APDE Seminar@UPV/EHU: A pathological set regarding the propagation of almost sure properties of Gaussian measures

Abstract

Given the periodic and cubic wave equation in 3d, we deal with two questions:

• From a deterministic point of view, given p > 2 and σ ≥ 0 large enough, what can we say about the propagation of spatial derivative regularity, in terms of Wσ,p(T^3), for the linear and the nonlinear flow of the aforementioned equation?

• From a probabilistic point of view, given the loss of derivative regularity that we will encounter for the previous question, can we randomize the initial data in any way such that we do not see such regularity loss in Wσ,p(T^3), almost surely with respect certain meaningful measure?

I will talk about a complementary result to the quasi-invariance of Gaussian measures supported on Sobolev spaces under the dynamics of this equation, proved by T. S. Gunaratnam, T. Oh, N. Tzvetkov and H. Weber (2022).

Namely, we will discuss about the existence of dense sets of general Sobolev spaces Wσ,p(T^3), for large p and σ, which do not preserve the regularity σ throughout the aforementioned dynamics, as long as t ̸= 0. This is in sharp contrast with the propagation of almost sure properties of the Gaussian measure along the flow. This is a joint work with Nikolay Tzvetkov.