Sobolev spaces of Wiener functionals and Malliavin’s calculus

Abstract

The notion of Sobolev spaces of Wiener f unctionals was first introduced by D. W. Stroock [10] a n d I. S h ig e k aw a [7 ] to form ulate M alliavin's calculus rigorously a n d study it systematically.However it cannot be denied that their Sobolev spaces sometimes appeared very complicated.It is mainly because they should have dealt with both th e derivative operator D a n d its d u al D * , or the Ornstein-Uhlenbeck operator L (=-D*D), not only o n L 2 b u t also on a ll LPspaces over the W iener sp ace.But in 1982, P. A. M eyer pointed out the possibility to remove those apparent complications ; that is, he proved th e equivalence o f th e two norms defined in terms o f L and D respectively.In the present paper, w e first aim to develop M eyer's results a n d prove th e equivalence among several Sobolev-type norm s.In doing this, there are two useful tools ; the W iener chaos decomposition o f L 2 and the hypercontractivity of the Ornstein-Uhlenbeck semigroup.Combining these two, we follow Shigek aw a's idea to prove Theorem 1 .1 , which offers a sufficient condition for a linear operator to be bounded o n D'.Next we construct the Sobolev spaces of Wiener f unctionals and discuss their properties.In particular, our definition allows o f negative indices and such spaces contain what we call generalized W iener f unctionals.In this context, we consider the com position of Schwartz's distributions and W iener functionals, which was first studied by S .Watanabe [ 1 1 ] .This presents another approach to Malliavin's calculus.Here,

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