On the Misra-Prigogine-Courbage theory of irreversibility 2. The existence of the nonunitary similarity

Nicolae Suciu
(original), paper

Abstract

Stochastic processes and dynamical systems in measure spaces are defined as classes of random variables in the Doob sense. Markov processes which are ergodic iato a “strong”  sense are shoiwn to be suitable models for the thermodynamic irreverisilitiy. These processes are isomorphic, in the Doob sense, with Kolmogorov dynamical systems into the speces of trajectories. In this approach, we show that the Misra-Prigogine-Courbage theory of irreversibility can be formulated as a change of representation, from strong  egodic. Markov processes to dynamical systems into the space of trajectories. The physical meaning is that all strong ergodic Markov processes, describing experimentally observed irreversibility, can be formally presented as unitary “superdynamics”.

Authors

N. Suciu
Tiberiu Popoviciu Institute of Numerical Analysis

Adelina Georgescu
University of Pitești

Keywords

Paper coordinates

N. Suciu, (2000), On the Misra-Prigogine-Courbage theory of irreversibility 2. The existence of the nonunitary similarity, Buletin ştiintific, Seria Mat.şi Inf. (Univ. Piteşti), 6, 213-222.

References

see the expansion block below.

PDF

soon

About this paper

Journal

Buletin ştiintific, Seria Mat. Inf.

Publisher Name

Univ. Piteşti

DOI

Not available yet.

Print ISSN

Not available yet.

Online ISSN

Not available yet.

Google Scholar Profile

soon

[1]  I. Antoniou and S. Tasaki,  Generalized Spectral Decomposition of the β-adic Baker’s Transformation and Intrinsec Irreveribility, Physica A, 190, 303-329, 1992.
[2] I. Antoniou and K. E. Gustafson,  From Probabilistic Description to Deterministic Dynamics, Physica A 179, 153-166, 1993.
[3] I. Antoniou and I. Prigogine,  Intrinsec Irreversibility and Integrability of Dynamics, Physica A 192, 443-464, 1993.
[4] I Antoniou and S. Tasaki,  Generalized Spectral Decomposition of Mixing Dynamical Systems,  Int. J. Quantum Chem., 46, 425-474, 1993.
[5] I. Antoniou and K. E. Gustafson, I From Irreversibile Markov Semigroups to Chaotic Dynamics, Physica A 236, 296-308, 1997.
[6] I. Antoniou, K. E. Gustafson and Z. Suchanecki,  On the Inverse Problem of Statistical Physics: From Irreversible Semigroups to Chaotic Dynamics, Physica A 256, 345-361, 1998.
[7] M. Courbage,  On Boltzmann Entropy and Coarse-Graining for Classical Dynamical Systems,  in Information Dynamics, ed. H. Atmanspacher and H. Scheingraber, Plenum New York, 1991.
[8] I. Cornfeld, S. Fomin and  Ya. G. Sinai,  Ergodic Theory,  Springer, Berlin, 1982.
[9] C. Gardiner,  Handbook of Stochastic Methods (for Physic, Chemstry and Natural Science), Springer, Berlin, 1983.
[10] N. A Friedman and D. S. Ornstein, On Isomorphism of Weak Bernoulli Transformations,  Adv. Math. 5 (1971), 365-394.
[11] S. Goldstein, J. L. Lebowitz and E. Presutti,  Stationary Markov Chains,  in Fritz, J. J. L. Lebowitz and D. Szasz, editors, Rigorous results in statistical mechanics and quantum field theory, Colloquia Matematica Societatis Janos Bolyai, 27, North Holland, Amsterdam, 1981.
[12] M. Iosifescu and P. Tăutu, Stochastic Processes and Applicationsin Biology and Medicine. I. Theory,  Editura Academiei, București and Springer, Berlin, 1973, 1970.
[13] A. Kolmogorov and S. Fomine,  Elementes de la theorie des fonctions et de l’analyse fonctionelle,  Mir. Moscou. 1975.
[14] A. Lasota and M. C. Meckey,  Probabilistic Properties of Deterministic Systems,  Springer, New York, 1985; Second edition and A. Lasota and M. C. Meckey, Chaos Fractals, and Noise, Stochastic Aspects of Dynamics, Springer, New York, 1994.
[15] M. C. Mackey,  The Dynamic Origin of Increasing Entropy,  Rev. Modern Pgys., 61, 4 (1989), 981-1016.
[16] P. Malliavin,  Integration and Probability,  Springer, New York, 1995.
[17] R. McCabe and P. Schields,  A Class of Markov Shifts which are Bernoullu Shifts,  Adv. Mat., 6 (1971), 323-328.
[18] B. Misra, I. Prigogine and M. Courbage, I From Deterministic Dynamics to Probabilistic Description, Physica 98A (1979), 1-26.
[19] G. Nicolis and C. Nicolis,  Chaotic Dynamics, Markovian Coarse-Graining and Information, Physica A, 163 (1990), 215-231.
[20] D. S. Ornstein and P. C. Shields, Mixing Markov Shifts of Kernel Type are Bernoulli,  Advances in Mathematics, 10 (1973), 143-146.
[21] V. A. Rokhlin,  Exact Endomorphisms of Lebesque Spaces,  Am. Math. Soc. Transl., (2) 39 (1964), 1-36.
[22] Ya. G. Sinai,  Introduction to Ergodic Theory, Princeton, Univ. Press., 1976.
[23] N. Suciu and A. Georgescu, On the Misra-Prigogine-Courbage Theory of Irreversibility,  Bul. Șt. Univ. Pitești, Seria Mat. Inf. 2 (1990), 169-188.
[24] B. Sz-Nagy and C. Foiaș,  Harmonic Analyse of Operators in Hilbert Spaces,  North-Holland, Amsterdam, 1970.
[25] N. G. van Kampen,  Stochastic Processes in Physics and Chemistry,  North-Holland, Amsterdam, 1981.
[26] A. D. Wentzell,  A Course in the Theory of Stochastic Processes,  McGraw-Hill, New York, 1981.

2000c Suciu - On the Misra-Prigogine-Courbage theory of irreversibility 2

ON THE MISRA-PRIGOGINE-COURBAGE THEORY OF IRREVERSIBILITY 2. THE EXISTENCE OF THE NONUNITARY SIMILARITY

Nicolae SUCIU"T.Popoviciu" Institute of Numerical Analysis, Cluj-Napocansuciu@ictp-acad.math.ubbcluj.roAdelina GEORGESCUUniversity of Pitesti, Faculty of Sciences, Department of Applied Mathematicsadgeorg@electra.upit.ro

Abstract

Stochastic processes and dynamical systems in measure spaces are defined as classes of random variables in the Doob sense. Markov processes which are ergodic into a "strong" sense are shown to be suitable models for the thermodynamic irreversibility. These processes are isomorphic, in the Doob sense, with Kolmogorov dynamical systems into the space of trajectories. In this approach, we show that the Misra-Prigogine-Courbage theory of irreversibility can be formulated as a change of representation, from strong ergodic Markov processes to dynamical systems into the space of trajectories. The physical meaning is that all strong ergodic Markov processes, describing experimentally observed irreversibility, can be formally presented as unitary "superdynamics".

1. RECENT DEVELOPMENTS OF THE MPC-THEORY

The paper of Misra, Prigogine and Courbage, entitled From Deterministic Dynamics to Probabilistic Description, [18] was followed by many attempts to obtain thermodynamic irreversibility using dynamical systems as models of physical processes. Herein after, such approaches will be referred to as MPC-theory. The irrevesibility models are given by "strong" Markov semigroups of operators (i.e. with an unique fixed point attracting the whole definition function space and, consequently yielding H-theorem and increasing entropy law). The aim of the MPC-Theory is to find suitable
("unstable" or "chaotic") dynamical systems whose evolution group of operators (Frobenius-Perron or Koopman), { U t } t ∈ R U t t ∈ R {U_(t)}_(t inR)\left\{U_{t}\right\}_{t \in \mathbb{R}}{Ut}t∈R, may be related to strong Markov semigroups, { K t } t ≥ 0 K t t ≥ 0 {K_(t)}_(t >= 0)\left\{K_{t}\right\}_{t \geq 0}{Kt}t≥0, by the intertwining relation
K t Φ = Φ U t , t ≥ 0 K t Φ = Φ U t , t ≥ 0 K_(t)Phi=PhiU_(t),t >= 0K_{t} \Phi=\Phi U_{t}, t \geq 0KtΦ=ΦUt,t≥0
In [18], pp.13-17, Φ Φ Phi\PhiΦ is an invertible mapping between the corresponding L 2 L 2 L^(2)L^{2}L2 function spaces and { U t } t ∈ R U t t ∈ R {U_(t)}_(t inR)\left\{U_{t}\right\}_{t \in \mathbb{R}}{Ut}t∈R the Frobenius-Perron group of operators corresponding to a Bernoulli shift (the symbolic dynamical system associated with the "baker mapping", i.e. a Bernoulli translation on a space of double infinite sequences). The relation (1) is said to be a nonunitary similarity. When Φ Φ Phi\PhiΦ is not invertible, (1) defines a coarse graining projection, which was proved for the larger class of K-systems (i.e. dynamical systems carring a given partition into the finest one-point partition for t → ∞ t → ∞ t rarr oot \rightarrow \inftyt→∞ and into the coarsest one-cell partition for t → − ∞ t → − ∞ t rarr-oot \rightarrow-\inftyt→−∞ ). The model example is a square mapping obtained by projecting "baker mapping" onto the cells of a K-partition [3, 7, 19].
While the coarse-graining rises no problems (irreversibility obtained by projecting a dynamical system could be interpreted as a loss of information in passing from microscopic to macroscopic description), the "similarity seems to be paradoxical because all dynamical systems preserve the entropy constant ([14], chap.9). To avoid that, we note that the irreversible evolution operators K t K t K_(t)K_{t}Kt act in L 2 ( Y ) L 2 ( Y ) L^(2)(Y)L^{2}(Y)L2(Y) (a space of functions defined in a state space Y Y YYY, where the usual reversible description is done by Liouville equation). The reversible dynamical system acts itself in a function space (for instance the space of real functions Y R Y R Y^(R)Y^{\mathbb{R}}YR ) and the corresponding operators U t U t U_(t)U_{t}Ut in L 2 ( Y R ) L 2 Y R L^(2)(Y^(R))L^{2}\left(Y^{\mathbb{R}}\right)L2(YR) and does not provide a model of any classical statistical mechanics system. Through (1) only a formal "dynamical" representation (in Y R Y R Y^(R)Y^{\mathbb{R}}YR ) of irreversible process is aimed, which does not contradict the entropy law of classical system evolving in Y Y YYY. That is why, most recent developments of MPC-theory go backwards, from probabilistic description to deterministic dynamics. So, Antoniou and Gustafson [2] try to answer the more physical relevant question "when experimentally observed strong Markov semigroups, like diffusion or chemical reactions, can be lifted to some unitary superdynamics". These approaches are made, following the theory of Sz.-Nagy and Foias [24], in terms of "positive unitary dilations". Results are obtained for "exact Markov semigroup" associated with square maps (using also the Rokhlin theorem on "natural extension of exact dynamical systems" [21]) in [2] and for more generally stationary Markov processes (by the extension of probability measures) in [ 5 , 6 ] [ 5 , 6 ] [5,6][5,6][5,6].
In a previous paper [23], we have proved the existence of positive unitary dilations for all stationary Markov processes. This result comes from the idea that unambiguous comparation between dynamical systems and
on group of opelated to strong
e corresponding up of operators 1 system associon on a space of nonunitary simning projection, namical systems a for t → ∞ t → ∞ t rarr oot \rightarrow \inftyt→∞ and del example is a onto the cells of
ility obtained by ss of information , the "similarity preserve the ent the irreversible defined in a state y Liouville equanction space (for ding operators U t U t U_(t)U_{t}Ut atistical mechansentation (in Y R Y R Y^(R)Y^{\mathbb{R}}YR ) t t ttt the entropy law ent developments tion to determinanswer the more ed strong Markov ifted to some uniwing the theory of ilations". Results with square maps of exact dynamical Markov processes
ce of positive uniresult comes from mical systems and
stochastic processes is possible when both are defined as random variables in suitable "phase" spaces. Now we prove that nonunitary similarity also exists for all strong-ergodic Markov processes defined by Gardiner [9].

2. DEFINITIONS AND PRELIMINARY RESULTS

Some definitions and three Lemmas from [23] are necessary in order to prove the Theorem from the next section.
Both stochastic processes and dynamical systems are equivalence classes of random variables. The spaces necessary to define random variables are: the complete probability space ( Ω , A , P ) ( Ω , A , P ) (Omega,A,P)(\Omega, \mathcal{A}, P)(Ω,A,P), and the measurable space ( X , B ) ( X , B ) (X,B)(X, \mathcal{B})(X,B), where A A A\mathcal{A}A and B B B\mathcal{B}B are the corresponding σ σ sigma\sigmaσ-algebra and P P PPP is a probability measure.
A random variable is a ( A , B ) ( A , B ) (A,B)(\mathcal{A}, \mathcal{B})(A,B) measurable function η : Ω → X η : Ω → X eta:Omega rarr X\eta: \Omega \rightarrow Xη:Ω→X. The measurability condition is, usually, written as { η ∈ B } ∈ A , ∀ B ∈ B { η ∈ B } ∈ A , ∀ B ∈ B {eta in B}inA,AA B inB\{\eta \in B\} \in \mathcal{A}, \forall B \in \mathcal{B}{η∈B}∈A,∀B∈B, where { η ∈ B } = { ω ∈ Ω ∣ η ( ω ) ∈ B } { η ∈ B } = { ω ∈ Ω ∣ η ( ω ) ∈ B } {eta in B}={omega in Omega∣eta(omega)in B}\{\eta \in B\}=\{\omega \in \Omega \mid \eta(\omega) \in B\}{η∈B}={ω∈Ω∣η(ω)∈B}. Hence the preimages through η η eta\etaη of the elements of B B B\mathcal{B}B are elements of A A A\mathcal{A}A. A measurable function is, necessarily, surjective because { η ∈ X } ∈ A { η ∈ X } ∈ A {eta in X}inA\{\eta \in X\} \in \mathcal{A}{η∈X}∈A.
The distribution of the random variable η η eta\etaη is the measure on B B B\mathcal{B}B given by
P η ( B ) = P ( { η ∈ B } ) , ∀ B ∈ B P η ( B ) = P ( { η ∈ B } ) , ∀ B ∈ B P_(eta)(B)=P({eta in B}),AA B inBP_{\eta}(B)=P(\{\eta \in B\}), \forall B \in \mathcal{B}Pη(B)=P({η∈B}),∀B∈B
The space ( Ω , A , P Ω , A , P Omega,A,P\Omega, \mathcal{A}, PΩ,A,P ) is usually called the basic probability space and the sets A ∈ A A ∈ A A inAA \in \mathcal{A}A∈A are rcferred to as events [26]. Since the measure P η P η P_(eta)P_{\eta}Pη satisfies the equalities P η ( X ) = P ( { η ∈ X } ) = P ( Ω ) = 1 P η ( X ) = P ( { η ∈ X } ) = P ( Ω ) = 1 P_(eta)(X)=P({eta in X})=P(Omega)=1P_{\eta}(X)=P(\{\eta \in X\})=P(\Omega)=1Pη(X)=P({η∈X})=P(Ω)=1, it follows that ( X , B , P η ) X , B , P η (X,B,P_(eta))\left(X, \mathcal{B}, P_{\eta}\right)(X,B,Pη) is also a probability space, called the phase space. For the sake of simplicity X X XXX is also called the phase space. The sets B ∈ B B ∈ B B inBB \in \mathcal{B}B∈B are called the realizations of the random variable.
The useful probabilistic description is done by the expectations of quantities defined by composed functions of random variables, i.e. the Lebesgue integral
(1) M f ( η ) = ∫ Ω f ( η ( ω ) ) P ( d ω ) (1) M f ( η ) = ∫ Ω   f ( η ( ω ) ) P ( d ω ) {:(1)Mf(eta)=int_(Omega)f(eta(omega))P(d omega):}\begin{equation*} M f(\eta)=\int_{\Omega} f(\eta(\omega)) P(d \omega) \tag{1} \end{equation*}(1)Mf(η)=∫Ωf(η(ω))P(dω)
Due to a change of variables formula, it follows that the random variables whose distribution are identical have identical expectations [16], p. 180.
Two probability spaces are isomorphic, ( Ω 1 , A 1 , P 1 ) ∼ ( Ω 2 , A 2 , P 2 ) Ω 1 , A 1 , P 1 ∼ Ω 2 , A 2 , P 2 (Omega_(1),A_(1),P_(1))∼(Omega_(2),A_(2),P_(2))\left(\Omega_{1}, \mathcal{A}_{1}, P_{1}\right) \sim\left(\Omega_{2}, \mathcal{A}_{2}, P_{2}\right)(Ω1,A1,P1)∼(Ω2,A2,P2), if there exists a bijective and bimeasurable mapping Θ : A 1 ⟼ A 2 Θ : A 1 ⟼ A 2 Theta:A_(1)longmapstoA_(2)\Theta: \mathcal{A}_{1} \longmapsto \mathcal{A}_{2}Θ:A1⟼A2 which preserves the probability measure, i.e. P 1 ( A 1 ) = P 2 ( Θ ( A 1 ) ) ( [ 8 ] , § I , 1 ) P 1 A 1 = P 2 Θ A 1 ( [ 8 ] , § I , 1 ) P_(1)(A_(1))=P_(2)(Theta(A_(1)))([8],§I,1)P_{1}\left(A_{1}\right)=P_{2}\left(\Theta\left(A_{1}\right)\right)([8], § \mathrm{I}, 1)P1(A1)=P2(Θ(A1))([8],§I,1).
Let A η = σ { { η ∈ B } ∣ B ∈ B } A η = σ { { η ∈ B } ∣ B ∈ B } A_(eta)=sigma{{eta in B}∣B inB}\mathcal{A}_{\eta}=\sigma\{\{\eta \in B\} \mid B \in \mathcal{B}\}Aη=σ{{η∈B}∣B∈B} be the σ σ sigma\sigmaσ-algebra on Ω Ω Omega\OmegaΩ generated by sets { η ∈ B } { η ∈ B } {eta in B}\{\eta \in B\}{η∈B}. The ( A , B A , B A,B\mathcal{A}, \mathcal{B}A,B ) measurability of η η eta\etaη implies A η ⊂ A A η ⊂ A A_(eta)subA\mathcal{A}_{\eta} \subset \mathcal{A}Aη⊂A. The set Ω Ω Omega\OmegaΩ endowed with the σ σ sigma\sigmaσ-algebra A η A η A_(eta)\mathcal{A}_{\eta}Aη and the measure P A η P A η P_(A_(eta))P_{\mathcal{A}_{\eta}}PAη, defined as the restriction of P P PPP to A η A η A_(eta)\mathcal{A}_{\eta}Aη is referred to as the minimal probability space, ( Ω , A η , P A η Ω , A η , P A η Omega,A_(eta),P_(A_(eta))\Omega, \mathcal{A}_{\eta}, P_{\mathcal{A}_{\eta}}Ω,Aη,PAη ), of the random variable η η eta\etaη. By the construction of the minimal probability space the
mapping η ~ : B ⟼ A η , η ~ ( B ) = { η ∈ B } = A ~ , ∀ B ∈ B η ~ : B ⟼ A η , η ~ ( B ) = { η ∈ B } = A ~ , ∀ B ∈ B tilde(eta):BlongmapstoA_(eta), tilde(eta)(B)={eta in B}= tilde(A),AA B inB\tilde{\eta}: \mathcal{B} \longmapsto \mathcal{A}_{\eta}, \tilde{\eta}(B)=\{\eta \in B\}=\tilde{A}, \forall B \in \mathcal{B}η~:B⟼Aη,η~(B)={η∈B}=A~,∀B∈B, is bijective, bimeasurable and measure preserving, thus we have ( Ω , A η , P A η ) ∼ ( X , B , P η ) Ω , A η , P A η ∼ X , B , P η (Omega,A_(eta),P_(A_(eta)))∼(X,B,P_(eta))\left(\Omega, \mathcal{A}_{\eta}, P_{\mathcal{A}_{\eta}}\right) \sim\left(X, \mathcal{B}, P_{\eta}\right)(Ω,Aη,PAη)∼(X,B,Pη). From this isomorphism it follows that an equivalence class of random variables is defined by a given phase space ( X , B , P B X , B , P B X,B,P_(B)X, \mathcal{B}, P_{\mathcal{B}}X,B,PB ). Consequently, when a random variable is defined by "a probability space called phase space" [3], pp.178, 188, its meaning is this equivalence class.
A random function in the Doob sense is a random variable valued into a functions space,
η : Ω ⟼ Y Λ , η ( ω ) ∈ Y Λ , ω ∈ Ω . η : Ω ⟼ Y Λ , η ( ω ) ∈ Y Λ , ω ∈ Ω . eta:Omega longmapstoY^(Lambda),quad eta(omega)inY^(Lambda),omega in Omega.\eta: \Omega \longmapsto Y^{\Lambda}, \quad \eta(\omega) \in Y^{\Lambda}, \omega \in \Omega .η:Ω⟼YΛ,η(ω)∈YΛ,ω∈Ω.
For fixed ω ω omega\omegaω, the graph of the function y ω : Λ ⟼ Y , η ( ω ) = y ω y ω : Λ ⟼ Y , η ( ω ) = y ω y^(omega):Lambda longmapsto Y,eta(omega)=y^(omega)y^{\omega}: \Lambda \longmapsto Y, \boldsymbol{\eta}(\omega)=y^{\omega}yω:Λ⟼Y,η(ω)=yω, is a trajectory and its values, y ω ( λ ) = y λ y ω ( λ ) = y λ y^(omega)(lambda)=y_(lambda)y^{\omega}(\lambda)=y_{\lambda}yω(λ)=yλ, are points in Y Y YYY. In order to avoid misunderstandings, we call Y Y YYY the state space and use the "phase space" for X = Y Λ X = Y Λ X=Y^(Lambda)X=Y^{\Lambda}X=YΛ only. Thus the realizations of the random function η η eta\etaη are sets of trajectories in Y Λ Y Λ Y^(Lambda)Y^{\Lambda}YΛ. For fixed λ λ lambda\lambdaλ, the function η λ : Ω ⟼ Y , η λ ( ω ) = y ω ( λ ) η λ : Ω ⟼ Y , η λ ( ω ) = y ω ( λ ) eta_(lambda):Omega longmapsto Y,eta_(lambda)(omega)=y^(omega)(lambda)\eta_{\lambda}: \Omega \longmapsto Y, \eta_{\lambda}(\omega)=y^{\omega}(\lambda)ηλ:Ω⟼Y,ηλ(ω)=yω(λ), is a random variable whose phase space coincides with the state space Y Y YYY. If Λ ⊆ R Λ ⊆ R Lambda subeR\Lambda \subseteq \mathbb{R}Λ⊆R and λ λ lambda\lambdaλ means time, then the random function is a stochastic process. If Λ ⊆ R d Λ ⊆ R d Lambda subeR^(d)\Lambda \subseteq \mathbb{R}^{d}Λ⊆Rd the random function is a d-dimensional random field.
In order to define the distribution P η P η P_(eta)P_{\eta}Pη in infinite dimensional function spaces one uses the joint distributions of finite dimensional random vectors ( η λ 1 , … , η λ n η λ 1 , … , η λ n eta_(lambda_(1)),dots,eta_(lambda_(n))\eta_{\lambda_{1}}, \ldots, \eta_{\lambda_{n}}ηλ1,…,ηλn ) on Y n Y n Y^(n)Y^{n}Yn. They are called finite dimensional distributions and are measures on the σ σ sigma\sigmaσ-algebra B n B n B^(n)\mathcal{B}^{n}Bn of Y n Y n Y^(n)Y^{n}Yn defined by
(2) P λ 1 … λ n ( B ) = P ( { η λ 1 ∈ B 1 , … , η λ n ∈ B n } ) , B ∈ B n , B i ∈ B . (2) P λ 1 … λ n ( B ) = P η λ 1 ∈ B 1 , … , η λ n ∈ B n , B ∈ B n , B i ∈ B . {:(2)P_(lambda_(1)dotslambda_(n))(B)=P({eta_(lambda_(1))inB_(1),dots,eta_(lambda_(n))inB_(n)})","B inB^(n)","B_(i)inB.:}\begin{equation*} P_{\lambda_{1} \ldots \lambda_{n}}(B)=P\left(\left\{\eta_{\lambda_{1}} \in B_{1}, \ldots, \eta_{\lambda_{n}} \in B_{n}\right\}\right), B \in \mathcal{B}^{n}, B_{i} \in \mathcal{B} . \tag{2} \end{equation*}(2)Pλ1…λn(B)=P({ηλ1∈B1,…,ηλn∈Bn}),B∈Bn,Bi∈B.
When (2) satisfy the consistency conditions P λ i 1 , … , λ i n ( B i 1 × … × B i n ) = P λ 1 … λ n ( B 1 × … × B n ) P λ i 1 , … , λ i n B i 1 × … × B i n = P λ 1 … λ n B 1 × … × B n P_(lambda_(i_(1)),dots,lambda_(i_(n)))(B_(i_(1))xx dots xxB_(i_(n)))=P_(lambda_(1)dotslambda_(n))(B_(1)xx dots xxB_(n))P_{\lambda_{i_{1}}, \ldots, \lambda_{i_{n}}}\left(B_{i_{1}} \times \ldots \times B_{i_{n}}\right)= P_{\lambda_{1} \ldots \lambda_{n}}\left(B_{1} \times \ldots \times B_{n}\right)Pλi1,…,λin(Bi1×…×Bin)=Pλ1…λn(B1×…×Bn), for any permutation { i 1 , … , i n } i 1 , … , i n {i_(1),dots,i_(n)}\left\{i_{1}, \ldots, i_{n}\right\}{i1,…,in} of { 1 , … , n } { 1 , … , n } {1,dots,n}\{1, \ldots, n\}{1,…,n} and any B 1 , … , B n ∈ B B 1 , … , B n ∈ B B_(1),dots,B_(n)inBB_{1}, \ldots, B_{n} \in \mathcal{B}B1,…,Bn∈B, and P λ 1 … λ n λ n + 1 ( B 1 × … × B n × Y ) = P λ 1 … λ n ( B 1 × … × B n ) P λ 1 … λ n λ n + 1 B 1 × … × B n × Y = P λ 1 … λ n B 1 × … × B n P_(lambda_(1)dotslambda_(n)lambda_(n+1))(B_(1)xx dots xxB_(n)xx Y)=P_(lambda_(1)dotslambda_(n))(B_(1)xx dots xxB_(n))P_{\lambda_{1} \ldots \lambda_{n} \lambda_{n+1}}\left(B_{1} \times \ldots \times B_{n} \times Y\right)=P_{\lambda_{1} \ldots \lambda_{n}}\left(B_{1} \times \ldots \times B_{n}\right)Pλ1…λnλn+1(B1×…×Bn×Y)=Pλ1…λn(B1×…×Bn), the 'Kolmogorov Theorem on finite dimensional distributions' ensures the existence of a probability measure obeying P η ( C n ) = P λ 1 … λ n ( B ) P η C n = P λ 1 … λ n ( B ) P_(eta)(C_(n))=P_(lambda_(1)dotslambda_(n))(B)P_{\eta}\left(C_{n}\right)=P_{\lambda_{1} \ldots \lambda_{n}}(B)Pη(Cn)=Pλ1…λn(B), for any C n ∈ B Λ C n ∈ B Λ C_(n)inB^(Lambda)C_{n} \in \mathcal{B}^{\Lambda}Cn∈BΛ, where B Λ B Λ B^(Lambda)\mathcal{B}^{\Lambda}BΛ is the smallest σ σ sigma\sigmaσ-algebra containing all the cylindrical sets, C n = { y λ ∣ ( y λ 1 , … , y λ n ) ∈ B , B ∈ B n , λ 1 , … , λ n ∈ Λ } C n = y λ ∣ y λ 1 , … , y λ n ∈ B , B ∈ B n , λ 1 , … , λ n ∈ Λ C_(n)={y_(lambda)∣(y_(lambda_(1)),dots,y_(lambda_(n)))in B,B inB^(n),lambda_(1),dots,lambda_(n)in Lambda}C_{n}=\left\{y_{\lambda} \mid\left(y_{\lambda_{1}}, \ldots, y_{\lambda_{n}}\right) \in B, B \in \mathcal{B}^{n}, \lambda_{1}, \ldots, \lambda_{n} \in \Lambda\right\}Cn={yλ∣(yλ1,…,yλn)∈B,B∈Bn,λ1,…,λn∈Λ}, and B n B n B^(n)\mathcal{B}^{n}Bn is the σ σ sigma\sigmaσ-algebra in R Λ R Λ R^(Lambda)\mathbb{R}^{\Lambda}RΛ. Then an equivalence class of random functions in the Doob sense η η eta\etaη is defined, up to an isomorphism by the space ( R Λ , B Λ , P η R Λ , B Λ , P η R^(Lambda),B^(Lambda),P_(eta)\mathbb{R}^{\Lambda}, \mathcal{B}^{\Lambda}, P_{\eta}RΛ,BΛ,Pη ) [12], p. 166.
Lemma 1. If η : Ω ⟼ Y I , I ⊆ R η : Ω ⟼ Y I , I ⊆ R eta:Omega longmapstoY^(I),I subeR\eta: \Omega \longmapsto Y^{I}, I \subseteq \mathbb{R}η:Ω⟼YI,I⊆R, is a stochastic process, defined on the state space ( Y , B ) , Y ⊆ R ( Y , B ) , Y ⊆ R (Y,B),Y subeR(Y, \mathcal{B}), Y \subseteq \mathbb{R}(Y,B),Y⊆R, density of the n n nnn-dimensional distributions are functions p ∈ L 1 ( Y n ) , Y ⊆ R p ∈ L 1 Y n , Y ⊆ R p inL^(1)(Y^(n)),Y subeRp \in L^{1}\left(Y^{n}\right), Y \subseteq \mathbb{R}p∈L1(Yn),Y⊆R, given by
(3) p ( y 1 , λ 1 ; … ; y n , λ n ) = M [ δ ( y 1 − η λ 1 ( ω ) ) … δ ( y n − η λ n ( ω ) ) ] . (3) p y 1 , λ 1 ; … ; y n , λ n = M δ y 1 − η λ 1 ( ω ) … δ y n − η λ n ( ω ) . {:(3)p(y_(1),lambda_(1);dots;y_(n),lambda_(n))=M[delta(y_(1)-eta_(lambda1)(omega))dots delta(y_(n)-eta_(lambda n)(omega))].:}\begin{equation*} p\left(y_{1}, \lambda_{1} ; \ldots ; y_{n}, \lambda_{n}\right)=M\left[\delta\left(y_{1}-\eta_{\lambda 1}(\omega)\right) \ldots \delta\left(y_{n}-\eta_{\lambda n}(\omega)\right)\right] . \tag{3} \end{equation*}(3)p(y1,λ1;…;yn,λn)=M[δ(y1−ηλ1(ω))…δ(yn−ηλn(ω))].
In (3), δ δ delta\deltaδ is the the Dirac functional and M M MMM the expectation defined by (1). The generalization for Y ⊆ R d , d ≥ 2 Y ⊆ R d , d ≥ 2 Y subeR^(d),d >= 2Y \subseteq \mathbb{R}^{d}, d \geq 2Y⊆Rd,d≥2 is straightforward ([13], p.209).
The conditional probability density is defined by
jective, bimea ) ∼ ( X , B , P η ) ) ∼ X , B , P η )∼(X,B,P_(eta))) \sim\left(X, \mathcal{B}, P_{\eta}\right))∼(X,B,Pη). of random variquently, when a hase space" [3],
able valued into
η ( ω ) = y ω η ( ω ) = y ω eta(omega)=y^(omega)\eta(\omega)=y^{\omega}η(ω)=yω, is a a order to avoid phase space" for ion η η eta\etaη are sets of η λ ( ω ) = y ω ( λ ) η λ ( ω ) = y ω ( λ ) eta_(lambda)(omega)=y^(omega)(lambda)\eta_{\lambda}(\omega)=y^{\omega}(\lambda)ηλ(ω)=yω(λ), state space Y Y YYY. If tochastic process. i field. ensional function 1 random vectors distributions and a, B i ∈ B B i ∈ B B_(i)inBB_{i} \in \mathcal{B}Bi∈B.
(2) B i 1 × … × B i n ) = (2) B i 1 × … × B i n = {:(2){:B_(i_(1))xx dots xxB_(i_(n)))=:}\begin{equation*} \left.B_{i_{1}} \times \ldots \times B_{i_{n}}\right)= \tag{2} \end{equation*}(2)Bi1×…×Bin)=
{ 1 , … , n } { 1 , … , n } {1,dots,n}\{1, \ldots, n\}{1,…,n} and any n ( B 1 × … × B n ) n B 1 × … × B n _(n)(B_(1)xx dots xxB_(n)){ }_{n}\left(B_{1} \times \ldots \times B_{n}\right)n(B1×…×Bn), tions' ensures the λ 1 … λ n ( B ) λ 1 … λ n ( B ) lambda_(1)dotslambda_(n)(B)\lambda_{1} \ldots \lambda_{n}(B)λ1…λn(B), for any g g ggg all the cylindri n ∈ Λ } n ∈ Λ {:_(n)in Lambda}\left.{ }_{n} \in \Lambda\right\}n∈Λ}, and B n B n B^(n)\mathcal{B}^{n}Bn is m functions in the pace ( R Λ , B Λ , P η ) R Λ , B Λ , P η (R^(Lambda),B^(Lambda),P_(eta))\left(\mathbb{R}^{\Lambda}, \mathcal{B}^{\Lambda}, P_{\boldsymbol{\eta}}\right)(RΛ,BΛ,Pη)
process, defined on ional distributions
η λ n ( ω ) ) ] η λ n ( ω ) {:eta_(lambda n)(omega))]\left.\left.\eta_{\lambda n}(\omega)\right)\right]ηλn(ω))].
ectation defined by ard ([13], p.209).
p ( y 1 , t 1 ; … ; y r , t r ∣ y r + 1 , t r + 1 ; … ; y n , t n ) = = p ( y 1 , t 1 ; … ; y n , t n ) / p ( y r + 1 , t r + 1 ; … ; y n , t n ) . p y 1 , t 1 ; … ; y r , t r ∣ y r + 1 , t r + 1 ; … ; y n , t n = = p y 1 , t 1 ; … ; y n , t n / p y r + 1 , t r + 1 ; … ; y n , t n . {:[p(y_(1),t_(1);dots;y_(r),t_(r)∣y_(r+1),t_(r+1);dots;y_(n),t_(n))=],[=p(y_(1),t_(1);dots;y_(n),t_(n))//p(y_(r+1),t_(r+1);dots;y_(n),t_(n)).]:}\begin{aligned} & p\left(y_{1}, t_{1} ; \ldots ; y_{r}, t_{r} \mid y_{r+1}, t_{r+1} ; \ldots ; y_{n}, t_{n}\right)= \\ = & p\left(y_{1}, t_{1} ; \ldots ; y_{n}, t_{n}\right) / p\left(y_{r+1}, t_{r+1} ; \ldots ; y_{n}, t_{n}\right) . \end{aligned}p(y1,t1;…;yr,tr∣yr+1,tr+1;…;yn,tn)==p(y1,t1;…;yn,tn)/p(yr+1,tr+1;…;yn,tn).
Markov processes depend only on one of the earlier states and not on the whole process history. By the Kolmogorov Theorem 1.1, Markov processes are uniquely defined, in the sense of Doob, by the 1 -dimensional density and the two states conditional density (called transition probability density). Transition probabilities obey the Chapman-Kolmogorov equation
(4) p ( y 1 , t 1 ∣ y 3 , t 3 ) = ∫ R p ( y 1 , t 1 ∣ y 2 , t 2 ) p ( y 2 , t 2 ∣ y 3 , t 3 ) d y 2 (4) p y 1 , t 1 ∣ y 3 , t 3 = ∫ R   p y 1 , t 1 ∣ y 2 , t 2 p y 2 , t 2 ∣ y 3 , t 3 d y 2 {:(4)p(y_(1),t_(1)∣y_(3),t_(3))=int_(R)p(y_(1),t_(1)∣y_(2),t_(2))p(y_(2),t_(2)∣y_(3),t_(3))dy_(2):}\begin{equation*} p\left(y_{1}, t_{1} \mid y_{3}, t_{3}\right)=\int_{\mathbf{R}} p\left(y_{1}, t_{1} \mid y_{2}, t_{2}\right) p\left(y_{2}, t_{2} \mid y_{3}, t_{3}\right) d y_{2} \tag{4} \end{equation*}(4)p(y1,t1∣y3,t3)=∫Rp(y1,t1∣y2,t2)p(y2,t2∣y3,t3)dy2
Stationary Markov processes are defined by p ( y 1 , t 1 ∣ y 2 , t 2 ) = p s ( y 1 , t 1 − t 2 ∣ y 2 ) p y 1 , t 1 ∣ y 2 , t 2 = p s y 1 , t 1 − t 2 ∣ y 2 p(y_(1),t_(1)∣y_(2),t_(2))=p_(s)(y_(1),t_(1)-:}{:t_(2)∣y_(2))p\left(y_{1}, t_{1} \mid y_{2}, t_{2}\right)=p_{s}\left(y_{1}, t_{1}-\right. \left.t_{2} \mid y_{2}\right)p(y1,t1∣y2,t2)=ps(y1,t1−t2∣y2) and p ( y , t ) = p s ( y ) p ( y , t ) = p s ( y ) p(y,t)=p_(s)(y)p(y, t)=p_{s}(y)p(y,t)=ps(y).
The stationary Markov operator of kernel type is the linear operator K τ : L 1 ( Y ) ⟼ L 1 ( Y ) K τ : L 1 ( Y ) ⟼ L 1 ( Y ) K^(tau):L^(1)(Y)longmapstoL^(1)(Y)K^{\tau}: L^{1}(Y) \longmapsto L^{1}(Y)Kτ:L1(Y)⟼L1(Y) defined by
(5) K τ f ( y ) = ∫ Y p ( y , τ ∣ y 0 ) f ( y 0 ) d y 0 , where τ = t 1 − t 2 , τ ≥ 0 (5) K τ f ( y ) = ∫ Y   p y , τ ∣ y 0 f y 0 d y 0 ,  where  τ = t 1 − t 2 , τ ≥ 0 {:(5)K^(tau)f(y)=int_(Y)p(y,tau∣y_(0))f(y_(0))dy_(0)","" where "tau=t_(1)-t_(2)","tau >= 0:}\begin{equation*} K^{\tau} f(y)=\int_{Y} p\left(y, \tau \mid y_{0}\right) f\left(y_{0}\right) d y_{0}, \text { where } \tau=t_{1}-t_{2}, \tau \geq 0 \tag{5} \end{equation*}(5)Kτf(y)=∫Yp(y,τ∣y0)f(y0)dy0, where τ=t1−t2,τ≥0
The operators K τ K τ K^(tau)K^{\tau}Kτ preserve the positivity and are isometric, i.e. they have the properties: (M1) K τ f ≥ 0 , ∀ f > 0 , f ∈ L 1 ( Y ) K τ f ≥ 0 , ∀ f > 0 , f ∈ L 1 ( Y ) K^(tau)f >= 0,AA f > 0,f inL^(1)(Y)K^{\tau} f \geq 0, \forall f>0, f \in L^{1}(Y)Kτf≥0,∀f>0,f∈L1(Y) and (M2) ‖ K τ f ‖ L 1 = ‖ f ‖ L 1 , ∀ f ∈ L 1 ( Y ) K τ f L 1 = ‖ f ‖ L 1 , ∀ f ∈ L 1 ( Y ) ||K^(tau)f||_(L^(1))=||f||_(L^(1)),AA f inL^(1)(Y)\left\|K^{\tau} f\right\|_{L^{1}}= \|f\|_{L^{1}}, \forall f \in L^{1}(Y)‖Kτf‖L1=‖f‖L1,∀f∈L1(Y). The evolution of densities is given by p ( y 1 , t 1 ) = ∫ Y p ( y 1 , t 1 − t 2 ∣ y 2 , t 2 ) p ( y 2 , t 2 ) d y 2 = ( K t 1 − t 2 p ) ( y 1 , t 1 ) p y 1 , t 1 = ∫ Y   p y 1 , t 1 − t 2 ∣ y 2 , t 2 p y 2 , t 2 d y 2 = K t 1 − t 2 p y 1 , t 1 p(y_(1),t_(1))=int_(Y)p(y_(1),t_(1)-t_(2)∣y_(2),t_(2))p(y_(2),t_(2))dy_(2)=(K^(t_(1)-t_(2))p)(y_(1),t_(1))p\left(y_{1}, t_{1}\right)= \int_{Y} p\left(y_{1}, t_{1}-t_{2} \mid y_{2}, t_{2}\right) p\left(y_{2}, t_{2}\right) d y_{2}=\left(K^{t_{1}-t_{2}} p\right)\left(y_{1}, t_{1}\right)p(y1,t1)=∫Yp(y1,t1−t2∣y2,t2)p(y2,t2)dy2=(Kt1−t2p)(y1,t1). By the ChapmanKolmogorov equation (4) the operators (5) have the properties K τ 1 + τ 2 f = K τ 2 K τ 1 f , K 0 f = f , ∀ f ∈ L 1 ( Y ) K τ 1 + τ 2 f = K τ 2 K τ 1 f , K 0 f = f , ∀ f ∈ L 1 ( Y ) K_(tau_(1)+tau_(2))f=K_(tau_(2))K_(tau_(1))f,K_(0)f=f,AA f inL^(1)(Y)K_{\tau_{1}+\tau_{2}} f= K_{\tau_{2}} K_{\tau_{1}} f, K_{0} f=f, \forall f \in L^{1}(Y)Kτ1+τ2f=Kτ2Kτ1f,K0f=f,∀f∈L1(Y) and ∀ τ 1 , τ 2 ≥ 0 ∀ τ 1 , τ 2 ≥ 0 AAtau_(1),tau_(2) >= 0\forall \tau_{1}, \tau_{2} \geq 0∀τ1,τ2≥0, which define the semigourp { K τ } τ ≥ 0 K τ τ ≥ 0 {K_(tau)}_(tau >= 0)\left\{K_{\tau}\right\}_{\tau \geq 0}{Kτ}τ≥0 of Markov operators.
A stationary Markov process is strongly ergodic ([9], p. 60) if p s ( y 1 , t 1 − t 2 ∣ y 2 ) ⟶ p s ( y 1 ) p s y 1 , t 1 − t 2 ∣ y 2 ⟶ p s y 1 p_(s)(y_(1),t_(1)-:}{:t_(2)∣y_(2))longrightarrowp_(s)(y_(1))p_{s}\left(y_{1}, t_{1}-\right. \left.t_{2} \mid y_{2}\right) \longrightarrow p_{s}\left(y_{1}\right)ps(y1,t1−t2∣y2)⟶ps(y1), for t 1 − t 2 ⟶ ∞ t 1 − t 2 ⟶ ∞ t_(1)-t_(2)longrightarrow oot_{1}-t_{2} \longrightarrow \inftyt1−t2⟶∞. Correspondingly, the strongly ergodic Markov operators obey
(6) ‖ K τ p − p s ‖ L 1 → 0 , for τ → ∞ (6) K τ p − p s L 1 → 0 ,  for  τ → ∞ {:(6)||K_(tau)p-p_(s)||_(L^(1))rarr0","" for "tau rarr oo:}\begin{equation*} \left\|K_{\tau} p-p_{s}\right\|_{L^{1}} \rightarrow 0, \text { for } \tau \rightarrow \infty \tag{6} \end{equation*}(6)‖Kτp−ps‖L1→0, for τ→∞
i.e. they are just the "strong Markov semigroups" [13] or "irreversible semigroups" [1] used as model of irreversibility in Misra-Prigogine-Courbage theory. For them it was easy shown that the Gibbs entropy, − ∫ p ( y , t ) log ⁡ p ( y , t ) d y − ∫ p ( y , t ) log ⁡ p ( y , t ) d y -int p(y,t)log p(y,t)dy-\int p(y, t) \log p(y, t) d y−∫p(y,t)log⁡p(y,t)dy, (and, also, any convex functional of p p ppp ) monotonically increases to the maximum value corresponding to thermodynamical equilibrium [18].
Lemma 2. The strong Markov semigroups { K τ } τ ≥ 0 K τ τ ≥ 0 {K_(tau)}_(tau >= 0)\left\{K_{\tau}\right\}_{\tau \geq 0}{Kτ}τ≥0, have the properties of mixing,
∫ 0 1 g ( y ) ( K τ f ) ( y ) d y → ∫ 0 1 g ( y ) d y ∫ 0 1 f ( y ′ ) d y ′ ∫ 0 1   g ( y ) K τ f ( y ) d y → ∫ 0 1   g ( y ) d y ∫ 0 1   f y ′ d y ′ int_(0)^(1)g(y)(K^(tau)f)(y)dy rarrint_(0)^(1)g(y)dyint_(0)^(1)f(y^('))dy^(')\int_{0}^{1} g(y)\left(K^{\tau} f\right)(y) d y \rightarrow \int_{0}^{1} g(y) d y \int_{0}^{1} f\left(y^{\prime}\right) d y^{\prime}∫01g(y)(Kτf)(y)dy→∫01g(y)dy∫01f(y′)dy′
and exactness,
‖ K τ f − 1 ‖ L 1 → 0 , for τ → ∞ K τ f − 1 L 1 → 0 ,  for  τ → ∞ ||K_(tau)f-1||_(L^(1))rarr0," for "tau rarr oo\left\|K_{\tau} f-1\right\|_{L^{1}} \rightarrow 0, \text { for } \tau \rightarrow \infty‖Kτf−1‖L1→0, for τ→∞
for any f , g ∈ L 1 ( Y , B , μ ′ ) f , g ∈ L 1 Y , B , μ ′ f,g inL^(1)(Y,B,mu^('))f, g \in L^{1}\left(Y, \mathcal{B}, \mu^{\prime}\right)f,g∈L1(Y,B,μ′), where
Y ⊆ R , μ ( Y ) < ∞ , μ ′ ( B ) = ( 1 / ∫ Y d y ) ∫ B d y Y ⊆ R , μ ( Y ) < ∞ , μ ′ ( B ) = 1 / ∫ Y   d y ∫ B   d y Y subeR,mu(Y) < oo,mu^(')(B)=(1//int_(Y)dy)int_(B)dyY \subseteq \mathbb{R}, \mu(Y)<\infty, \mu^{\prime}(B)=\left(1 / \int_{Y} d y\right) \int_{B} d yY⊆R,μ(Y)<∞,μ′(B)=(1/∫Ydy)∫Bdy
is the probability measure in B B B\mathcal{B}B, and p s ( y ) ≡ 1 p s ( y ) ≡ 1 p_(s)(y)-=1p_{s}(y) \equiv 1ps(y)≡1 is the stationary density with respect to μ ′ μ ′ mu^(')\mu^{\prime}μ′.
Usually, measure dynamical systems on ( Y , B , μ Y , B , μ Y,B,muY, \mathcal{B}, \muY,B,μ ) are defined as oneparameter groups of transformations of the state space, { S t ∣ S t : Y ⟼ Y } t ∈ R (or Z ) S t ∣ S t : Y ⟼ Y } t ∈ R  (or  Z  )  {S_(t)∣S_(t):Y longmapsto:}Y}_(t inR" (or "Z" ) ")\left\{S_{t} \mid S_{t}: Y \longmapsto\right. Y\}_{t \in \mathbb{R} \text { (or } \mathbb{Z} \text { ) }}{St∣St:Y⟼Y}t∈R (or Z ) , which preserve the Lebesgue measure, i.e. μ ( B ) = μ ( S − t B ) , ∀ t ∈ R μ ( B ) = μ S − t B , ∀ t ∈ R mu(B)=mu(S_(-t)B),AA t inR\mu(B)=\mu\left(S_{-t} B\right), \forall t \in \mathbb{R}μ(B)=μ(S−tB),∀t∈R (or Z Z Z\mathbb{Z}Z ) and ∀ B ∈ B ∀ B ∈ B AA B inB\forall B \in \mathcal{B}∀B∈B. They are groups of automorphisms on a measure space [8]. Semigroups of endomorphisms on measure spaces are called semidynamical systems.
If Y ⊆ R , P Y ⊆ R , P Y subeR,PY \subseteq \mathbb{R}, PY⊆R,P is an absolutely-continuous measure with respect to the Lebesgue measure μ μ mu\muμ, if μ ( S − t B ) = 0 ∀ B ∈ B μ S − t B = 0 ∀ B ∈ B mu(S_(-t)B)=0AA B inB\mu\left(S_{-t} B\right)=0 \forall B \in \mathcal{B}μ(S−tB)=0∀B∈B with μ ( B ) = 0 μ ( B ) = 0 mu(B)=0\mu(B)=0μ(B)=0, and P ( B ) = P ( S − t B ) , ∀ t ∈ R P ( B ) = P S − t B , ∀ t ∈ R P(B)=P(S_(-t)B),AA t inRP(B)= P\left(S_{-t} B\right), \forall t \in \mathbb{R}P(B)=P(S−tB),∀t∈R, then the dynamical system { S t } t ∈ R S t t ∈ R {S_(t)}_(t inR)\left\{S_{t}\right\}_{t \in \mathbb{R}}{St}t∈R defines a random function isomorphic in the sense of Doob with the state space ( Y , B , P Y , B , P Y,B,PY, \mathcal{B}, PY,B,P ). Thus, the phase space of dynamical systems (i.e. the space of trajectories) is isomorphic to the state space. More generally, this isomorphism defines the deterministic processes [11], in opposition with the genuine stochastic processes (where the state space is a projection, at given time value, of the phase space).
Dynamical systems are degenerated Markov processes, in the sense that a point y 0 y 0 y_(0)y_{0}y0 of every trajectory of the process is transported forward, on the same trajectory, at the point y y yyy. The degenerate transition probabilities are given by p ( y , t ∣ y 0 ) = δ ( y − S t ( y 0 ) ) p y , t ∣ y 0 = δ y − S t y 0 p(y,t∣y_(0))=delta(y-S_(t)(y_(0)))p\left(y, t \mid y_{0}\right)=\delta\left(y-S_{t}\left(y_{0}\right)\right)p(y,t∣y0)=δ(y−St(y0)). The corresponding Markov operators (8), called Frobenius-Perron operators, are defined by U t f ( y ) = ∫ Y δ ( y − S t ( y 0 ) ) f ( y 0 ) d y 0 U t f ( y ) = ∫ Y   δ y − S t y 0 f y 0 d y 0 U_(t)f(y)=int_(Y)delta(y-S_(t)(y_(0)))f(y_(0))dy_(0)U_{t} f(y)= \int_{Y} \delta\left(y-S_{t}\left(y_{0}\right)\right) f\left(y_{0}\right) d y_{0}Utf(y)=∫Yδ(y−St(y0))f(y0)dy0. The adjoint of the Frobenius-Perron operator, acting on bounded Lebesgue integrable functions on Y Y YYY whose norm is given by the essential supremum, i.e. g ∈ L ∞ ( Y ) ( [ 14 ] g ∈ L ∞ ( Y ) ( [ 14 ] g inL^(oo)(Y)([14]g \in L^{\infty}(Y)([14]g∈L∞(Y)([14], p.43), defined by ∫ Y f ( y ) U t ∗ g ( y ) d y = ∫ Y g ( y ) U t f ( y ) d y ∫ Y   f ( y ) U t ∗ g ( y ) d y = ∫ Y   g ( y ) U t f ( y ) d y int_(Y)f(y)U_(t)^(**)g(y)dy=int_(Y)g(y)U_(t)f(y)dy\int_{Y} f(y) U_{t}^{*} g(y) d y=\int_{Y} g(y) U_{t} f(y) d y∫Yf(y)Ut∗g(y)dy=∫Yg(y)Utf(y)dy, for any f ∈ L 1 ( Y ) f ∈ L 1 ( Y ) f inL^(1)(Y)f \in L^{1}(Y)f∈L1(Y) and g ∈ L ∞ ( Y ) g ∈ L ∞ ( Y ) g inL^(oo)(Y)g \in L^{\infty}(Y)g∈L∞(Y), is the Koopman operator,
(7) U t ∗ g ( y ) = g ( S ( y ) ) , ∀ g ∈ L ∞ ( Y ) (7) U t ∗ g ( y ) = g ( S ( y ) ) , ∀ g ∈ L ∞ ( Y ) {:(7)U_(t)^(**)g(y)=g(S(y))","quad AA g inL^(oo)(Y):}\begin{equation*} U_{t}^{*} g(y)=g(S(y)), \quad \forall g \in L^{\infty}(Y) \tag{7} \end{equation*}(7)Ut∗g(y)=g(S(y)),∀g∈L∞(Y)
The measure preserving property μ ( B ) = μ ( S − t B ) μ ( B ) = μ S − t B mu(B)=mu(S_(-t)B)\mu(B)=\mu\left(S_{-t} B\right)μ(B)=μ(S−tB) implies
( U t ∗ ) − 1 f ( y ) = f ( S − t ( y ) ) = U t f ( y ) U t ∗ − 1 f ( y ) = f S − t ( y ) = U t f ( y ) (U_(t)^(**))^(-1)f(y)=f(S_(-t)(y))=U_(t)f(y)\left(U_{t}^{*}\right)^{-1} f(y)=f\left(S_{-t}(y)\right)=U_{t} f(y)(Ut∗)−1f(y)=f(S−t(y))=Utf(y)
i.e. the operator adjoint to U t ∗ U t ∗ U_(t)^(**)U_{t}^{*}Ut∗ is also its inverse. Thus, a measure dynamical system in ( Y , B , μ Y , B , μ Y,B,muY, \mathcal{B}, \muY,B,μ ) induces an unitary group { U t } t ∈ R U t t ∈ R {U_(t)}_(t inR)\left\{U_{t}\right\}_{t \in \mathbb{R}}{Ut}t∈R which invariates
the Hilbert space L 2 ( Y ) L 2 ( Y ) L^(2)(Y)L^{2}(Y)L2(Y) (Lemma at p. 26 in [8]). This result allows for the thermodynamic behavior of dynamical systems to be described in terms of unitary groups of operators [18].
Lemma 3. A stationary Markov process can be embedded into the equivalence class of dynamical systems defined by the phase space ( Y R , B R , P η Y R , B R , P η Y^(R),B^(R),P_(eta)Y^{\mathbb{R}}, \mathcal{B}^{\mathbb{R}}, P_{\eta}YR,BR,Pη ), where P η P η P_(eta)P_{\eta}Pη is the extension by the Kolmogorov Theorem of the measure defined on cylindrical sets by
P η ( C n + 1 ) = P t 0 , ⋯ , ι n ( B 0 × ⋯ × B n ) = (8) = ∫ B 0 p s ( y 0 ) d y 0 ∫ B 1 p s ( y 1 , t 1 − t 0 ∣ y 0 ) d y 1 ⋯ ⋯ ∫ B n p s ( y n , t n − t n − 1 ∣ y n − 1 ) d y n . P η C n + 1 = P t 0 , ⋯ , ι n B 0 × ⋯ × B n = (8) = ∫ B 0   p s y 0 d y 0 ∫ B 1   p s y 1 , t 1 − t 0 ∣ y 0 d y 1 ⋯ ⋯ ∫ B n   p s y n , t n − t n − 1 ∣ y n − 1 d y n . {:[P_(eta)(C_(n+1))=P_(t_(0),cdots,iota_(n))(B_(0)xx cdots xxB_(n))=],[(8)=int_(B_(0))p_(s)(y_(0))dy_(0)int_(B_(1))p_(s)(y_(1),t_(1)-t_(0)∣y_(0))dy_(1)cdots],[cdotsint_(B_(n))p_(s)(y_(n),t_(n)-t_(n-1)∣y_(n-1))dy_(n).]:}\begin{gather*} P_{\eta}\left(C_{n+1}\right)=P_{t_{0}, \cdots, \iota_{n}}\left(B_{0} \times \cdots \times B_{n}\right)= \\ =\int_{B_{0}} p_{s}\left(y_{0}\right) d y_{0} \int_{B_{1}} p_{s}\left(y_{1}, t_{1}-t_{0} \mid y_{0}\right) d y_{1} \cdots \tag{8}\\ \cdots \int_{B_{n}} p_{s}\left(y_{n}, t_{n}-t_{n-1} \mid y_{n-1}\right) d y_{n} . \end{gather*}Pη(Cn+1)=Pt0,⋯,ιn(B0×⋯×Bn)=(8)=∫B0ps(y0)dy0∫B1ps(y1,t1−t0∣y0)dy1⋯⋯∫Bnps(yn,tn−tn−1∣yn−1)dyn.
The natural representative of this class of dynamical systems is the shift along the trajectories of the Markov process, { Σ τ } τ ∈ R , Σ τ : Y I ⟼ Y I Σ τ τ ∈ R , Σ τ : Y I ⟼ Y I {Sigma_(tau)}_(tau inR),Sigma_(tau):Y^(I)longmapstoY^(I)\left\{\Sigma_{\tau}\right\}_{\tau \in \mathbb{R}}, \Sigma_{\tau}: Y^{I} \longmapsto Y^{I}{Στ}τ∈R,Στ:YI⟼YI, Σ τ ( y ω ( t ) ) = y ω ( t + τ ) Σ τ y ω ( t ) = y ω ( t + τ ) Sigma_(tau)(y^(omega)(t))=y^(omega)(t+tau)\Sigma_{\tau}\left(y^{\omega}(t)\right)=y^{\omega}(t+\tau)Στ(yω(t))=yω(t+τ), which invariates the measure of cylindrical sets
P η ( C n + 1 ) = P η ( Σ τ ( C n + 1 ) ) = P η ( Σ − τ ( C n + 1 ) ) . P η C n + 1 = P η Σ τ C n + 1 = P η Σ − τ C n + 1 . P_(eta)(C_(n+1))=P_(eta)(Sigma_(tau)(C_(n+1)))=P_(eta)(Sigma_(-tau)(C_(n+1))).P_{\boldsymbol{\eta}}\left(C_{n+1}\right)=P_{\boldsymbol{\eta}}\left(\Sigma_{\tau}\left(C_{n+1}\right)\right)=P_{\boldsymbol{\eta}}\left(\Sigma_{-\tau}\left(C_{n+1}\right)\right) .Pη(Cn+1)=Pη(Στ(Cn+1))=Pη(Σ−τ(Cn+1)).
So, the phase space of the stationary Markov process, is also the state space of this dynamical system.

3. THE EXISTENCE OF NONUNITARY SIMILARITY

Following the definition given by Sz.-Nagy and Foias [24], pp.10, 31, the group of operators { U τ } τ ⊂ R U τ τ ⊂ R {U_(tau)}_(tau subR)\left\{U_{\tau}\right\}_{\tau \subset \mathbb{R}}{Uτ}τ⊂R, defined in Hilbert space L 2 ( Y R ) L 2 Y R L^(2)(Y^(R))L^{2}\left(Y^{\mathbb{R}}\right)L2(YR), is the unitary dilation of the semigroup { K τ } τ ≥ 0 K τ τ ≥ 0 {K_(tau)}_(tau >= 0)\left\{K_{\tau}\right\}_{\tau \geq 0}{Kτ}τ≥0, defined in L 2 ( Y ) L 2 ( Y ) L^(2)(Y)L^{2}(Y)L2(Y), if they are related by
(9) K τ = Pr U τ , τ ≥ 0 , (9) K τ = Pr U τ , τ ≥ 0 , {:(9)K_(tau)=PrU_(tau)","quad tau >= 0",":}\begin{equation*} K_{\tau}=\operatorname{Pr} U_{\tau}, \quad \tau \geq 0, \tag{9} \end{equation*}(9)Kτ=PrUτ,τ≥0,
where Pr Pr Pr\operatorname{Pr}Pr is the orthogonal projection of L 2 ( Y R ) L 2 Y R L^(2)(Y^(R))L^{2}\left(Y^{\mathbb{R}}\right)L2(YR) on L 2 ( Y ) L 2 ( Y ) L^(2)(Y)L^{2}(Y)L2(Y) (by its construction L 2 ( Y ) L 2 ( Y ) L^(2)(Y)L^{2}(Y)L2(Y) is a sub-space of L 2 ( Y R ) L 2 Y R L^(2)(Y^(R))L^{2}\left(Y^{\mathbb{R}}\right)L2(YR) ). The Theorem 8.1, p. 31 in [24] proves that any continuous group of contractions can be dilated to a minimal unitary and continuous group, uniquely defined up to an isomorphism. The property (M2) from § 2 shows that Markov operators are isometric (and thus, contractions) the quoted theorem ensures the existence of unitary dilation.
The dilated group of MPC theory acts on probabilities densities (which are positively defined), so it is also necessary that the dilation preserves the positivity, i.e. for any positive function from L 2 ( Y R ) L 2 Y R L^(2)(Y^(R))L^{2}\left(Y^{\mathbb{R}}\right)L2(YR), the value of Pr U τ Pr U τ PrU_(tau)\operatorname{Pr} U_{\tau}PrUτ
must be a positive function from L 2 ( Y ) L 2 ( Y ) L^(2)(Y)L^{2}(Y)L2(Y). By the Theorem proved in [2], Markov semigroups induced by exact dynamical systems possess unitary dilations to groups induced by K -systems. In [5, 6] the result is extended to constant preserving stationary Markov semigroups (obeying K τ 1 = 1 K τ 1 = 1 K_(tau)1=1K_{\tau} 1=1Kτ1=1 ). The dilations groups are obtained by Rokhlin theorem [21] on natural extension of exact dynamical systems in [2], and extending probability measures in [ 5 , 6 ] [ 5 , 6 ] [5,6][5,6][5,6].
Now we prove that all stationary Markov semigroups possess unitary dilations. For strongly ergodic and constant preserving Markov processes, we also find the results of Antoniou et al. Moreover, for these processes, we prove the existence of the 'nonunitary equivalence' of the MPC theory.

Theorem 1.

  1. The adjoint Markov semigroups { K τ ∗ } τ ≥ 0 K τ ∗ τ ≥ 0 {K_(tau)^(**)}_(tau >= 0)\left\{K_{\tau}^{*}\right\}_{\tau \geq 0}{Kτ∗}τ≥0 corresponding to a stationary Markov process possess positive dilation to unitary groups;
  2. For strongly ergodic and constant preserving Markov semigroups, both { K τ ∗ } τ ≥ 0 K τ ∗ τ ≥ 0 {K_(tau)^(**)}_(tau >= 0)\left\{K_{\tau}^{*}\right\}_{\tau \geq 0}{Kτ∗}τ≥0 and { K τ } τ ≥ 0 K τ τ ≥ 0 {K_(tau)}_(tau >= 0)\left\{K_{\tau}\right\}_{\tau \geq 0}{Kτ}τ≥0 possess positive dilations to unitary groups induced by K-systems;
  3. Between the operators K τ K τ K_(tau)K_{\tau}Kτ of strong ergodic semigroup and those of the dilation group there exists the intertwining relation K τ Φ = Φ U τ K τ Φ = Φ U τ K_(tau)Phi=PhiU_(tau)K_{\tau} \Phi=\Phi U_{\tau}KτΦ=ΦUτ.
    Proof. 1) From the definition of the adjoint K τ ∗ K τ ∗ K_(tau)^(**)K_{\tau}^{*}Kτ∗, in L 2 ( Y ) L 2 ( Y ) L^(2)(Y)L^{2}(Y)L2(Y), of the Markov operator (5), using the Lemma 1 and the form (3) of the finite-dimensional distributions, we have
( K τ ∗ f ) ( y ) = ∫ Y p ( y ′ , τ ∣ y ) f ( y ′ ) d y ′ = 1 p ( y ) ∫ Y p ( y ′ , t + τ ; y , t ) f ( y ′ ) d y ′ = = 1 p ( y ) ∫ Y f ( y ′ ) d y ′ ∫ Ω δ ( y ′ − η ( t + τ , ω ) ) δ ( y − η ( t , ω ) ) P ( d ω ) = = 1 p ( y ) ∫ Ω f ( η ( t + τ , ω ) ) δ ( y − η ( t , ω ) ) P ( d ω ) = = 1 p ( y ) ∫ Ω f ( Σ τ y ω ( t ) ) δ ( y − η ( t , ω ) ) P ( d ω ) = = 1 p ( y ) ∫ Ω U τ ∗ f ( y ω ( t ) ) δ ( y − η ( t , ω ) ) P ( d ω ) K τ ∗ f ( y ) = ∫ Y   p y ′ , τ ∣ y f y ′ d y ′ = 1 p ( y ) ∫ Y   p y ′ , t + τ ; y , t f y ′ d y ′ = = 1 p ( y ) ∫ Y   f y ′ d y ′ ∫ Ω   δ y ′ − η ( t + τ , ω ) δ ( y − η ( t , ω ) ) P ( d ω ) = = 1 p ( y ) ∫ Ω   f ( η ( t + τ , ω ) ) δ ( y − η ( t , ω ) ) P ( d ω ) = = 1 p ( y ) ∫ Ω   f Σ τ y ω ( t ) δ ( y − η ( t , ω ) ) P ( d ω ) = = 1 p ( y ) ∫ Ω   U τ ∗ f y ω ( t ) δ ( y − η ( t , ω ) ) P ( d ω ) {:[(K_(tau)^(**)f)(y)=int_(Y)p(y^('),tau∣y)f(y^('))dy^(')=(1)/(p(y))int_(Y)p(y^('),t+tau;y,t)f(y^('))dy^(')=],[=(1)/(p(y))int_(Y)f(y^('))dy^(')int_(Omega)delta(y^(')-eta(t+tau,omega))delta(y-eta(t","omega))P(d omega)=],[=(1)/(p(y))int_(Omega)f(eta(t+tau","omega))delta(y-eta(t","omega))P(d omega)=],[=(1)/(p(y))int_(Omega)f(Sigma_(tau)y^(omega)(t))delta(y-eta(t","omega))P(d omega)=],[=(1)/(p(y))int_(Omega)U_(tau)^(**)f(y^(omega)(t))delta(y-eta(t","omega))P(d omega)]:}\begin{aligned} \left(K_{\tau}^{*} f\right)(y) & =\int_{Y} p\left(y^{\prime}, \tau \mid y\right) f\left(y^{\prime}\right) d y^{\prime}=\frac{1}{p(y)} \int_{Y} p\left(y^{\prime}, t+\tau ; y, t\right) f\left(y^{\prime}\right) d y^{\prime}= \\ & =\frac{1}{p(y)} \int_{Y} f\left(y^{\prime}\right) d y^{\prime} \int_{\Omega} \delta\left(y^{\prime}-\eta(t+\tau, \omega)\right) \delta(y-\eta(t, \omega)) P(d \omega)= \\ & =\frac{1}{p(y)} \int_{\Omega} f(\eta(t+\tau, \omega)) \delta(y-\eta(t, \omega)) P(d \omega)= \\ & =\frac{1}{p(y)} \int_{\Omega} f\left(\Sigma_{\tau} y^{\omega}(t)\right) \delta(y-\eta(t, \omega)) P(d \omega)= \\ & =\frac{1}{p(y)} \int_{\Omega} U_{\tau}^{*} f\left(y^{\omega}(t)\right) \delta(y-\eta(t, \omega)) P(d \omega) \end{aligned}(Kτ∗f)(y)=∫Yp(y′,τ∣y)f(y′)dy′=1p(y)∫Yp(y′,t+τ;y,t)f(y′)dy′==1p(y)∫Yf(y′)dy′∫Ωδ(y′−η(t+τ,ω))δ(y−η(t,ω))P(dω)==1p(y)∫Ωf(η(t+τ,ω))δ(y−η(t,ω))P(dω)==1p(y)∫Ωf(Στyω(t))δ(y−η(t,ω))P(dω)==1p(y)∫ΩUτ∗f(yω(t))δ(y−η(t,ω))P(dω)
where y ω ( t ) = η ( t , ω ) y ω ( t ) = η ( t , ω ) y^(omega)(t)=eta(t,omega)y^{\omega}(t)=\eta(t, \omega)yω(t)=η(t,ω). From Lemma 4 it follows that the Markov shift Σ τ Σ τ Sigma_(tau)\Sigma_{\tau}Στ preserves the mcasure from the space of trajectories. Thus, the Koopman operator, defined by (7), U T ∗ f ( y ω ) = f ( Σ τ y ω ) U T ∗ f y ω = f Σ τ y ω U_(T)^(**)f(y^(omega))=f(Sigma_(tau)y^(omega))U_{T}^{*} f\left(y^{\omega}\right)=f\left(\Sigma_{\tau} y^{\omega}\right)UT∗f(yω)=f(Στyω). is a unitary operator in L 2 ( Y r ) L 2 Y r  L^(2)(Y^("r "))L^{2}\left(Y^{\text {r }}\right)L2(Yr ). The previous relation becomes
(10) ( K τ ∗ f ) ( y ) = ( Pr y U τ ∗ f ) ( y ) , (10) K τ ∗ f ( y ) = Pr y U τ ∗ f ( y ) , {:(10)(K_(tau)^(**)f)(y)=(Pr_(y)U_(tau)^(**)f)(y)",":}\begin{equation*} \left(K_{\tau}^{*} f\right)(y)=\left(\operatorname{Pr}_{y} U_{\tau}^{*} f\right)(y), \tag{10} \end{equation*}(10)(Kτ∗f)(y)=(PryUτ∗f)(y),
where Pr y : L 2 ( Y R ) ⟼ L 2 ( Y ) , ( Pr y f ) ( y ) = ∫ Ω f ( y ω ) P y ( d ω ) Pr y : L 2 Y R ⟼ L 2 ( Y ) , Pr y f ( y ) = ∫ Ω   f y ω P y ( d ω ) Pr_(y):L^(2)(Y^(R))longmapstoL^(2)(Y),(Pr_(y)f)(y)=int_(Omega)f(y^(omega))P_(y)(d omega)\operatorname{Pr}_{y}: L^{2}\left(Y^{\mathbb{R}}\right) \longmapsto L^{2}(Y),\left(\operatorname{Pr}_{y} f\right)(y)=\int_{\Omega} f\left(y^{\omega}\right) P_{y}(d \omega)Pry:L2(YR)⟼L2(Y),(Pryf)(y)=∫Ωf(yω)Py(dω) is the conditional expectation with respect to the measure
P y ( d ω ) = 1 p ( y ) δ ( y − η ( t , ω ) ) P ( d ω ) P y ( d ω ) = 1 p ( y ) δ ( y − η ( t , ω ) ) P ( d ω ) P_(y)(d omega)=(1)/(p(y))delta(y-eta(t,omega))P(d omega)P_{y}(d \omega)=\frac{1}{p(y)} \delta(y-\eta(t, \omega)) P(d \omega)Py(dω)=1p(y)δ(y−η(t,ω))P(dω)
proved in [2], possess unitary t is extended to K τ 1 = 1 K τ 1 = 1 K_(tau)1=1K_{\tau} 1=1Kτ1=1 ). The atural extension lity measures in
possess unitary arkov processes, these processes, the MPC theory.
ding to a stationoups; semigroups, both ry groups induced
roup and those of ζ τ Φ = Φ U τ ζ τ Φ = Φ U τ zeta_(tau)Phi=PhiU_(tau)\zeta_{\tau} \Phi=\Phi U_{\tau}ζτΦ=ΦUτ.
Y Y YYY ), of the Markov finite-dimensional
y , t ) f ( y ′ ) d y ′ = η ( t , ω ) ) P ( d ω ) = y , t ) f y ′ d y ′ = η ( t , ω ) ) P ( d ω ) = {:[y","t)f(y^('))dy^(')=],[eta(t","omega))P(d omega)=]:}\begin{array}{r} y, t) f\left(y^{\prime}\right) d y^{\prime}= \\ \eta(t, \omega)) P(d \omega)= \end{array}y,t)f(y′)dy′=η(t,ω))P(dω)=
he Markov shift Σ τ Σ τ Sigma_(tau)\Sigma_{\tau}Στ Thus, the Koopman unitary operator in
P y ( d ω ) P y ( d ω ) P_(y)(d omega)P_{y}(d \omega)Py(dω) is the condi-
Hence, according to the definition (9), the group { U τ ∗ } τ ∈ R U τ ∗ τ ∈ R {U_(tau)^(**)}_(tau inR)\left\{U_{\tau}^{*}\right\}_{\tau \in \mathbb{R}}{Uτ∗}τ∈R is the unitary dilation of the semigroup { K τ ∗ } τ ≥ 0 K τ ∗ τ ≥ 0 {K_(tau)^(**)}_(tau >= 0)\left\{K_{\tau}^{*}\right\}_{\tau \geq 0}{Kτ∗}τ≥0. Because U τ ∗ U τ ∗ U_(tau)^(**)U_{\tau}^{*}Uτ∗ are also Markov operators, the property (M1) implies the positivity of the dilation.
2) Strongly ergodic Markov processes (6) are mixing, according to Lemma 2. Then the Markov shift { Σ τ } τ ∈ R Σ τ τ ∈ R {Sigma_(tau)}_(tau inR)\left\{\Sigma_{\tau}\right\}_{\tau \in \mathbb{R}}{Στ}τ∈R which preserves the measure (8) is a Ksystem (see [8], p.181and [22]). Measure preserving property and (10) also imply ( K τ f ) ( y ) = ( Pr y U τ f ) ( y ) K τ f ( y ) = Pr y U τ f ( y ) (K_(tau)f)(y)=(Pr_(y)U_(tau)f)(y)\left(K_{\tau} f\right)(y)=\left(\operatorname{Pr}_{y} U_{\tau} f\right)(y)(Kτf)(y)=(PryUτf)(y). Here, Pr y Pr y Pr_(y)\operatorname{Pr}_{y}Pry has the meaning of a projection on the cells of the K-partition consisting of all the trajectories containing the point ( y , t ) [ 2 ] ( y , t ) [ 2 ] (y,t)[2](y, t)[2](y,t)[2].
3) We define the canonical injection I ω : L 2 ( Y ) ⟼ L 2 ( Y R ) I ω : L 2 ( Y ) ⟼ L 2 Y R I_(omega):L^(2)(Y)longmapstoL^(2)(Y^(R))I_{\omega}: L^{2}(Y) \longmapsto L^{2}\left(Y^{\mathbb{R}}\right)Iω:L2(Y)⟼L2(YR), I ω f ( y ω ) ≡ f ( y ) , ∀ y ω ∈ { y ω ∣ y ω ( t ) = y I ω f y ω ≡ f ( y ) , ∀ y ω ∈ y ω ∣ y ω ( t ) = y I_(omega)f(y^(omega))-=f(y),AAy^(omega)in{y^(omega)∣y^(omega)(t)=y:}\mathrm{I}_{\omega} f\left(y^{\omega}\right) \equiv f(y), \forall y^{\omega} \in\left\{y^{\omega} \mid y^{\omega}(t)=y\right.Iωf(yω)≡f(y),∀yω∈{yω∣yω(t)=y, for fixed- t } t {:t}\left.t\right\}t} (so that I ω f I ω f I_(omega)f\mathrm{I}_{\omega} fIωf takes constant values on cylindrical sets of the K-partition). Obviously, Pr y I ω = 1 Pr y I ω = 1 Pr_(y)I_(omega)=1\operatorname{Pr}_{y} \mathrm{I}_{\omega}=1PryIω=1. For each set of the K -partition the relation I ω Pr y = 1 I ω Pr y = 1 I_(omega)Pr_(y)=1\mathrm{I}_{\omega} \operatorname{Pr}_{y}=1IωPry=1 also holds. By representing the corresponding L 2 L 2 L^(2)L^{2}L2 functions as limits of linear combinations of step functions one finds that I ω = ( Pr y ) − 1 : L 2 ( Y ) ⟼ L 2 ( Y R ) I ω = Pr y − 1 : L 2 ( Y ) ⟼ L 2 Y R I_(omega)=(Pr_(y))^(-1):L^(2)(Y)longmapstoL^(2)(Y^(R))\mathrm{I}_{\omega}=\left(\operatorname{Pr}_{y}\right)^{-1}: L^{2}(Y) \longmapsto L^{2}\left(Y^{\mathbb{R}}\right)Iω=(Pry)−1:L2(Y)⟼L2(YR). Thus the projection relation (8) becomes ( K τ f ) ( y ) = ( Pr y U τ I ω f ) ( y ) , f ∈ L 2 ( Y ) K τ f ( y ) = Pr y U τ I ω f ( y ) , f ∈ L 2 ( Y ) (K_(tau)f)(y)=(Pr_(y)U_(tau)I_(omega)f)(y),f inL^(2)(Y)\left(K_{\tau} f\right)(y)=\left(\operatorname{Pr}_{y} U_{\tau} \mathrm{I}_{\omega} f\right)(y), f \in L^{2}(Y)(Kτf)(y)=(PryUτIωf)(y),f∈L2(Y), and renaming Pr y Pr y Pr_(y)\operatorname{Pr}_{y}Pry by Φ Φ Phi\PhiΦ, it just the intertwining relation of the MPC theory, K τ Φ = Φ U τ K τ Φ = Φ U τ K_(tau)Phi=PhiU_(tau)K_{\tau} \Phi=\Phi U_{\tau}KτΦ=ΦUτ. ◻ ◻ ◻\square◻
The formalism of the ergodic statistical mechanics deals with abstract irreversible dynamical systems [ 1 − 6 , 18 ] [ 1 − 6 , 18 ] [1-6,18][1-6,18][1−6,18]. The use of the unitary operators formalism seems to be a "technical" requirement in developing present day statistical mechanics. The previous theorem shows that realistic irreversible processes with strong ergodicity property (as Brownian motions with reflecting boundaries [11,15,20], modelling ideal gases or OrnsteinUhlenbeck process, useful in, and model example of fluctuation-dissipation theorems [9,25]) also may be represented by dynamical systems and unitary operators.

References

[1] I. Antoniou and S. Tasaki, Generalized Spectral Decomposition of the β β beta\betaβ-adic Baker's Transformation and Intrinsec Irreversibility, Physica A, 190, 303-329, 1992.
[2] I. Antoniou and K. E. Gustafson, From Probabilistic Description to Deterministic Dynamics, Physica A. 179, 153-166, 1993.
[3] I. Antoniou and I. Prigogine, Intrinsec Irreversibility and Integrability of Dynamics, Physica A 192, 443-464, 1993.
[4] I. Antoniou and S. Tasaki, Generalized Spectral Decomposition of Mixing Dynamical Systems, Int. J. Quantum Chem., 46, 425-474, 1993.
[5] I. Antoniou and K. E. Gustafson, From Irreversible Markov Semigroups to Chaotic Dynamics, Physica A 236, 296-308, 1997.
[6] I. Antonion, K. E. Gustafson and Z. Suchanecki, On the Inverse Problem of Statistical Physics: From Irreversible Semigroups to Chaotic Dynamics, Physica A 256, 345-361, 1998.
[7] M. Courbage, On Boltzmann Entropy and Coarse-Graining for Classical Dynamical Systems, in Information Dynamics, ed. H. Atmanspacher and H. Scheingraber, Plenum, New York, 1991.
[8] I. Cornfeld, S. Fomin and Ya. G. Sinai, Ergodic Theory. Springer, Berlin, 1982.
[9] C. Gardiner, Handbook of Stochastic Methods (for Physics, Chemistry and Natural Science), Springer, Eerlin, 1983.
[10] N. A. Friedman and D. S. Ornstein, On Isomorphism of Weak Bernoulli Transformations, Adv. Math. 5 (1971), 365-394.
[11] S. Goldstein, J. L. Lebowitz and E. Presutti, Stationary Markov Chains. in Fritz, J., J. L. Lebowitz and D. Szasz, editors, Rigorous results in statistical mechanics and quantum field theory, Colloquia Matematica Societatis Janos Bolyai, 27, North Holland, Amsterdam, 1981.
[12] M. Iosifescu and P. Tăutu, Stochastic Processes and Applications in Biology and Medicine. I Theory, Editura Academiei, Bucureşti and Springer, Berlin, 1973, 1970.
[13] A. Kolmogorov and S. Fomine, Élémentes de la théorie des fonctions et de l'analyse fonctionelle, Mir, Moscou, 1975.
[14] A. Lasota and M. C. Mackey, Probabilistic Properties of Deterministic Systems, Springer, New York, 1985; Second edition, and A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise, Stochastic Aspects of Dynamics, Springer, New York, 1994.
[15] M. C. Mackey, The Dynamic Origin of Increasing Entropy, Rev. Modern Phys., 61, 4 (1989), 981-1016.
[16] P. Malliavin, Integration and Probability, Springer, New York, 1995.
[17] R. McCabe and P. Shields, A Class of Markov Shifts which are Bernoulli Shifts, Adv. Math. 6 (1971), 323-328.
[18] B. Misra, I. Prigogine and M. Courbage, From Deterministic Dynamics to Probabilistic Description, Physica 98A (1979), 1-26.
[19] G. Nicolis and C. Nicolis, Chaotic Dynamics, Markovian Coarse-Graining and Information, Physica A, 163 (1990), 215-231.
[20] D. S. Ornstein and P. C. Shields, Mixing Markov Shifts of Kernel Type are Bernoulli, Advances in Mathematics, 10 (1973), 143-146.
[21] V. A. Rokhlin, Exact Endomorphisms of Lebesgue Spaces, Am. Math. Soc. Transl., (2) 39 (1964), 1-36.
[22] Ya. G. Sinai, Introduction to Ergodic Theory, Princeton Univ. Press., 1976.
[23] N. Suciu and A. Georgescu, On the Misra-Prigogine-Courhage Theory of Irreversibility, Bul. Şt.Univ. Piteşti, Seria Mat. Inf. 2, (1990), 169-188.
[24] B. Sz-Nagy and C. Foiaş, Harmonic Analysis of Operators in Hilbert Spaces, NorthHolland, Amsterdam, 1970.
[25] N. G. van Kampen, Stochastic Processes in Physics and Chemistry, North-Holland, Amsterdam, 1981.
[26] A. D. Wentzell, A Course in the Theory of Stochastic Processes, McGraw-Hill, New York, 1981.
2000

Related Posts