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

### Lucian Beznea

Romanian Academy, Bucharest, and University of Bucharest, Romania### Liviu I. Ignat

Romanian Academy, Bucharest, and University of Bucharest, Romania### Julio D. Rossi

Universidad de Buenos Aires, Argentina

## Abstract

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.

## Cite this article

Lucian Beznea, Liviu I. Ignat, Julio D. Rossi, From Gaussian estimates for nonlinear evolution equations to the long time behavior of branching processes. Rev. Mat. Iberoam. 35 (2019), no. 3, pp. 823–846

DOI 10.4171/RMI/1071