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30
Weak Order for the Discretization of the Stochastic Heat Equation Driven by Impulsive Noise
, 2009
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WEAK ORDER FOR THE DISCRETIZATION OF THE STOCHASTIC HEAT EQUATION
, 2008
"... Abstract. In this paper we study the approximation of the distribution of Xt Hilbert–valued stochastic process solution of a linear parabolic stochastic partial differential equation written in an abstract form as dXt + AXt dt = Q 1/2 dW(t), X0 = x ∈ H, t ∈ [0,T], driven by a Gaussian space time noi ..."
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Cited by 34 (4 self)
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Abstract. In this paper we study the approximation of the distribution of Xt Hilbert–valued stochastic process solution of a linear parabolic stochastic partial differential equation written in an abstract form as dXt + AXt dt = Q 1/2 dW(t), X0 = x ∈ H, t ∈ [0,T], driven by a Gaussian space time noise whose covariance operator Q is given. We assume that A−α is a finite trace operator for some α>0andthatQis bounded from H into D(Aβ)forsomeβ≥0. It is not required to be nuclear or to commute with A. The discretization is achieved thanks to finite element methods in space (parameter h>0) and a θmethod in time (parameter ∆t = T/N). We define a discrete solution Xn h and for suitable functions ϕ defined on H, we show that E ϕ(X N h) − E ϕ(XT)  = O(h 2γ +∆t γ) where γ<1 − α + β. Let us note that as in the finite dimensional case the rate of convergence is twice the one for pathwise approximations. 1.
Weak approximation of stochastic partial differential equations: the non linear case
 Math. Comp
, 2011
"... Abstract. We study the error of the Euler scheme applied to a stochastic partial differential equation. We prove that as it is often the case, the weak order of convergence is twice the strong order. A key ingredient in our proof is Malliavin calculus which enables us to get rid of the irregular ter ..."
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Cited by 31 (0 self)
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Abstract. We study the error of the Euler scheme applied to a stochastic partial differential equation. We prove that as it is often the case, the weak order of convergence is twice the strong order. A key ingredient in our proof is Malliavin calculus which enables us to get rid of the irregular terms of the error. We apply our method to the case a semilinear stochastic heat equation driven by a spacetime white noise.
Weak and strong order of convergence of a semidiscrete scheme for the stochastic nonlinear Schrödinger equation
 Appl. Math. Optim
"... Abstract. In this article we analyze the error of a semidiscrete scheme for the stochastic nonlinear Schrödinger equation with power nonlinearity. We consider supercritical or subcritical nonlinearity and the equation can be either focusing or defocusing. Allowing sufficient spatial regularity we p ..."
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Cited by 30 (1 self)
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Abstract. In this article we analyze the error of a semidiscrete scheme for the stochastic nonlinear Schrödinger equation with power nonlinearity. We consider supercritical or subcritical nonlinearity and the equation can be either focusing or defocusing. Allowing sufficient spatial regularity we prove that the numerical scheme has strong order 1 2 in general and order 1 if the noise is additive. Furthermore, we also prove that the weak order is always 1.
Large deviations for stochastic evolution equations with small multiplcative noise
"... The FreidlinWentzell large deviation principle is established for the distributions of stochastic evolution equations with general monotone drift and small multiplicative noise. Roughly speaking, besides the assumptions for existence and uniqueness of the solution, one only need assume some additio ..."
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Cited by 24 (10 self)
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The FreidlinWentzell large deviation principle is established for the distributions of stochastic evolution equations with general monotone drift and small multiplicative noise. Roughly speaking, besides the assumptions for existence and uniqueness of the solution, one only need assume some additional assumptions on diffusion coefficient in order to obtain Large deviation principle for the distribution of solution. As applications we can apply the main result to different type examples of SPDEs (e.g. stochastic reactiondiffusion equation, stochastic porous media and fast diffusion equations, stochastic pLaplacian equation) in Hilbert space. The weak convergence approach is employed to verify the Laplace principle, which is equivalent to large deviation principle in our framework. AMS subject Classification: 60F10, 60H15.
Rate of Convergence of Space Time Approximations for stochastic evolution equations
, 2007
"... Abstract. Stochastic evolution equations in Banach spaces with unbounded nonlinear drift and diffusion operators are considered. Under some regularity condition assumed for the solution, the rate of convergence of various numerical approximations are estimated under strong monotonicity and Lipschitz ..."
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Cited by 17 (1 self)
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Abstract. Stochastic evolution equations in Banach spaces with unbounded nonlinear drift and diffusion operators are considered. Under some regularity condition assumed for the solution, the rate of convergence of various numerical approximations are estimated under strong monotonicity and Lipschitz conditions. The abstract setting involves general consistency conditions and is then applied to a class of quasilinear stochastic PDEs of parabolic type.
Rate of Convergence of Implicit Approximations for stochastic evolution equations
, 2006
"... Abstract. Stochastic evolution equations in Banach spaces with unbounded nonlinear drift and diffusion operators are considered. Under some regularity condition assumed for the solution, the rate of convergence of implicit Euler approximations is estimated under strong monotonicity and Lipschitz con ..."
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Cited by 13 (1 self)
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Abstract. Stochastic evolution equations in Banach spaces with unbounded nonlinear drift and diffusion operators are considered. Under some regularity condition assumed for the solution, the rate of convergence of implicit Euler approximations is estimated under strong monotonicity and Lipschitz conditions. The results are applied to a class of quasilinear stochastic PDEs of parabolic type.
ON THE BACKWARD EULER APPROXIMATION OF THE STOCHASTIC ALLENCAHN EQUATION
"... Abstract. We consider the stochastic AllenCahn equation perturbed by smooth additive Gaussian noise in a spatial domain with smooth boundary in dimension d ≤ 3, and study the semidiscretization in time of the equation by an implicit Euler method. We show that the method converges pathwise with a ra ..."
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Cited by 12 (0 self)
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Abstract. We consider the stochastic AllenCahn equation perturbed by smooth additive Gaussian noise in a spatial domain with smooth boundary in dimension d ≤ 3, and study the semidiscretization in time of the equation by an implicit Euler method. We show that the method converges pathwise with a rate O(∆tγ) for any γ < 1 2. We also prove that the scheme converges uniformly in the strong Lpsense but with no rate given. 1.
On stochastic evolution equations with nonLipschitz coefficients
 Stoc. Dyna
"... Abstract. In this paper, we study the existence and uniqueness of solutions for several classes of stochastic evolution equations with nonLipschitz coefficients, that is, backward stochastic evolution equations, stochastic Volterra type evolution equations and stochastic functional evolution equati ..."
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Cited by 10 (0 self)
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Abstract. In this paper, we study the existence and uniqueness of solutions for several classes of stochastic evolution equations with nonLipschitz coefficients, that is, backward stochastic evolution equations, stochastic Volterra type evolution equations and stochastic functional evolution equations. In particular, the results can be used to treat a large class of quasilinear stochastic equations, which includes the reaction diffusion and porous medium equations.