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

Lucian Beznea, Liviu I. Ignat , Julio D. Rossi

• We study solutions to the evolution equation ut=Δu−u+∑k⩾1qkuk, t>0, in Rd. Here the coefficients qk⩾0 satisfy ∑k⩾1qk=1<∑k⩾1kqk<∞. First, we deal with existence, uniqueness, and the asymptotic behavior of the solutions as t→+∞. We then deduce results on the long time behavior of the associated branching process, with state space the set of all finite configurations of Rd. 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 Rd. 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.