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### A Maximal Theorem

C. S. Herz
1961 Proceedings of the American Mathematical Society
Suppose there is a 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

C. S. Herz
1961 Proceedings of the American Mathematical Society
Suppose there is a 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]

Michel Benaim, Jean-Yves Le Boudec
2011 arXiv   pre-print
Assume that a family 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]

Tamara Kucherenko, Daniel J. Thompson
2021 arXiv   pre-print
continuous roof function such that the set 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]

Godofredo Iommi, Anibal Velozo
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

Yongluo Cao
2006 Journal of Differential Equations
Let M P (φ) and E P (φ) denote the set 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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