Well-posedness of the transport equation by stochastic perturbation

Flandoli, Franco; Gubinelli, Massimiliano; Priola, Enrico

Flandoli, Franco; Gubinelli, Massimiliano; Priola, Enrico (2010), Well-posedness of the transport equation by stochastic perturbation, Inventiones Mathematicae, 180, 1, p. 1-53. http://dx.doi.org/10.1007/s00222-009-0224-4

Type

Article accepté pour publication ou publié

External document link

http://hal.archives-ouvertes.fr/hal-00359723/en/

Date

2010

Journal name

Inventiones Mathematicae

Volume

180

Number

Publisher

Springer

Pages

1-53

Abstract (EN)

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 Tanaka type.

Subjects / Keywords

Probability