A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs
Abstract
We present a theory of hypoellipticity and unique ergodicity for semilinear parabolic
stochastic PDEs with "polynomial" nonlinearities and additive noise, considered as
abstract evolution equations in some Hilbert space. It is shown that if Hörmander's
bracket condition holds at every point of this Hilbert space, then a lower bound on
the Malliavin covariance operatorμt can be obtained. Informally, this bound can be
read as "Fix any finite-dimensional projection on a subspace of sufficiently regular
functions. Then the eigenfunctions of μt with small eigenvalues have only a very small
component in the image of Π." We also show how to use a priori bounds on the solutions
to the equation to obtain good control on the dependency of the bounds on the Malliavin
matrix on the initial condition. These bounds are sufficient in many cases to obtain
the asymptotic strong Feller property introduced in [HM06]. One of the main novel
technical tools is an almost sure bound from below on the size of "Wiener polynomials,"
where the coefficients are possibly non-adapted stochastic processes satisfying a
Lips chitz condition. By exploiting the polynomial structure of the equations, this
result can be used to replace Norris' lemma, which is unavailable in the present context.
We conclude by showing that the two-dimensional stochastic Navier-Stokes equations
and a large class of reaction-diffusion equations fit the framework of our theory.
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https://hdl.handle.net/10161/9521Collections
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Jonathan Christopher Mattingly
Kimberly J. Jenkins Distinguished University Professor of New Technologies
Jonathan Christopher Mattingly grew up in Charlotte, NC where he attended Irwin Ave
elementary and Charlotte Country Day. He graduated from the NC School of Science
and Mathematics and received a BS is Applied Mathematics with a concentration in physics
from Yale University. After two years abroad with a year spent at ENS Lyon studying
nonlinear and statistical physics on a Rotary Fellowship, he returned to the US to
attend Princeton University where he obtained a PhD in Applied and

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