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A Maximal Theorem

1961
*
Proceedings of the American Mathematical Society
*

Suppose there is a

doi:10.2307/2034312
fatcat:slg7ngpxmrg5zndp6gxsbavzxi
*measure*-preserving*flow**on*a*measure**space*X inducing*probability**semi*-groups {Q(t)}, t>0,*on*the Lebesgue*spaces*Lp with respect to an invariant*measure*. ... Next, we observe that / is simply the supremum*of*the averages*of*|/| with respect to a*probability**semi*-group corresponding to a*measure*-preserving*flow**on*X, in this case the*flow*is Brownian motion ...##
###
A maximal theorem

1961
*
Proceedings of the American Mathematical Society
*

Suppose there is a

doi:10.1090/s0002-9939-1961-0151861-0
fatcat:qabshiprx5db5padbbriykikgq
*measure*-preserving*flow**on*a*measure**space*X inducing*probability**semi*-groups {Q(t)}, t>0,*on*the Lebesgue*spaces*Lp with respect to an invariant*measure*. ... Next, we observe that / is simply the supremum*of*the averages*of*|/| with respect to a*probability**semi*-group corresponding to a*measure*-preserving*flow**on*X, in this case the*flow*is Brownian motion ...##
###
Page 975 of Mathematical Reviews Vol. 45, Issue 4
[page]

1973
*
Mathematical Reviews
*

given),

*one*for proper conservative*measure*preserving*semi*-*flows**on*¢o-finite*measure**spaces*(an application*of*this theorem is given in proving that h(7',)=th(7',), where A is the entropy as introduced ... The author studies*semi*-groups {7',: t= 0}*of*null-preserving mappings, to be called*semi*-*flows*, and groups {7',: —co<t<0o}*of*bi-*measurable**one*-to-*one*and null-preserving mappings, to be called non- ...##
###
Page 231 of American Mathematical Society. Proceedings of the American Mathematical Society Vol. 12, Issue 2
[page]

1961
*
American Mathematical Society. Proceedings of the American Mathematical Society
*

Let {P(t)} be the

*semi*-group P(#)=exp(—tA). This is a*probability**semi*-group corresponding to Brownian motion*on*the sphere which is a*measure*-preserving*flow*with respect to the uniform*measure*. ... Let P(t)} be a*probability**semi*-group defined*on*a meas- ure*space*X and suppose {Q(s)}, s>0, is a*one*-parameter family*of*operators subordinate to } P(t) - i.e., Q(s)=Joo(s, t)P(i)dt. ...##
###
Page 1620 of Mathematical Reviews Vol. 48, Issue 5
[page]

1974
*
Mathematical Reviews
*

Theorem 2: If the

*flow*(X, R) is strictly ergodic, meaning that there is exactly*one**probability**measure**on*X which is invariant under the action*of*R, then I is indeed a Dirichlet algebra. ... (m) vanishes*on*a set*of*positive*measure*. Theorem 4: If the*flow*(X,R) is strictly ergodic then each representing*measure*for I*on*X, other than a point mass, is ergodic. ...##
###
On Mean Field Convergence and Stationary Regime
[article]

2011
*
arXiv
*
pre-print

Assume that a family

arXiv:1111.5710v1
fatcat:sjd3iorxwva5bfvour3bsvnddq
*of*stochastic processes*on*some Polish*space*E converges to a deterministic process; the convergence is in distribution (hence in*probability*) at every fixed point in time. ... We show that any limit point*of*an invariant*probability**of*the stochastic process is an invariant*probability**of*the deterministic process. The results are valid in discrete and in continuous time. ... Assumptions and Notation A Collection*of*Random Processes Let (E, d) be a Polish*space*and P(E) the set*of**probability**measures**on*E, endowed with the topology*of*weak convergence. ...##
###
Page 5010 of Mathematical Reviews Vol. , Issue 2000g
[page]

2000
*
Mathematical Reviews
*

Let L(Q, H) be the Hilbert

*space**of*all H-valued random variables*on*a*probability**space*(Q, Fu) which are u-square integrable. ... Summary: “For a random function depending*on*time and*on*a point*of*a*measure**space*, we find an asymptotic expression for the*measure**of*the region in which the values*of*the function do not exceed a given ...##
###
Page 1221 of Mathematical Reviews Vol. 47, Issue 5
[page]

1974
*
Mathematical Reviews
*

./’ and the closed-open subsets

*of*its Stone*space*X (a compact metric*space*) determines a*measure*Y*on*the algebra #*of*Borel subsets*of*X, and the action*of*7*on*»’ induces a homeomorphism U*of*X. ... (He has already proved the existence*of*these*flows*[Contributions to ergodic theory and*probability*(Proc. ...##
###
Page 6990 of Mathematical Reviews Vol. , Issue 2003i
[page]

2003
*
Mathematical Reviews
*

The process y, determines a stochastic

*flow**on*any compact homogeneous*space*M*of*G. In a previous pa- per [Proc. London Math. ... a*probability**measure*Py. ...##
###
Measures of maximal entropy on subsystems of topological suspension semi-flows
[article]

2021
*
arXiv
*
pre-print

continuous roof function such that the set

arXiv:1909.07317v3
fatcat:qojf7xknz5brbm5zwngls6ubpm
*of**measures**of*maximal entropy for the suspension*semi*-*flow*over (X,f) consists precisely*of*the lifts*of**measures*which maximize entropy*on*Y. ... In particular, for a suspension*flow**on*the full shift*on*a finite alphabet, the set*of*ergodic*measures**of*maximal entropy may be countable, uncountable, or have any finite cardinality. ... We remark that the existence*of*suspension*flows*in the above class for which the set*of*ergodic MMEs is uncountable has been independently obtained in a preprint by Iommi and Velozo [9] . ...##
###
Measures of maximal entropy for suspension flows
[article]

2019
*
arXiv
*
pre-print

*measures*

*of*maximal entropy, and that the same can be arranged so that the new

*flow*has a unique

*measure*

*of*maximal entropy. ... We study suspension

*flows*defined over sub-shifts

*of*finite type with continuous roof functions. We prove the existence

*of*suspension

*flows*with uncountably many ergodic

*measures*

*of*maximal entropy. ... Let X be a compact metric

*space*and T : X Ñ X a continuous map. Denote by M T the

*space*

*of*invariant

*probability*

*measures*

*of*pX, T q and by E T the subset

*of*ergodic

*ones*. Definition 3.10. ...

##
###
On growth rates of sub-additive functions for semi-flows: Determined and random cases

2006
*
Journal of Differential Equations
*

Let M P (φ) and E P (φ) denote the set

doi:10.1016/j.jde.2006.08.016
fatcat:flz2mojuozemvcx4erht6wctay
*of*all φ-invariant*measures**on*Ω × M and the set*of*all ergodic φ-invariant*measures*whose marginal*on*Ω coincide with P respectively. ... Let φ : R + × Ω × M → Ω × M be a*measurable*random dynamical systems*on*the compact metric*space*M over (Ω, F , P, (σ (t)) t∈R + ) with time R + . ... This work is partially supported by NSFC (10571130), NCET, SFMSBRP and SRFDP*of*China. ...##
###
Page 2183 of Mathematical Reviews Vol. 56, Issue 6
[page]

1978
*
Mathematical Reviews
*

By using the result obtained, the author proves that, if a process is a

*semi*-martingale with respect to an arbitrary family*of**probability**measures*indexed by points*of*a*probability**space*, then it is ... Stricker who used it to show that any*semi*-martingale with respect to a*flow**of*o-algebras adapted to a narrower*flow*is also a*semi*-martingale with respect to this*flow*. ...##
###
Page 1618 of Mathematical Reviews Vol. , Issue 82d
[page]

1982
*
Mathematical Reviews
*

Let (J,,XnsYn), n20, be a J-X process with the state

*space*Z X[0, 0) X R', where (Z, F,) is an arbitrary*measurable**space*and (X,,,¥,) is the “ X-component”. ... He concludes that the*flows*in branches*of*the classical exponential networks studied by Jackson, for which the state*probability*vector has the form*of*the product*of*the*probabilities**of*the states at ...##
###
Page 3463 of Mathematical Reviews Vol. , Issue 2003e
[page]

2003
*
Mathematical Reviews
*

The
authors prove that the

*spaces**of*all*probability*Borel*measures**of*the*flow*and*of*the*semi*-*flow*are homeomorphic. ... A continuous*semi*-*flow**on*a compact metric*space*and a*flow*which is the inverse limit*of*the*semi*-*flow*are considered. ...
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