally, an illustrative example is provided to demonstrate the applicability of the theoretical result. [ABSTRACT FROM AUTHOR]

In this paper, we analyze multidimensional generalized backward stochastic differential equations with jumps in a filtration that supports a Brownian motion and an independent integer-valued random measure. Under monotonicity and linear growth assumptions on the coefficients, we give the existence and uniqueness of 핃 2 -solutions provided that the generators and the terminal condition satisfy some suitable integrability conditions. [ABSTRACT FROM AUTHOR]

We study the sharp interface limit of the two dimensional stochastic Cahn-Hilliard equation driven by two types of singular noise: a space-time white noise and a space-time singular divergence-type noise. We show that with appropriate scaling of the noise the solutions of the stochastic problems converge to the solutions of the determinisitic Mullins-Sekerka/Hele-Shaw problem. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish the equivalence between Lagrangian and classical formulations for the stochastic incompressible Euler equations, the proof is based on Ito–Wentzell–Kunita formula and stochastic analysis techniques. Moreover, we prove a local existence result for the Lagrangian formulation in suitable Sobolev Spaces. [ABSTRACT FROM AUTHOR]

This article establishes a large deviation principle for a non-Newtonian fluid of differential type, filling a two-dimensional non-axisymmetric bounded domain with slip boundary conditions. More precisely, we show that the solutions of small stochastic white noise perturbations of the third grade fluid equations converges to the deterministic solution, as the intensity of the noise goes to zero. Moreover, this convergence has an exponential rate given by a suitable rate function. To establish such asymptotic result, we follow the weak convergence approach introduced by Budhiraja, Dupuis and Ellis. [ABSTRACT FROM AUTHOR]

We study in this article the existence and uniqueness of solutions to a class of stochastic transport equations with irregular coefficients and unbounded divergence. In the first result, we assume the drift is L 2 ([ 0 , T ] × R d) ∩ L ∞ ([ 0 , T ] × R d) , and the divergence is the locally integrable. In the second result, we show that the smoothing acts as a selection criterion when the drift is in L 2 ([ 0 , T ] × R d) ∩ L ∞ ([ 0 , T ] × R d) without any condition on the divergence. [ABSTRACT FROM AUTHOR]

We analyse the performance of a data-assimilation algorithm based on a linear feedback control when used with observational data that contains measurement errors. Our model problem consists of dynamics governed by the two-dimensional incompressible Navier-Stokes equations, observational measurements given by finite volume elements or nodal points of the velocity field and measurement errors which are represented by stochastic noise. Under these assumptions, the data-assimilation algorithm consists of a system of stochastically forced Navier-Stokes equations. The main result of this paper provides explicit conditions on the observation density (resolution) which guarantee explicit asymptotic bounds, as the time tends to infinity, on the error between the approximate solution and the actual solutions which is corresponding to these measurements, in terms of the variance of the noise in the measurements. Specifically, such bounds are given for the limit supremum, as the time tends to infinity, of the expected value of the L2-norm and of the H1 Sobolev norm of the difference between the approximating solution and the actual solution. Moreover, results on the average time error in mean are stated.

A well-known Itô formula for finite-dimensional processes, given in terms of stochastic integrals with respect to Wiener processes and Poisson random measures, is revisited and is revised. The revised formula, which corresponds to the classical Itô formula for semimartingales with jumps, is then used to obtain a generalisation of an important infinite-dimensional Itô formula for continuous semimartingales from Krylov (Probab Theory Relat Fields 147:583–605, 2010) to a class of L p -valued jump processes. This generalisation is motivated by applications in the theory of stochastic PDEs. [ABSTRACT FROM AUTHOR]

We study the Hardy-Hénon parabolic equations on R N (N=2 or 3) under the effect of an additive fractional Brownian noise with Hurst parameter H > max 1 / 2 , N / 4. We show local existence and uniqueness of a mid L q -solution under suitable assumptions on q. [ABSTRACT FROM AUTHOR]

We are concerned with a stochastic mean curvature flow of graphs over a periodic domain of any space dimension. For the first time, we are able to construct martingale solutions which satisfy the equation pointwise and not only in a generalized (distributional or viscosity) sense. Moreover, we study their large-time behavior. Our analysis is based on a viscous approximation and new global bounds, namely, an L ω , x , t ∞ estimate for the gradient and an L ω , x , t 2 bound for the Hessian. The proof makes essential use of the delicate interplay between the deterministic mean curvature part and the stochastic perturbation, which permits to show that certain gradient-dependent energies are supermartingales. Our energy bounds in particular imply that solutions become asymptotically spatially homogeneous and approach a Brownian motion perturbed by a random constant. [ABSTRACT FROM AUTHOR]

In this article, we construct a global martingale solution to a general nonlinear Schrödinger equation with linear multiplicative noise in the Stratonovich form. Our framework includes many examples of spatial domains like R d , non-compact Riemannian manifolds, and unbounded domains in R d with different boundary conditions. The initial value belongs to the energy space H 1 and we treat subcritical focusing and defocusing power nonlinearities. The proof is based on an approximation technique which makes use of spectral theoretic methods and an abstract Littlewood–Paley-decomposition. In the limit procedure, we employ tightness of the approximated solutions and Jakubowski's extension of the Skorohod theorem to nonmetric spaces. [ABSTRACT FROM AUTHOR]

The limiting behavior of stochastic evolution processes with small noise intensity ϵ is investigated in distribution-based approaches. Let μϵ be a stationary measure for stochastic process Xϵ with small ϵ and X0 be a semiflow on a Polish space. Assume that {μϵ: 0 < ϵ ⩽ ϵ0} is tight. Then all their limits in the weak sense are X0-invariant and their supports are contained in the Birkhoff center of X0. Applications are made to various stochastic evolution systems, including stochastic ordinary differential equations, stochastic partial differential equations, and stochastic functional differential equations driven by Brownian motion or Lévy processes. [ABSTRACT FROM AUTHOR]

Stochastic non-local conservation law equation in the presence of discontinuous flux functions is considered in an L 1 ∩ L 2 setting. The flux function is assumed bounded and integrable (spatial variable). Our result is to prove existence and uniqueness of weak solutions. The solution is strong solution in the probabilistic sense. The proofs are constructive and based on the method of characteristics (in the presence of noise), Itô–Wentzell–Kunita formula and commutators. Our results are new , to the best of our knowledge, and are the first nonlinear extension of the seminar paper (Flandoli et al. in Invent Math 180:1–53, 2010) where the linear case was addressed. [ABSTRACT FROM AUTHOR]

We present a simple uniqueness argument for a collection of McKean–Vlasov problems that have seen recent interest. Our first result shows that, in the weak feedback regime, there is global uniqueness for a very general class of random drivers. By weak feedback we mean the case where the contagion parameters are small enough to prevent blow‐ups in solutions. Next, we specialise to a Brownian driver and show how the same techniques can be extended to give short‐time uniqueness after blow‐ups, regardless of the feedback strength, by exploiting a localised form of weak feedback. The heart of our approach is a surprisingly simple probabilistic comparison argument that is robust in the sense that it does not ask for any regularity of the solutions. [ABSTRACT FROM AUTHOR]

Under the framework of sublinear expectation, we introduce a new type of G-Gaussian random fields, which contains a type of spatial white noise as a special case. Based on this result, we also introduce a spatial-temporal G-white noise. Different from the case of linear expectation, in which the probability measure needs to be known, under the uncertainty of probability measures, spatial white noises are intrinsically different from temporal cases. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give the predictable representation property associated with Lévy process in the non-homogeneous case. In this latter, we establish the existence and uniqueness of solution for the Backward Stochastic Differential Equations and its relation with Partial integro-differential equations. [ABSTRACT FROM AUTHOR]

We consider a stochastic nonlinear Schrödinger equation with multiplicative noise in an abstract framework that covers subcritical focusing and defocusing Stochastic NLSE in H 1 on compact manifolds and bounded domains. We construct a martingale solution using a modified Faedo–Galerkin-method based on the Littlewood–Paley-decomposition. For the 2d manifolds with bounded geometry, we use the Strichartz estimates to show the pathwise uniqueness of solutions. [ABSTRACT FROM AUTHOR]

We study the long-time behavior of solutions to a stochastically driven Navier–Stokes system describing the motion of a compressible viscous fluid driven by a temporal multiplicative white noise perturbation. The existence of stationary solutions is established in the framework of Lebesgue–Sobolev spaces pertinent to the class of weak martingale solutions. The methods are based on new global-in-time estimates and a combination of deterministic and stochastic compactness arguments. An essential tool in order to obtain the global-in-time estimate is the stationarity of solutions on each approximation level, which provides a certain regularizing effect. In contrast with the deterministic case, where related results were obtained only under rather restrictive constitutive assumptions for the pressure, the stochastic case is tractable in the full range of constitutive relations allowed by the available existence theory, due to the underlying martingale structure of the noise. [ABSTRACT FROM AUTHOR]

A linear stochastic continuity equation with non-regular coefficients is considered. We prove existence and uniqueness of strong solution, in the probabilistic sense, to the Cauchy problem when the vector field has low regularity, in which the classical DiPerna-Lions-Ambrosio theory of uniqueness of distributional solutions does not apply. We solve partially the open problem that is the case when the vector-field has random dependence. In addition, we prove a stability result for the solutions. [ABSTRACT FROM AUTHOR]

We consider the 3D stochastic Navier–Stokes–Voigt equations in bounded domains with the homogeneous Dirichlet boundary condition and infinite-dimensional Wiener process. First, we prove the existence and uniqueness of solutions to the problem. Then we investigate the mean square exponential stability and the almost sure exponential stability of the stationary solutions. [ABSTRACT FROM AUTHOR]

We prove local well-posedness in regular spaces and a Beale–Kato–Majda blow-up criterion for a recently derived stochastic model of the 3D Euler fluid equation for incompressible flow. This model describes incompressible fluid motions whose Lagrangian particle paths follow a stochastic process with cylindrical noise and also satisfy Newton's second law in every Lagrangian domain. [ABSTRACT FROM AUTHOR]

We consider the Navier–Stokes equation on the 2D torus, with a stochastic forcing term which is a cylindrical fractional Wiener noise of Hurst parameter H. Following Albeverio and Ferrario (Ann Probab 32(2):1632–1649, 2004) and Da Prato and Debussche (J Funct Anal 196(1):180–210, 2002) which dealt with the case H = 1 2 , we prove a local existence and uniqueness result when 7 16 < H < 1 2 and a global existence and uniqueness result when 1 2 < H < 1 . [ABSTRACT FROM AUTHOR]

Objectives: In the paper, two new reliable analytical methods have been devised for getting new exact analytical solutions of wick-type stochastic time-fractional Benjamin-Bona-Mahony (BBM) equation. Moreover, the Hermite transform and inverse Hermite transform have been utilized for converting fractional stochastic differential equation to deterministic fractional partial differential equation and vice versa respectively. Here for reducing fractional partial differential equations (FPDE) to the ordinary differential equation (ODE), fractional complex transform has been utilized. Methods: The authors have used a newly proposed method and Kudryshov method for getting the solutions for wick-type stochastic time-fractional Benjamin-Bona-Mahony (BBM) equation. Results: By using two reliable methods, here, the authors find the new exact solutions for the governing equations. Conclusion: Two new approaches to find solutions of the aforementioned equation have been established. Also, the new exact solutions have been obtained for stochastic differential equation by using two methods. [ABSTRACT FROM AUTHOR]

A comparison between two numerical approximation methods i.e., Euler-Murayama and 1.5 strong Taylor methods have been established in this article. The stochastic point reactor kinetics equations consist of a system of Itô stochastic differential equations (SDEs) and this system is solved over each time-step size using Euler-Murayama and 1.5 strong Taylor methods. The obtained results establish the accuracy of both the methods in solving the stochastic point reactor kinetics equations in presence of Newtonian temperature feedback. [ABSTRACT FROM AUTHOR]

We study the Cauchy problem for an abstract quasilinear stochastic parabolic evolution equation on a Banach space driven by a cylindrical Brownian motion. We prove existence and uniqueness of a local strong solution up to a maximal stopping time, that is characterized by a blow-up alternative. The key idea is an iterative application of the theory about maximal Lp- regularity for semilinear stochastic evolution equations by Van Neerven, Veraar and Weis. We apply our local well-posedness result to a convection-diffusion equation on a bounded domain with Dirichlet, Neumann or mixed boudary conditions and to a generalized Navier-Stokes equation describing non-Newtonian fluids. In the first example, we can even show that the solution exists globally. [ABSTRACT FROM AUTHOR]

The stochastic partial differential equation analyzed in this work, is motivated by a simplified mesoscopic physical model for phase separation. It describes pattern formation due to adsorption and desorption mechanisms involved in surface processes, in the presence of a stochastic driving force. This equation is a combination of Cahn–Hilliard and Allen–Cahn type operators with a multiplicative, white, space–time noise of unbounded diffusion. We apply Malliavin calculus, in order to investigate the existence of a density for the stochastic solution u . In dimension one, according to the regularity result in [5] , u admits continuous paths a.s. Using this property, and inspired by a method proposed in [8] , we construct a modified approximating sequence for u , which properly treats the new second order Allen–Cahn operator. Under a localization argument, we prove that the Malliavin derivative of u exists locally, and that the law of u is absolutely continuous, establishing thus that a density exists. [ABSTRACT FROM AUTHOR]

In this paper, we study a grade-two fluid driven by multiplicative Gaussian noise. Under appropriate assumptions on the initial condition and the noise, we prove a large and moderate deviations principle in the space C([0,T];Hm), m∈{2,3}, of the solution of our stochastic model as the viscosity ε converges to 0 and the coefficient of the noise is multiplied by ε12. We present a unifying approach to the proof of the two deviations principles and express the rate function in term of the solution of the inviscid grade-two fluid which is also known as Lagrangian averaged Euler equations. Our proof is based on the weak convergence approach to large deviations principle. [ABSTRACT FROM AUTHOR]

We study the Cauchy problem for a semilinear stochastic Maxwell equation with Kerr-type nonlinearity and a retarded material law. We show existence and uniqueness of strong solutions using a refined Faedo-Galerkin method and spectral multiplier theorems for the Hodge-Laplacian. We also make use of a rescaling transformation that reduces the problem to an equation with additive noise to get an appropriate a priori estimate for the solution. [ABSTRACT FROM AUTHOR]

We consider the stochastic NLS with nonlinear Stratonovich noise for initial values in L2(Rd) and prove local existence and uniqueness of a mild solution for subcritical and critical nonlinearities. The proof is based on deterministic and stochastic Strichartz estimates. In the subcritical case we prove that the solution is global, if we impose an additional assumption on the nonlinear noise. [ABSTRACT FROM AUTHOR]

Two generalizations of Itô formula to infinite-dimensional spaces are given. The first one, in Hilbert spaces, extends the classical one by taking advantage of cancellations when they occur in examples and it is applied to the case of a group generator. The second one, based on the previous one and a limit procedure, is an Itô formula in a special class of Banach spaces having a product structure with the noise in a Hilbert component; again the key point is the extension due to a cancellation. This extension to Banach spaces and in particular the specific cancellation are motivated by path-dependent Itô calculus. [ABSTRACT FROM AUTHOR]

We study the question of invertibility of the stochastic Fourier transformation of a random function f(t,ω). By the inversion of the SFT (short for “stochastic Fourier transformation”) we understand the two different meanings as follows; the reconstruction of the random function f(t,ω) itself by only the SFT or the reconstruction of the f(t,ω) with some supplemental information. The former we will call the inversion in strong sense and the latter the inversion in wide sense. The question of inversion in wide sense having been studied extensively in the previous papers, now in this note we are to study the question of invertibility in strong sense and show some typical results. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of noise effects on blow-up solutions to stochastic nonlinear Schrödinger equations. It is a continuation of our recent work [2] , where the (local) well-posedness is established in H 1 , also in the non-conservative critical case. Here we prove that in the non-conservative focusing mass-(super)critical case, by adding a large multiplicative Gaussian noise, with high probability one can prevent the blow-up on any given bounded time interval [ 0 , T ] , 0 < T < ∞ . Moreover, in the case of spatially independent noise, the explosion even can be prevented with high probability on the whole time interval [ 0 , ∞ ) . The noise effects obtained here are completely different from those in the conservative case studied in [5] . [ABSTRACT FROM AUTHOR]