Spatial variation for the solution to the stochastic linear wave equation driven by additive space-time white noise

2018 ◽  
Vol 18 (05) ◽  
pp. 1850036 ◽  
Author(s):  
M. Khalil ◽  
C. A. Tudor ◽  
M. Zili
Keyword(s):  
Asymptotic Behavior ◽  
Wave Equation ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
White Noise ◽  
Space Time ◽  
Linear Wave ◽  
Linear Wave Equation ◽  
Stochastic Wave Equation ◽  
Quadratic Variations

We study the asymptotic behavior of the spatial quadratic variation for the solution to the stochastic wave equation driven by additive space-time white noise. We prove that the sequence of its renormalized quadratic variations satisfies a central limit theorem (CLT for short). We obtain the rate of convergence for this CLT via the Stein–Malliavin calculus and we also discuss some consequences.

Central limit theorem for the solution to the heat equation with moving time

2016 ◽  
Vol 19 (01) ◽  
pp. 1650005 ◽  
Author(s):  
Junfeng Liu ◽  
Ciprian A. Tudor
Keyword(s):  
Asymptotic Behavior ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Heat Equation ◽  
White Noise ◽  
Limit Distribution ◽  
Malliavin Calculus ◽  
Time Variable ◽  
Time Space ◽  
Quadratic Variations

We consider the solution to the stochastic heat equation driven by the time-space white noise and study the asymptotic behavior of its spatial quadratic variations with “moving time”, meaning that the time variable is not fixed and its values are allowed to be very big or very small. We investigate the limit distribution of these variations via Malliavin calculus.

Asymptotic expansion for the quadratic variations of the solution to the heat equation with additive white noise

2020 ◽  
Vol 21 (02) ◽  
pp. 2150010
Author(s):  
Héctor Araya ◽  
Ciprian A. Tudor
Keyword(s):  
Asymptotic Expansion ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Heat Equation ◽  
White Noise ◽  
Malliavin Calculus ◽  
Order Term ◽  
Sharp Estimates ◽  
Additive White Noise ◽  
Quadratic Variations

We consider the sequence of spatial quadratic variations of the solution to the stochastic heat equation with space-time white noise. This sequence satisfies a Central Limit Theorem. By using Malliavin calculus, we refine this result by proving the convergence of the sequence of densities and by finding the second-order term in the asymptotic expansion of the densities. In particular, our proofs are based on sharp estimates of the correlation structure of the solution, which may have their own interest.

scholarly journals Parameter identification for the linear wave equation with Robin boundary condition

2019 ◽  
Vol 27 (1) ◽  
pp. 25-41
Author(s):  
Valeria Bacchelli ◽  
Dario Pierotti ◽  
Stefano Micheletti ◽  
Simona Perotto
Keyword(s):  
Wave Equation ◽  
Mixed Boundary ◽  
Stability Result ◽  
Interval Length ◽  
Left Endpoint ◽  
Linear Wave ◽  
Reconstruction Procedure ◽  
Element Discretization ◽  
Linear Wave Equation ◽  
Bounded Interval

Abstract We consider an initial-boundary value problem for the classical linear wave equation, where mixed boundary conditions of Dirichlet and Neumann/Robin type are enforced at the endpoints of a bounded interval. First, by a careful application of the method of characteristics, we derive a closed-form representation of the solution for an impulsive Dirichlet data at the left endpoint, and valid for either a Neumann or a Robin data at the right endpoint. Then we devise a reconstruction procedure for identifying both the interval length and the Robin parameter. We provide a corresponding stability result and verify numerically its performance moving from a finite element discretization.

Sign in / Sign up

Export Citation Format

Share Document