10. chen.kim:17:on#
On comparison principle and strict positivity of solutions to the nonlinear stochastic fractional heat equations
Le Chen and Kunwoo Kim
Abstract: In this paper, we prove a sample-path comparison principle for the nonlinear stochastic fractional heat equation on \(\mathbb{R}\) with measure-valued initial data. We give quantitative estimates about how close to zero the solution can be. These results extend Mueller’s comparison principle on the stochastic heat equation to allow more general initial data such as the (Dirac) delta measure and measures with heavier tails than linear exponential growth at \(\pm\infty\). These results generalize a recent work by Moreno Flores [MF14], who proves the strict positivity of the solution to the stochastic heat equation with the delta initial data. As one application, we establish the full intermittency for the equation. As an intermediate step, we prove the Hölder regularity of the solution starting from measure-valued initial data, which generalizes, in some sense, a recent work by Chen and Dalang [CD14a].
MSC 2010 subject classifications: Primary 60H15. Secondary 60G60, 35R60.
Keywords: nonlinear stochastic fractional heat equation, parabolic Anderson model, comparison principle, measure-valued initial data, stable processes.
[CK17] Chen, Le & Kim, Kunwoo (2017) ‘On comparison principle and strict positivity of solutions to the nonlinear stochastic fractional heat equations’, Ann. Inst. Henri Poincar’e Probab. Stat. 53, 358–388. <https://doi.org/10.1214/15-AIHP719>
@article{chen.kim:17:on,
author = {Chen, Le and Kim, Kunwoo},
title = {On comparison principle and strict positivity of solutions to the nonlinear stochastic fractional heat equations},
journal = {Ann. Inst. Henri Poincar\'{e} Probab. Stat.},
fjournal = {Annales de l'Institut Henri Poincar\'{e} Probabilit\'{e}s et Statistiques},
volume = {53},
year = {2017},
number = {1},
pages = {358--388},
issn = {0246-0203},
mrclass = {60H15 (35B51 35R11 35R60 60G60)},
mrnumber = {3606745},
mrreviewer = {Mohammud Foondun},
doi = {10.1214/15-AIHP719},
url = {https://doi.org/10.1214/15-AIHP719}
}
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