scholarly journals Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise II. Fully discrete schemes

2012 ◽  
Author(s):  
Mihály Kovács ◽  
Stig Larsson ◽  
Fredrik Lindgren
Keyword(s):  
Finite Element ◽  
Weak Convergence ◽  
Additive Noise ◽  
Evolution Equations ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Fully Discrete ◽  
Finite Element Approximations

scholarly journals Weak convergence of fully discrete finite element approximations of semilinear hyperbolic SPDE with additive noise

2020 ◽  
Vol 54 (6) ◽  
pp. 2199-2227
Author(s):  
Mihály Kovács ◽  
Annika Lang ◽  
Andreas Petersson
Keyword(s):  
Finite Element ◽  
Weak Convergence ◽  
Additive Noise ◽  
Convergence Rates ◽  
Fully Discrete ◽  
Finite Element Approximations ◽  
Negative Order ◽  
Stochastic Wave Equation ◽  
Discrete Finite Element ◽  
Spatial Approximation

The numerical approximation of the mild solution to a semilinear stochastic wave equation driven by additive noise is considered. A standard finite element method is employed for the spatial approximation and a a rational approximation of the exponential function for the temporal approximation. First, strong convergence of this approximation in both positive and negative order norms is proven. With the help of Malliavin calculus techniques this result is then used to deduce weak convergence rates for the class of twice continuously differentiable test functions with polynomially bounded derivatives. Under appropriate assumptions on the parameters of the equation, the weak rate is found to be essentially twice the strong rate. This extends earlier work by one of the authors to the semilinear setting. Numerical simulations illustrate the theoretical results.

INTEGRATION BY PARTS AND SMOOTHNESS OF THE LAW FOR A CLASS OF STOCHASTIC EVOLUTION EQUATIONS

2004 ◽  
Vol 07 (01) ◽  
pp. 89-129 ◽  
Author(s):  
STEFANO BONACCORSI ◽  
MARCO FUHRMAN
Keyword(s):  
Additive Noise ◽  
Evolution Equations ◽  
Transition Probabilities ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Integration By Parts ◽  
Infinite Dimensional ◽  
Integration By Parts Formula ◽  
Derivatives Of ◽  
Logarithmic Derivatives

We consider a Markov process X in a Hilbert space H, solution of a semilinear stochastic evolution equation driven by an infinite-dimensional Wiener process, occurring in the equation as an additive noise. Using techniques of the Malliavin calculus, under suitable assumptions, we prove an integration by parts formula for the transition probabilities νt, t>0 (the laws of Xt). We deduce results on differentiability (i.e. existence of logarithmic derivatives) of νt along a set of directions h∈H which can be described in terms of the coefficients of the equation. The general results are then applied to various classes of non linear stochastic partial differential equations and systems.

scholarly journals Integrated solutions of stochastic evolution equations with additive noise

2001 ◽  
Vol 64 (2) ◽  
pp. 281-290 ◽  
Author(s):  
A. Filinkov ◽  
I. Maizurna
Keyword(s):  
Evolution Equation ◽  
Additive Noise ◽  
Evolution Equations ◽  
Stochastic Evolution Equation ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Existence Of A Solution ◽  
Integrated Semigroup

We investigate the existence of a solution to the abstract stochastic evolution equation with additive noise: in the case when A is the generator of an n-times integrated semigroup.

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