Reminiscences of some of Paul Lévy’s ideas in Brownian Motion and in Markov Chains

  • Kai Lai Chung
Chapter
Part of the Progress in Probability book series (PRPR, volume 17)

Abstract

We begin with a resume. Let {P(t), t ≥ 0} be a semigroup of stochastic matrices with elements p ij (t), (i,j) ∈ I ×I, where I is a countable set, satisfying the condition
$$\mathop {\lim }\limits_{t \downarrow 0} p_{ii} (t) = 1 $$
(1)
. It is known that p’ ij (0) = q ij exists and
$$ 0 \leqslant {q_i} = - {q_{{ii}}} \leqslant + \infty, \;0 \leqslant {q_{{ij}}} < \infty, i \ne j; $$
(2)
$$\sum\limits_{{j \ne i}} {{q_{{ij}}} \leqslant {q_i}.} $$
(3)
The state i is called stable if q i < +∞, and instantaneous if q i = +∞ (Lévy’s terminology). The matrix Q = (q ij ) is called conservative when equality holds in (3) for all i.

Keywords

Dust 

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Copyright information

© Birkhäuser Boston 1989

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  • Kai Lai Chung

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