
显示样式: 排序: IF: - GO 导出
-
On the energy transfer to high frequencies in the damped/driven nonlinear Schrödinger equation Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2021-01-02 Guan Huang, Sergei Kuksin
We consider a damped/driven nonlinear Schrödinger equation in \(\mathbb {R}^n\), where n is arbitrary, \({\mathbb {E}}u_t-\nu \Delta u+i|u|^2u=\sqrt{\nu }\eta (t,x), \quad \nu >0,\) under odd periodic boundary conditions. Here \(\eta (t,x)\) is a random force which is white in time and smooth in space. It is known that the Sobolev norms of solutions satisfy \( \Vert u(t)\Vert _m^2 \le C\nu ^{-m}, \)
-
Stochastic homogenization of the Landau–Lifshitz–Gilbert equation Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2021-01-02 François Alouges, Anne de Bouard, Benoît Merlet, Léa Nicolas
Following the ideas of Zhikov and Piatnitski (Izv Math 70(1):19–67, 2006), and more precisely the stochastic two-scale convergence, this paper establishes a homogenization theorem in a stochastic setting for two nonlinear equations: the equation of harmonic maps into the sphere and the Landau–Lifschitz–Gilbert equation. These equations have strong nonlinear features, and in general their solutions
-
A K -rough path above the space-time fractional Brownian motion Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2021-01-01 Xia Chen, Aurélien Deya, Cheng Ouyang, Samy Tindel
We construct a K-rough path [along the terminology of Deya (Probab Theory Relat Fields 166:1–65, 2016)] above either a space-time or a spatial fractional Brownian motion, in any space dimension d. This allows us to provide an interpretation and a unique solution for the corresponding parabolic Anderson model, understood in the renormalized sense. We also consider the case of a spatial fractional noise
-
An extension of the sewing lemma to hyper-cubes and hyperbolic equations driven by multi-parameter Young fields Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2020-11-23 Fabian A. Harang
This article extends the celebrated Sewing lemma, known from the theory of rough paths, to multi-parameter fields on hyper-cubes. We use this to construct Young integrals for multi-parameter Hölder fields on general domains \(\left[ 0,T\right] ^{k}\) with \(k\ge 1\) taking values in \({\mathbb {R}}^{d}\). Moreover, we show existence, uniqueness and stability of some particular types of hyperbolic SPDEs
-
Almost sure well-posedness for the cubic nonlinear Schrödinger equation in the super-critical regime on $$\mathbb {T}^d$$ T d , $$d\ge 3$$ d ≥ 3 Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2020-10-11 Haitian Yue
In this paper we prove almost sure local (in time) well-posedness in both adapted atomic spaces and Fourier restriction spaces for the periodic cubic nonlinear Schrödinger equation on \(\mathbb {T}^d\) (\(d\ge 3\)) in the super-critical regime.
-
Existence and uniqueness for the mild solution of the stochastic heat equation with non-Lipschitz drift on an unbounded spatial domain Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2020-09-10 M. Salins
We prove the existence and uniqueness of the mild solution for a nonlinear stochastic heat equation defined on an unbounded spatial domain. The nonlinearity is not assumed to be globally, or even locally, Lipschitz continuous. Instead the nonlinearity is assumed to satisfy a one-sided Lipschitz condition. First, a strengthened version of the Kolmogorov continuity theorem is introduced to prove that
-
Initial-boundary value problem for stochastic transport equations Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2020-09-07 Wladimir Neves, Christian Olivera
This paper concerns the Dirichlet initial-boundary value problem for stochastic transport equations with non-regular coefficients. First, the existence and uniqueness of the strong stochastic traces is proved. The existence of weak solutions relies on the strong stochastic traces, and also on the passage from the Stratonovich into Itô’s formulation for bounded domains. Moreover, the uniqueness is established
-
Uniqueness for nonlinear Fokker–Planck equations and weak uniqueness for McKean–Vlasov SDEs Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2020-09-07 Viorel Barbu, Michael Röckner
One proves the uniqueness of distributional solutions to nonlinear Fokker–Planck equations with monotone diffusion term and derive as a consequence (restricted) uniqueness in law for the corresponding McKean–Vlasov stochastic differential equation (SDE).
-
More on the long time stability of Feynman–Kac semigroups Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2020-08-30 Grégoire Ferré, Mathias Rousset, Gabriel Stoltz
Feynman–Kac semigroups appear in various areas of mathematics: non-linear filtering, large deviations theory, spectral analysis of Schrödinger operators among others. Their long time behavior provides important information, for example in terms of ground state energy of Schrödinger operators, or scaled cumulant generating function in large deviations theory. In this paper, we propose a simple and natural
-
Existence of dynamical low rank approximations for random semi-linear evolutionary equations on the maximal interval Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2020-08-05 Yoshihito Kazashi, Fabio Nobile
An existence result is presented for the dynamical low rank (DLR) approximation for random semi-linear evolutionary equations. The DLR solution approximates the true solution at each time instant by a linear combination of products of deterministic and stochastic basis functions, both of which evolve over time. A key to our proof is to find a suitable equivalent formulation of the original problem
-
Strong approximation of monotone stochastic partial differential equations driven by multiplicative noise Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2020-08-03 Zhihui Liu, Zhonghua Qiao
We establish a general theory of optimal strong error estimation for numerical approximations of a second-order parabolic stochastic partial differential equation with monotone drift driven by a multiplicative infinite-dimensional Wiener process. The equation is spatially discretized by Galerkin methods and temporally discretized by drift-implicit Euler and Milstein schemes. By the monotone and Lyapunov
-
A central limit theorem and moderate deviation principle for the stochastic 2D Oldroyd model of order one Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2020-07-16 Manil T. Mohan
In this work, we discuss some asymptotic behavior of the stochastic two-dimensional viscoelastic fluid flow equations, arising from the Oldroyd model of order one, for the non-Newtonian fluid flows. We prove a central limit theorem and establish the moderate deviation principle for such models using a weak convergence method.
-
Two-scale coupling for preconditioned Hamiltonian Monte Carlo in infinite dimensions Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2020-06-15 Nawaf Bou-Rabee, Andreas Eberle
We derive non-asymptotic quantitative bounds for convergence to equilibrium of the exact preconditioned Hamiltonian Monte Carlo algorithm (pHMC) on a Hilbert space. As a consequence, explicit and dimension-free bounds for pHMC applied to high-dimensional distributions arising in transition path sampling and path integral molecular dynamics are given. Global convexity of the underlying potential energies
-
Decay of the stochastic linear Schrödinger equation in $$d \ge 3$$d≥3 with small multiplicative noise Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2020-05-12 Chenjie Fan, Weijun Xu
We give decay estimates of the solution to the linear Schrödinger equation in dimension \(d \ge 3\) with a small noise which is white in time and colored in space. As a consequence, we also obtain certain asymptotic behaviour of the solution. The proof relies on the bootstrapping argument used by Journé–Soffer–Sogge for decay of deterministic Schrödinger operators.
-
Continuity equation in LlogL for the 2D Euler equations under the enstrophy measure Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2020-05-12 Giuseppe Da Prato, Franco Flandoli, Michael Röckner
The 2D Euler equations with random initial condition has been investigates by Albeverio and Cruzeiro (Commun Math Phys 129:431–444, 1990) and other authors. Here we prove existence of solutions for the associated continuity equation in Hilbert spaces, in a quite general class with LlogL densities with respect to the enstrophy measure.
-
Stochastic regularization for transport equations Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2020-05-08 Jinlong Wei, Jinqiao Duan, Hongjun Gao, Guangying Lv
We investigate a stochastic transport equation driven by a multiplicative noise. For drift coefficients in \(L^q(0,T;{\mathcal {C}}^\alpha _b({\mathbb {R}}^d))\) (\(\alpha >2/q\)) and initial data in \(W^{1,r}({\mathbb {R}}^d)\), we show the existence and uniqueness of stochastic strong solutions. Opposite to the deterministic case where the same assumptions on drift coefficients and initial data induce
-
2D-defocusing nonlinear Schrödinger equation with random data on irrational tori Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2020-05-07 Chenjie Fan, Yumeng Ou, Gigliola Staffilani, Hong Wang
We revisit the work of Bourgain on the invariance of the Gibbs measure for the cubic, defocusing nonlinear Schrödinger equation in 2D on a square torus, and we prove the equivalent result on any tori.
-
On the Cauchy problem for stochastic integro-differential equations with radially O-regularly varying Lévy measure Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2020-03-29 R. Mikulevicius, C. Phonsom
Parabolic integro-differential nondegenerate Cauchy problem is considered in the scale of \(L_{p}\) spaces of functions whose regularity is defined by a Lévy measure with O-regularly varying radial profile. Existence and uniqueness of a solution is proved by deriving apriori estimates. Some probability density function estimates of the associated Lévy process are used as well.
-
Convergent numerical approximation of the stochastic total variation flow Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2020-03-29 L’ubomír Baňas, Michael Röckner, André Wilke
We study the stochastic total variation flow (STVF) equation with linear multiplicative noise. By considering a limit of a sequence of regularized stochastic gradient flows with respect to a regularization parameter \(\varepsilon \) we obtain the existence of a unique variational solution of the STVF equation which satisfies a stochastic variational inequality. We propose an energy preserving fully
-
Wong–Zakai approximation and support theorem for semilinear stochastic partial differential equations with finite dimensional noise in the whole space Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2020-03-27 Timur Yastrzhembskiy
In this paper we consider the following stochastic partial differential equation (SPDE) in the whole space: \( du (t, x) = [a^{i j} (t, x) D_{i j} u(t, x) + f(u, t, x)]\, dt + \sum _{k = 1}^m g^k (u(t, x)) dw^k (t). \) We prove the convergence of a Wong–Zakai type approximation scheme of the above equation in the space \( C^{\theta } ([0, T], H^{\gamma }_p ({\mathbb {R}}^d)) \) in probability, for
-
Stochastic analysis with modelled distributions Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2020-03-03 Chong Liu, David J. Prömel, Josef Teichmann
Using a Besov topology on spaces of modelled distributions in the framework of Hairer’s regularity structures, we prove the reconstruction theorem on these Besov spaces with negative regularity. The Besov spaces of modelled distributions are shown to be UMD Banach spaces and of martingale type 2. As a consequence, this gives access to a rich stochastic integration theory and to existence and uniqueness
-
Drift estimation for discretely sampled SPDEs Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2020-01-08 Igor Cialenco, Francisco Delgado-Vences, Hyun-Jung Kim
The aim of this paper is to study the asymptotic properties of the maximum likelihood estimator (MLE) of the drift coefficient for fractional stochastic heat equation driven by an additive space-time noise. We consider the traditional for stochastic partial differential equations statistical experiment when the measurements are performed in the spectral domain, and in contrast to the existing literature
-
On the stochastic nonlinear Schrödinger equations at critical regularities Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2020-01-03 Tadahiro Oh, Mamoru Okamoto
We consider the Cauchy problem for the defocusing stochastic nonlinear Schrödinger equations (SNLS) with an additive noise in the mass-critical and energy-critical settings. By adapting the probabilistic perturbation argument employed in the context of the random data Cauchy theory by Bényi et al. (Trans Am Math Soc Ser B 2:1–50, 2015) to the current stochastic PDE setting, we present a concise argument
-
An order approach to SPDEs with antimonotone terms Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2020-01-03 Luca Scarpa, Ulisse Stefanelli
We consider a class of parabolic stochastic partial differential equations featuring an antimonotone nonlinearity. The existence of unique maximal and minimal variational solutions is proved via a fixed-point argument for nondecreasing mappings in ordered spaces. This relies on the validity of a comparison principle.
-
On the convergence of stochastic transport equations to a deterministic parabolic one Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2020-01-02 Lucio Galeati
A stochastic transport linear equation (STLE) with multiplicative space-time dependent noise is studied. It is shown that, under suitable assumptions on the noise, a multiplicative renormalization leads to convergence of the solutions of STLE to the solution of a deterministic parabolic equation. Existence and uniqueness for STLE are also discussed. Our method works in dimension \(d\ge 2\); the case
-
Uniform estimate of an iterative method for elliptic problems with rapidly oscillating coefficients Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2019-12-21 Chenlin Gu
We study the iterative algorithm proposed by Armstrong et al. (An iterative method for elliptic problems with rapidly oscillating coefficients, 2018. arXiv preprint arXiv:1803.03551) to solve elliptic equations in divergence form with stochastic stationary coefficients. Such equations display rapidly oscillating coefficients and thus usually require very expensive numerical calculations, while this
-
A high order time discretization of the solution of the non-linear filtering problem Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2019-12-09 Dan Crisan, Salvador Ortiz-Latorre
The solution of the continuous time filtering problem can be represented as a ratio of two expectations of certain functionals of the signal process that are parametrized by the observation path. We introduce a class of discretization schemes of these functionals of arbitrary order. The result generalizes the classical work of Picard, who introduced first order discretizations to the filtering functionals
-
Classical and generalized solutions of fractional stochastic differential equations Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2019-12-07 S. V. Lototsky, B. L. Rozovsky
For stochastic evolution equations with fractional derivatives, classical solutions exist when the order of the time derivative of the unknown function is not too small compared to the order of the time derivative of the noise; otherwise, there can be a generalized solution in suitable weighted chaos spaces. Presence of fractional derivatives in both time and space leads to various modifications of
-
Higher-order pathwise theory of fluctuations in stochastic homogenization Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2019-11-28 Mitia Duerinckx, Felix Otto
We consider linear elliptic equations in divergence form with stationary random coefficients of integrable correlations. We characterize the fluctuations of a macroscopic observable of a solution to relative order \(\frac{d}{2}\), where d is the spatial dimension; the fluctuations turn out to be Gaussian. As for previous work on the leading order, this higher-order characterization relies on a pathwise
-
Correction to: Kernel-based collocation methods for Zakai equations Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2019-11-20 Yumiharu Nakano
The claim of Lemma 3.7 in [1] needs to be modified. This lemma states that the sum of the absolute value of the cardinal functions is bounded with respect to the approximation parameters and the spatial variables.
-
Accelerated finite elements schemes for parabolic stochastic partial differential equations Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2019-11-01 István Gyöngy, Annie Millet
For a class of finite elements approximations for linear stochastic parabolic PDEs it is proved that one can accelerate the rate of convergence by Richardson extrapolation. More precisely, by taking appropriate mixtures of finite elements approximations one can accelerate the convergence to any given speed provided the coefficients, the initial and free data are sufficiently smooth.
-
Exponential convergence of solutions for random Hamilton–Jacobi equations Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2019-10-28 Renato Iturriaga, Konstantin Khanin, Ke Zhang
We show that for a family of randomly kicked Hamilton–Jacobi equations on the torus, almost surely, the solution of an initial value problem converges exponentially fast to the unique stationary solution. Combined with the earlier results of the authors, this completes the program in the multi-dimensional setting started by E, Khanin, Mazel and Sinai in the one-dimensional case.
-
On the density of the supremum of the solution to the linear stochastic heat equation Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2019-09-23 Robert C. Dalang, Fei Pu
We study the regularity of the probability density function of the supremum of the solution to the linear stochastic heat equation. Using a general criterion for the smoothness of densities for locally nondegenerate random variables, we establish the smoothness of the joint density of the random vector whose components are the solution and the supremum of an increment in time of the solution over an
-
Support theorem for an SPDE with multiplicative noise driven by a cylindrical Wiener process on the real line Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2019-09-18 Timur Yastrzhembskiy
We prove a Stroock–Varadhan’s type support theorem for a stochastic partial differential equation on the real line with a noise term driven by a cylindrical Wiener process on \(L_2 ({\mathbb {R}})\). The main ingredients of the proof are V. Mackevičius’s approach to support theorem for diffusion processes and N.V. Krylov’s \(L_p\)-theory of SPDEs.
-
$$L^p$$Lp -estimates and regularity for SPDEs with monotone semilinearity Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2019-09-11 Neelima; David Šiška
Semilinear stochastic partial differential equations on bounded domains \({\mathscr {D}}\) are considered. The semilinear term may have arbitrary polynomial growth as long as it is continuous and monotone except perhaps near the origin. Typical examples are the stochastic Allen–Cahn and Ginzburg–Landau equations. The first main result of this article are \(L^p\)-estimates for such equations. The \(L^p\)-estimates
-
Gaussian fluctuations for the stochastic heat equation with colored noise Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2019-07-13 Jingyu Huang; David Nualart; Lauri Viitasaari; Guangqu Zheng
In this paper, we present a quantitative central limit theorem for the d-dimensional stochastic heat equation driven by a Gaussian multiplicative noise, which is white in time and has a spatial covariance given by the Riesz kernel. We show that the spatial average of the solution over an Euclidean ball is close to a Gaussian distribution, when the radius of the ball tends to infinity. Our central limit
-
Optimal scaling of the MALA algorithm with irreversible proposals for Gaussian targets Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2019-07-10 Michela Ottobre; Natesh S. Pillai; Konstantinos Spiliopoulos
It is well known in many settings that reversible Langevin diffusions in confining potentials converge to equilibrium exponentially fast. Adding irreversible perturbations to the drift of a Langevin diffusion that maintain the same invariant measure accelerates its convergence to stationarity. Many existing works thus advocate the use of such non-reversible dynamics for sampling. When implementing
-
Normal approximation of the solution to the stochastic heat equation with Lévy noise Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2019-07-10 Carsten Chong; Thomas Delerue
Given a sequence \({\dot{L}}^\varepsilon \) of Lévy noises, we derive necessary and sufficient conditions in terms of their variances \(\sigma ^2(\varepsilon )\) such that the solution to the stochastic heat equation with noise \(\sigma (\varepsilon )^{-1} {\dot{L}}^\varepsilon \) converges in law to the solution to the same equation with Gaussian noise. Our results apply to both equations with additive
-
Convergence of the Finite Volume method for scalar conservation laws with multiplicative noise: an approach by kinetic formulation Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2019-07-10 Sylvain Dotti; Julien Vovelle
Under a standard CFL condition, we prove the convergence of the explicit-in-time Finite Volume method with monotone fluxes for the approximation of scalar first-order conservation laws with multiplicative, compactly supported noise. In Dotti and Vovelle (Arch Ration Mech Anal 230(2):539–591, 2018), a framework for the analysis of the convergence of approximations to stochastic scalar first-order conservation
-
Numerical methods for conservation laws with rough flux Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2019-06-14 H. Hoel; K. H. Karlsen; N. H. Risebro; E. B. Storrøsten
Finite volume methods are proposed for computing approximate pathwise entropy/kinetic solutions to conservation laws with flux functions driven by low-regularity paths. For a convex flux, it is demonstrated that driving path oscillations may lead to “cancellations” in the solution. Making use of this property, we show that for \(\alpha \)-Hölder continuous paths the convergence rate of the numerical
-
Gaussian fluctuations from the 2D KPZ equation Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2019-06-11 Yu Gu
We prove the two dimensional KPZ equation with a logarithmically tuned nonlinearity and a small coupling constant, scales to the Edwards–Wilkinson equation with an effective variance.
-
Multilevel methods for uncertainty quantification of elliptic PDEs with random anisotropic diffusion Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2019-05-27 Helmut Harbrecht; Marc Schmidlin
We consider elliptic diffusion problems with a random anisotropic diffusion coefficient, where, in a notable direction given by a random vector field, the diffusion strength differs from the diffusion strength perpendicular to this notable direction. The Karhunen–Loève expansion then yields a parametrisation of the random vector field and, therefore, also of the solution of the elliptic diffusion problem
-
Semilinear evolution equations for the Anderson Hamiltonian in two and three dimensions Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2019-05-24 M. Gubinelli; B. Ugurcan; I. Zachhuber
We analyze nonlinear Schrödinger and wave equations whose linear part is given by the renormalized Anderson Hamiltonian in two and three dimensional periodic domains.
-
Weak martingale solutions for the stochastic nonlinear Schrödinger equation driven by pure jump noise Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2019-05-16 Zdzisław Brzeźniak; Fabian Hornung; Utpal Manna
We construct a martingale solution of the stochastic nonlinear Schrödinger equation (NLS) 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
-
Large deviations and averaging for systems of slow-fast stochastic reaction–diffusion equations Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2019-05-11 Wenqing Hu; Michael Salins; Konstantinos Spiliopoulos
We study a large deviation principle for a system of stochastic reaction–diffusion equations (SRDEs) with a separation of fast and slow components and small noise in the slow component. The derivation of the large deviation principle is based on the weak convergence method in infinite dimensions, which results in studying averaging for controlled SRDEs. By appropriate choice of the parameters, the
-
A reflected moving boundary problem driven by space–time white noise Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2019-03-27 Ben Hambly; Jasdeep Kalsi
We study a system of two reflected SPDEs which share a moving boundary. The equations describe competition at an interface and are motivated by the modelling of the limit order book in financial markets. The derivative of the moving boundary is given by a function of the two SPDEs in their relative frames. We prove existence and uniqueness for the equations until blow-up, and show that the solution
-
Electro-rheological fluids under random influences: martingale and strong solutions Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2019-03-18 Dominic Breit; Franz Gmeineder
We study generalised Navier–Stokes equations governing the motion of an electro-rheological fluid subject to stochastic perturbation. Stochastic effects are implemented through (i) random initial data, (ii) a forcing term in the momentum equation represented by a multiplicative white noise and (iii) a random character of the variable exponent \(p=p(\omega ,t,x)\) (as a result of a random electric field)
-
Restoring uniqueness to mean-field games by randomizing the equilibria Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2019-03-06 François Delarue
We here address the question of restoration of uniqueness in mean-field games deriving from deterministic differential games with a large number of players. The general strategy for restoring uniqueness is inspired from earlier similar results on ordinary and stochastic differential equations. It consists in randomizing the equilibria through an external noise. As a main feature, we choose the external
-
Spatial asymptotic of the stochastic heat equation with compactly supported initial data Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2019-03-02 Jingyu Huang; Khoa Lê
We investigate the growth of the tallest peaks of random field solutions to the parabolic Anderson models over concentric balls as the radii approach infinity. The noise is white in time and correlated in space. The spatial correlation function is either bounded or non-negative satisfying Dalang’s condition. The initial data are Borel measures with compact supports, in particular, include Dirac masses
-
Stochastic maximal regularity for rough time-dependent problems Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2019-03-02 Pierre Portal; Mark Veraar
We unify and extend the semigroup and the PDE approaches to stochastic maximal regularity of time-dependent semilinear parabolic problems with noise given by a cylindrical Brownian motion. We treat random coefficients that are only progressively measurable in the time variable. For 2m-th order systems with VMO regularity in space, we obtain \(L^{p}(L^{q})\) estimates for all \(p>2\) and \(q\ge 2\)
-
Talagrand concentration inequalities for stochastic partial differential equations Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2019-02-22 Davar Khoshnevisan; Andrey Sarantsev
One way to define the concentration of measure phenomenon is via Talagrand inequalities, also called transportation-information inequalities. That is, a comparison of the Wasserstein distance from the given measure to any other absolutely continuous measure with finite relative entropy. Such transportation-information inequalities were recently established for some stochastic differential equations
-
Some results on the penalised nematic liquid crystals driven by multiplicative noise: weak solution and maximum principle Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2019-01-24 Zdzisław Brzeźniak; Erika Hausenblas; Paul André Razafimandimby
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,
-
Kernel-based collocation methods for Zakai equations Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2019-01-24 Yumiharu Nakano
We examine an application of the kernel-based interpolation to numerical solutions for Zakai equations in nonlinear filtering, and aim to prove its rigorous convergence. To this end, we find the class of kernels and the structure of collocation points explicitly under which the process of iterative interpolation is stable. This result together with standard argument in error estimation shows that the
-
Invariant measure and large time dynamics of the cubic Klein–Gordon equation in 3 D Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2018-11-29 Mouhamadou Sy
In this paper we construct an invariant probability measure concentrated on \(H^2(K)\times H^1(K)\) for a general cubic Klein–Gordon equation (including the case of the wave equation). Here K represents both the 3-dimensional torus or a bounded domain with smooth boundary in \({\mathbb {R}}^3\). That allows to deduce some corollaries on the long time behaviour of the flow of the equation in a probabilistic
-
Well-posedness of Hall-magnetohydrodynamics system forced by L $$\acute{\mathrm{e}}$$ e ´ vy noise Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2018-11-15 Kazuo Yamazaki; Manil T. Mohan
We establish the existence and uniqueness of a local smooth solution to the Cauchy problem for the Hall-magnetohydrodynamics system that is inviscid, resistive, and forced by multiplicative L\(\acute{\mathrm {e}}\)vy noise in the three dimensional space. Moreover, when the initial data is sufficiently small, we prove that the solution exists globally in time in probability; that is, the probability
-
On a doubly nonlinear PDE with stochastic perturbation Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2018-10-25 Niklas Sapountzoglou; Petra Wittbold; Aleksandra Zimmermann
We consider a doubly nonlinear evolution equation with multiplicative noise and show existence and pathwise uniqueness of a strong solution. Using a semi-implicit time discretization we get approximate solutions with monotonicity arguments. We establish a-priori estimates for the approximate solutions and show tightness of the sequence of image measures induced by the sequence of approximate solutions
-
Berry–Esseen theorem and quantitative homogenization for the random conductance model with degenerate conductances Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2018-10-12 Sebastian Andres; Stefan Neukamm
We study the random conductance model on the lattice \({\mathbb {Z}}^d\), i.e. we consider a linear, finite-difference, divergence-form operator with random coefficients and the associated random walk under random conductances. We allow the conductances to be unbounded and degenerate elliptic, but they need to satisfy a strong moment condition and a quantified ergodicity assumption in form of a spectral
-
Iterated stochastic integrals in infinite dimensions: approximation and error estimates Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2018-09-22 Claudine Leonhard; Andreas Rößler
Higher order numerical schemes for stochastic partial differential equations that do not possess commutative noise require the simulation of iterated stochastic integrals. In this work, we extend the algorithms derived by Kloeden et al. (Stoch Anal Appl 10(4):431–441, 1992. https://doi.org/10.1080/07362999208809281) and by Wiktorsson (Ann Appl Probab 11(2):470–487, 2001. https://doi.org/10.1214/aoap/1015345301)
-
Stochastic nonlinear Schrödinger equations on tori Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2018-08-29 Kelvin Cheung; Razvan Mosincat
We consider the stochastic nonlinear Schrödinger equations (SNLS) posed on d-dimensional tori with either additive or multiplicative stochastic forcing. In particular, for the one-dimensional cubic SNLS, we prove global well-posedness in \(L^2(\mathbb {T})\). As for other power-type nonlinearities, namely (i) (super)quintic when \(d = 1\) and (ii) (super)cubic when \(d \ge 2\), we prove local well-posedness
-
Path properties of the solution to the stochastic heat equation with Lévy noise Stoch. PDE Anal. Comp. (IF 1.39) Pub Date : 2018-08-17 Carsten Chong; Robert C. Dalang; Thomas Humeau
We consider sample path properties of the solution to the stochastic heat equation, in \({\mathbb {R}}^d\) or bounded domains of \({\mathbb {R}}^d\), driven by a Lévy space–time white noise. When viewed as a stochastic process in time with values in an infinite-dimensional space, the solution is shown to have a càdlàg modification in fractional Sobolev spaces of index less than \(-\frac{d}{2}\). Concerning