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Nonlinear Dirichlet problem of non-local branching processes

Lucian Beznea, Oana Lupascu-Stamate, Alexandra Teodor

Abstract

We present a method of solving a nonlinear Dirichlet problem with discontinuous boundary data and we give a probabilistic representation of the solution using the nonlocal branching process associated with the nonlinear term of the operator. Instead of the pointwise convergence of the solution to the given boundary data we use the controlled convergence which allows to have discontinuities at the boundary.

Nonlinear Dirichlet problem of non-local branching processes

Abstract

We present a method of solving a nonlinear Dirichlet problem with discontinuous boundary data and we give a probabilistic representation of the solution using the nonlocal branching process associated with the nonlinear term of the operator. Instead of the pointwise convergence of the solution to the given boundary data we use the controlled convergence which allows to have discontinuities at the boundary.

Paper Structure

This paper contains 6 sections, 5 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Preliminary results, controlled convergence, and branching processes
  3. Dirichlet problem of non-local branching processes
  4. The case of the Laplace operator
  5. The case of the gradient type operator
  6. Appendix

Key Result

Lemma 2.1

Let $(S_t)_{t\geqslant 0}$ be the transition function of a right Markov process $Y$ on the Lusin topological space $F$, $c\in b\mathcal{B}_+(F)$, and consider $( S^c_{t}) _{t\geqslant 0}$, the transition function of the process obtained from $Y$ by killing with the multiplicative functional induced

Theorems & Definitions (14)

  • Lemma 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Remark 2.4
  • Theorem 3.1
  • Remark 3.2
  • proof : Proof of Theorem \ref{['thm1.1']}
  • Example 3.3
  • Theorem 3.4
  • proof
  • ...and 4 more