Abstract.
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=∫δŴ sdL s is supported by ∂D. Then we represent the solution v of Δu=4u 2 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=4u 2 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.
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Mathematics Subject Classification (2000): 60J55, 60J80, 60H30, 60G57, 35C15, 35J65
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Abraham, R., Delmas, JF. Solutions of Δu=4u 2 with Neumann’s conditions using the Brownian snake. Probab. Theory Relat. Fields 128, 475–516 (2004). https://doi.org/10.1007/s00440-003-0302-2
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DOI: https://doi.org/10.1007/s00440-003-0302-2
Keywords or phrases
- Super Brownian motion
- Brownian snake
- Exit measure
- Neumann’s problem
- Reflected Brownian motion
- Semi-linear PDE