Solutions of Δu = 4u² with Neumann's conditions using the Brownian snake

We consider a Brownian snake (W_s, s ≥ 0) with underlying process a reflected Brownian motion in a bounded domain D. We construct a continuous additive functional (L_s, s ≥ 0) of the Brownian snake which counts the time spent by the end points Ŵ_s of the Brownian snake paths on ∂ D. The random measure Z = ∫ δ_ŴsdL_s is supported by ∂ D. Then we represent the solution v of u = 4u² in D with weak Neumann boundary condition φ ≥ 0 by using exponential moment of (Z, φ) under the excursion measure of the Brownian snake. We then derive an integral equation for v. For small φ it is then possible to describe negative solution of Δu = 4u² in D with weak Neumann boundary condition φ. In contrast to the exit measure of the Brownian snake out of D, the measure Z is more regular. In particular we show it is absolutely continuous with respect to the surface measure on ∂ D for dimension 2 and 3.