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We consider stochastic wave map equation on real line with solutions taking values in a $d$-dimensional compact Riemannian manifold. We show first that this equation has unique, global, strong in PDE sense, solution in local Sobolev spaces. The main result of the paper is a proof of the Large Deviations Principle for solutions in the case of vanishing noise., The current paper is an expanded and corrected version of the previous submission. Major change is the addition of Lemma 5.5. Martin Ondrej\'at's name has been added as a new author. The title of the paper has also been modified to a more suitable one to our results

We show that the stochastic 3D primitive equations with the Neumann boundary condition on the top, the lateral Dirichlet boundary condition and either the Dirichlet or the Neumann boundary condition on the bottom driven by multiplicative gradient-dependent white noise have unique maximal strong solutions both in the stochastic and PDE senses under certain assumptions on the growth of the noise. For the case of the Neumann boundary condition on the bottom, global existence is established by using the decomposition of the vertical velocity to the barotropic and baroclinic modes and an iterated stopping time argument. An explicit example of non-trivial infinite dimensional noise depending on the vertical average of the horizontal gradient of horizontal velocity is presented.

We study two dimensional Navier–Stokes equations driven by a multiplicative Gaussian noise in the Stratonovich form along with a constraint on the L 2 -norm of the solution. In the deterministic setting [5] , it was shown that the global solution exists only on a two dimensional torus and hence we focus on such a case here. The existence of a martingale solution is shown. Moreover, the pathwise uniqueness of the solution is proved by using Schmalfuss idea [24] , concluding the existence of a strong solution via a Yamada–Watanabe type result from Ondrejat [21] .

In this paper we consider the 2D Ericksen–Leslie equations which describe the hydrodynamics of nematic liquid crystal with external body forces and anisotropic energy modeling the energy of applied external control such as magnetic or electric field. Under general assumptions on the initial data, the external data and the anisotropic energy, we prove the existence and uniqueness of global weak solutions with finitely many singular times. If the initial data and the external forces are sufficiently small, then we establish that the global weak solution does not have any singular times and is regular as long as the data are regular.

We consider a stochastic Camassa-Holm equation driven by a one-dimensional Wiener process with a first order differential operator as diffusion coefficient. We prove the existence and uniqueness of local strong solutions of this equation. In order to do so, we transform it into a random quasi-linear partial differential equation and apply Kato's operator theory methods. Some of the results have potential to find applications to other nonlinear stochastic partial differential equations.

The Landau–Lifshitz–Bloch equation perturbed by a space-dependent noise was proposed in [10] as a model for evolution of spins in ferromagnetic materials at the full range of temperatures, including the temperatures higher than the Curie temperature. In the case of a ferromagnet filling a bounded domain D ⊂ R d , d = 1 , 2 , 3 , we show the existence of strong (in the sense of PDEs) martingale solutions. Furthermore, in cases d = 1 , 2 we prove uniqueness of pathwise solutions and the existence of invariant measures. 1

Incompressible Navier-Stokes equations on a thin spherical domain $Q_\varepsilon$ along with free boundary conditions under a random forcing are considered. The convergence of the martingale solution of these equations to the martingale solution of the stochastic Navier-Stokes equations on a sphere $\mathbb{S}^2$ as the thickness converges to zero is established., Comment: 54 Pages. Published Version. Includes additional sections corresponding to the analysis of deterministic NSEs on a thin spherical domain

We consider 1D Landau-Lifshitz-Gilbert equations with an external force. We first prove the existence and uniqueness of the strong solution, and then give a definition of the quasipotential and prove that the quasipotential is equal to the potential energy of the system.

In this work we study stochastic Landau–Lifshitz–Gilbert equations (SLLGEs) in one dimension, with non-zero exchange energy only. Firstly, by introducing a suitable transformation, we convert the SLLGEs to a highly nonlinear time dependent partial differential equation with random coefficients, which is not fully parabolic. We then prove that there exists a pathwise unique solution to this equation and that this solution enjoys the maximal regularity property. Following regular approximation of the Brownian motion and using reverse transformation, we show existence of strong solution of SLLGEs taking values in a two-dimensional unit sphere S 2 in R 3 . The construction of the solution and its corresponding convergence results are based on Wong–Zakai approximation.

In this work we study a stochastic three-dimensional Landau–Lifshitz–Gilbert equation with non-zero anisotropy energy, which is drive by pure jump noise. We show existence of weak martingale solutions taking values in a two-dimensional sphere S 2 . The construction of the solution is based on the classical Faedo–Galerkin approximation, the compactness method and the Jakubowski version of the Skorokhod Theorem for nonmetric spaces.

In this work we study a stochastic three-dimensional Landau–Lifshitz–Gilbert equation perturbed by pure jump noise in the Marcus canonical form. We show the existence of a weak martingale solution taking values in a two-dimensional sphere $${\mathbb{S}^2}$$ and discuss certain regularity results. The construction of a solution is based on the classical Faedo–Galerkin approximation, the compactness methods and the Jakubowski version of the Skorokhod Theorem for nonmetric spaces.

Röckner and Zhang (Probab Theory Relat Fields 145, 211–267, 2009) proved the existence of a unique strong solution to a stochastic tamed 3D Navier–Stokes equation in the whole space and for the periodic boundary case using a result from Stroock and Varadhan (Multidimensional diffusion processes, Springer, Berlin, 1979). In the latter case, they also proved the existence of an invariant measure. In this paper, we improve their results (but for a slightly simplified system) using a self-contained approach. In particular, we generalise their result about an estimate on the $$L^4$$ L 4 -norm of the solution from the torus to $${\mathbb {R}}^3$$ R 3 , see Lemma 5.1 and thus establish the existence of an invariant measure on $${\mathbb {R}}^3$$ R 3 for a time-homogeneous damped tamed 3D Navier–Stokes equation, given by (6.1).

The aim of this thesis is threefold. Firstly, we study the stochastic evolution equations (driven by an infinite dimensional cylindrical Wiener process) in a class of Banach spaces satisfying the so-called H-condition. In particular, we deal with the questions of the existence and uniqueness of solutions for such stochastic evolution equations. Moreover, we analyse the Markov property of the solution. Secondly, we apply the abstract results obtained in the first part to the so-called Heath-Jarrow-Morton-Musiela (HJMM) equation. In particular, we prove the existence and uniqueness of solutions to the HJMM equation in a large class of function spaces, such as the weighted Lebesgue and Sobolev spaces. Thirdly, we study the ergodic properties of the solution to the HJMM equation. In particular, we analyse the Markov property of the solution and we find a sufficient condition for the existence and uniqueness of an invariant measure for the Markov semigroup associated to the HJMM equation (when the coefficients are time independent) in the weighted Lebesgue spaces.

We establish the existence of weak martingale solutions to a class of second order parabolic stochastic partial differential equations. The equations are driven by multiplicative jump type noise, with a non-Lipschitz multiplicative functional. The drift in the equations contains a dissipative nonlinearity of polynomial growth., Comment: See journal reference for teh final published version

We study a stochastic Landau–Lifshitz equation on a bounded interval and with finite dimensional noise. We first show that there exists a pathwise unique solution to this equation and that this solution enjoys the maximal regularity property. Next, we prove the large deviations principle for the small noise asymptotic of solutions using the weak convergence method. An essential ingredient of the proof is the compactness, or weak to strong continuity, of the solution map for a deterministic Landau–Lifschitz equation when considered as a transformation of external fields. We then apply this large deviations principle to show that small noise can cause magnetisation reversal. We also show the importance of the shape anisotropy parameter for reducing the disturbance of the solution caused by small noise. The problem is motivated by applications from ferromagnetic nanowires to the fabrication of magnetic memories.

Existence and uniqueness of solutions to stochastic differential equation $$dX-\text {div}\,a(\nabla X)\,dt=\sum _{j=1}^N(b_j\cdot \nabla X)\circ d\beta _j$$ in $$(0,T)\times \mathcal O$$ ; $$X(0,\xi )=x(\xi )$$ , $$\xi \in \mathcal O$$ , $$X=0$$ on $$(0,T)\times \partial \mathcal O$$ is studied. Here $$\mathcal O$$ is a bounded and open domain of $$\mathbb R^d$$ , $$d\ge 1$$ , $$\{b_j\}$$ is a divergence free vector field, $$a:[0,T]\times \mathcal O\times \mathbb R^d\rightarrow \mathbb R^d$$ is a continuous and monotone mapping of subgradient type and $$\{\beta _j\}$$ are independent Brownian motions in a probability space $$(\Omega ,\mathcal F,\mathbb P)$$ . The weak solution is defined via stochastic optimal control problem.

We establish a new version of the stochastic Strichartz estimate for the stochastic convolution driven by jump noise which we apply to the stochastic nonlinear Schr\"{o}dinger equation with nonlinear multiplicative jump noise in the Marcus canonical form. With the help of the deterministic Strichartz estimates, we prove the existence and uniqueness of a global solution to stochastic nonlinear Schr\"{o}dinger equation in $L^2(\RR^n)$ with either focusing or defocusing nonlinearity in the full subcritical range of exponents as in the deterministic case., Comment: 27 pages

We announce a result on the existence of a unique local solution to a stochastic geometric wave equation on the one dimensional Minkowski space $\mathbb{R}^{1+1}$ with values in an arbitrary compact Riemannian manifold. We consider a rough initial data in the sense that its regularity is lower than the energy critical.

We present remarkably simple proofs of Burkholder-Davis-Gundy inequalities for stochastic integrals and maximal inequalities for stochastic convolutions in Banach spaces driven by L\'{e}vy-type processes. Exponential estimates for stochastic convolutions are obtained and two versions of It\^{o}'s formula in Banach spaces are also derived. Based on the obtained maximal inequality, the existence and uniqueness of mild solutions of stochastic quasi-geostrophic equation with L\'{e}vy noise is established., Comment: 39 pages

We study a stochastic two-dimensional nematic liquid crystal model with multiplicative Gaussian noise. We prove the Wentzell-Freidlin type large deviations principle for the small noise asymptotic of solutions using weak convergence method

We construct a martingale solution of the stochastic nonlinear Schr\"odinger equation with a multiplicative noise of jump type in the Marcus canonical form. The problem is formulated in a general framework that covers the subcritical focusing and defocusing stochastic NLS in $H^1$ on compact manifolds and on bounded domains with various boundary conditions. The proof is based on a variant of the Faedo-Galerkin method. In the formulation of the approximated equations, finite dimensional operators derived from the Littlewood-Paley decomposition complement the classical orthogonal projections to guarantee uniform estimates. Further ingredients of the construction are tightness criteria in certain spaces of cadlag functions and Jakubowski's generalization of the Skorohod-Theorem to nonmetric spaces., Comment: 47 pages, 1 figure

We study a stochastic Landau-Lifschitz-Gilbert Equations with non-zero anisotrophy energy and multidimensional noise. We prove the existence and some regularities of weak solution proved. Our paper is motivated by finite-dimensional study of stochastic LLGEs or general stochasric differential equations with constraints studied by Kohn et al \cite{Kohn_2005} and Leli{\`e}vre et al \cite{LeBris_2008}.

The main goal of this paper is to generalize to Banach spaces the well-known results for diffusions on Hilbert spaces obtained in [Peszat, S. and Zabczyk, J. (1995). Strong Feller property and irreducibility for diffusions on Hilbert spaces. Ann. Probab. 23(1): 157-172.]. More precisely, we are aiming to prove the strong Feller property and irreducibility of the solutions to a stochastic evolution equations (SEEs) in Banach spaces. We give sufficient conditions on the path space and the coefficients of the SEEs for these aforementioned properties to hold. We apply our result to investigate the long-time behavior of a stochastic nonlinear heat equations on $L^p$-space with $p>4$. Our result implies the uniqueness of the invariant measure, if it exists, for the stochastic nonlinear heat equations on $L^p$-space with $p>4$.

The objective of this paper is to prove the existence of a global solution to a certain stochastic partial differential equation subject to the [Formula: see text]-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 [Formula: see text] in a Hilbert space [Formula: see text].

We prove the existence and the uniqueness of a local maximal solution to an $H^1$-critical stochastic wave equation with multiplicative noise on a smooth bounded domain $\mathcal{D} \subset \mathbb{R}^2$ with exponential nonlinearity. First, we derive the appropriate deterministic and stochastic Strichartz inequalities in suitable spaces and, then use them in arguments based on fixed point method to show the local well-posedness result. We also present an explosion result for the constructed unique local maximal solution., added some explanation

We consider a stochastic nonlinear Schrodinger 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.

In this work we consider a stochastic evolution equation which describes the system governing the nematic liquid crystals driven by a pure jump noise in the Marcus canonical form. The existence of a martingale solution is proved for both 2D and 3D cases. The construction of the solution relies on a modified Faedo–Galerkin method based on the Littlewood–Paley-decomposition, compactness method and the Jakubowski version of the Skorokhod representation theorem for non-metric spaces. We prove that in the 2-D case the martingale solution is pathwise unique and hence deduce the existence of a strong solution.

In this paper, we prove several mathematical results related to a system of highly nonlinear stochastic partial differential equations (PDEs). These stochastic equations describe the dynamics of penalised nematic liquid crystals under the influence of stochastic external forces. Firstly, we prove the existence of a global weak solution (in the sense of both stochastic analysis and PDEs). Secondly, we show the pathwise uniqueness of the solution in a 2D domain. In contrast to several works in the deterministic setting we replace the Ginzburg–Landau function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {1}_{|{\mathbf {n}}|\le 1}(|{\mathbf {n}}|^2-1){\mathbf {n}}$$\end{document}1|n|≤1(|n|2-1)n by an appropriate polynomial \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f({\mathbf {n}})$$\end{document}f(n) and we give sufficient conditions on the polynomial f for these two results to hold. Our third result is a maximum principle type theorem. More precisely, if we consider \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f({\mathbf {n}})=\mathbb {1}_{|d|\le 1}(|{\mathbf {n}}|^2-1){\mathbf {n}}$$\end{document}f(n)=1|d|≤1(|n|2-1)n and if the initial condition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {n}}_0$$\end{document}n0 satisfies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|{\mathbf {n}}_0|\le 1$$\end{document}|n0|≤1, then the solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {n}}$$\end{document}n also remains in the unit ball.

In this paper we prove the existence of random attractors for the Navier--Stokes equations on 2 dimensional sphere under random forcing irregular in space and time. We also deduce the existence of an invariant measure.

In this paper we first prove the existence and uniqueness of the solution to the stochastic Navier–Stokes equations on the rotating 2-dimensional unit sphere. Then we show the existence of an asymptotically compact random dynamical system associated with the equations.

We prove pathwise uniqueness for solutions of the nonlinear Schr\"{o}dinger equation with conservative multiplicative noise on compact 3D manifolds. In particular, we generalize the result by Burq, G\'erard and Tzvetkov (N. Burq, P. G\'erard, and N. Tzvetkov. Strichartz inequalities and the nonlinear Schr\"{o}dinger equation on compact manifolds. American Journal of Mathematics, 126 (3):569--605, 2004) to the stochastic setting. The proof is based on deterministic and stochastic Strichartz estimates and the Littlewood-Paley decomposition.

We study 2D Navier–Stokes equations with a constraint forcing the conservation of the energy of the solution. We prove the existence and uniqueness of a global solution for the constrained Navier–Stokes equation on R 2 and T 2 , by a fixed point argument. We also show that the solution of the constrained equation converges to the solution of the Euler equation as the viscosity ν vanishes.

Building upon a recent work by two of the authors and J. Seidler on $bw$-Feller property for stochastic nonlinear beam and wave equations, we prove the existence of an invariant measure to stochastic 2-D Navier–Stokes (with multiplicative noise) equations in unbounded domains. This answers an open question left after the first author and Y. Li proved a corresponding result in the case of an additive noise.

Let $(E, \| \cdot\|)$ be a Banach space such that, for some $q\geq 2$, the function $x\mapsto \|x\|^q$ is of $C^2$ class and its first and second Fr\'{e}chet derivatives are bounded by some constant multiples of $(q-1)$-th power of the norm and $(q-2)$-th power of the norm and let $S$ be a $C_0$-semigroup of contraction type on $(E, \| \cdot\|)$. We consider the following stochastic convolution process \begin{align*} u(t)=\int_0^t\int_ZS(t-s)\xi(s,z)\,\tilde{N}(\mathrm{d} s,\mathrm{d} z), \;\;\; t\geq 0, \end{align*} where $\tilde{N}$ is a compensated Poisson random measure on a measurable space $(Z,\mathcal{Z})$ and $\xi:[0,\infty)\times\Omega\times Z\rightarrow E$ is an $\mathbb{F}\otimes \mathcal{Z}$-predictable function. We prove that there exists a c\`{a}dl\`{a}g modification a $\tilde{u}$ of the process $u$ which satisfies the following maximal inequality \begin{align*} \mathbb{E} \sup_{0\leq s\leq t} \|\tilde{u}(s)\|^{q^\prime}\leq C\ \mathbb{E} \left(\int_0^t\int_Z \|\xi(s,z) \|^{p}\,N(\mathrm{d} s,\mathrm{d} z)\right)^{\frac{q^\prime}{p}}, \end{align*} for all $ q^\prime \geq q$ and $1

We are dealing with the Navier-Stokes equation in a bounded regular domain $$\mathcal {O}$$ of $$\mathbb {R}^2$$ , perturbed by an additive Gaussian noise $$\partial w^{Q_\delta }/\partial t$$ , which is white in time and colored in space. We assume that the correlation radius of the noise gets smaller and smaller as $$\delta \searrow 0$$ , so that the noise converges to the white noise in space and time. For every $$\delta >0$$ we introduce the large deviation action functional $$S^\delta _{T}$$ and the corresponding quasi-potential $$U_\delta $$ and, by using arguments from relaxation and $$\Gamma $$ -convergence we show that $$U_\delta $$ converges to $$U=U_0$$ , in spite of the fact that the Navier-Stokes equation has no meaning in the space of square integrable functions, when perturbed by space-time white noise. Moreover, in the case of periodic boundary conditions the limiting functional $$U$$ is explicitly computed. Finally, we apply these results to estimate of the asymptotics of the expected exit time of the solution of the stochastic Navier-Stokes equation from a basin of attraction of an asymptotically stable point for the unperturbed system.

Motivated by applications to various semilinear and quasi-linear stochastic partial differential equations (SPDEs) appeared in real world models, we establish the existence and uniqueness of strong solutions to coercive and locally monotone SPDEs driven by Levy processes. We illustrate the main results of our paper by showing how it can be applied to a large class of SPDEs such as stochastic reaction–diffusion equations, stochastic Burgers type equations, stochastic 2D hydrodynamical systems and stochastic equations of non-Newtonian fluids, which generalize many existing results in the literature.

We study the existence and uniqueness of solutions of 2D Stochastic Navier–Stokes equation with space irregular jump noise for initial data in certain Sobolev spaces of negative order. Comparing with the Galerkin approximation method, the main advantage of this work is to use an L p -setting to obtain the solution under much weaker assumptions on the noise and the initial condition.

We show that the stochastic flow generated by the 2-dimensional Stochastic Navier–Stokes equations with rough noise on a Poincare-like domain has a unique random attractor. One of the technical problems associated with the rough noise is overcomed by the use of the corresponding Cameron–Martin (or reproducing kernel Hilbert) space. Our results complement the result by Brzeźniak and Li (2006) [10] who showed that the corresponding flow is asymptotically compact and also generalize Caraballo et al. (2006) [12] who proved existence of a unique attractor for the time-dependent deterministic Navier–Stokes equations.

In this article we present a different approach to some of the results published in our recent paper (Brzeźniak and Dawidowicz in Semigroup Forum, 78(1):118–137, 2009). This new approach is based on a deep result from a paper (Ergod. Theory Dyn. Syst. 17(4):793–819, 1997) by Desch Schappacher and Webb.

We prove the existence and the uniqueness of a solution to the stochastic NSLEs on a two-dimensional compact riemannian manifold. Thus we generalize (and improve) a recent work by Burq et al. (J Nonlinear Math Phys 10(1):12–27, 2003) and a series of papers by de Bouard and Debussche, see e.g. de Bouard and Debussche (Commun Math Phys 205(1):161–181, 1999 and Stoch Anal Appl 21(1):97–126, 2003) who have examined similar questions in the case of the flat euclidean space. We prove the existence and the uniqueness of a local maximal solution to stochastic nonlinear Schrodinger equations with multiplicative noise on a compact d-dimensional riemannian manifold. Under more regularity on the noise, we prove that the solution is global when the nonlinearity is of defocusing or of focusing type, d = 2 and the initial data belongs to the finite energy space. Our proof is based on improved stochastic Strichartz inequalities.

We prove the existence of a global solution for the filament equation with the inital condition given by a geometric rough path in the sense of Lyons (1998 Rev. Mat. Iberoamericana 14 215?310). Our work gives a positive answer to a question left open in recent publications: Berselli and Gubinelli (2007 Commun. Math. Phys. 269 693?713) showed the existence of a global solution for a smooth initial condition while Bessaih et al (2005 Ann. Probab. 33 1825?55) proved the existence of a local solution for a general initial condition given by a rough path.

The aim of this article is to study the asymptotic behavior for large times of solutions of linear stochastic partial differential equations of parabolic type. In particular, we will prove the backward uniqueness result and the existence of the spectral limit for abstract SPDEs and then show how these results can be applied to linear SPDEs.

In the paper, we are studying the existence and uniqueness of the solution of an abstract nonlinear equation driven by a multiplicative noise of Levy type. Our result is formulated in an abstract setting. This type of equation covers the stochastic 2D Navier–Stokes Equations, the 2D stochastic Magneto-Hydrodynamic Equations, the 2D stochastic Boussinesq Model for the Benard Convection, the 2D stochastic Magnetic Bernard Problem, the 3D stochastic Leray α -Model for the Navier–Stokes Equations and several stochastic Shell Models of turbulence.

Existence of an invariant measure for a stochastic extensible beam equation and for a stochastic damped wave equation with polynomial nonlinearities is proved. It is shown first that the corresponding transition semigroups map the space of all bounded sequentially weakly continuous functions on the state space into itself and then by a Lyapunov functions approach solutions bounded in probability are found.