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arXiv · 0809.1310

Well-posedness of the transport equation by stochastic perturbation

Franco Flandoli·Massimiliano Gubinelli·Enrico Priola
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Abstract

We consider the linear transport equation with a globally Holder continuous and bounded vector field. While this deterministic PDE may not be well-posed, we prove that a multiplicative stochastic perturbation of Brownian type is enough to render the equation well-posed. This seems to be the first explicit example of partial differential equation that become well-posed under the influece of noise. The key tool is a differentiable stochastic flow constructed and analysed by means of a special transformation of the drift of Ito-Tanaka type.

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Franco Flandoli, Massimiliano Gubinelli, Enrico Priola. 2009-05-08. Well-posedness of the transport equation by stochastic perturbation. https://doi.org/10.1007/s00222-009-0224-4

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