20. balan.chen.ea:22:exact#
Exact asymptotics of the stochastic wave equation with time-independent noise
Raluca M. Balan, Le Chen, and Xia Chen
Abstract: In this article, we study the stochastic wave equation in all dimensions \(d\leq 3\), driven by a Gaussian noise \(\dot{W}\) which does not depend on time. We assume that either the noise is white, or the covariance function of the noise satisfies a scaling property similar to the Riesz kernel. The solution is interpreted in the Skorohod sense using Malliavin calculus. We obtain the exact asymptotic behaviour of the \(p\)-th moment of the solution either when the time is large or when \(p\) is large. For the critical case, that is the case when \(d=3\) and the noise is white, we obtain the exact transition time for the second moment to be finite.
MSC 2010 subject classifications: Primary 60H15; Secondary 60H07, 37H15.
Keywords: stochastic partial differential equations, stochastic wave equation, Malliavin calculus, Lyapunov exponents, exact moment asymptotics, moment blowup.
[BCC22] Balan, Raluca, Chen, Le & Ma, Yiping (2022) ‘Parabolic Anderson model with rough noise in space and rough initial conditions’, Electron. Commun. Probab. 27, Paper No. 65, 12. <https://doi.org/10.1214/22-ecp506>
@article{balan.chen.ea:22:exact,
author = {Balan, Raluca M. and Chen, Le and Chen, Xia},
title = {Exact asymptotics of the stochastic wave equation with time-independent noise},
journal = {Ann. Inst. Henri Poincar\'{e} Probab. Stat.},
fjournal = {Annales de l'Institut Henri Poincar\'{e} Probabilit\'{e}s et Statistiques},
volume = {58},
year = {2022},
number = {3},
pages = {1590--1620},
issn = {0246-0203},
mrclass = {60H15 (37H15 60H07)},
mrnumber = {4452644},
doi = {10.1214/21-aihp1207},
url = {https://doi.org/10.1214/21-aihp1207}
}
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