Mathematics

I. V. Girsanov

On the Spectra of Dynamical Systems Generated by Stationary Gaussian Processes

(Presented by Academician A. N. Kolmogorov on 21 I 1958)

Let (x(t,\omega)=\int e^{i\lambda t}\Phi(d\lambda,\omega)) be a real stationary (in the narrow sense) process, (F(d\lambda)=M(|\Phi(d\lambda,\omega)|^2)) its spectral measure; the shift
(S_\tau x(t,\omega)=x(t+\tau,\omega)) preserves measure in the space of trajectories of the process and therefore defines a dynamical system, to which there corresponds a group of unitary operators (U^\tau) in the Hilbert space (H) of real functionals of the trajectories of the process.

The spectrum of (U^\tau) is described by the maximal spectral type (\rho) and the multiplicity function (\nu(\lambda)). (We shall everywhere denote a measure and its spectral type by the same letter.) If (\nu(\lambda)\equiv 1), then the spectrum is called simple; we shall say that the spectrum has a simple component if there exists a set (A) such that (\rho(A)>0), (\nu(\lambda)=1) for (\lambda\in A).

In the case of a discrete measure (F(d\lambda)), the structure of the spectrum of (U^\tau) has been studied completely. The aim of the present note is a partial solution of the problem proposed by A. N. Kolmogorov at the seminar on dynamical systems—to investigate the case of a Gaussian process with continuous (F(d\lambda))*, having support without commensurabilities, and also the more general case of a continuous measure whose support has a measurable basis (see § 3).

We shall rely on the following theorem of Ito, proved by him for processes with continuous measure (F(d\lambda)).

Theorem. (H=\oplus\sum\limits_0^\infty H_k), where (H_k) is an invariant subspace with respect to (U^\tau), isomorphic to the space (\widetilde L_k^2) of complex functions
(f(\lambda_1,\ldots,\lambda_k)), square-integrable with respect to the measure
(F(d\lambda_1)\ldots F(d\lambda_k)), symmetric with respect to permutations of
(\lambda_1,\ldots,\lambda_k), and such that**

[
f(\lambda_1,\ldots,\lambda_k)=\overline{f(-\lambda_1,\ldots,-\lambda_k)}.
]

The image of (U^\tau) in (\widetilde L_k^2) is (V_k^\tau), where

[
V_k^\tau f(\lambda_1,\ldots,\lambda_k)=e^{i(\lambda_1+\cdots+\lambda_k)\tau}f(\lambda_1,\ldots,\lambda_k).
]

With the aid of this theorem one can construct the following examples.

§ 1. A process with simple continuous spectrum*. Let (X) be a perfect set on the real line, symmetric

* Everywhere below we shall write “process,” meaning by this a stationary Gaussian process.

** This condition is caused by the reality of the process. The formulation and proof of Ito’s theorem apply to the complex case, but can be easily modified for a real process.

*** In the examples known so far the spectrum is countably multiple (i.e. (\nu(\lambda)=\aleph_0) (see (3)).

relative to zero, and such that among its positive elements there are no rational relations (2); (F_0(d\lambda)) is a continuous finite measure, symmetric with respect to zero, whose support is (X) (the distribution function of such a measure can be constructed after the manner of the “Cantor staircase”). Then the process with spectral measure (F(d\lambda)=F_0(d\lambda)) has a simple spectrum; its maximal spectral type will be, as was proved in (3),

[
\rho=\sum_{1}^{\infty} F_0^i,
]

where (F_0^i=F_0\ldotsF_0) is the (i)-fold composition of the measure (F_0); the cyclic vector in (H) can be expanded in a series of Itô stochastic integrals of all orders.

The proof of this follows from the following facts: first, the measures (F_0^i) are mutually singular (see Lemma 1); second, every function from (\bar L_k^2) can be replaced by an equivalent function of the sum (\lambda_1+\ldots+\lambda_k). This latter fact can be done because the set

[
\underbrace{X\otimes\ldots\otimes X}_{k}
]

in (k)-dimensional Euclidean space is projected onto the axis (\lambda_1=\lambda_2=\ldots=\lambda_k) one-to-one up to permutations of coordinates, except for a set of (F_0^k)-measure zero on this axis.

p. 2. A process with a nonhomogeneous spectrum. Put now

[
F(d\lambda)=F_0(d\lambda)+F_0^2(d\lambda).
]

The corresponding Gaussian process will have the same maximal type (\rho) and

[
\nu(\lambda)=
\sum_{k=\left[\frac{n+1}{2}\right]}^{n}
\frac{n!}{2^{\,n-k}(n-k)!(2k-n)!}
\quad
\text{for } \lambda\in X^n\setminus\sum_{1}^{n-1}X^k,
]

where (X^i) denotes the totality of points (\lambda=\lambda_1+\ldots+\lambda_i,\ \lambda_r\in X). An analogous, but more cumbersome, formula can be obtained for

[
F=\sum_{i=1}^{\infty} F_0^{p_i}.
]

Examples are easily constructed: 1) with (\nu(\lambda)) taking only the values (1) and (\aleph_0); 2) a process for which (\nu(\lambda)\equiv 1), but rational relations exist between the points of the support of the measure (F(d\lambda)).

p. 3. We shall say that a measure (F) has a measurable basis if there exists a set (X), among whose positive elements there are no rational relations, measurable together with all (X^i,\ i\geqslant 1), and such that

[
F(\Delta)=F\left(\Delta\cap\bigcup_{1}^{\infty} X^k\right)
]

for every measurable (\Delta). The set (X) is called a basis of the measure (F).

By the order of the measure (F) relative to the basis (X) we shall mean the largest (k) such that (F(X^i-h)=0) for (i<k) and all (h) ((X^i-h) denotes the set (X^i) shifted by (h)). If the measure is discrete, then we put its order equal to 0.

Theorem 1. If the spectral measure of the process (x(t,\omega)) is continuous and has finite order, then the spectrum of the dynamical system determined by this process contains a simple component.

In the proof of Theorem 1 the following lemmas are used; they are also of independent interest.

Lemma 1. Let (G_1,\ldots,G_s) be finite measures with orders (k_1,\ldots,k_s) relative to the basis (X). Then the order of (G=G_1\ldotsG_s) is equal to (k_1+\ldots+k_s).

Lemma 2. If the spectral measure (F) of the process (x(t,\omega)) is subordinated*

* Independently of the author, this result was obtained by Yu. Rozanov.

(\sum_2^\infty F^i), then (\nu(\lambda)\equiv \aleph_0); if the measure (F) contains a component singular with respect to (\sum_2^\infty F^i), then the spectrum contains a simple component.

To prove Theorem 1, denote by (k) the order of (F) and note that, by Lemma 1, the order of (\sum_2^\infty F^i) is (2k>k), since (k\geqslant 1) by the continuity of (F). Consequently, (F) contains a component singular with respect to (\sum_2^\infty F^i), and we may apply Lemma 2.

Corollary. A continuous measure with a measurable basis always has finite order; hence it follows that there is a simple component in the spectrum of the corresponding process.

Theorem 2. The multiplicity function (\nu(\lambda)) of the spectrum of the dynamical system of a process with continuous (F(d\lambda)) is either identically equal to 1 or increases without bound.

Indeed, if (\nu(\lambda)>1) on a set of positive measure
[
\rho=\sum_1^\infty F^i,
]
then one can find two elements (\varphi_1) and (\varphi_2), lying in (H_{k_1}) and (H_{k_2}), respectively (where, possibly, (k_1=k_2)), such that the subspaces spanned by (U^\tau\varphi_1) and (U^\tau\varphi_2) are orthogonal, while (U^\tau) has in them one and the same spectral type (\rho_0); by Itô’s theorem, (\varphi_1) and (\varphi_2) correspond to functions (f_1(\lambda_1,\ldots,\lambda_{k_1})) and (f_2(\lambda_1,\ldots,\lambda_{k_2})) in (\tilde L^2_{k_1}) and (\tilde L^2_{k_2}). Then all
[
f_{p,q}
=
f_1(\lambda_1^{(1)},\ldots,\lambda_{k_1}^{(1)})\cdots
f_1(\lambda_1^{(p)},\ldots,\lambda_{k_1}^{(p)})
f_2(\lambda_1^{(p+1)},\ldots,\lambda_{k_2}^{(p+1)})\cdots
f_2(\lambda_1^{(p+q)},\ldots,\lambda_{k_2}^{(p+q)})
]
for different (p,q) such that (p+q=s,\ p\geqslant 0,\ q\geqslant 0), generate orthogonal invariant subspaces with spectral type (\rho_0^s), and therefore (\nu(\lambda)\geqslant s+1) on some set of positive measure.

In conclusion, I thank A. N. Kolmogorov for advice and guidance in carrying out the present work.

Moscow State University
named after M. V. Lomonosov

Received
8 I 1958

CITED LITERATURE

(^{1}) K. Itô, Japan. J. Math., 22, 63 (1952).
(^{2}) J. Neymann, Math. Ann., 99, No. 1, 134 (1928).
(^{3}) S. V. Fomin, Ukr. matem. zhurn., 2, 2, 25 (1950).