Quantitative hypoelliptic regularity and the estimation of
Lyapunov exponents and other long-time dynamical properties of stochastic
differential equations
In the talk, we will discuss the connection between quantitative hypoelliptic
PDE methods and the long-time dynamics of stochastic differential equations
(SDEs). In a recent joint work with Alex Blumenthal and Sam Punshon-Smith, we
put forward a new method for obtaining quantitative lower bounds on the top
Lyapunov exponent of stochastic differential equations (SDEs). Our method
combines (i) an (apparently new) identity connecting the top Lyapunov exponent
to a degenerate Fisher information-like functional of the stationary density of
the Markov process tracking tangent directions with (ii) a novel, quantitative
version of Hörmander’s hypoelliptic regularity theory in an framework which
estimates this (degenerate) Fisher information from below by a Sobolev
norm using the associated Kolmogorov equation for the stationary density. As an
initial application, we prove the positivity of the top Lyapunov exponent for a
class of weakly-dissipative, weakly forced SDE and we prove that this class
includes the classical Lorenz 96 model in any dimension, provided the additive
stochastic driving is applied to any consecutive pair of modes. This is the
first mathematically rigorous proof of chaos (in the sense of positive Lyapunov
exponents) for Lorenz 96 (stochastically driven or otherwise), despite the
overwhelming numerical evidence. If time permits, I will also discuss joint work
with Kyle Liss, in which we obtain sharp, quantitative estimates on the spectral
gap of the Markov semigroup and uniform upper and lower bounds on the stationary
density in the weakly-dissipative, weakly forced class of SDEs studied above,
which includes finite dimensional models in fluid mechanics, including not only
the Lorenz 96 system but also Galerkin–Navier–Stokes and finite dimensional
truncations of the shell models GOY and SABRA. In both of these works, obtaining
various kinds of quantitative hypoelliptic regularity estimates that are uniform
in certain parameters play a pivotal role.