Measure-valued branching processes associated with Neumann nonlinear\n semiflows

2018/03/15 by Viorel Barbu, Barbu, Viorel, Lucian Beznea +1
Mathematics · Physics and Astronomy · #31B20 #35J25 #47D07 #60J35 #60J45 #60J50 #60J80 #Analysis of PDEs (math.AP) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.1803.05639

openalex publication_date 2018/03/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We construct a measure-valued branching Markov process associated with a\nnonlinear boundary value problem, where the boundary condition has a nonlinear\npseudo monotone branching mechanism term -\β, which includes as a limit\ncase \β(u) = - u^m, with 0 < m < 1. The process is then used in the\nprobabilistic representation of the solution of the parabolic problem\nassociated with a nonlinear Neumann boundary value problem. In this way the\nclassical association of the superprocesses to the Dirichlet boundary value\nproblems also holds for the nonlinear Neumann boundary value problems. It turns\nout that the obtained branching process behaves on the measures carried by the\ngiven open set like the linear continuous semiflow, induced by the reflected\nBrownian motion, while the branching occurs on the measures having non-zero\ntraces on the boundary of the open set, with the behavior of the\n(-\β)-superprocess, having as spatial motion the process on the boundary\nassociated to the reflected Brownian motion\n

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