Jacob Bedrossian

University of Maryland

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 \(L^1\) framework which estimates this (degenerate) Fisher information from below by a \(W^{s,1}\) 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.