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)

 
 

 

Markov processes with Lipschitz semigroups


Author: Richard Bass
Journal: Trans. Amer. Math. Soc. 267 (1981), 307-320
MSC: Primary 60J35; Secondary 47D07, 60H20, 60J60
DOI: https://doi.org/10.1090/S0002-9947-1981-0621990-0
MathSciNet review: 621990
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For $ f$ a function on a metric space, let

$\displaystyle \operatorname{Lip} f = \mathop {\sup }\limits_{x \ne y} \vert f(x) - f(y)\vert/d(x,\,y),$

and say that a semigroup $ {P_t}$ is Lipschitz if $ \operatorname{Lip} ({P_t}f) \leqslant {e^{Kt}}\operatorname{Lip} f$ for all $ f$, $ t$, where $ K$ is a constant. If one has two Lipschitz semigroups, then, with some additional assumptions, the sum of their infinitesimal generators will also generate a Lipschitz semigroup. Furthermore a sequence of uniformly Lipschitz semigroups has a subsequence which converges in the strong operator topology.

Examples of Markov processes with Lipschitz semigroups include all diffusions on the real line which are on natural scale whose speed measures satisfy mild conditions, as well as some jump processes. One thus gets Markov processes whose generators are certain integro-differential operators. One can also interpret the results as giving some smoothness conditions for the solutions of certain parabolic partial differential equations.


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

  • [1] L. Arnold, Stochastic differential equations: Theory and applications, Wiley, New York, 1974. MR 0443083 (56:1456)
  • [2] R. Bass, Adding and subtracting jumps from Markov processes, Trans. Amer. Math. Soc. 255 (1979), 363-376. MR 542886 (81b:60070)
  • [3] R. Blumenthal and R. Getoor, Markov processes and potential theory, Academic Press, New York, 1968. MR 0264757 (41:9348)
  • [4] L. Breiman, Probability, Addison-Wesley, Reading, Mass., 1968. MR 0229267 (37:4841)
  • [5] H. Brezis, W. Rosenkrantz and B. Singer, On a degenerate elliptic-parabolic equation occurring in the theory of probability, Comm. Pure Appl. Math. 24 (1971), 395-416. MR 0284717 (44:1941)
  • [6] P. Chernoff, Note on product formulas for operator semigroups, J. Funct. Anal. 2 (1968), 238-242. MR 0231238 (37:6793)
  • [7] G. Fichera, Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine, Atti Accad. Naz. Lincei Mem. Ser. (8) 5 (1956), 1-30. MR 0089348 (19:658a)
  • [8] M. I. Freidlin, A priori estimates of solutions of degenerate elliptic equations, Dokl. Akad. Nauk SSSR 158 (1964), 281-283 = Soviet Math. Dokl. 5 (1964), 1231-1234. MR 0173075 (30:3290)
  • [9] K. Ito and H. P. McKean, Jr., Diffusion processes and their sample paths, Academic Press, New York, 1965. MR 0199891 (33:8031)
  • [10] T. Kato, Perturbation theory for linear operators, Springer-Verlag, New York, 1966. MR 0203473 (34:3324)
  • [11] J. J. Kohn and L. Nirenberg, Degenerate elliptic-parabolic equations of second order, Comm. Pure Appl. Math. 20 (1967), 797-872. MR 0234118 (38:2437)
  • [12] T. Komatsu, Markov processes associated with certain integro-differential operators, Osaka J. Math. 10 (1973), 271-303. MR 0359017 (50:11472)
  • [13] V. A. Kostin, Smoothness of solutions of certain parabolic equations. I, II, III, Differential Equations 12 (1976), 1054-1063; ibid. 12 (1976), 1139-1143; ibid. 12 (1976), 1470-1473.
  • [14] J. P. Lepeltier and B. Marchal, Problème des martingales et equations différentielles stochastiques associées à un opérateur intégro-différentiel, Ann. Inst. H. Poincaré 12 (1976), 43-103. MR 0413288 (54:1403)
  • [15] O. A. Oleinik, On the smoothness of solutions of degenerate elliptic and parabolic equations, Dokl. Akad. Nauk SSSR 163 (1965), 577-580 = Soviet Math. Dokl. 6 (1965), 972-976. MR 0200595 (34:486)
  • [16] A. Skorokhod, Studies in the theory of random processes, Addison-Wesley, Reading, Mass., 1965. MR 0185620 (32:3082b)
  • [17] D. Stroock, Diffusion processes associated with Lévy generators, Z. Wahrsch. Verw. Gebiete 32 (1975), 209-244. MR 0433614 (55:6587)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 60J35, 47D07, 60H20, 60J60

Retrieve articles in all journals with MSC: 60J35, 47D07, 60H20, 60J60


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1981-0621990-0
Keywords: Semigroups, infinitesimal generators, jump processes, diffusions, parabolic partial differential equations
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society