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

Abstract

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