Non-linear neumann's condition for the heat equation: A probabilistic representation using catalytic super-Brownian motion

Let D be a bounded domain in ℝ^d with smooth boundary ∂D. We give a probabilistic representation formula for the non-negative solution of the mixed Dirichlet non-linear Neumann boundary value problem (DNP) {Δu = 0 in D, u = φ on F₂ ∂_nu + 2u² = 0 on F₁, where F₁, F₂ is a non-trivial partition of ∂D, φ is a non-negative, bounded and continuous function defined on F₂, and ∂F_n denotes the outward normal derivative on the boundary of D. To solve the DNP, we consider a catalytic super-Brownian motion with underlying motion a Brownian motion reflected on ∂D, killed when it reaches F₂ and catalysed by the set F₁, i.e. the branching rate is given by the local time of the paths on F₁. Then we prove that the log-Laplace transform of φ integrated with respect to the exit measure of the catalytic process on F₂, is a non-negative weak solution of the DNP. In a second part we show that we still have a probabilistic representation formula if the Dirichlet condition on F₂ is replaced by a Neumann condition.