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Probability Theory and Related Fields

, Volume 171, Issue 3–4, pp 653–684 | Cite as

A probabilistic Harnack inequality and strict positivity of stochastic partial differential equations

  • Zhenan Wang
Article

Abstract

Under general conditions we show an a priori probabilistic Harnack inequality for the non-negative solution of a stochastic partial differential equation of the following form
$$\begin{aligned} \partial _t u=\mathrm {div}{\; (}{\mathbb {A}}\nabla u)+f(t,x,u;\omega )+g_i(t,x,u;\omega )\dot{w}_t^i. \end{aligned}$$
We also show that the solutions of the above equation are almost surely strictly positive if the initial condition is non-negative and not identically vanishing.

Mathematics Subject Classification

60H15 

Notes

Acknowledgements

The project was initially started by the author and Doctor Yu Wang (currently in Goldman Sachs) in 2014 upon the completion of [6]. Although the collaboration ended after the departure of Yu Wang, the discussion with him has helped clarify many confusions. His contribution to this project is greatly appreciated.

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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.University of WashingtonSeattleUSA

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