Measure-valued branching processes associated with Neumann nonlinear semiflows

TL;DR

This paper constructs a measure-valued branching process linked to nonlinear Neumann boundary problems, extending classical superprocess associations to nonlinear boundary conditions and providing probabilistic representations of solutions.

Contribution

It introduces a new measure-valued branching process for nonlinear Neumann problems, generalizing superprocess connections to include nonlinear boundary conditions.

Findings

The process models solutions to nonlinear Neumann boundary value problems.

Branching occurs on boundary measures with behavior akin to a $(-\beta)$-superprocess.

Classical superprocess associations are extended to nonlinear boundary conditions.

Abstract

We construct a measure-valued branching Markov process associated with a nonlinear boundary value problem, where the boundary condition has a nonlinear pseudo monotone branching mechanism term $-\beta$, which includes as a limit case $\beta(u) = - u^{m}$, with $0 < m < 1$. The process is then used in the probabilistic representation of the solution of the parabolic problem associated with a nonlinear Neumann boundary value problem. In this way the classical association of the superprocesses to the Dirichlet boundary value problems also holds for the nonlinear Neumann boundary value problems. It turns out that the obtained branching process behaves on the measures carried by the given open set like the linear continuous semiflow, induced by the reflected Brownian motion, while the branching occurs on the measures having non-zero traces on the boundary of the open set, with the behavior of the $(-\beta)$-superprocess, having as spatial motion the process on the boundary associated to the reflected Brownian motion