Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA
Department Mathematik, FAU Erlangen–Nürnberg, Cauerstrasse 11, 91058-Erlangen, Germany
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA
Author to whom correspondence should be addressed.
Received: 7 May 2018 / Accepted: 20 May 2018 / Published: 1 June 2018
In this article we study the connection of fractional Brownian motion, representation theory and reflection positivity in quantum physics. We introduce and study reflection positivity for affine isometric actions of a Lie group on a Hilbert space and show in particular that fractional Brownian motion for Hurst index is reflection positive and leads via reflection positivity to an infinite dimensional Hilbert space if . We also study projective invariance of fractional Brownian motion and relate this to the complementary series representations of . We relate this to a measure preserving action on a Gaussian -Hilbert space .
fractional brownian motion; reflection positivity; reflection negative kernels; representations of
In this paper we continue our investigations of the representation theoretic aspects of reflection positivity and its relations to stochastic processes ([1,2]). This is a basic concept in constructive quantum field theory [3,4,5,6], where it arises as a requirement on the euclidean side to establish a duality between euclidean and relativistic quantum field theories . It is closely related to “Wick rotations” or “analytic continuation” in the time variable from the real to the imaginary axis.
The underlying structure is that of a reflection positive Hilbert space, introduced in . This is a triple , where is a Hilbert space, is a unitary involution and is a closed subspace of which is -positive in the sense that the hermitian form is positive semidefinite on . We write for the corresponding Hilbert space and for the canonical map.
To relate this to group representations, let us call a triple a symmetric semigroup if G is a Lie group, is an involutive automorphism of G and a subsemigroup invariant under the involution . The Lie algebra of G decomposes into -eigenspaces and we obtain the Cartan dual Lie algebra . We write for a Lie group with Lie algebra . The prototypical pair consists of the euclidean motion group and the orthochronous Poincaré group . If is a symmetric Lie group and a reflection positive Hilbert space, then we say that a unitary representation is reflection positive with respect to if
If is a reflection positive representation of G on , then defines a representation of the involutive semigroup by contractions ( Lemma 1.4,  or , Prop. 3.3.3). However, if S has interior points, we would like to have a unitary representation of a Lie group with Lie algebra on whose derived representation is compatible with the representation of S. If such a representation exists, then we call a euclidean realization of the representation of . Sufficient conditions for the existence of have been developed in .
Althought this is a rather general framework, the present paper is only concerned with very concrete aspect of reflection positivity. The main new aspect we introduce is a notion of reflection positivity for affine isometric actions of a symmetric semigroup on a real Hilbert space. Here is naturally defined by the closed subspace generated by the S-orbit of the origin. On the level of positive definite functions, this leads to the notion of a reflection negative function. For , reflection negative functions are easily determined because reflection negativity is equivalent to being a Bernstein function (). An announcement of some of the results in the present paper appeared in .
For a group G, affine isometric actions on a real Hilbert space are encoded in real-valued negative definite functions satisfying (cf. [12,13]). Especially for , these structures have manifold applications in various fields of mathematics (see for instance [14,15,16], and also  for the generalization to spirals which corresponds to actions of by affine conformal maps). For the group , the homogeneous function is negative definite if and only if , and this leads to the positive definite kernels
which for are the covariance kernels of fractional Brownian motion with Hurst index H ([18,19,20,21,22]).
One of the central results of this paper is an extension of the well-known projective invariance of Brownian motion in the sense of P. Lévy (cf.  I.2, and ) to fractional Brownian motion. Here we use the identification of with the real projective line, which leads to the action of by Möbius transformations for . Starting from a realization of fractional Brownian motion with Hurst index in a suitable Hilbert space by the functions
we associate to every pair of distinct points in a normalized process whose covariance kernels transform naturally under Möbius transformations in the sense that
Here the normalized fractional Brownian motion has the covariance kernel and the transformed process is equivalent to the original one.
The structure of this paper is as follows. In Section 2 we briefly recall the general background of reflection positive Hilbert spaces and representations and in Section 3 we introduce reflection positive affine isometric actions on real Hilbert spaces . Since the group has a natural unitary representation on the Fock space , the -space of the canonical Gaussian measure of , affine isometric representations are closely linked with symmetries of Gaussian stochastic processes for which G acts on the corresponding index set. This is made precise in Appendix B.1, where we discuss the measure preserving G-action corresponding to a stochasic process with stationary increments. For square integrable processes, this connects with affine isometric actions on Hilbert spaces.
To pave the way for the analysis of the interaction of fractional Brownian motion with unitary representations, we introduce in Section 4 a family of unitary representations of , respectively its projective quotient , i.e., the group of Möbius transformations on the real projective line. For this is the natural representation on (belonging to the principal series), whereas for it belongs to the complementary series ([4,25]). The Hilbert spaces are obtained from positive definite distribution kernels by completion of with respect to the scalar product
In Section 5 we realize fractional Brownian motion in a very natural way in terms of the cocycle (2) defining an affine isometric action of the translation group on . Acting with the group on these functions leads naturally to the projective invariance of fractional Brownian motion, both on the level of the normalized kernels as in (3), and with respect to our concrete realization (Theorem 1).
Reflection positivity is then explored in Section 6. For in we consider a reflection with a fixed point and exchanging and . Here our main result is Theorem 2, asserting that the normalized kernels on the complement of the two-element set in is reflection positive with respect to if and only if . In particular, this implies reflection positivity for a Brownian bridge on a real interval with respect to the reflection in the midpoint. Reflection positivity for the complementary series representations of has already been observed in , where the representation is identified as a holomorphic discrete series representation.
Reflection positivity for the affine action of the translation group in defined by the cocycle realizing fractional Brownian motion is studied in Section 7. Although we always have involutions that lead to reflection positive Hilbert spaces in a natural way, only for we obtain reflection positive affine actions of . We conclude Section 7 with a discussion of the increments of a 1-cocycle defining an affine isometric action. In particular, we characterize cocycles with orthogonal increments as those corresponding to multiples of Brownian motion. Note that the increments of fractional Brownian motion are positively correlated for and negatively correlated for . We conclude this paper with a brief discussion of some related results concerning higher dimensional spaces in Section 8. We plan to return to the corresponding representation theoretic aspects in the near future.
In order not to distract the reader from the main line of the paper, we moved several auxiliary tools and some definitions and calculations into appendices: Appendix A deals with affine isometries and positive definite kernels and Appendix B reviews some properties of stochastic processes. In particular, we provide in Proposition A3 a representation theoretic proof for the Lévy–Khintchine formula for the real line, which represents a negative definite function in terms of its spectral measure ([19,20,23,26] Thm. 32). Appendix C briefly recalls the measure theoretic perspective on Second Quantization, Appendix D contains the verification that the representations mentioned above are unitary, and Appendix E contains a calculation of the spectral measure for fractional Brownian motion.
A different kind of projective invariance, in the path parameter t, for one-dimensional Brownian motion has been observed by S. Takenaka in : For a Brownian motion , the process
also is a Brownian motion, and the relation leads to a unitary representation of on the realization space. From that he derives the projective invariance in the sense of Lévy, and he argues that his method does not extend to fractional Brownian motion. In , Takenaka shows that the representation of he obtains belongs to the discrete series, so that it is differet from ours. He also hints at the possibility of extending Hida’s method  to fractional Brownian motion, and in a certain sense this is carried out in the present paper.
2. Reflection Positive Functions and Representations
Since our discussion is based on positive definite kernels and the associated Hilbert spaces ([29,30] Ch. I, ), we first recall the pertinent definitions. As customary in physics, we follow the convention that the inner product of a complex Hilbert space is linear in the second argument.
(a) Let X be a set. A kernel is called hermitian if . A hermitian kernel Q is called positive definite if for , we have . It is called negative definite if holds for and with ().
(b) If is an involutive semigroup, then is called positive (negative) definite if the kernel is positive (negative) definite. If G is a group, then we consider it as an involutive semigroup with and definite positive/negative definite functions accordingly.
We shall use the following lemma to translate between positive definite and negative definite kernels (, Lemma 3.2.1):
Let X be a set, and be a hermitian kernel. Then the kernel
is positive definite if and only if Q is negative definite.
According to Schoenberg’s Theorem ( Thm. 3.2.2), a kernel is negative definite if and only if, for every , the kernel is positive definite.
Let X be a set, be a positive definite kernel and be the corresponding reproducing kernel Hilbert space. This is the unique Hilbert subspace of on which all point evaluations are continuous and given by
Then the map has total range and satisfies . The latter property determines the pair up to unitary equivalence ( Ch. I).
A reflection positive Hilbert space is a triple , where is a Hilbert space, θ a unitary involution and is a closed subspace which is θ-positive in the sense that the hermitian form is positive semidefinite on .
For a reflection positive Hilbert space , let and write for the completion of with respect to the inner product . We write for the canonical map.
Suppose that is a positive definite kernel and is an involution leaving K invariant and that is a subset with the property that the kernel is also positive definite on . We call such kernels K reflection positive with respect to . Then the closed subspace generated by is θ-positive for . We thus obtain a reflection positive Hilbert space .
In this context, the space can be identified with the reproducing kernel space , where q corresponds to the map
For a symmetric semigroup , we obtain natural classes of reflection positive kernels:
A function on a group G is called reflection positive () if the kernel is reflection positive with respect to in the sense of Example 1 with and . These are two simultaneous positivity conditions, namely that the kernel is positive definite on G and that the kernel is positive definite on S.
The usal Gelfand–Naimark–Segal construction naturally extends to reflection positive functions and provides a correspondence with reflection positive representations (see  Thm. 3.4.5).
For a symmetric semigroup , a unitary representation U of G on a reflection positive Hilbert space is called reflection positive if for and for every .
(a) If is a reflection positive representation of on , then we obtain contractions on , determined by
and this leads to an involutive representation of S by contractions (cf.  Cor. 3.2,  or ). We then call a euclidean realization of .
(b) For , continuous reflection positive unitary one-parameter groups lead to a strongly continuous semigroup of hermitian contractions and every such semigroup has a natural euclidean realization obtained as the GNS representation associated to the positive definite operator-valued function , ( [Prop. 6.1]).
On , we have:
For , the function on is negative definite by  Cor. 3.2.10 because is obviously negative definite.
For , the function is reflection negative if and only if ( Ex. 4.3(a)).
The function is reflection negative for ( Ex. 6.5.15,  Ex. 4.4(a)).
3. Reflection Positivity for Affine Actions
In this section we introduce reflection positive affine isometric actions on real Hilbert spaces and relate it to the corresponding measure preserving action on the Gaussian -space .
Let be a symmetric semigroup and be a real Hilbert space, endowed with an isometric involution . We consider an affine isometric action
where is an orthogonal representation and a 1-cocycle, i.e.,
Note that (5) in particular implies and thus . We further assume that which is equivalent to
If is total in , then we can realize as a reproducing kernel Hilbert space with kernel
For the function
we then obtain
In view of (7), is negative definite by Lemma 1. Equation (7) implies that, if is total, then the affine action can be recovered completely from the function and every real-valued negative definite function with is of this form (cf. [12,13]). We also note that implies that .
(Reflection positive affine actions) The closed subspace generated by is invariant under the affine action of S on because for . We call the affine action reflection positive with respect to if is θ-positive.
(A universal example) Let be a reflection positive real Hilbert space, and write for its motion group. We define an involution on by . For we put . Then
is a ♯-invariant subsemigroup of with
By construction, the affine action of on is reflection positive in the sense of Definition 5.
For , the relation is equivalent to and . This shows that is equivalent to (because of θ-positivity) and to the condition that the restrictionss of g to are unitary.
The positive definite kernel (Appendix C) is reflection positive with respect to because the kernel is positive definite on (cf. Example 1). From the -invariance of Q, we thus obtain a reflection positive representation of on the corresponding reflection positive Hilbert space .
It is instructive to make the corresponding space more explicit and to see how it identifies with .
we derive that
For , this leads to
In particular, the cyclic subrepresentation generated by the constant function is determined for by the positive definite function
It follows that the function on is reflection positive for .
The following lemma provides a characterization of reflection positive affine actions in terms of kernels.
Let be a symmetric semigroup and be an affine isometric action of on the real Hilbert space . We write for the closed subspace generated by . Then the following are equivalent:
The kernel is positive definite on .
is reflection positive with respect to , i.e., is θ-positive.
The kernel is positive definite on .
The function is negative definite on .
(a) ⇔ (b): In view of the kernel is positive definite on if and only if the kernel is positive definite on , but this is equivalent to being -positive ( Rem. 2.8).
(b) ⇔ (c): Since is generated by , this follows from (7) and the definition of C.
(c) ⇔ (d): By Lemma 1, the kernel is positive definite if and only if the kernel is negative definite, which is (d). ☐
This leads us to the following concept:
We call a continuous function reflection negative with respect to if ψ is a negative definite function on G and is a negative definite function on the involutive semigroup (Definition 1).
From Schoenberg’s Theorem for kernels (Remark 1) we immediately obtain from Lemma 2:
Let be a reflection positive affine action of . Then, for every , the function is reflection positive, i.e., the function is reflection negative.
(a) Let be a real Hilbert space. For , the function on is positive definite. A corresponding cyclic representation can be realized as follows. We consider the unitary representation of on given by
where is the Gaussian measure on with Fourier transform and as in Definition A6 (see also Remark A3). Then the constant function 1 is a cyclic vector, and the corresponding positive definite function is
(b) We conclude that, for every reflection positive affine action , for , a cyclic reflection positive representation of corresponding to is obtained on the cyclic subspace of generated by the constant function 1.
4. Some Unitary Representations of
In this section we introduce a family of unitary representations of , respectively of the projective group .
We identify the real projective line of one-dimensional linear subspaces of with . On this space the group acts naturally by fractional linear maps
which shows that g acts on the circle in an orientation preserving fashion if and only if .
For two different elements , we write for the open interval between α and γ with respect to the cyclic order. For in this means that
For the action of G on , Lebesgue measure λ on , resp., the corresponding measure on with is quasi-invariant with A unitary representation of (resp., of ) on is given by
We could as well work without the -factor, but we shall see below that it is more natural this way when it comes to the relation with fractional Brownian motion.
We now explain how this representation can be embedded into a family of unitary representations . For , these representations belong to the so-called complementary series (cf. [4,8,25]). For , the corresponding Hilbert space is the completion of the Schwartz space with respect to the inner product
Note that , so that the kernel is locally integrable and defines a positive definite distribution kernel on . This implies in particular that (12) makes sense for any pair of compactly supported bounded measurable functions on and that any such function defines an element of . In Appendix D we show that
As we have seen in Example 2(a), the continuous function on is negative definite for . Therefore (13) defines for a positive semidefinite form on . We write for the corresponding Hilbert space. Here we use that the total integrals of and vanish (cf. Remark 12). Note that this definition also makes sense for and , but and is one-dimensional.
We obtain unitary representations of (resp., the quotient ) on , by
For the verification of unitarity we refer to Appendix D. For , we obtain the representation on from (11).
(a) Considering the singularities of the factors in the formula for , we see that the operators preserve the class of locally bounded measurable functions for which
For , all these functions are contained in , so that we obtain a dense subspace of invariant under the operators .
(b) We note that the representation is equivalent to , as can be seen by realizing these representations on (see  Ch. 7). We will not use this duality here.
The unitary representations of yield in particular three important one-parameter groups:
Translations: for .
Dilations: for .
Inverted translations: for .
For with , we have
5. Fractional Brownian Motion
In this section we introduce fractional Brownian motion in terms of its covariance kernel. We then show that the unitary representations of and a realization of fractional Brownian motion in the Hilbert space , resp., on its Fock space, can be used to obtain in a very direct and simple fashion the projective invariance of fractional Brownian motion.
5.1. A Realization of Fractional Brownian Motion
Fractional Brownian motion with Hurst index is a real-valued Gaussian process with zero means and covariance kernel
(cf.  Satz 7 for the determination of those parameters for which this kernel is positive definite). A curve with values in a Hilbert space satisfying is called a fractional Wiener spiral.
Brownian motion arises for , and in this case
We refer to the monograph  for a stochastic calculus for fractional Brownian motion.
(Bifractional Brownian motion) For and , the kernel
on is positive definite (Lemma 1). The corresponding centered Gaussian process is called bifractional Brownian motion (). For we obtain fractional Brownian motion which has stationary increments, but for the process does not have this property since the kernel
on is not translation invariant.
For a concept of trifractional Brownian motion and decompositions of fractional Brownian motion into independent bifractional and trifractional components we refer to .
For , the kernel satisfies
These transformation rules show that:
For a fractional Brownian motion with Hurst index H, the centered Gaussian process defined by
also is a fractional Brownian motion with Hurst index H.
For and the process is a fractional Brownian motion with Hurst index H.
For , the process also is a fractional Brownian motion with Hurst index H.
For and , consider the random variables
Then , i.e., is a realization of fractional Brownian motion with Hurst index H.
Case : As , we only have to show that
This is an elementary calculation.
Case : In this case we can calculate the scalar product (A12) by using the formula (a difference of two point evaluations). This leads to
For , the preceding lemma follows from  (p. 168) and for it is already contained in  (p. 117). Other realizations of fractional Brownian motion are discussed in . ☐
For Brownian motion , an alternative realization is obtained by for (see  p. 130). For this sign change does no longer work because for .
5.2. Projective Invariance of the Covariance Kernels
Recall the cross ratio
and that it is invariant under the action of . As , we obtain for the relation
expressing g as a cross ratio. Accordingly, we obtain for each triple of mutually different elements of the following kernel
where the last expression only makes sense for . By construction we then have
Note that and that, for , and , we obtain in particular for the dilation :
which is a multiple of . In particular, normalization of and leads on to the same kernels. We also observe that
implies the equality of the normalized kernels
From (19) and the preceding discussion we obtain immediately:
For in , the normalized kernel on does not depend on β and satisfies the symmetry condition
For we have
In particular, if preserves the 2-element set , then the kernel on is g-invariant.
Equation (21) follows directly from (20) and the remainder is a consequence of Lemma 4.
The preceding proposition expresses the projective invariance of fractional Brownian motion in the sense of P. Lévy. For , this is classical ( I.2, [24,37] Thm. 5.2). ☐
The identity component of the stabilizer of in is a (hyperbolic) one-parameter group of whose fixed points are α and β (these are the orientation preserving transformations mapping the interval onto itself). The full stabilizer of the pair is isomorphic to . It also contains an involution in exchanging the two connected components of .
Moreover, there exists for each a unique involution exchanging α and γ and fixing β. It satisfies
The subgroup has four connected components.
5.3. Projective Invariance of the Realization
We now link the projective invariance of fractional Brownian motion to the specific realization in the Hilbert space . Formula (b) in the theorem below connects the normalized projective transforms of the kernel to the unitary representation of on .
For a triple of mutually different points in , there exists a uniquely determined Möbius transformation with We thus obtain functions of the form
Then the following assertions hold:
All functions are unit vectors in .
As and the representation is unitary, the functions are unit vectors in .
For with ,
This relation is the reason for the -factor in the definition of . ☐
For , the element fixes 0 and ∞, hence is linear and given by multiplication with . We thus obtain with Remark 7, (18) and (23)
Since the kernel on the left hand side is normalized, (b) follows.
From Proposition 1 and Theorem 1, we obtain:
The normalized stochastic process defined by is stationary with respect to the stabilizer of the two-point set in .
Since the representations of are irreducible,  (Prop. 5.20) implies that the space of smooth vectors is nuclear. Therefore  (Cor. 5.19) shows that the Gaussian measure can be realized on the space of distribution vectors for this representation (cf. Appendix C). Therefore our construction leads to a realization of fractional Brownian motion on the topological dual space of the -invariant subspace of smooth vectors.
From the proof of  (Prop. 5.20(b)), we further derive that an element is a smooth vector if and only if it is a smooth vector for the compact subgroup . Considering as a space of distributions on the circle , it is not hard to see that and hence that is the space of distributions on the circle.
6. Fractional Brownian Motion and Reflection Positivity
We now turn to reflection positivity in connection with fractional Brownian motion. Our main result is Theorem 2 on the reflection positivity of the normalized kernels for . We start with the normalization of the kernel , which corresponds to the pair .
is invariant under the involution . It is reflection positive on if and only of . If this is the case, then , with the measure
For we have , and is one-dimensional.
Reflection positivity with respect to is equivalent to the positive definiteness of the kernel
for (cf. Example 1). This kernel is positive definite on if and only if the function
on the multiplicative semigroup is positive definite. For we have
For , we have and all other factors are positive. As is increasing, it is positive definite if and only if there exists a positive Radon measure on with
(use  (Prop. 4.4.2) or apply  (Cor. VI.2.11) to ). Therefore is positive definite if and only if . In this case the description of follows from the proof of  (Thm. VI.2.10). ☐
(Reflection positivity of fractional Brownian motion) For with , we have In particular,
Therefore is not the unique unitary involution of transforming into (cf. Remark 7).
With the kernels (Lemma 4), we obtain a family of normalized Gaussian processes, covariant with respect to the action of on . The following proposition shows that, for , these kernels are reflection positive with respect to involutions exchanging and .
Let in be mutually different and let be the projective involution exchanging α and γ and fixing β. Let and be the intersection of X with the two connected components of the complement of the fixed point set of θ (which consists of two points). Then the kernel is reflection positive with respect to if and only if .
Since the family of kernels is invariant under the action of and
it suffices to verify the assertion for . Then and we may put . Hence the assertion follows from Proposition 2. ☐
For , the involution is given by . It has the two fixed points 0 and ∞. From (19) we obtain
Theorem 2 now implies that the kernel is reflection positive with respect to .
As we shall see below, for , the covariance kernels turn out to correspond to Brownian bridges.
( Def. 2.8, p. 109) (a) A Gaussian process is called a Brownian bridge if is an affine function and
If for every t, then is called a pinned Brownian motion.
(b) A normalized Brownian bridge is a Brownian bridge whose variance is normalized to 1, so that its covariance kernels is
(Reflection positivity of the Brownian bridge) For in and , the kernel is the covariance of a normalized Brownian bridge on the interval . This kernel is reflection positive for , where , and is the reflection in the midpoint. The corresponding Hilbert space is one-dimensional.
First we observe that
so that we obtain for the associated normalized kernel
This is the kernel (24) of a normalized Brownian bridge on .
For , the reflection is given by , which leaves the kernel invariant by Proposition 1. For , we have
This is a positive definite kernel defining a one-dimensional Hilbert space. We conclude that the kernel C is reflection positive for , where and . ☐
7. Affine Actions and Fractional Brownian Motion
In this section we discuss reflection positivity for the affine isometric action of corresponding to fractional Brownian motion . In Subsection 7.2 we shall encounter the curious phenomenon that, for every H there exists a natural unitary involution that leads to a reflection positive Hilbert space, but only for it can be implemented in such a way that , so that we obtain a reflection positive affine action of . In a third subsection we discuss increments of a 1-cocycle defining an affine isometric action and characterize cocycles with orthogonal increments as those corresponding to multiples of Brownian motion.
If with is an affine isometric action of on the complex Hilbert space , then Proposition A2 implies that, up to unitary equivalence, for a Borel measure on . We may assume that
Then second quantization leads to the centered Gaussian process whose covariance kernel is given by
(Proposition A3). Below we only consider real Hilbert spaces. This corresponds to the situation where the measure is symmetric. Then the function is also real and given by
Let be a continuous positive definite kernel with . For , we put
Then the subspace
is dense in the corresponding reproducing kernel Hilbert space .
From the existence of the -valued integral defining , it follows that these are elements of . Let denote the closed subspace generated by the elements , .
If is a -sequence, we obtain . For with , we have , so that and imply that . Using a sequences of the form , which converges to , we see that for , hence that . ☐
In the following we write .
Let define a continuous affine isometric action of on the real Hilbert space . For we put . Then generates the same closed subspace as .
(a) A function with is negative definite if and only if
is a positive definite kernel (Lemma 1). Then yields
(cf.  (p. 222) for a corresponding statement on more general homogeneous spaces).
(b) Write and for a cocycle of an orthogonal representation of on the real Hilbert space (see Section 3 and [12,13]). We also assume that the family is total in . Then
As is generated by the , the Hilbert space can be identified with the reproducing kernel Hilbert space corresponding to the positive definite distribution C, but the preceding argument adds another picture. It can also be identified with the Hilbert space obtained by completing with respect to the scalar product (27). Taking into account that , this is how we introduced the Hilbert space in (A12).
7.2. Fractional Brownian Motion
For fractional Brownian motion with Hurst index , we have
As in (25), the spectral measure (a Borel measure on ) of fractional Brownian motion is determined by
The corresponding realization is obtained by (Proposition A2). According to  p. 40, the measure is given by
For , the spectral measure σ is a multiple of Lebesgue measure:
This leads to a natural realization of Brownian motion by a cocycle of the multiplication representation of on and, by Fourier transform, to the realization of Brownian motion as a cocycle for the translation representation of on .
Combining Remark 12 with (A12) in Section 4, we see that the Hilbert space can alternatively be constructed from the scalar product
as the completion of . This implies that the map
so that D extends to a unitary operator . Here we write for the Hilbert space of distributions defined by , obtained by completing with respect to the scalar product , defined by the positive definite distribution kernel and associating to the distribution . Note that
where we consider the δ-functionals as elements of with
As a distribution, corresponds to the function which corresponds to the evaluation in s in the reproducing kernel Hilbert space . Compare also with the corresponding discussion in [8,9] Ch. 7.
The inverse of the unitary operator is given by
Consider the realization of fractional Brownian motion in the Hilbert space , the affine isometric -action defined by
where denotes the translation by t on , and the closed subspace generated by . Then
defines a unitary involution with for . Now is reflection positive if and only if , so that we obtain in this case a reflection positive affine action. For , the triple is also reflection positive, but it does not lead to a reflection positive affine action because for .
By Example 2(a), the function is negative definite on the additive group and it is reflection positive for if and only if by Example 2(b). Accordingly, the reflection on leads to the twisted kernel
on which is positive definite if and only if is negative definite on (Lemma 1), which in turn is equivalent to .
For , the unitary involution satisfies , which leads to the twisted kernel
As is negative definite on the semigroup if (Example 2(c)), the assertion follows. ☐
We conclude that the affine actions of corresponding to fractional Brownian motion with Hurst parameter leads to a reflection positive affine action, and from the calculation in Example 3 we derive that the reflection positive function on corresponding to the constant function is given by . The preceding proposition also explains why we obtain trivial reflection positivity for since in this case are both reflection positive.
For and and the kernel on , we obtain with the same arguments the positive definite functions on .
7.3. Cocycles with Orthogonal Increments
In this subsection we discuss the question when a cocycle for an orthogonal representation of has orthogonal increments in the sense that, for we have
There exists a such that for all . If , then realizes a two-sided Brownian motion in .
(cf.  Satz 8 for a variant of this observation)
(i) ⇒ (ii) follows with and .
(ii) ⇒ (i): Let be the closed subspaces generated by the for . Then (ii) means that . For we now observe that
and similary Therefore
(ii) ⇒ (iii): Put and note that this function is increasing for . For we have
so that translation invariance of this kernel leads to
From the orthogonality of the increments, we further derive for the relation
Since is continuous, there exists a with for , and therefore (28) yields for .
We likewise find some with
Now implies that , and this completes the proof.
(iii) ⇒ (ii) follows from the fact that for holds for the covariance kernel of Brownian motion. ☐
If is reflection positive for an affine -action, is generated by and , then the space is trivial if and only if
which in turn means that has orthogonal increments by Proposition 5. In view pf Proposition 5(iii), Brownian motion can, up to positive multiples, be characterized as a process with stationary orthogonal increments.
Consider the stochastic process associated to the cocycle in . We say that the increments of this process are positively (negatively) correlated if, for , we have
we may w.l.o.g. assume that , i.e., . Therefore the process has positively (negatively) correlated increments if and only if, for every , the functions
are increasing (decreasing) on . Note that this implies in particular that , resp., for .
For fractional Brownian motion, we have for
is non-negative for and non-positive for , it follows that fractional Brownian motion has positively correlated increments for and negatively correlated increments for .
In this final section we briefly discuss some results that are possibly related to far reaching generalizations of what we discuss in the present paper on the real line, resp., on its conformal compactification .
8.1. Helices and Hilbert Distances
Let G be a Lie group, be a closed subgroup. We write for the corresponding homogeneous space and for the base point in X.
In  Def. 2.3, a kernel on is called a Lévy–Schoenberg kernel if
C is positive definite and .
The kernel is G-invariant.
Then , where , defines a function on G with
so that is a negative definite K-biinvariant function with on G (Lemma 1). Conversely, every such function defines by (29) a Lévy–Schoenberg kernel on .
For a Lévy–Schoenberg kernel C, there exists a map into a real Hilbert space with , unique up to orthogonal equivalence (Lemma A1), such that
Then is called a helix and is called an invariant Hilbert distance on X. The uniqueness of further implies the existence of an affine isometric action for which is equivariant. Writing , we then have for . In particular, any helix specifies an orthogonal representation of G.
Classification results for Lévy–Schoenberg kernels, resp., invariant Hilbert distances, resp., affine isometric actions of G with a K-fixed points, are mostly stated in terms of integral formulas (Lévy–Khintchine formulas). Results are nown in various contexts:
for G locally compact and K compact (); see [40,41] for locally compact abelian groups.
for G compact ( Thm. 3.15); see  for and .
for Riemannian symmetric ( (Thm. 3.31) and  (Thm. 4.1) for )
for G the additive group of a Hilbert space and K a closed subspace ().
for and and X the infinite dimensional hyberbolic space ( Thm. 8.1). It is shown in particular that the kernel is negative definite, so that all kernels are positive definite; they correspond to the spherical functions of X (cf.  Thm. 21, p. 79).
(a) If is a real Hilbert space and its isometry group, then defines for a negative definite -biinvariant function on G with ( Ex. 3.2.13(b)). The corresponding Lévy–Schoenberg kernel on is
on the sphere , where d denotes the Riemannian metric on and is fixed. Here and is the stabilizer of ( p. 174, ). This means that the Riemannian metric on is a negative definite kernel.
(c) From (a) it follows that for any real-valued negative definite function ψ satisfying on the group G, the functions , , are negative definite as well ( p. 189).
8.2. Brownian Motion on Metric Spaces
Let be a metric space. In  a real-valued Gaussian process is called a Brownian motion with parameter space if there exists a point with
For a metric space a Brownian motion with parameter space exists if and only if the metric is a negative definite kernel, which is equivalent to the kernels
being positive definite for every (Lemma 1; see also  and  (Cor. 58)). This is verified for in  (Thm. 7) and for in  (Thm. 5).
If d is negative definite, then there exists an isometric embedding into a real Hilbert space with and then
Example 7(a) then implies that the kernels
are positive definite as well. This suggests to call a Gaussian process a fractional Brownian motion with parameter space and Hurst index if there exists an with for . Note that yields a realization of fractional Brownian motion with parameter space in the Fock space . If is a homogeneous space, then the map η is called a fractional Brownian helix (cf.  for the terminology).
For various aspects of fractional Brownian motion on , we refer to  and .
The natural analog of the function which generates the realization of the fractional Brownian motion in has a natural higher dimensional analog in , the characteristic function of a half space. Does this correspond to some “fractional Brownian motion” on ?
8.3. Complementary Series of the Conformal Group
The function on is locally integrable if and only if , and it defines a positive definite distribution if and only if ( Lemma 2.13). We thus obtain a family of Hilbert subspaces for . For this space is one-dimensional, consisting of constant functions.
From  (Prop. 6.1) we also know that, for , the distribution is reflection positive with respect to if and only if or .
Let be the conformal group of , considered as a group of diffeomorphisms of the conformal compactification (implemented by a stereographic projection). We consider the kernels
We then have
In fact, this relation is obvious for affine maps , . As the conformal group is generated by the affine conformal group and the inversion in the unit sphere, it now suffices to verify the relation also for . It is a consequence of
where is the reflection in , so that .
As a consequence, we obtain
(see  (Lemma 5.8) for the corresponding relation on the sphere ).
The transformation Formula (31) implies in particular that the conformal cross ratio
on test functions. The same calculation as in  (Lemma 5.8) now implies that defines a unitary representation of G on the space , specified by the scalar product
For , we have in particular
and for the involution we have
Up to the factor , this specializes for and to the representation for . We refer to Appendix D for more detailed discussion of this case.
As the kernel , , on is negative definite, the corresponding kernel
is positive definite. We thus obtain on a positive semidefinite hermitian form by
For , we have
so that the corresponding reproducing kernel Hilber space is . For , we then have
is the center of mass of the measure .
In  (Thm. 7) Takenaga derives some “conformal invariance” of Brownian motion in but it seems that his method only works on the parabolic subgroups of the conformal group stabilizing either 0 or ∞. So it would be interesting to use the complementary series representations of the conformal group to derive a more complete conformal invariance in the spirit of the present paper for .
Similar arguments as in Remark 10 apply in the higher dimensional context: Since the complementary series representations of are irreducible,  (Prop. 5.20) implies that the space of smooth vectors is nuclear. From the proof of  (Prop. 5.20(b)), we further derive that an element is a smooth vector if and only if it is a smooth vector for the maximal compact subgroup . Considering as a space of distributions on the sphere , it is not hard to see that and hence that is the space of distributions on the sphere.
8.4. The Ornstein—Uhlenbeck Process
In this section we describe shortly the connection to the Ornstein–Uhlenbeck process. For that let . Then , , is a stationary Gaussian process realized in . It is the Ornstein–Uhlenbeck process. The corresponding covariance kernel is
which is reflection positive with respect to because the kernel
on is positive definite leading to a one-dimensional Hilbert space via the Osterwalder-Schrader construction (cf. Example 1).
For , we also obtain by , , in a stationary Gaussian process. The corresponding covariance kernel is
We then have
In Proposition 2 we have seen that this function is positive definite if and only if . Hence C is reflection positive for .
Now let so that corresponds to the Ornstein–Uhlenbeck process. In this case is invariant under the reflection . As is cyclic in for the dilation group, there exists a unique unitary isometry V on with for . The latter relation is is equivalent to resp.,
Therefore V coincides with the unitary involution corresponding to the symmetry of Brownian motion under inverstion of t (see Remark 7(a), and also Lemma 6 below).
Note that also defines an isometric involution on having the same intertwining properties with the dilation group as , but this involution does not fix .
To derive a formula for the involution , we recall the Sobolev space .
Let denote the Sobolev space of all absolutely continuous functions satisfying and . Then
is a bijection. We define a real Hilbert space structure on in such a way that I is isometric. The inverse isometry is then given by .
(a) From the relation , it follows that is the real reproducing kernel Hilbert space with kernel , i.e., the covariance kernel of Brownian motion .
(b) We observe that for and follows immediately from the Cauchy–Schwarz inequality and .
There exists a uniquely determined isometric involution on satisfying
It is given by
First we observe that the family , , is total in . Since the family satisfies (Remark 7), there exists a uniquely determined isometry with for . As is also total, is surjective. Now for implies .
Let denote the involutive isometry of specified by , resp., . Then we obtain
(a) Note that (35) has a striking similarity with the formula for one finds in  p. 273. This suggests these operators correspond to a discrete series representation of , hence cannot be implemented in the complementary series representation that we consider.
(b) From the explicit formula for , we can also make the natural map q from to the space more explicit. It is given by
This follows from
Here spans the one-dimensional subspace of -fixed points, so that can be identified with the projection onto .
defines for each on a unitary operator which acts on the point evaluations by
where we put for (), for () and for (). On general functions , the operator acts by
Note that all summands are well defined for , , resp., because implies
The relation for now leads to
and hence to . We thus obtain on a continuous unitary representation of . Note that , so that this representation does NOT factor through a representation of . The most economical way to verify the assertion that the operators are unitary is to do that for and and then to verify that holds for all and or . As is generated by elements of this form, it follows that U defines a unitary representation on .
With the aforementioned conventions concerning expressions of the form for , the representation is given by
Since we want to express this in terms of the derivatives, we observe that, formally, we expect something like
In particular, we have
For , this reads
This formula describes the unique unitary involution on mapping to (cf. Lemma 6).
These authors contribute equally to this work.
Conflicts of Interest
There is no conflict of interest.
Appendix A. Existence of Affine Isometries
For a map into a Hilbert space, the closed subspace generated by all differences , , is called the chordal space of γ (cf. ). The following lemma is an abstraction of  Satz 1,3.
Let X be a non-empty set, be a real or complex Hilbert space and and be maps with and . For , consider the kernel
Then the following are equivalent:
There exists an affine isometry with .
for every .
for some .
If and are real, then these conditions are equivalent to
If (i)–(iii) are satisfied, then the affine isometry V in (i) is uniquely determined by the relation .
(ii) ⇒ (iii) is trivial.
(iii) ⇒ (i): From  (Ch. I) it follows that there exists a unique unitary operator with
Then we put .
(i) ⇒ (ii): If is an affine isometry with , then
(iv) ⇔ (iii): The kernel satisfies
Therefore (iv) is equivalent to . If is real, this is equivalent to (iii). ☐
Appendix B. Stochastic Processes
Let be a probability space and be a measurable space. A stochastic process with state space is a family of measurable functions , where T is a set.
(a) We call the stochastic process full if, up to sets of measure 0, Σ is the smallest σ-algebra for which all functions are measurable.
(b) For or , we say that is square integrable if every is square integrable. Then the covariance kernel
on T is positive definite. If for every , then is called the associated normalized process. Its covariance kernel is
(c) On the product space of all maps , there exists a unique probability measure ν with the property that, for , the image of ν under the evaluation map is the image of μ under the map . We call ν the distribution of the process (, Thm. 1.5).
Let be a centered -valued stochastic process and be a group acting on T.
(a) The process is called stationary if, for every , the process has the same distribution. Then we obtain a measure preserving G-action on the underlying path space by , resp., .
(b) The process is said to have stationary increments if, for , the random vectors
( Def. 2.8.1) A square integrable process is said to be wide sense stationary if the function is constant and there exists a function such that .
Appendix B.1. Processes with Stationary Increments
(The flow of a process with stationary increments) Let be a -valued stochastic process and be a G-action on T. Then the following are equivalent:
has stationary increments.
For every ,
defines a measure preserving flow on the path space satisfying
(i) ⇒ (ii): For each , we consider the map
Since, for every finite subset , the random vector has the same distribution as , the flow on defined by is measure preserving.
(ii) ⇒ (i): If there exists a measure preserving G-action on satisfying (A2), then the distribution of is the same as the distribution of . Subtracting , it follows that the distribution of is the same as the distribution of , i.e., that has stationary increments. ☐
(a) If the -valued process on is square integrable, then generates a closed linear subspace . The existence of a unitary representation on with for , is equivalent to the invariance of the covariance kernel
(cf.  Ch. I). This condition is in particular satisfied if the process is stationary.
(b) For a square integrable process, it likewise follows that the existence of an action of G by affine isometries on the closed affine subspace generated by satisfying
is equivalent to the independence from of the kernel
for some (and hence for all )) (Lemma A1). For a real-valued process (), this condition is equivalent to the G-invariance of the kernel
on (Lemma A1).
Let be a continuous isometric affine -action of the form
on the real or complex Hilbert space . If is total in , then the unitary representation is cyclic.
First proof: We write , where is the closed subspace of U-fixed vectors and . Accordingly, we write . Then is a continuous homomorphism, hence of the form for some . We conclude that , so that it suffices to show that the representation on is cyclic. We may therefore assume from now on that .
Step 1: First we assume that is compact and does not contain 0. Then there exists an such that the operators are invertible for . For , we then have
is independent of t. Now the relation holds for , but since is a continuous cocycle, it follows for all . Clearly, is a U-cyclic vector.
Step 2: Now we consider the general case where is complex. We write , where is relatively compact with . If P is the spectral measure of U, we accordingly obtain a U-invariant decomposition into subspace on which U has compact spectrum not containing 0. Now our assumption implies that every is generated by the values of the -component of . Step 1 now implies that each is cyclic, and since representations on the subspaces are mutually disjoint, the representation on is cyclic.
Step 3: Finally, we consider the general case where is real. Then we may choose the sets such that they are symmetric, i.e., . Then the corresponding spectral subspaces of are invariant under complex conjugation and we can proceed as in Step 2.
Alternative proof: A more direct argument can be derived from the work of P. Masani (; see also ). For the element
one shows that the shift operators
satisfy . Here the main point is to verify first the switching property ( Lemma 2.18)
and that ( Thm. A.2). Then the assertion follows from
(Normal form of cocycles) Let be a continuous unitary one-parameter group and be a continuous cocycle. Then there exists a Borel measure σ on such that the triple is unitarily equivalent to the triple with
In view of Lemma A2, we may assume that the representation is cyclic.
Step 1: First we assume that . According to Bochner’s Theorem, any cyclic unitary one-parameter group with is equivalent to the multiplication representation on some space by . For this representation it is easy to determine the cocycles. They are of the form
where is a measurable function with the property that, for every , the function is square integrable. Replacing by the measure
we may assume that , which leads to .
Step 2: If , then is a continuous homomorphism, hence of the form for some . The cyclicity assumption implies that for the measure . Here the vector v corresponds to the constant function 1, so that .
The assertion now follows by applying Steps 1 and 2 to the summands of the decomposition . ☐
The following theorem is basically the Lévy–Khintchine Theorem for the group (cf.  (Thm 5.5.1),  (Thm. 32), and  for a different form).
Let be a complex-valued zero mean Gaussian process on with and stationary quadratic increments. Then there exists a uniquely determined Borel measure σ on such that
A measure σ on arises for such a process if and only if
is negative definite and satisfies
All other negative definite continuous functions satisfying (A7) are of the form for some .
The measure is called the spectral measure of the process .
Let be the closed affine subspace generated by . As , this is actually a linear subspace. Now Lemma A1 implies the existence of an affine isometric action of on satisfying . In particular, is a corresponding cocycle. Now the existence of follows from Proposition A2.
Now we show that (A6) is equivalent to the square integrability of all (Definition A1) and the continuity of the function .
it follows that the square integrability of all with respect to is equivalent to
If and t is sufficiently small, then the integrand has a positive infimum on the interval . Therefore the finiteness of all implies that all compact subsets of have finite -measure. Since the function is continuous, for every , we have
As the function has a positive infimum on , it follows that . This implies that .
Suppose, conversely, (cf.  Lemma 5.5.1). We claim that we obtain a continuous negative definite function
We first show that the integrals exist. To this end, we observe that
Since all three summands are bounded, the existence of the integral (A8) defining follows. The first two summands are bounded independently of t, and the third summand can also be written as
where the function is bounded. We conclude that all summands are locally uniformly bounded in t. Therefore the continuity of the function r follows from Lebesgue’s Dominated Convergence Theorem. Moreover, r is negative definite because the functions and are.
We further have the relation
showing that is the positive definite kernel associated to the continuous negative definite function r, hence in particular continuous.
Appendix C. Second Quantization and Gaussian Processes
( Def. 1.6) Let T be a set and or . A -valued stochastic process is said to be Gaussian if, for all finite subsets , the corresponding distribution of the random vector with values in is Gaussian.
Let be a -Hilbert space. A Gaussian random process indexed by is a random process on a probability space indexed by such that
is full, i.e., the random variables generate the σ-algebra Σ modulo zero sets.
Each is a Gaussian random variable of mean zero.
If T is a set, a map and is a Gaussian process indexed by , then is a Gaussian process indexed by T with zero means and covariance kernel
For any function , we obtain a Gaussian process with mean vector by
If is total in , then the corresponding Gaussian process is full.
Conversely, every Gaussian process with mean vector is of this form. Here we may choose as the subspace of generated by the ( Thm. 1.10).
(Second quantization; ) For a real Hilbert space , we write for its algebraic dual, i.e., the set of all linear functionals , continuous or not. Let denote the canonical Gaussian measure space on . This measure is defined on the smallest σ-algebra for which all evaluations , , are measurable. It is determined uniquely by
Considering the as random variables, we thus obtain the canonical centered Gaussian process over . It satisfies
(The unitary representation of on ) The group of bijective isometries of has a natural unitary representation on given by
In particular, the map
is -equivariant with total range. The canonical Gaussian process over the real Hilbert space satisfies
The canonical Gaussian measure γ on the algebraic dual is called white noise measure. The space of smooth vectors of the unitary representation of (see  (Ch. 7) for this concept) can be naturally identified with the space , considered as a subspace of . It coincides with the space in Hida’s book  (p. 304).
Appendix D. The Hilbert Spaces ,
Appendix D.1. The Scalar Product on
In this section we give a short discussion about the complementary series representation in the one dimensional case. For detailed discussion see  (p. 28) and  (Sect. 9).
For and , we have
We accordingly obtain
It therefore makes sense to put , so that we have Hilbert spaces for .
In the form (A11), the scalar product is defined by a distribution kernel which is locally integrable for any . We shall use this observation to define Hilbert spaces for . To find a more symmetric form of the scalar product, we calculate
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