2. chen.dalang:15:moments#
Moments and growth indices for the nonlinear stochastic heat equation with rough initial conditions
Le Chen and Robert C. Dalang
Abstract: We study the nonlinear stochastic heat equation in the spatial domain \(\mathbb{R}\), driven by space-time white noise. A central special case is the parabolic Anderson model. The initial condition is taken to be a measure on \(\mathbb{R}\), such as the Dirac delta function, but this measure may also have non-compact support and even be non-tempered (for instance with exponentially growing tails). Existence and uniqueness of a random field solution is proved without appealing to Gronwall’s lemma, by keeping tight control over moments in the Picard iteration scheme. Upper bounds on all \(p\)-th moments \((p\ge 2)\) are obtained as well as a lower bound on second moments. These bounds become equalities for the parabolic Anderson model when \(p=2\). We determine the growth indices introduced by Conus and Khoshnevisan [CK12].
MSC 2010 subject classifications: Primary 60H15. Secondary 60G60, 35R60.
Keywords: nonlinear stochastic heat equation, parabolic Anderson model, rough initial data, growth indices.
[CD15b] Chen, Le & Dalang, Robert C. (2015) ‘Moments and growth indices for the nonlinear stochastic heat equation with rough initial conditions’, Ann. Probab. 43, 3006–3051. <https://doi.org/10.1214/14-AOP954>
@article{chen.dalang:15:moments,
author = {Chen, Le and Dalang, Robert C.},
title = {Moments and growth indices for the nonlinear stochastic heat equation with rough initial conditions},
journal = {Ann. Probab.},
fjournal = {The Annals of Probability},
volume = {43},
year = {2015},
number = {6},
pages = {3006--3051},
issn = {0091-1798},
mrclass = {60H15 (35R60 60G60)},
mrnumber = {3433576},
mrreviewer = {Mathew Joseph},
doi = {10.1214/14-AOP954},
url = {https://doi.org/10.1214/14-AOP954}
}
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