| 概要 | The general theme of these two lectures is the
regularization introduced by noise in ordinary and partial
differential equations. The final examples we have in mind arise from
fluid dynamics.
The first lecture will be devoted to a review of definitions and
results of uniqueness for SDEs with additive noise and non smooth
drift with emphasis on the fact that the deterministic equations with
the same drift may have non unique solutions. We sketch the proof of
the construction of a stochastic flow of diffeomorphisms in the case
of Hölder continuous drift. We also describe the zero-noise limit
problem when the limit deterministic equation is not well posed and we
recall a known result in dimension 1.
The second lecture is devoted to examples of SPDEs where similar
regularization occurs. The role of bilinear multiplicative noise is
discussed. At the PDE level the regularization due to noise may appear
both for the problem of uniqueness and for the problem of
singularities. We show in particular examples of linear transport
equations and linear vector advection equations where noise prevents
singularities which otherwise would emerge for the corresponding
deterministic PDE. We give also an example where we may control the
zero-noise limit.
Part of the literature related to these lecture is:
F. Flandoli, Random Perturbation of PDEs and Fluid Dynamic Models, Saint
Flour summer school lectures 2010, Lecture Notes in Mathematics n. 2015,
Springer, Berlin 2011.
F. Flandoli, M. Gubinelli, E. Priola, Well-posedness of the transport
equation by stochastic perturbation, Invent. Math. 180 (2010), no. 1,
1--53. |