Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Adding and subtracting jumps from Markov processes


Author: Richard F. Bass
Journal: Trans. Amer. Math. Soc. 255 (1979), 363-376
MSC: Primary 60J25
DOI: https://doi.org/10.1090/S0002-9947-1979-0542886-X
MathSciNet review: 542886
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: If $ {X_t}$ is a continuous Markov process with infinitesimal generator A, if n is a kernel satisfying certain conditions, and if B is an operator given by

$\displaystyle Bg(x)\, = \,\int {[ {g( y)\, - \,g(x)}]} \,n({x,\,dy}),$

then $ A\, + \,B$ will be the generator of a Markov process that has Lévy system $ (n,\,dt)$. Conversely, if $ {X_t}$ has Lévy system $ (n,\,dt)$, n satisfies certain conditions, and B is defined as above, then $ A\, - \,B$ will be the generator of a continuous Markov process.

References [Enhancements On Off] (What's this?)

  • [1] R. Bass, Perturbations of Hunt processes (to appear).
  • [2] A. Benveniste and J. Jacod, Systèmes de Lévy des processus de Markov, Invent. Math. 21 (1973), 183-198. MR 49 #8117. MR 0343375 (49:8117)
  • [3] R. M. Blumenthal and R. K. Getoor, Markov processes and potential theory, Academic Press, New York, 1968. MR 41 #9348. MR 0264757 (41:9348)
  • [4] J. M. Cook, Weak infinitesimal generators of a class of jump perturbed Markov processes, J. Math. Anal. Appl. 44 (1973), 676-700. MR 51 #4419. MR 0368178 (51:4419)
  • [5] M. G. Crandall and T. M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math. 93 (1971), 265-298. MR 44 #4563. MR 0287357 (44:4563)
  • [6] N. Ikeda, M. Nagasawa and S. Watanabe, Branching Markov processes. I, II, III, J. Math. Kyoto Univ. 8 (1969), 233-278, 365-410; 9 (1969), 95-160. MR 38 #764; #6677. MR 0232439 (38:764)
  • [7] T. Kato, Perturbation theory for linear operations, Springer-Verlag, New York, 1976. MR 34 #3324. MR 0407617 (53:11389)
  • [8] T. Leviatan, Perturbations of Markov processes, J. Functional Analysis 10 (1972), 309-325. MR 53 #4243. MR 0400409 (53:4243)
  • [9] P. A. Meyer, Integrales stochastiques. I, II, III, IV, Séminaire de Probabilitiés I, Springer-Verlag, New York, 1967. MR 37 #7000. MR 0231445 (37:7000)
  • [10] -, Renaissance, recollectments, mélanges, ralentissement de processus de Markov, Ann. Inst. Fourier (Grenoble) 25 (1975), 465-497. MR 54 #3862. MR 0415784 (54:3862)
  • [11] M. Motoo and S. Watanabe, On a class of additive functionals of Markov processes, J. Math Kyoto Univ. 4 (1965), 429-469. MR 33 #4994. MR 0196808 (33:4994)
  • [12] S. A. Sawyer, A formula for semigroups, with an application to branching diffusion processes, Trans. Amer. Math. Soc. 152 (1970), 1-38. MR 42 # 1226. MR 0266319 (42:1226)
  • [13] D. W. Stroock, Diffusion processes associated with Lévy generators, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 32 (1975), 209-244. MR 55 #6587. MR 0433614 (55:6587)
  • [14] S. Watanabe, On discontinuous additive functionals and Lévy measures of a Markov process, Japan J. Math. 34 (1964), 53-70. MR 32 #3137. MR 0185675 (32:3137)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 60J25

Retrieve articles in all journals with MSC: 60J25


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1979-0542886-X
Keywords: Jump processes, Hunt processes, Lévy systems, semigroups, infinitesimal generators
Article copyright: © Copyright 1979 American Mathematical Society

American Mathematical Society