Superprocesses on ultradistributions

Abstract

Stochastic solutions provide new rigorous results for nonlinear PDE’s and, through its local nongrid nature, are a natural tool for parallel computation. There are two different approaches for the construction of stochastic solutions: McKean’s and superprocesses. In favour of superprocesses is the fact that they handle arbitrary boundary conditions. However, when restricted to measures, superprocesses can only be used to generate solutions for a limited class of nonlinear PDE’s. A new class of superprocesses, namely superprocesses on ultradistributions, is proposed to extend the stochastic solution approach to a wider class of PDE’s.

Keywords:

• Superprocesses
• ultradistributions
• nonlinear PDE’s

Notes

No potential conflict of interest was reported by the author.

1 For a detailed account of the nature of the limitations of superprocesses on measures as related to the positivity of the coefficients in the offspring generating function, see [26].

2 Distributions which locally are , f bounded and continuous.

3 These two examples had been described at an heuristic level in the conference paper [24]. However, their rigorous proof depended on the establishment of Proposition 1.

Additional information

Funding

This work was supported by Fundação para a Ciência e a Tecnologia.