Picture of Egor Pifagorov
Franco Flandoli (Scuola Normale Superiore di Pisa) «Regularization and selection by noise»
by Egor Pifagorov - Tuesday, 25 September 2018, 01:10 PM
пятница 5 октября 15:25 – 17:00, понедельник 8 октября 17:00 – 18.30
(ауд. 413, 14-я линия В.О., 29)

Franco Flandoli (Scuola Normale Superiore di Pisa)

«Regularization and selection by noise»

The most classical properties we ask on a differential equation are
existence and uniqueness. Existence has its own philosophy, going for
instance in the direction of enlarging the meaning of the equation and
thus the class of potential solutions; even if some of the topics of
this minicourse intersect the question of existence, it will not be the
main task. Uniqueness is our main concern. For certain equation and
classes of solutions it is not fulfilled, or it is an open problem;
examples will be recalled, mainly for ordinary differential equations
for sake of simplicity, but also for certain partial differential equations.

Probability has several links with the question of uniqueness. In the
minicourse we shall discuss three of them: restoring uniqueness by
noise; selecting special solutions of a deterministic equation by means
of a zero-noise limit; detecting the existence of intrinsically
stochastic solutions of certain deterministic equations. The reason why
noise in able to restore uniqueness for equations with very singular
behavior is understood in a number of cases; one could even say that
there is nowadays a theory. However, it is still a very active field of
research, especially for nonlinear partial differential equations. The
zero-noise limit is, on the other side, almost not understood, except
for one-dimensional differential equations, which will be our main
example. It is a striking open problem even for certain simple 2D
examples that will be shown. Its potentiality is therefore still not
completely explored.

The topic of intrinsic stochasticity is related: we look for examples
where uniqueness fails for a deterministic equation and also the
zero-noise limit does not identify a single special solution, but some
truly stochastic process emerges as interesting solution of the
deterministic equation.