Malliavin calculus for infinite-dimensional systems with additive noise

@article{Bakhtin2006MalliavinCF,
  title={Malliavin calculus for infinite-dimensional systems with additive noise},
  author={Yuri Bakhtin and Jonathan C. Mattingly},
  journal={Journal of Functional Analysis},
  year={2006},
  volume={249},
  pages={307-353}
}
Hypoellipticity in Infinite Dimensions
We consider semilinear parabolic stochastic PDEs driven by additive noise. The question addressed in this note is that of the regularity of transition probabilities. If the equation satisfies a
Hörmander’s theorem for semilinear SPDEs
We consider a broad class of semilinear SPDEs with multiplicative noise driven by a finite-dimensional Wiener process. We show that, provided that an infinite-dimensional analogue of Hormander's
An Introduction to Stochastic PDEs
These notes are based on a series of lectures given first at the University of Warwick in spring 2008 and then at the Courant Institute in spring 2009. It is an attempt to give a reasonably
Stochastic integrals and Brownian motion on abstract nilpotent Lie groups
  • T. Melcher
  • Mathematics
    Journal of the Mathematical Society of Japan
  • 2021
We construct a class of iterated stochastic integrals with respect to Brownian motion on an abstract Wiener space which allows for the definition of Brownian motions on a general class of
A Theory of Hypoellipticity and Unique Ergodicity for Semilinear Stochastic PDEs
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
Absolutely Continuous Laws of Jump-Diffusions in Finite and Infinite Dimensions with Applications to Mathematical Finance
TLDR
It is shown that for the construction of numerical procedures for the calculation of the Greeks in fairly general jump-diffusion cases one can proceed as in a pure diffusion case, and how the given results apply to infinite dimensional questions in mathematical Finance is shown.
A version of the Hörmander–Malliavin theorem in 2-smooth Banach spaces
We consider a stochastic evolution equation in a 2-smooth Banach space with a densely and continuously embedded Hilbert subspace. We prove that under Hormander's bracket condition, the image measure
Spectral gaps in Wasserstein distances and the 2D stochastic Navier–Stokes equations
We develop a general method to prove the existence of spectral gaps for Markov semigroups on Banach spaces. Unlike most previous work, the type of norm we consider for this analysis is neither a
...
...

References

SHOWING 1-10 OF 37 REFERENCES
Uniqueness of the Invariant Measure¶for a Stochastic PDE Driven by Degenerate Noise
Abstract: We consider the stochastic Ginzburg–Landau equation in a bounded domain. We assume the stochastic forcing acts only on high spatial frequencies. The low-lying frequencies are then only
Stochastic calculus of variations for stochastic partial differential equations
Malliavin calculus for the stochastic 2D Navier—Stokes equation
We consider the incompressible, two‐dimensional Navier‐Stokes equation with periodic boundary conditions under the effect of an additive, white‐in‐time, stochastic forcing. Under mild restrictions on
Smoothing properties of transition semigroups relative to SDEs with values in Banach spaces
Abstract. In the present paper we consider the transition semigroup Pt related to some stochastic reaction-diffusion equations with the nonlinear term f having polynomial growth and satisfying some
Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing
The stochastic 2D Navier-Stokes equations on the torus driven by degenerate noise are studied. We characterize the smallest closed invariant subspace for this model and show that the dynamics
Hypoellipticity in infinite dimensions and an application in interest rate theory
We apply methods from Malliavin calculus to prove an infinite- dimensional version of Hormander's theorem for stochastic evolution equations in the spirit of Da Prato-Zabczyk. This result is used to
Ergodicity for Infinite Dimensional Systems: Invariant measures for stochastic evolution equations
Part I. Markovian Dynamical Systems: 1. General dynamical systems 2. Canonical Markovian systems 3. Ergodic and mixing measures 4. Regular Markovian systems Part II. Invariant Measures For
Ergodicity for the Navier‐Stokes equation with degenerate random forcing: Finite‐dimensional approximation
We study Galerkin truncations of the two‐dimensional Navier‐Stokes equation under degenerate, large‐scale, stochastic forcing. We identify the minimal set of modes that has to be forced in order for
Ergodicity of the Finite Dimensional Approximation of the 3D Navier–Stokes Equations Forced by a Degenerate Noise
We prove ergodicity of the finite dimensional approximations of the three dimensional Navier–Stokes equations, driven by a random force. The forcing noise acts only on a few modes and some algebraic
...
...