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# 7 Results

[635.1.1] The time evolution of almost invariant states can be defined by the addition of random recurrence times. [635.1.2] Let be the probability density of the sum

 (40)

of independent and identically-distributed random recurrence times . [635.1.3] Let from Equation (39) be the common probability density of all . [635.1.4] Then, with and ,

 (41)

is an -fold convolution of the discrete recurrence time density in Equation (39). [635.1.5] The family of distributions obeys

 (42)

for all , and the discrete analogue of Equation (5)

 (43)

holds for all . [635.1.6] Because the individual states in are indistinguishable within the given accuracy , but may evolve very differently in time, it is natural to define the duration of time needed for the first recurrence (a single time step) as an average

 (44)

over recurrence times. [635.1.7] If a macroscopic time evolution with a rescaled time exists, then one has to rescale the sums and the iterations

 (45)

in the limit with suitable norming constants .

###### Theorem 3.

[635.1.8] Let be the probability density of specified above in (41). [635.1.9] If the distributions of converge to a limit as for suitable norming constants , then there exist constants and , such that

 (46)

[page 636, §0]    where:

 (47)

if diverges, while

 (48)

if converges. [636.0.1] For , the function is . [636.0.2] For , the function for and

 (49)

for .

###### Proof.

[636.1.1] The existence of a limiting distribution for is known to be equivalent to the stability of the limit [18]. [636.1.2] If the limit distribution is nondegenerate, this implies that the rescaling constants have the form

 (50)

where is a slowly varying function [19], defined by the requirement that

 (51)

holds for all . [636.1.3] That the number obeys Equation (47) is proven in [18] (p. 179). [636.1.4] It is bounded as , because the rescaled random variables are positive.

[636.2.1] To prove Equation (46), note that the characteristic function of is the -th power

 (52)

because the characteristic functions of are identical for all . [636.2.2] Inverse Fourier transformation gives:

 (53)

where:

 (54)

and was substituted with . [636.2.3] Let denote the characteristic function of , so that

 (55)

holds.

[page 637, §1]    [637.1.1] Following [20], the difference in (46) can be decomposed and bounded from above as

 (56)

with constants to be specified below. [637.1.2] The terms involving from the second and third integral have been absorbed in the fourth integral. [637.1.3] The four integrals are now discussed further individually.

[637.2.1] The first integral converges uniformly to zero for , because belongs to the domain of attraction of a stable law with index , as already noted above.

[637.3.1] To estimate the second integral, note that the characteristic function belongs to the domain of attraction for index if and only if it behaves for as [20]

 (57)

where and is a slowly varying function at infinity obeying

 (58)

[637.3.2] By the representation theorem for slowly varying functions ([21] p. 12), there exist functions and , such that the function can be represented as

 (59)

[page 638, §0]    for some where is measurable and , as well as hold for . [638.0.1] As a consequence

 (60)

so that with and

 (61)

is obtained for . [638.0.2] Therefore, there exists for any a positive number independent of , such that

 (62)

for sufficiently large . [638.0.3] If is sufficiently large, it is then possible to choose an (and find ), such that

 (63)

and this converges to zero for .

[638.1.1] The third integral is estimated by noting that for . [638.1.2] Hence, there is a positive constant , such that

 (64)

for . [638.1.3] Consequently, with Equation (50),

 (65)

converges to zero as .

[638.2.1] Finally, the fourth integral converges to zero, because the characteristic function is integrable on . [638.2.2] In summary, all four terms in Equation (56) vanish for , and Equation (46) holds. ∎

[638.3.1] Equation (46) implies

 (66)

for sufficiently large and all . [638.3.2] Inserting this into Equation (45) gives

 (67)

[page 639, §0]    [639.0.1] For , the average return time is proportional to the discretization . [639.0.2] In the case , the average time for return into the set in a single step diverges. [639.0.3] This suggests an infinite rescaling of time as for . [639.0.4] This rescaling of time combined with was called the ultra-long-time limit in [22]. [639.0.5] In the ultra-long-time limit with:

 (68)

one finds from Equation (67) the result

 (69)

for sufficiently large and . [639.0.6] The limit gives rise to a family of one-parameter semigroups (with family index and parameter ) of ultra-long-time evolution operators

 (70)

which are convolutions instead of translations. [639.0.7] Note that because and . [639.0.8] The rescaled age evolutions are called fractional time evolutions, because their infinitesimal generators are fractional time derivatives [22, 23].

[639.1.1] The result shows that a proper mathematical formulation of local stationarity requires a generalization of the left-hand side in Equation (1), because Equation (1) assumes implicitly a translation along the orbit. [639.1.2] In general, the integration of infinitesimal system changes leads to convolutions instead of just translations along the orbit [22, 23]. [639.1.3] Of course, translations are a special case of convolutions, to which they reduce in the case when the parameter approaches unity. [639.1.4] For , one finds

 (71)

and therefore

 (72)

is a right translation. [639.1.5] Here, is an age or duration. [639.1.6] This shows that also the special case of induced right translations does not give a group, but only a semigroup.