Regularity of probability laws by using an interpolation method

@article{Bally2016RegularityOP,
  title={Regularity of probability laws by using an interpolation method},
  author={Vlad Bally and Lucia Caramellino},
  journal={arXiv: Probability},
  year={2016},
  pages={83-114}
}
One of the outstanding applications of Malliavin calculus is the criterion of regularity of the law of a functional on the Wiener space (presented in Section 2.3). The functional involved in such a criterion has to be regular in Malliavin sense, i.e., it has to belong to the domain of the differential operators in this calculus. As long as solutions of stochastic equations are concerned, this amounts to regularity properties of the coefficients of the equation. 
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References

SHOWING 1-10 OF 58 REFERENCES
Regularity of Wiener functionals under a Hörmander type condition of order one
We study the local existence and regularity of the density of the law of a functional on the Wiener space which satisfies a criterion that generalizes the Hormander condition of order one (that is,
Absolute Continuity of the Law of the Solution of a Parabolic SPDE
Let {u(t, x); t ≥ 0, 0 < x < 1} denote the solution of a white noise driven parabolic stochastic partial differential equation with Dirichlet boundary conditions. Using Malliavin′s calculus, we give
Riesz transform and integration by parts formulas for random variables
Smoothness of the law of some one-dimensional jumping S.D.E.s with non-constant rate of jump
We consider a one-dimensional jumping Markov process, solving a Poisson-driven stochastic differential equation. We prove that the law of this process admits a smooth density for all positive times,
Convergence and regularity of probability laws by using an interpolation method
Fournier and Printems [Bernoulli 16 (2010) 343–360] have recently established a methodology which allows to prove the absolute continuity of the law of the solution of some stochastic equations with
Existence of densities for the 3D Navier–Stokes equations driven by Gaussian noise
We prove three results on the existence of densities for the laws of finite dimensional functionals of the solutions of the stochastic Navier–Stokes equations in dimension $$3$$. In particular, under
Integration by parts formula and applications to equations with jumps
We establish an integration by parts formula in an abstract framework in order to study the regularity of the law for processes arising as the solution of stochastic differential equations with
On the distances between probability density functions
We give estimates of the distance between the densities of the laws of two functionals $F$ and $G$ on the Wiener space in terms of the Malliavin-Sobolev norm of $F-G.$ We actually consider a  more
Introduction to stochastic partial differential equations
We introduce the Hilbert space-valued Wiener process and the corresponding stochastic integral of Ito type. This is then used together with semigroup theory to obtain existence and uniqueness of weak
An optimal control variance reduction method for density estimation
...
1
2
3
4
5
...