Regularity of laws and ergodicity of hypoelliptic SDEs driven by rough paths

Abstract

We consider differential equations driven by rough paths and study the regularity of the laws and their long time behavior. In particular, we focus on the case when the driving noise is a rough path valued fractional Brownian motion with Hurst parameter $H\in (\frac{1}{3},\frac{1}{2}]$. Our contribution in this work is twofold. First, when the driving vector fields satisfy Hörmander's celebrated "Lie bracket condition," we derive explicit quantitative bounds on the inverse of the Malliavin matrix. En route to this, we provide a novel "deterministic" version of Norris's lemma for differential equations driven by rough paths. This result, with the added assumption that the linearized equation has moments, will then yield that the transition laws have a smooth density with respect to Lebesgue measure. Our second main result states that under Hörmander's condition, the solutions to rough differential equations driven by fractional Brownian motion with $H\in (\frac{1}{3},\frac{1}{2}]$ enjoy a suitable version of the strong Feller property. Under a standard controllability condition, this implies that they admit a unique stationary solution that is physical in the sense that it does not "look into the future."

Published in at http://dx.doi.org/10.1214/12-AOP777 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (6)

• Formulae for the derivatives of heat semigroups
• Integrability and tail estimates for Gaussian rough differential equations
• Ergodic properties of a class of non-Markovian processes
• Smoothness of Density for the Area Process of Fractional Brownian Motion
• Integrability of (non-)linear rough differential equations and integrals
• Smooth density for some nilpotent rough differential equations

Cited by in corpus (29)

• A theory of regularity structures
• KPZ reloaded
• Smoothness of the density for solutions to Gaussian rough differential equations
• Integrability and tail estimates for Gaussian rough differential equations
• Rough paths, Signatures and the modelling of functions on streams
• Pathwise integration with respect to paths of finite quadratic variation
• From rough path estimates to multilevel Monte Carlo
• Singular stochastic PDEs
• Stochastic Calculus with respect to G-Brownian Motion Viewed through Rough Paths
• Controlled viscosity solutions of fully nonlinear rough PDEs
• Symplectic Runge-Kutta Methods for Hamiltonian Systems Driven by Gaussian Rough Paths
• Doob--Meyer for rough paths
• C-infinity regularization of ODEs perturbed by noise
• High order paracontrolled calculus
• The Hörmander condition for delayed stochastic differential equations
• Smoothing effect of rough differential equations driven by fractional Brownian motions
• Statistical Inference for Stochastic Differential Equations with Memory
• Asymptotic expansion of the density for hypoelliptic rough differential equation
• Integrability of (non-)linear rough differential equations and integrals
• Varadhan Estimates for rough differential equations driven by fractional Brownian motions
• Local times of stochastic differential equations driven by fractional Brownian motions
• Smooth density for some nilpotent rough differential equations
• A Stratonovich-Skorohod integral formula for Volterra Gaussian rough paths
• Smoothness of densities for path-dependent SDEs under Hörmander's condition
• Optimal convergence rate of modified Milstein scheme for SDEs with rough fractional diffusions
• Martingale solutions for the three-dimensional stochastic nonhomogeneous incompressible Navier-Stokes equations driven by Levy processes
• Fractal Dimensions of Rough Differential Equations Driven by Fractional Brownian Motions
• Random dynamical systems, rough paths and rough flows
• Hypoelliptic diffusions with singular drifts