Temporal Hölder continuity of the parabolic Anderson model driven by a class of time-independent Gaussian fields with rough initial conditions
In this paper, we considered the parabolic Anderson model with a class of time-independent generalized Gaussian fields on $ \mathbb{R}^d $, which included fractional white noise, Bessel field, massive free field, and other nonstationary Gaussian fields. Under the rough initial conditions, we constru...
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AIMS Press
2024-12-01
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.20241659 |
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| author | Hui Sun Yangyang Lyu |
| author_facet | Hui Sun Yangyang Lyu |
| author_sort | Hui Sun |
| collection | DOAJ |
| description | In this paper, we considered the parabolic Anderson model with a class of time-independent generalized Gaussian fields on $ \mathbb{R}^d $, which included fractional white noise, Bessel field, massive free field, and other nonstationary Gaussian fields. Under the rough initial conditions, we constructed the Feynman-Kac formula as a solution in the Stratonovich integral by Brownian bridge, and then proved the Hölder continuity of the solution with respect to the time variable. As a comparison, we also studied the Hölder continuity under the regular initial conditions that $ u_0\equiv C $ and $ u_0\in C^\kappa(\mathbb{R}^d) $ with $ \kappa\in(0, 1] $. |
| format | Article |
| id | doaj-art-a27f482da903453c95278c1cf88bb386 |
| institution | Kabale University |
| issn | 2473-6988 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj-art-a27f482da903453c95278c1cf88bb3862025-01-23T07:53:25ZengAIMS PressAIMS Mathematics2473-69882024-12-01912348383486210.3934/math.20241659Temporal Hölder continuity of the parabolic Anderson model driven by a class of time-independent Gaussian fields with rough initial conditionsHui Sun0Yangyang Lyu1School of Mathematics and Statistics, Minnan Normal University, Zhangzhou 363000, ChinaSchool of Mathematics and Statistics, Minnan Normal University, Zhangzhou 363000, ChinaIn this paper, we considered the parabolic Anderson model with a class of time-independent generalized Gaussian fields on $ \mathbb{R}^d $, which included fractional white noise, Bessel field, massive free field, and other nonstationary Gaussian fields. Under the rough initial conditions, we constructed the Feynman-Kac formula as a solution in the Stratonovich integral by Brownian bridge, and then proved the Hölder continuity of the solution with respect to the time variable. As a comparison, we also studied the Hölder continuity under the regular initial conditions that $ u_0\equiv C $ and $ u_0\in C^\kappa(\mathbb{R}^d) $ with $ \kappa\in(0, 1] $.https://www.aimspress.com/article/doi/10.3934/math.20241659parabolic anderson modelfeynman-kac formulageneralized gaussian fieldbrownian bridgehölder continuitymeasure-valued initial data |
| spellingShingle | Hui Sun Yangyang Lyu Temporal Hölder continuity of the parabolic Anderson model driven by a class of time-independent Gaussian fields with rough initial conditions AIMS Mathematics parabolic anderson model feynman-kac formula generalized gaussian field brownian bridge hölder continuity measure-valued initial data |
| title | Temporal Hölder continuity of the parabolic Anderson model driven by a class of time-independent Gaussian fields with rough initial conditions |
| title_full | Temporal Hölder continuity of the parabolic Anderson model driven by a class of time-independent Gaussian fields with rough initial conditions |
| title_fullStr | Temporal Hölder continuity of the parabolic Anderson model driven by a class of time-independent Gaussian fields with rough initial conditions |
| title_full_unstemmed | Temporal Hölder continuity of the parabolic Anderson model driven by a class of time-independent Gaussian fields with rough initial conditions |
| title_short | Temporal Hölder continuity of the parabolic Anderson model driven by a class of time-independent Gaussian fields with rough initial conditions |
| title_sort | temporal holder continuity of the parabolic anderson model driven by a class of time independent gaussian fields with rough initial conditions |
| topic | parabolic anderson model feynman-kac formula generalized gaussian field brownian bridge hölder continuity measure-valued initial data |
| url | https://www.aimspress.com/article/doi/10.3934/math.20241659 |
| work_keys_str_mv | AT huisun temporalholdercontinuityoftheparabolicandersonmodeldrivenbyaclassoftimeindependentgaussianfieldswithroughinitialconditions AT yangyanglyu temporalholdercontinuityoftheparabolicandersonmodeldrivenbyaclassoftimeindependentgaussianfieldswithroughinitialconditions |