Wittmann, Rainer Natural densities of Markov transition probabilities. (English) Zbl 0581.60057 Probab. Theory Relat. Fields 73, 1-10 (1986). Let \((P_ t)\), \((P^*\!_ t)\) be two measurable submarkovian semigroups on a measurable space E which are absolutely continuous and in duality with respect to a \(\sigma\)-finite measure \(\mu\). Then it is shown that there exists a unique measurable function \(p: (0,\infty)\times E\times E\to {\bar {\mathbb{R}}}_+\) satisfying \[ (i)\quad P_ tf(x)=\int p(t,x,y)f(y)\mu (dy),\quad P^*\!_ tf(x)=\int p(t,y,x)f(y)\mu (dy)\quad (f\in E_+,\quad x\in E) \]\[ (ii)\quad p(s+t,x,y)=\int p(s,x,z)p(t,z,y)\mu (dz),\quad s,t>0,\quad x,y\in E. \] In particular, for a symmetric semigroup, there exists a unique symmetric density satisfying (i), (ii). A more general result for inhomogeneous transition probabilities is also given. Cited in 1 ReviewCited in 2 Documents MSC: 60J35 Transition functions, generators and resolvents 47D07 Markov semigroups and applications to diffusion processes Keywords:submarkovian semigroups; symmetric density; inhomogeneous transition probabilities PDFBibTeX XMLCite \textit{R. Wittmann}, Probab. Theory Relat. Fields 73, 1--10 (1986; Zbl 0581.60057) Full Text: DOI References: [1] Blumenthal, R. M.; Getoor, R. K., Markov processes and potential theory (1968), New York London Amsterdam: Academic Press, New York London Amsterdam · Zbl 0169.49204 [2] Dynkin, E. B., Minimal excessive measures and functions, Trans. Am. Math. Soc., 258, 217-244 (1980) · Zbl 0422.60057 [3] Dynkin, E. B., Markov processes and random fields, Bull. Am. Math. Soc., 3, 975-999 (1980) · Zbl 0519.60046 [4] Dynkin, E. B., Additive functionals of seversible Markov processes, J. Funct. Anal., 42, 64-101 (1981) · Zbl 0467.60069 [5] Dynkin, E. B.; Kuznecov, S. E., Determining functions of Markov processes and corresponding dual regular classes, Soviet. Math. Dokl., 15, 20-23 (1974) · Zbl 0334.60032 [6] Hunt, G. A., Markoff processes and potentials III. Ill, J. Math., 2, 151-213 (1958) [7] Kunita, H.; Watanabe, T., Markov processes and Martin boundaries. Ill, J. Math., 9, 485-526 (1965) · Zbl 0147.16505 [8] Meyer, P. A., Probability and potentials (1966), Waltham (Mass.): Blaisdell Publishing Company, Waltham (Mass.) · Zbl 0138.10401 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.