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For clarity, we will sometimes speak of the Green distribution G(x,t) or Green density g(x,t) where appropriate. For homogeneous lattice processes, discussed in Section 4, we will speak of Green probabilities. 4 Lattice processes When the initial distribution F 0 (x) has its support on the lattice of integers and transitions terminate in integral values, the discrete time process Xk of Section 2 is confined to the lattice. It will then be denoted by Nk . Markov processes in discrete or continuous time on a denumerable state space are called Markov chains.

6 The characteristic function and generating function For the homogeneous processes discussed in Sections 3 and 4, analysis is facilitated by the introduction of characteristic functions and generating functions. The characteristic function of a distribution F(x), is defined by the expectation ¢(z) = E(eizx) = J_~ eizx dF(x). 1) is a Fourier-Stieltjes transform of F(x). 2) eizxf(x)d~. Characteristic functions will be denoted by corresponding lower-case Greek letters. A good derivation of the properties of characteristic functions may be found in a companion volume of this series by E.

F(x,t) at = -vF(x,t)+v f 00 A(x-x')dF(x',t). 3) and (3. 7) are homogeneous in x, and the additive processes with independent random increments are said to be spatially homogeneous processes. 8) The jump process X(t) of frequency v and increment distribution A(x) will then also have for its transition distribution G(x,t) a mean and variance, given by µ,(t) = vt µ, 1 a 2 (t) = vt(a12 + µ,~). 9) We note that the distribution G(x,t) for a jump process of frequency v and increment distribution A(x) acquires some of the simple properties of A(x).

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Actes du Congres International Des Mathematiciens (1970)(fr)(520s)

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