# Malliavin Calculus

• Hiroshi Kunita
Chapter
Part of the Probability Theory and Stochastic Modelling book series (PTSM, volume 92)

## Abstract

We will discuss the Malliavin calculus on the Wiener space and on the space of Poisson random measures, called the Poisson space. There are extensive works on the Malliavin calculus on the Wiener space. Also, there are some interesting works on the Malliavin calculus on the Poisson space. These two types of calculus have often been discussed separately, since the Wiener space and that of the Poisson space are quite different. Even so, we are interested in the Malliavin calculus on the product of these two spaces; we want to develop these two theories in a compatible way.

In Sects. 5.1, 5.2, and 5.3, we study the Malliavin calculus on the Wiener space. We define ‘H-derivative’ operator Dt and its adjoint δ (Skorohod integral by Wiener process) on the Wiener space. Then, after introducing Sobolev norms for Wiener functionals, we get a useful estimate of the adjoint operator with respect to Sobolev norms (Theorem 5.2.1). In Sect. 5.3, we give Malliavin’s criterion for which the law of a given Wiener functional has a smooth density. It will be stated in terms of the Malliavin covariance.

In Sects. 5.4, 5.5, 5.6, and 5.7, we study the Malliavin calculus on the space of Poisson random measure. Difference operator $${\tilde D}_u$$ and its adjoint $$\tilde \delta$$ (Skorohod integral by Poisson random measure) are defined for Poisson functionals following Picard [92]. We will define a family of Sobolev norms conditioned to a family of star-shaped neighborhoods {A(ρ), 0 < ρ < 1}. Then a criterion for the smooth density of the law of a Poisson functional will be given using this family of Sobolev norms.

In Sects. 5.8, 5.9, and 5.10, we study the Malliavin calculus on the product of the Wiener space and Poisson space. Sobolev norms for Wiener–Poisson functionals are studied in Sect. 5.9. In Sect. 5.10, we study criteria for the smooth density. Finally in Sect. 5.11, we discuss the composition of a ‘nondegenerate’ Wiener–Poisson functional and Schwartz’s distribution.

Results of this chapter will be applied in the next chapter for getting the fundamental solution of heat equations discussed in Chap. .

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