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} }
20 Citations
Hypoellipticity in Infinite Dimensions
- Mathematics
- 2009
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
- MathematicsElectronic Journal of Probability
- 2019
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
- Mathematics
- 2009
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
- MathematicsJournal 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…
Ergodic and mixing properties of the Boussinesq equations with a degenerate random forcing
- Mathematics
- 2013
A Theory of Hypoellipticity and Unique Ergodicity for Semilinear Stochastic PDEs
- Mathematics
- 2008
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
- MathematicsSIAM J. Math. Anal.
- 2008
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
- Mathematics
- 2014
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
- Mathematics
- 2008
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
- Mathematics
- 2001
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…
Malliavin calculus for the stochastic 2D Navier—Stokes equation
- Mathematics
- 2004
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…
On finite-dimensional projections of distributions for solutions of randomly forced PDE's
- Mathematics
- 2006
Smoothing properties of transition semigroups relative to SDEs with values in Banach spaces
- Mathematics
- 1999
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
- Mathematics
- 2004
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
- Mathematics
- 2005
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
- Mathematics
- 1996
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
- Mathematics
- 2001
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
- Mathematics
- 2002
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…