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A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs

Publication ,  Journal Article
Hairer, M; Mattingly, JC
Published in: Electronic Journal of Probability
2011

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.

Duke Scholars

Published In

Electronic Journal of Probability

ISSN

1083-6489

Publication Date

2011

Volume

16

Start / End Page

658 / 738

Related Subject Headings

  • Statistics & Probability
  • 4905 Statistics
  • 0105 Mathematical Physics
  • 0104 Statistics
 

Citation

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Hairer, M., & Mattingly, J. C. (2011). A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs. Electronic Journal of Probability, 16, 658–738.
Hairer, M., and J. C. Mattingly. “A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs.” Electronic Journal of Probability 16 (2011): 658–738.
Hairer M, Mattingly JC. A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs. Electronic Journal of Probability. 2011;16:658–738.
Hairer, M., and J. C. Mattingly. “A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs.” Electronic Journal of Probability, vol. 16, 2011, pp. 658–738.
Hairer M, Mattingly JC. A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs. Electronic Journal of Probability. 2011;16:658–738.

Published In

Electronic Journal of Probability

ISSN

1083-6489

Publication Date

2011

Volume

16

Start / End Page

658 / 738

Related Subject Headings

  • Statistics & Probability
  • 4905 Statistics
  • 0105 Mathematical Physics
  • 0104 Statistics