On a continuous analog of the parabolic Anderson model
UDC 519.21; 517.9 We consider a stochastic equation in $\mathbb{R}^{d},$ whose nonlocal part is a convolution operator with nonnegative symbol and the local part is an operator of multiplication by an ergodic field in $\mathbb{R}^{d}.$ We present the upper and lower bounds for its solutions correspo...
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| Datum: | 2026 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Institute of Mathematics, NAS of Ukraine
2026
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/8539 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860513152187236352 |
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| author | Kondratiev, Yu. Pastur, L. Kondratiev, Yu. Pastur, L. |
| author_facet | Kondratiev, Yu. Pastur, L. Kondratiev, Yu. Pastur, L. |
| author_sort | Kondratiev, Yu. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2026-03-21T13:31:39Z |
| description | UDC 519.21; 517.9
We consider a stochastic equation in $\mathbb{R}^{d},$ whose nonlocal part is a convolution operator with nonnegative symbol and the local part is an operator of multiplication by an ergodic field in $\mathbb{R}^{d}.$ We present the upper and lower bounds for its solutions corresponding to the constant initial data and present an example of the field for which these bounds coalesce as $t\rightarrow \infty $ resulting in an asymptotic formula for the logarithm of the solution. We also briefly discuss the spectral aspect of our results. |
| doi_str_mv | 10.3842/umzh.v77i4.8539 |
| first_indexed | 2026-03-24T03:40:08Z |
| format | Article |
| fulltext | |
| id | umjimathkievua-article-8539 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:40:08Z |
| publishDate | 2026 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
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| spelling | umjimathkievua-article-85392026-03-21T13:31:39Z On a continuous analog of the parabolic Anderson model On a continuous analog of the parabolic Anderson model Kondratiev, Yu. Pastur, L. Kondratiev, Yu. Pastur, L. - UDC 519.21; 517.9 We consider a stochastic equation in $\mathbb{R}^{d},$ whose nonlocal part is a convolution operator with nonnegative symbol and the local part is an operator of multiplication by an ergodic field in $\mathbb{R}^{d}.$ We present the upper and lower bounds for its solutions corresponding to the constant initial data and present an example of the field for which these bounds coalesce as $t\rightarrow \infty $ resulting in an asymptotic formula for the logarithm of the solution. We also briefly discuss the spectral aspect of our results. УДК 519.21; 517.9 Про неперервний аналог параболічної моделі Андерсона Розглянуто стохастичне рівняння в $\mathbb{R}^d,$ нелокальна частина якого є оператором згортки з невід’ємним символом, а локальна частина — оператором множення на ергодичне поле в $\mathbb{R}^d.$ Отримано верхню та нижню оцінки розв'язків, що відповідають сталим початковим даним, а також наведено приклад поля, для якого ці оцінки збігаються при $t \to \infty,$ що дає змогу записати асимптотичну формулу для логарифма розв'язку. Крім того, стисло обговорено спектральний аспект отриманих результатів. Institute of Mathematics, NAS of Ukraine 2026-03-21 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/8539 10.3842/umzh.v77i4.8539 Ukrains’kyi Matematychnyi Zhurnal; Vol. 77 No. 4 (2025); 284–285 Український математичний журнал; Том 77 № 4 (2025); 284–285 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/8539/10444 Copyright (c) 2025 Yu. Kondratiev, L. Pastur |
| spellingShingle | Kondratiev, Yu. Pastur, L. Kondratiev, Yu. Pastur, L. On a continuous analog of the parabolic Anderson model |
| title | On a continuous analog of the parabolic Anderson model |
| title_alt | On a continuous analog of the parabolic Anderson model |
| title_full | On a continuous analog of the parabolic Anderson model |
| title_fullStr | On a continuous analog of the parabolic Anderson model |
| title_full_unstemmed | On a continuous analog of the parabolic Anderson model |
| title_short | On a continuous analog of the parabolic Anderson model |
| title_sort | on a continuous analog of the parabolic anderson model |
| topic_facet | - |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/8539 |
| work_keys_str_mv | AT kondratievyu onacontinuousanalogoftheparabolicandersonmodel AT pasturl onacontinuousanalogoftheparabolicandersonmodel AT kondratievyu onacontinuousanalogoftheparabolicandersonmodel AT pasturl onacontinuousanalogoftheparabolicandersonmodel |