Malliavin calculus

In infinite-dimensional vector spaces, translation-invariant measures like the Lebesgue measure do not exist. Therefore, to construct a Sobolev differential calculus in which one can work with the measure-equivalence classes of functions instead of the functions themselves, one should use other measures. A very pleasant candidate for this is the Gaussian measure, since it shares several nice properties with the Lebesgue measure in finite dimensions and inherits some of them in infinite dimensions. In particular, for any Gaussian measure $ \mu $ on a separable Banach space $ W $ there exists a separable Hilbert space $ H $, called a Cameron–Martin space, which is densely and continuously injected in $ W $, such that the measure is quasi-invariant under the translations with respect to the elements of $ H $. In other words, for any given $ h \in H $, there exists a strictly positive random variable $ L _ {h} $ such that $ L _ {h} \in L _ {p} ( \mu ) $( where $ L _ {p} ( \mu ) $ denotes the space of $ \mu $- equivalence classes of random variables having finite moments up to order $ p $) for any $ p > 1 $ and such that

$$ \int\limits _ { W } {F ( w + h ) } {\mu ( dw ) } = \int\limits _ { W } {F ( w ) L _ {h} ( w ) } {\mu ( dw ) } , $$

for any continuous, bounded function $ F $ on $ W $. Furthermore, the mapping $ h \mapsto L _ {h} $ is infinitely differentiable as a mapping from $ H $ into $ L _ {p} ( \mu ) $.

The quasi-invariance of the Gaussian measure $ \mu $ allows one to define the directional derivatives of the $ \mu $- equivalence classes of the functions defined on $ W $, in the directions of the Cameron–Martin space $ H $. The $ L _ {p} $- differentiability of the mapping $ h \mapsto L _ {h} $ implies that the derivative operator $ \nabla $, first defined on the cylindrical functions, is closeable in $ L _ {p} ( \mu ) $ for $ p > 1 $. Consequently, one can define the Sobolev spaces $ {\mathcal D} _ {p,k } ( X ) $ of equivalence classes of $ X $- valued functions (or random variables; cf. also Sobolev space; Sobolev classes (of functions)), where $ X $ is any separable Hilbert space, $ p > 1 $, $ k \in \mathbf N $, as the completion of $ X $- valued cylindrical functions with respect to the norm

$$ \left \| \phi \right \| _ {p,k } = \left \| \phi \right \| _ {L _ {p} ( \mu,X ) } + \sum _ {j = 1 } ^ { k } \left \| {\nabla ^ {j} \phi } \right \| _ {L _ {p} ( \mu,H ^ {\otimes j } \otimes X ) } , $$

where $ L _ {p} ( \mu,Y ) $ denotes the equivalence classes of $ Y $- valued random variables with finite moments up to order $ p $ and $ Y = X $ or $ Y = H ^ {\otimes j } \otimes X $, $ H ^ {\otimes j } \otimes X = H \otimes \dots \otimes H \otimes X $, and $ \otimes $ is the Hilbert–Schmidt tensor product (cf. also Tensor product).

Since $ \nabla $ and its iterates are closeable, $ {\mathcal D} _ {p,k } ( X ) $ injects continuously and densely into $ {\mathcal D} _ {q,l } ( X ) $ for $ q \leq p $ and $ l \leq k $. Hence there is a sequence of spaces

$$ L _ {p} ( \mu,X ) \supset {\mathcal D} _ {p,1 } ( X ) \supset {\mathcal D} _ {p,2 } ( X ) \supset \dots $$

as in the finite-dimensional case, and $ \nabla ^ {j} $ extends as a linear, continuous mapping from $ {\mathcal D} _ {p,k } ( X ) $ into $ {\mathcal D} _ {p,k - j } ( H ^ {\otimes j } \otimes X ) $ for any $ p > 1 $, $ k \geq j $ and $ j \geq 1 $.

To define Sobolev spaces of negative order one uses the Ornstein–Uhlenbeck operator $ {\mathcal L} $, known as the number operator in physics, which is defined for any cylindrical function $ \psi $ on $ W $ as

$$ \left . {\mathcal L} \psi ( w ) = - { \frac{d}{dt } } \int\limits _ { W } {\psi ( e ^ {- t } w + \sqrt {1 - e ^ {- 2t } } y ) } {\mu ( dy ) } \right | _ {t = 0 } . $$

The inequalities of P.A. Meyer (cf. [a6]) say that the norms of $ {\mathcal D} _ {p,k } ( X ) $, defined above, are equivalent to those defined by

$$ \left \| {\left | \phi \right | } \right \| _ {p,k } = \left \| {( I + {\mathcal L} ) ^ { {k / 2 } } \phi } \right \| _ {L _ {p} ( \mu,X ) } , $$

for any $ p > 1 $ and $ k \in \mathbf N $. Moreover, such $ \| {| \cdot | } \| _ {p,k } $ norms can also be defined for negative $ k $, and in this way one can complete the Sobolev scale as

$$ \dots \supset {\mathcal D} _ {p, - 2 } ( X ) \supset {\mathcal D} _ {p, - 1 } ( X ) \supset $$

$$ \supset L _ {p} ( \mu,X ) \supset {\mathcal D} _ {p,1 } ( X ) \supset \dots . $$

It follows from the construction that $ {\mathcal D} _ {p,k } ( X ) $ is the dual of $ {\mathcal D} _ {q, - k } ( X ^ \prime ) $, where $ q ^ {- 1 } = 1 - p ^ {- 1 } $ and $ X ^ \prime $ is the dual of $ X $. The adjoint of $ \nabla $, denoted by $ \delta $ and called the divergence operator, is then a linear, continuous mapping from $ {\mathcal D} _ {p,k } ( X \otimes H ) $ into $ {\mathcal D} _ {p,k - 1 } ( X ) $ for any $ p > 1 $, $ k \in \mathbf Z $. If $ H = H _ {1} ( [ 0,1 ] ) $, i.e., the space of absolutely continuous functions on $ [ 0,1 ] $ with square-integrable derivatives, one can take $ W = C ( [ 0,1 ] ) $, and if $ u \in {\mathcal D} _ {p,1 } ( H ) $ is such that the Lebesgue density of $ u $ is adapted to the family of sigma-algebras $ {\mathcal F} _ {t} $, $ t \in [ 0,1 ] $, generated by the mappings $ \{ {s \mapsto w ( s ) } : {s \leq t } \} $( cf. also Optional random process), then the Itô stochastic integral of the Lebesgue density of $ u $ coincides with the divergence of $ u $, i.e., with $ \delta u $. This is a key observation which explains the applicability of all this theory in the Itô stochastic calculus.

Although the quasi-invariance properties of the Gaussian measures were well-known since the 1940s, the subject has become very popular after important work of P. Malliavin (cf. [a9]), who showed that an integral of the form

$$ \int\limits _ { W } {\partial ^ \alpha f ( \phi ( w ) ) M ( w ) } {\mu ( dw ) } , $$

where $ \phi \in \cap _ {p,k } {\mathcal D} _ {p,k } ( \mathbf R ^ {d} ) $, $ M \in \cap _ {p,k } {\mathcal D} _ {p,k } ( \mathbf R ) $, $ f : {\mathbf R ^ {d} } \rightarrow \mathbf R $ is a smooth function, and $ \alpha \in \mathbf N ^ {d} $, can be written as

$$ \int\limits _ { W } {f ( \phi ( w ) ) K _ \alpha ( \phi,M ) } {\mu ( dw ) } , $$

where $ K _ \alpha ( \phi,M ) $ is a function of $ \nabla ^ {i} \phi $, $ \nabla ^ {j} M $, with $ i,j \leq | \alpha | $, which, however, is independent of $ f $, provided that the inverse of the determinant of the matrix $ \{ {( \nabla \phi ^ {i} , \nabla \phi ^ {j} ) _ {H} } : {i,j \leq d } \} $ is in $ \cap _ {p} L _ {p} ( \mu ) $. Consequently, the measure $ f \mapsto \int _ {W} {f ( \phi ) M } {d \mu } $ can be written as $ f \mapsto \int _ {\mathbf R ^ {d} } {f ( x ) p _ {M, \phi } ( x ) } {dx } $ with $ p _ {M, \phi } $ an infinitely differentiable function (it is even rapidly decreasing) (cf. [a12]). Malliavin has applied this observation to prove the regularity of the fundamental solutions of second-order, degenerate, parabolic partial differential operators satisfying the Hörmander condition (cf. [a1], [a11], [a17]), which correspond to the infinitesimal generators of certain diffusion processes defined by the Itô stochastic differential equations. This result is accepted as a cornerstone in probability theory, since before it was known one always used the theory of partial differential equations to show the regularity of the densities of the solutions of Itô stochastic differential equations.

Since then, the theory has developed in several different directions; for instance, the notion of causality has been better understood and non-causal problems have been attacked (cf. [a3], [a8], [a16]). The Girsanov theorem has been extended to the general non-causal case (cf. [a4], [a10], [a14]) and then applied to the proof of the existence of solutions of non-linear stochastic partial differential equations via degree theory on the Wiener space (cf. [a15] and Wiener space, abstract). Some of these results have further been extended to the path spaces of compact Riemannian manifold-valued Brownian motion and the corresponding loop spaces (cf. [a2], [a5], [a13]).

References

[a1]	J.-M. Bismut, "Martingales, Malliavin calculus and hypoellipticity under general Hörmander's condition" Z. Wahrscheinlichkeitsth. verw. Gebiete , 63 (1981) pp. 469–505
[a2]	A.-B. Cruzeiro, P. Malliavin, "Renormalized differential geometry on path space: structural equation, curvature" J. Funct. Anal. , 139 (1996) pp. 119–181
[a3]	L. Decreusefond, A.S. Üstünel, "Stochastic analysis of fractional Brownian motion" Preprint
[a4]	S. Kusuoka, "The non-linear transformation of Gaussian measure on Banach space and its absolute continuity I" J. Fac. Sci. Univ. Tokyo, IA , 29 (1982) pp. 567–597
[a5]	L. Gross, "Uniqueness of ground states for Schrödinger operators over loop groups" J. Funct. Anal. , 112 (1993) pp. 373–441
[a6]	P.A. Meyer, "Transformations de Riesz pour les lois gaussiennes" , Sem. Probab. XVIII , Lecture Notes in Mathematics , 1059 , Springer (1984) pp. 179–193
[a7]	S. Kusuoka, "The nonlinear transformation of Gaussian measures on Banach space and its absolute continuity, I" J. Fac. Sci. Univ. Tokyo Sect.IA, Math. , 29 (1982) pp. 567–598
[a8]	D. Nualart, "The Malliavin calculus and related topics. Probability and its applications" , Springer (1995)
[a9]	P. Malliavin, "Stochastic calculus of variations and hypoelliptic operators" , Proc. Int. Symp. Stochastic Diff. Eq. (Kyoto, 1976) , Wiley (1978) pp. 195–263
[a10]	R. Ramer, "On nonlinear transformations of Gaussian measures" J. Funct. Anal. , 15 (1974) pp. 166–187
[a11]	D.W. Stroock, "Some applications of stochastic calculus to partial differential equations" , Ecole d'Eté de Probab. de Saint-Flour , Lecture Notes in Mathematics , 976 , Springer (1983) pp. 267–382
[a12]	A.S. Üstünel, "An introduction to analysis on Wiener space" , Lecture Notes in Mathematics , 1610 , Springer (1995)
[a13]	A.S. Üstünel, "Stochastic analysis on Lie groups" , Proc. Sixth Workshop Oslo–Silivri on Stochastic Anal. , Progress in Math. , Birkhäuser (to appear)
[a14]	A.S. Üstünel, M. Zakai, "Transformation of the Wiener measure under non-invertible shifts" Probab. Th. Rel. Fields , 99 (1994) pp. 485–500
[a15]	A.S. Üstünel, M. Zakai, "The degree theory on the Wiener space" Probab. Th. Rel. Fields (to appear)
[a16]	A.S. Üstünel, M. Zakai, "The constructions of filtrations on abstract Wiener spaces" J. Funct. Anal. (to appear)
[a17]	S. Watanabe, "Lectures on stochastic differential equations and Malliavin calculus" , Tata Inst. Fundam. Res. and Springer (1984)