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

Citations:

Zbl 0635.00013; Zbl 0581.60057
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