A remark on the generator of a right-continuous Markov process

Röckner M, Trutnau G (2007)
Infinite Dimensional Analysis, Quantum Probability and Related Topics 10(4): 633-640.

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Zeitschriftenaufsatz | Veröffentlicht | Englisch
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Abstract / Bemerkung
Given a right-continuous Markov process (X t) t >= 0 on a second countable metrizable space E with transition semigroup (pt) t >= 0, we prove that there exists a sigma- finite Borel measure mu with full support on E, and a closed and densely defined linear operator (L-p; D(L-p)) generating (pt)(t) >= 0 on L-p(E; mu). In particular, we solve the corresponding Cauchy problem in L-p (E; mu) for any initial condition u is an element of D(Lp). Furthermore, for any real beta > 0 we show that there exists a generalized Dirichlet form which is associated to (e(-beta t) pt) t >= 0. If the beta-subprocess of (X-t)(t >= 0) corresponding to (e(-beta t) pt) t >= 0, beta > 0, is mu-special standard then all results from generalized Dirichlet form theory become available, and Fukushima's decomposition holds for u is an element of D (L-2). If (X-t)(t >= 0) is transient, then beta can be chosen to be zero.
Erscheinungsjahr
Zeitschriftentitel
Infinite Dimensional Analysis, Quantum Probability and Related Topics
Band
10
Ausgabe
4
Seite(n)
633-640
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Röckner M, Trutnau G. A remark on the generator of a right-continuous Markov process. Infinite Dimensional Analysis, Quantum Probability and Related Topics. 2007;10(4):633-640.
Röckner, M., & Trutnau, G. (2007). A remark on the generator of a right-continuous Markov process. Infinite Dimensional Analysis, Quantum Probability and Related Topics, 10(4), 633-640. doi:10.1142/S0219025707002907
Röckner, M., and Trutnau, G. (2007). A remark on the generator of a right-continuous Markov process. Infinite Dimensional Analysis, Quantum Probability and Related Topics 10, 633-640.
Röckner, M., & Trutnau, G., 2007. A remark on the generator of a right-continuous Markov process. Infinite Dimensional Analysis, Quantum Probability and Related Topics, 10(4), p 633-640.
M. Röckner and G. Trutnau, “A remark on the generator of a right-continuous Markov process”, Infinite Dimensional Analysis, Quantum Probability and Related Topics, vol. 10, 2007, pp. 633-640.
Röckner, M., Trutnau, G.: A remark on the generator of a right-continuous Markov process. Infinite Dimensional Analysis, Quantum Probability and Related Topics. 10, 633-640 (2007).
Röckner, Michael, and Trutnau, Gerald. “A remark on the generator of a right-continuous Markov process”. Infinite Dimensional Analysis, Quantum Probability and Related Topics 10.4 (2007): 633-640.