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Natural densities of Markov transition probabilities
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  • Published: March 1986

Natural densities of Markov transition probabilities

  • Rainer Wittmann1 

Probability Theory and Related Fields volume 73, pages 1–10 (1986)Cite this article

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  • 5 Citations

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Summary

Let (P t ) t>0 , (P * t ) t>0 be two measurable submarkovian semigroups on a measurable spaceE which are absolutely continuous and in duality with respect to a σ-finite measure μ. Then we show that there exists a unique measurable function\(p:]0,\infty [ \times E \times E \to \mathop \mathbb{R}\limits^{\_} _ +\) satisfying

$$\begin{gathered} P_t f(x) = \smallint p(t,x,y)f(y)\mu (dy), \hfill \\ P_t^* f(x) = \smallint p(t,y,x)f(y)\mu (dy)(f \in E_ + ,x \in E) \hfill \\ \end{gathered} $$
((i))
$$p(s + t,x,y) = \smallint p(s,x,z)p(t,z,y)\mu (dz)(s,t > 0,x,y \in E).$$
((ii))

A similar result is shown for inhomogeneous transition functions.

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References

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Author information

Authors and Affiliations

  1. Mathematisch-Geographische Fakultät, Katholische Universität Eichstätt, Ostenstraße 26-28, D-8078, Eichstätt, Federal Republic of Germany

    Rainer Wittmann

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  1. Rainer Wittmann
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Cite this article

Wittmann, R. Natural densities of Markov transition probabilities. Probab. Th. Rel. Fields 73, 1–10 (1986). https://doi.org/10.1007/BF01845990

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  • Received: 23 July 1983

  • Revised: 22 May 1985

  • Accepted: 07 January 1986

  • Issue Date: March 1986

  • DOI: https://doi.org/10.1007/BF01845990

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Keywords

  • Stochastic Process
  • Probability Theory
  • Measurable Function
  • Transition Function
  • Mathematical Biology
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