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

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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.

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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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  1. University of Washington, Seattle, WA, USA

    Zhenan Wang

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  1. Zhenan Wang
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Correspondence to Zhenan Wang.

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Wang, Z. A probabilistic Harnack inequality and strict positivity of stochastic partial differential equations. Probab. Theory Relat. Fields 171, 653–684 (2018). https://doi.org/10.1007/s00440-017-0789-6

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  • DOI: https://doi.org/10.1007/s00440-017-0789-6

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