From Gaussian estimates for nonlinear evolution equations to the long time behavior of branching processes
| Autor: | Beznea, L., Ignat, L. I., Rossi, J. D. |
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| Rok vydání: | 2017 |
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| Druh dokumentu: | Working Paper |
| Popis: | We study solutions to the evolution equation $u_t=\Delta u-u +\sum_{k\geqslant 1}q_ku^k$, $t>0$, in $\mathbf{R}^d$. Here the coefficients $q_k\geqslant 0$ verify $ \sum_{k\geqslant 1}q_k=1< \sum_{k\geqslant 1}kq_k<\infty$. First, we deal with existence, uniqueness, and the asymptotic behavior of the solutions as $t\to +\infty$. We then deduce results on the long time behavior of the associated branching process, with state space the set of all finite configurations of $\mathbf{R}^d$, under the assumption that $\sum_{k\geq 1} k^2q_k<\infty$. It turns out that the distribution of the branching process behaves when the time tends to infinity like that of the Brownian motion on the set of all finite configurations of $\mathbf{R}^d$. However, due to the lack of conservation of the total mass of the initial non linear equation, a deformation with a multiplicative coefficient occurs. Finally, we establish asymptotic properties of the occupation time of this branching process. Comment: 22 pages |
| Databáze: | arXiv |
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