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

## TL;DR

This paper analyzes the long-term behavior of solutions to a nonlinear evolution equation and its associated branching process, revealing convergence to Brownian motion behavior and detailing occupation time asymptotics.

## Contribution

It provides new results on existence, uniqueness, and asymptotic behavior of solutions, linking nonlinear PDEs to branching processes and their long-time probabilistic properties.

## Key findings

- Solutions exhibit asymptotic behavior similar to Brownian motion.
- Branching process distribution converges to that of Brownian motion.
- Occupation time of the process has well-characterized asymptotics.

## Abstract

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.