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A formula for densities of transition functions. (English) Zbl 0662.60081

Séminaire de probabilités XXII, Strasbourg/France, Lect. Notes Math. 1321, 92-100 (1988).
[For the entire collection see Zbl 0635.00013.]
Let \((P_{s,t})\) (resp. (\^P\({}_{st}))\) be a measurable forward (resp. backward) transition function on a measurable space (E,\(\underset {=} E)\) and let \((\mu_ t)\) be a family of \(\sigma\)-finite measures on E. Under suitable assumptions of absolute continuity and duality, R. Wittmann [Probab. Theory Relat. Fields 73, 1-10 (1986; Zbl 0581.60057)] has constructed densities \[ P_{s,t}(x,dy)=p(s,t,x,y)\mu_ t(dy),\quad \hat P_{s,t}(y,dx)=p(s,t,x,y)\mu_ s(dx), \] satisfying identically the Chapman-Kolmogorv equation \[ p(s,t,x,y,)=\int p(s,r,x,z)p(r,t,z,y)\mu_ r(dz)\quad for\quad s<r<t. \] In the present paper we give a simpler proof of this result under weaker hypotheses. We give an explicit formula for the density. In the last section we construct a good density for any transition function, without duality hypotheses.

MSC:

60J35 Transition functions, generators and resolvents
60G17 Sample path properties
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