Branching diffusion representation of semi-linear elliptic PDEs and estimation using Monte Carlo method - Archive ouverte HAL Access content directly
Journal Articles Stochastic Processes and their Applications Year : 2020

Branching diffusion representation of semi-linear elliptic PDEs and estimation using Monte Carlo method

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Abstract

We study semi-linear elliptic PDEs with polynomial non-linearity and provide a probabilistic representation of their solution using branching diffusion processes. When the non-linearity involves the unknown function but not its derivatives, we extend previous results in the literature by showing that our probabilistic representation provides a solution to the PDE without assuming its existence. In the general case, we derive a new representation of the solution by using marked branching diffusion processes and automatic differentiation formulas to account for the non-linear gradient term. In both cases, we develop new theoretical tools to provide explicit sufficient conditions under which our probabilistic representations hold. As an application, we consider several examples including multi-dimensional semi-linear elliptic PDEs and estimate their solution by using the Monte Carlo method.
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Dates and versions

hal-01709033 , version 1 (14-02-2018)

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Ankush Agarwal, Julien Claisse. Branching diffusion representation of semi-linear elliptic PDEs and estimation using Monte Carlo method. Stochastic Processes and their Applications, 2020, ⟨10.1016/j.spa.2020.02.009⟩. ⟨hal-01709033⟩
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