Continuous flows driving branching processes and their nonlinear evolution equations
We consider on M(ℝd) (the set of all finite measures on ℝd) the evolution equation associated with the nonlinear operator F↦ΔF′+∑k⩾1bkFkF \mapsto \Delta F' + \sum\nolimits_{k \geqslant 1} b_k F^k , where F′ is the variational derivative of F and we show that it has a solution represented by me...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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De Gruyter
2022-02-01
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| Series: | Advances in Nonlinear Analysis |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/anona-2021-0229 |
| _version_ | 1811280162715598848 |
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| author | Beznea Lucian Vrabie Cătălin Ioan |
| author_facet | Beznea Lucian Vrabie Cătălin Ioan |
| author_sort | Beznea Lucian |
| collection | DOAJ |
| description | We consider on M(ℝd) (the set of all finite measures on ℝd) the evolution equation associated with the nonlinear operator
F↦ΔF′+∑k⩾1bkFkF \mapsto \Delta F' + \sum\nolimits_{k \geqslant 1} b_k F^k , where F′ is the variational derivative of F and we show that it has a solution represented by means of the distribution of the d-dimensional Brownian motion and the non-local branching process on the finite configurations of M(ℝd), induced by the sequence (bk)k⩾1 of positive numbers such that
∑k⩾1bk⩽1\sum\nolimits_{k \geqslant 1} b_k \leqslant 1. It turns out that the representation also holds with the same branching process for the solution to the equation obtained replacing the Laplace operator by the generator of a Markov process on ℝd instead of the d-dimensional Brownian motion; more general, we can take the generator of a right Markov process on a Lusin topological space. We first investigate continuous flows driving branching processes. We show that if the branching mechanism of a superprocess is independent of the spatial variable, then the superprocess is obtained by introducing the branching in the time evolution of the right continuous flow on measures, canonically induced by a right continuous flow as spatial motion. A corresponding result holds for non-local branching processes on the set of all finite configurations of the state space of the spatial motion. |
| first_indexed | 2024-04-13T01:08:46Z |
| format | Article |
| id | doaj.art-43313c2ead58420d8b0b155b339ea575 |
| institution | Directory Open Access Journal |
| issn | 2191-9496 2191-950X |
| language | English |
| last_indexed | 2024-04-13T01:08:46Z |
| publishDate | 2022-02-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Advances in Nonlinear Analysis |
| spelling | doaj.art-43313c2ead58420d8b0b155b339ea5752022-12-22T03:09:15ZengDe GruyterAdvances in Nonlinear Analysis2191-94962191-950X2022-02-0111192193610.1515/anona-2021-0229Continuous flows driving branching processes and their nonlinear evolution equationsBeznea Lucian0Vrabie Cătălin Ioan1Simion Stoilow Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, RO-014700Bucharest, Romania, and Faculty of Mathematics and Computer Science, University of BucharestSimion Stoilow Institute of Mathematics of the Romanian Academy, Research unit No. 2, P.O. Box 1-764, RO-014700Bucharest, RomaniaWe consider on M(ℝd) (the set of all finite measures on ℝd) the evolution equation associated with the nonlinear operator F↦ΔF′+∑k⩾1bkFkF \mapsto \Delta F' + \sum\nolimits_{k \geqslant 1} b_k F^k , where F′ is the variational derivative of F and we show that it has a solution represented by means of the distribution of the d-dimensional Brownian motion and the non-local branching process on the finite configurations of M(ℝd), induced by the sequence (bk)k⩾1 of positive numbers such that ∑k⩾1bk⩽1\sum\nolimits_{k \geqslant 1} b_k \leqslant 1. It turns out that the representation also holds with the same branching process for the solution to the equation obtained replacing the Laplace operator by the generator of a Markov process on ℝd instead of the d-dimensional Brownian motion; more general, we can take the generator of a right Markov process on a Lusin topological space. We first investigate continuous flows driving branching processes. We show that if the branching mechanism of a superprocess is independent of the spatial variable, then the superprocess is obtained by introducing the branching in the time evolution of the right continuous flow on measures, canonically induced by a right continuous flow as spatial motion. A corresponding result holds for non-local branching processes on the set of all finite configurations of the state space of the spatial motion.https://doi.org/10.1515/anona-2021-0229nonlinear evolution equationsuperprocessnon-local branching processright continuous flowweak generatorlog-potential35j6060j3560j8060j6860j4547d07 |
| spellingShingle | Beznea Lucian Vrabie Cătălin Ioan Continuous flows driving branching processes and their nonlinear evolution equations Advances in Nonlinear Analysis nonlinear evolution equation superprocess non-local branching process right continuous flow weak generator log-potential 35j60 60j35 60j80 60j68 60j45 47d07 |
| title | Continuous flows driving branching processes and their nonlinear evolution equations |
| title_full | Continuous flows driving branching processes and their nonlinear evolution equations |
| title_fullStr | Continuous flows driving branching processes and their nonlinear evolution equations |
| title_full_unstemmed | Continuous flows driving branching processes and their nonlinear evolution equations |
| title_short | Continuous flows driving branching processes and their nonlinear evolution equations |
| title_sort | continuous flows driving branching processes and their nonlinear evolution equations |
| topic | nonlinear evolution equation superprocess non-local branching process right continuous flow weak generator log-potential 35j60 60j35 60j80 60j68 60j45 47d07 |
| url | https://doi.org/10.1515/anona-2021-0229 |
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