View source for Malliavin calculus

In infinite-dimensional vector spaces, translation-invariant measures like the [[Lebesgue measure|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m1100501.png" /> on a separable [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m1100502.png" /> there exists a separable [[Hilbert space|Hilbert space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m1100503.png" />, called a Cameron–Martin space, which is densely and continuously injected in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m1100504.png" />, such that the measure is quasi-invariant under the translations with respect to the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m1100505.png" />. In other words, for any given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m1100506.png" />, there exists a strictly positive [[Random variable|random variable]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m1100507.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m1100508.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m1100509.png" /> denotes the space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005010.png" />-equivalence classes of random variables having finite moments up to order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005011.png" />) for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005012.png" /> and such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005013.png" /></td> </tr></table>

for any continuous, bounded function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005014.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005015.png" />. Furthermore, the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005016.png" /> is infinitely differentiable as a mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005017.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005018.png" />.

The quasi-invariance of the Gaussian measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005019.png" /> allows one to define the directional derivatives of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005020.png" />-equivalence classes of the functions defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005021.png" />, in the directions of the Cameron–Martin space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005022.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005023.png" />-differentiability of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005024.png" /> implies that the derivative operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005025.png" />, first defined on the cylindrical functions, is closeable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005026.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005027.png" />. Consequently, one can define the Sobolev spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005028.png" /> of equivalence classes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005029.png" />-valued functions (or random variables; cf. also [[Sobolev space|Sobolev space]]; [[Sobolev classes (of functions)|Sobolev classes (of functions)]]), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005030.png" /> is any separable Hilbert space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005032.png" />, as the completion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005033.png" />-valued cylindrical functions with respect to the norm

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005034.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005035.png" /> denotes the equivalence classes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005036.png" />-valued random variables with finite moments up to order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005038.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005040.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005041.png" /> is the Hilbert–Schmidt tensor product (cf. also [[Tensor product|Tensor product]]).

Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005042.png" /> and its iterates are closeable, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005043.png" /> injects continuously and densely into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005044.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005046.png" />. Hence there is a sequence of spaces

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005047.png" /></td> </tr></table>

as in the finite-dimensional case, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005048.png" /> extends as a linear, continuous mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005049.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005050.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005053.png" />.

To define Sobolev spaces of negative order one uses the Ornstein–Uhlenbeck operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005054.png" />, known as the number operator in physics, which is defined for any cylindrical function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005055.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005056.png" /> as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005057.png" /></td> </tr></table>

The inequalities of P.A. Meyer (cf. [[#References|[a6]]]) say that the norms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005058.png" />, defined above, are equivalent to those defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005059.png" /></td> </tr></table>

for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005061.png" />. Moreover, such <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005062.png" /> norms can also be defined for negative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005063.png" />, and in this way one can complete the Sobolev scale as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005064.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005065.png" /></td> </tr></table>

It follows from the construction that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005066.png" /> is the dual of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005067.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005068.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005069.png" /> is the dual of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005070.png" />. The adjoint of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005071.png" />, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005072.png" /> and called the divergence operator, is then a linear, continuous mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005073.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005074.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005076.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005077.png" />, i.e., the space of absolutely continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005078.png" /> with square-integrable derivatives, one can take <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005079.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005080.png" /> is such that the Lebesgue density of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005081.png" /> is adapted to the family of sigma-algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005082.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005083.png" />, generated by the mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005084.png" /> (cf. also [[Optional random process|Optional random process]]), then the Itô [[Stochastic integral|stochastic integral]] of the Lebesgue density of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005085.png" /> coincides with the divergence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005086.png" />, i.e., with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005087.png" />. 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. [[#References|[a9]]]), who showed that an integral of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005088.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005089.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005090.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005091.png" /> is a smooth function, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005092.png" />, can be written as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005093.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005094.png" /> is a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005095.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005096.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005097.png" />, which, however, is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005098.png" />, provided that the inverse of the determinant of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m11005099.png" /> is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m110050100.png" />. Consequently, the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m110050101.png" /> can be written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m110050102.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110050/m110050103.png" /> an infinitely differentiable function (it is even rapidly decreasing) (cf. [[#References|[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. [[#References|[a1]]], [[#References|[a11]]], [[#References|[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. [[#References|[a3]]], [[#References|[a8]]], [[#References|[a16]]]). The Girsanov theorem has been extended to the general non-causal case (cf. [[#References|[a4]]], [[#References|[a10]]], [[#References|[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. [[#References|[a15]]] and [[Wiener space, abstract|Wiener space, abstract]]). Some of these results have further been extended to the path spaces of compact Riemannian manifold-valued [[Brownian motion|Brownian motion]] and the corresponding loop spaces (cf. [[#References|[a2]]], [[#References|[a5]]], [[#References|[a13]]]).

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