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And Appl. 5(1971), 19-21 (in Russian). Hennion, Sur le mouvement d'une particule }ourde soummise a des callisions dane un systeme infini de particules legres, Zeitschrift fur Wahrsch. 25(1972/73), 123-154. WINTER SCHOOL ON PROBABILITY Kar2acz q975 Sprlnger's LNM 472 s ON LEVY'S BROWNIAN MOTION WITH SEVERAL - DIMENSIONAL TIME By Z. Ciesielski Institute of Mathematics of the Polish Academy of Sciences 1. Introduction. The purpose of this lecture is to present an approach to Levy s B r o w n i a n M o t i o n based on Chentsov's construction [3] and on the author's note [2].

G. gives E(X(a) X) I Rd - d,1 -~ I ~ (b) db Ibl 1/2 where ~(b) (~d+1) = E L2(Rd; C) = T r 1 6 2H k . 1 Let ~ . Since ;(b) 2 d ; C) is in Lo(R ~,~ r C~o(Rd) = ~(-b) and and therefore be real valued and such that 48 ~(0) = ~(0) = 0 . ,a n) . i=I Proof. f. [ ]) then d+l D-~-- are understood in the generalized d+1 IblT d+1 ~(b) -_ #L (D-x- 9) (b) and d+1 Ibl d$1 ~(b) = (D-~- @)" (b) . ol. 5. The covariance operator. We assume again that on (G,F,P) ~X(a) : a E Rd~ implies the continmous LBM is given.

1 clearly ~Y(a), Yn n=o ~Y(a), ~ 9 fn d~ a E Rd~ . is a continuous LBM and H c H where H is the Gaussian subspace generated by Yd . x a E R ~ . We are golng to show that Hy = H x . Suppose that Y E H x - H y . 4) W(f) = ~ Yn(f, fn ) n=O It follows immediately that Clearly, E(W(f) W(g)) = (f,g)~ . e. such that E(Y(a) W(f)) = O give YY(a) = W(l,) E for and H x . Consequently, and this element is orthogonal a E R d . 1 we find that unitary operator a E R d . 4) whence Y(a) W(f) W(f) = Y = 0 .

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