29. chen.eisenberg:23:interpolating#
Interpolating the stochastic heat and wave equations with time-independent noise: solvability and exact asymptotics
Le Chen and Nicholas Eisenberg
Abstract: In this article, we study a class of stochastic partial differential equations with fractional differential operators subject to some time-independent multiplicative Gaussian noise. We derive sharp conditions, under which a unique global \(L^p(\Omega)\)-solution exists for all \(p\ge 2\). In this case, we derive exact moment asymptotics following the same strategy as that in a recent work by Balan et al [BCC22]. In the case when there exists only a local solution, we determine the precise deterministic time, \(T_2\), before which a unique \(L^2(\Omega)\)-solution exists, but after which the series corresponding to the \(L^2(\Omega)\) moment of the solution blows up. By properly choosing the parameters, results in this paper interpolate the known results for both stochastic heat and wave equations.
MSC 2010 subject classifications: Primary 60H15; Secondary 60H07, 37H15.
Keywords: stochastic partial differential equations, Caputo derivatives, Riemann-Liouville fractional integral, fractional Laplacian, Malliavin calculus, Skorohod integral, exact moment asymptotics, time-independent Gaussian noise, white noise, global and local solutions.
[CE23] Chen, Le & Eisenberg, Nicholas (2023) ‘Interpolating the stochastic heat and wave equations with time-independent noise: solvability and exact asymptotics’, Stoch. Partial Differ. Equ. Anal. Comput. 11, 1203–1253. <https://doi.org/10.1007/s40072-022-00258-6>
@article{chen.eisenberg:23:interpolating,
author = {Chen, Le and Eisenberg, Nicholas},
title = {Interpolating the stochastic heat and wave equations with time-independent noise: solvability and exact asymptotics},
journal = {Stoch. Partial Differ. Equ. Anal. Comput.},
fjournal = {Stochastic Partial Differential Equations. Analysis and Computations},
volume = {11},
year = {2023},
number = {3},
pages = {1203--1253},
issn = {2194-0401},
mrclass = {60H15 (35R11 37H15 60H07)},
mrnumber = {4624137},
doi = {10.1007/s40072-022-00258-6},
url = {https://doi.org/10.1007/s40072-022-00258-6}
}
References: Balan et al. [BCC22]; Balan and Song [BS17]; Bass et al. [BCR09]; Chen [Che17a]; Chen et al. [CHN19]; Chen et al. [CHHH17]; Chen [Che12]; Chen [Che17b]; Chen [Che19]; Chen and Li [CL04]; Chen et al. [CHSX15]; Chen et al. [CHSS18]; Chen et al. [CDOT21]; Dalang [Dal09]; Hairer and Labbé [HL15]; Hairer and Labbé [HL18]; Hu [Hu01]; Hu [Hu02]; Lê [Le16]; Lieb and Loss [LL01]; Mijena and Nane [MN15]; Nualart and Nualart [NN18];