# Additive functionals of Markov processes and stochastic systems

Annales de l'institut Fourier (1975)

- Volume: 25, Issue: 3-4, page 177-200
- ISSN: 0373-0956

## Access Full Article

top## Abstract

top## How to cite

topDynkin, Evgeny B.. "Additive functionals of Markov processes and stochastic systems." Annales de l'institut Fourier 25.3-4 (1975): 177-200. <http://eudml.org/doc/74240>.

@article{Dynkin1975,

abstract = {Intuitively, an additive functional of a stochastic process $(x_t,P)$ gives a method to measure time taking into account the development of the process. We associate with any set of states $C$ the mathematical expectation of time $x_t$ belongs to $C$. In this way, we establish to one-to-one correspondence between all the normal additive functionals of a Markov process and all the $\delta $-finite measures on the state space which charge no inaccessible set. This is proved under the condition that transition probabilities are almost all paths. According to the previous results of the author, if two-dimensional probability distributions of a Markov process are absolutely continuous with respect to products of corresponding one-dimensional distributions, then the process can be modified in such a way that the set of additive functionals does not change and transition and cotransition probabilities acquire the above-mentioned properties.},

author = {Dynkin, Evgeny B.},

journal = {Annales de l'institut Fourier},

language = {eng},

number = {3-4},

pages = {177-200},

publisher = {Association des Annales de l'Institut Fourier},

title = {Additive functionals of Markov processes and stochastic systems},

url = {http://eudml.org/doc/74240},

volume = {25},

year = {1975},

}

TY - JOUR

AU - Dynkin, Evgeny B.

TI - Additive functionals of Markov processes and stochastic systems

JO - Annales de l'institut Fourier

PY - 1975

PB - Association des Annales de l'Institut Fourier

VL - 25

IS - 3-4

SP - 177

EP - 200

AB - Intuitively, an additive functional of a stochastic process $(x_t,P)$ gives a method to measure time taking into account the development of the process. We associate with any set of states $C$ the mathematical expectation of time $x_t$ belongs to $C$. In this way, we establish to one-to-one correspondence between all the normal additive functionals of a Markov process and all the $\delta $-finite measures on the state space which charge no inaccessible set. This is proved under the condition that transition probabilities are almost all paths. According to the previous results of the author, if two-dimensional probability distributions of a Markov process are absolutely continuous with respect to products of corresponding one-dimensional distributions, then the process can be modified in such a way that the set of additive functionals does not change and transition and cotransition probabilities acquire the above-mentioned properties.

LA - eng

UR - http://eudml.org/doc/74240

ER -

## References

top- [1]R. M. BLUMENTHAL and R. K. GETOOR, Markov Processes and Potential Theory, Academic Press, New York-London, 1968. Zbl0169.49204MR41 #9348
- [2]C. DELLACHERIE, Ensembles aléatoires I, Séminaire de Probabilités III, Université de Strasbourg, Lecture Notes in Math., 88, Springer-Verlag, Berlin-Heidelberg-New York, 1969, 97-114. Zbl0184.41202MR41 #2767
- [3]C. DELLACHERIE, Capacités et processus stochastiques, Springer-Verlag, Berlin-Heidelberg-New York, 1972. Zbl0246.60032MR56 #6810
- [4]E. B. DYNKIN, Markovskie processy, Fizmatgiz, Moskva, 1963, Markov Processes Vol. I-II, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1965.
- [5]E. B. DYNKIN, Regular Markov processes, Uspehi Matem. Nauk, 28, 2 (170), (1973) (English translation in Russian Mathematical Surveys, London). Zbl0385.60059MR53 #4244
- [6]E. B. DYNKIN, Additive functionals of Markov processes and their spectral measures, Dokl. Akad. Nauk SSSR, 214 (1974), 1241-1244 (English translation in Soviet Math. Dokl.). Zbl0306.60047MR49 #6376
- [7]E. B. DYNKIN, Markov representations of stochastic systems, Dokl. Akad. Nauk SSSR, 218 (1974), 1009-1012. Zbl0323.60070MR50 #14945
- [8]E. B. DYNKIN, Markov representations of stochastic systems, Uspehi Matem. Nauk, 30, 1 (1975), 61-99. Zbl0316.60019MR53 #6701
- [9]P. A. MEYER, Fonctionnelles multiplicatives et additives de Markov, Ann. Inst. Fourier, 12 (1962), 125-230. Zbl0138.40802MR25 #3570
- [10]P. A. MEYER, Processus de Markov, Lecture Notes in Math., 26, Springer-Verlag, Berlin-Heidelberg-New York, 1967. Zbl0189.51403MR36 #2219
- [11]P. A. MEYER, Probability and Potentials, Blaisdell, Waltham Mass., 1966. Zbl0138.10401MR34 #5119
- [12]M. METIVIER, Mesures vectorielles et intégrale stochastique II, Séminaire de probabilités, année 1972, Preprint.
- [13]D. REVUZ, Mesures associées aux fonctionnelles additives de Markov I, Trans. Amer. Math. Soc., 148 (1970), 501-531. Zbl0266.60053MR43 #5611
- [14]M. ŠUR, Continuous additive functionals of Markov processes and excessive functions, Dokl, Akad. Nauk SSSR, 137 (1961), 800-803 [Soviet Math. 2 (1961), 365-368]. Zbl0111.33204MR26 #7023
- [15]M. ŠUR, On approximation of additive functionals of Markov processes Uspehi Mat. Nauk, 29, 6 (1974), 183-184. Zbl0326.60092
- [16]A. D. WENTZELL, Nonnegative additive functionals of Markov processes, Dokl. Akad. Nauk SSSR, 137 (1961), 17-20 [Soviet Math., 2 (1961), 218-221].

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.