Revista Matemática Iberoamericana


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Volume 35, Issue 3, 2019, pp. 823–846
DOI: 10.4171/rmi/1071

Published online: 2019-04-08

From Gaussian estimates for nonlinear evolution equations to the long time behavior of branching processes

Lucian Beznea[1], Liviu I. Ignat[2] and Julio D. Rossi[3]

(1) Romanian Academy, Bucharest, and University of Bucharest, Romania
(2) Romanian Academy, Bucharest, and University of Bucharest, Romania
(3) Universidad de Buenos Aires, Argentina

We study solutions to the evolution equation $u_t=\Delta u-u +\sum _{k\geqslant 1} q_k u^k$, $t > 0$, in ${\mathbb R}^d$. Here the coefficients $q_k\geqslant 0$ satisfy $\sum_{k\geqslant 1}q_k=1 < \sum_{k\geqslant 1}k q_k < \infty$. 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 ${\mathbb R}^d$. 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 ${\mathbb R}^d$. 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.

Keywords: Branching process, occupation time, long time behavior, space of finite configurations, branching kernel, nonlinear PDE

Beznea Lucian, Ignat Liviu, Rossi Julio: From Gaussian estimates for nonlinear evolution equations to the long time behavior of branching processes. Rev. Mat. Iberoam. 35 (2019), 823-846. doi: 10.4171/rmi/1071