[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 .

[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 .

[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 .

[638.3.1] 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.