Well-posedness of the transport equation by stochastic perturbation

2008/09/30 by Franco Flandoli, Massimiliano Gubinelli, Enrico Priola · 1 citation
Mathematics · #math.AP #math.PR #msc:60H15 #msc:35L #msc:35F10

paper · pdf · doi:10.1007/s00222-009-0224-4

Addition of new parts

arxiv created 2009/05/08 · arxiv updated 2015/05/12

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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