Franco Flandoli, Massimiliano Gubinelli and Enrico Priola
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.
Franco Flandoli, Massimiliano Gubinelli and Enrico Priola. Well-posedness of the transport equation by stochastic perturbation. Papers of Massimiliano Gubinelli, 2008, version 2. https://mgubi.org/papers/paper/transport-noise-wellposedness/
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