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

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.