Recurrent and sweep-out sets for Markov processes

Abstract

Let P be a Markov process on a probability space (X,Z,m). Roughly speaking, a sweep-out set is a set which is reached with probability 1 under the action of the process for m-almost all starting points. Obviously, in a finite state space no sweep-out sets of arbitrary small measure exist. The authors show that in general arbitrary small sweep-out sets exist, unless there is an invariant subset of the state space on which the process behaves as on a finite state space. Moreover, if there exist arbitrary small sweep-out sets and if ~ is countably generated, then there exists an algebra of sweepout sets generating 2L The main tool to obtain these results is the use of embedded processes. Some properties of these processes are collected, and as a side-result a short and elementary proof of the decomposition theorem of E. HoPF of the state space in a conservative and a dissipative part is given. w 1. Introduction and SummaryIn this note we shall use the description of a Markov process by means of a positive linear operator. To this end, let (X, Z,m) be a fixed probability space, the state space of the process. Let J//+ be the set of all (equivalence classes of m-almost everywhere equal) non negative extended real valued measurable functions. A Markov process Y on (X, Z,m) is a mapping of Js into itself such thatHere, as in the sequel, all statements on sets and functions have to be interpreted modulo m-null sets. Moreover, all sets and functions are understood to be measurable. Note that if for every A e Z we choose a representative P (., A) from the equivalence class P1A, then P(x,A) can be interpreted

Terbutaline Slow‐Release Tablets in Children with Bronchial Asthma

“…The children visited the clinic at the end of each period and stayed there during the 6th night and the following day. At various times (0,2,4,6,8,10, and 12 h after the morning dose) during the 7th day, heart rate, blood pressure, PEF, FEVj and tremor were recorded. Finger tremor was measured by an optoelectronic tremorgraph (10).…”

$\sigma$-finite invariant densities for eventually conservative Markov operators

“…Set A = [h * > 0]. Then A ⊂ [h > 0] ∩ C (see the conservative part of the induced operator for [34]) and it follows from A ⊂ E that…”

“…Then if P E h 0 = h 0 for some h 0 ∈ L 1 + we have alsoP (P h 0 ) = P h 0 . Now we assume [h 0 > 0] = E as the condition (3) and writê [34] implies that…”

“…Then a Markov operator on L 1 space can be extended to a positive linear operator on the space M σ + . Further, we extend a Markov operator to a positive linear operator on M + where M + stands for the space of all non-negative measurable functions on (X, F , m) (see [20] or [34] for more information). Therefore, we can consider a fixed point of a Markov operator in M σ + as the density of a σ-finite (infinite) invariant measure.…”

“…From the above lemma, we naturally give the following definition of sweep-out sets. See [34] for more precise properties of sweep-out sets.…”