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)



Explosion problems for symmetric diffusion processes

Author: Kanji Ichihara
Journal: Trans. Amer. Math. Soc. 298 (1986), 515-536
MSC: Primary 60J60
MathSciNet review: 860378
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We discuss the explosion problem for a symmetric diffusion process. Hasminskii's idea cannot be applied to this case. Instead, the theory of Dirichlet forms is employed to obtain criteria for conservativeness and explosion of the process. The fundamental criteria are given in terms of the $ \alpha $-equilibrium potentials and $ \alpha $-capacities of the unit ball centered at the origin. They are applied to obtain sufficient conditions on the coefficients of the infinitesimal generator for conservativeness and explosion.

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

  • [1] W. Feller, The parabolic differential equations and the associated semigroups of transformations, Ann. of Math. (2) 55 (1952), 468-519. MR 0047886 (13:948a)
  • [2] M. Fukushima, Dirichlet forms and Markov processes, Kodansha, Tokyo, 1980. MR 569058 (81f:60105)
  • [3] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Springer-Verlag, Berlin and New York, 1983. MR 737190 (86c:35035)
  • [4] R. Z. Hasminskii, Ergodic properties of recurrent diffusion processes and stabilization of the problem to the Cauchy problem for parabolic equations, Teor. Veroyatnost. i Primenen. 5 (1960), 196-214. MR 0133871 (24:A3695)
  • [5] G. A. Hunt, On positive Green functions, Proc. Nat. Acad. Sci. U.S.A. 40 (1954), 816-818. MR 0063538 (16:135b)
  • [6] K. Ichihara, Some global properties of symmetric diffusion processes, Publ. Res. Inst. Math. Sci. 14 (1978), 441-486. MR 509198 (80d:60099)
  • [7] -, Explosion problems for symmetric diffusion processes, Proc. Japan Acad. Ser. A 60 (1984), 243-245. MR 774562 (86i:60200)
  • [8] K. Ichihara and H. Watanabe, Double points for diffusion processes, Japan. J. Math. 6 (1980), 267-281. MR 615172 (82g:60102)
  • [9] K. Itô and H. P. McKean, Diffusion processes and their sample paths, Springer-Verlag, Berlin and New York, 1965. MR 0199891 (33:8031)
  • [10] W. Littman, G. Stampacchia, and H. F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa 17 (1963), 43-77. MR 0161019 (28:4228)
  • [11] H. P. McKean, Stochastic integrals, Academic Press, New York, 1969. MR 0247684 (40:947)
  • [12] H. P. McKean and H. Tanaka, Additive functionals of the Brownian path, Memoirs Coll. Sci. Univ. Kyoto Ser. A Math. 33 (1961), 479-506. MR 0131295 (24:A1147)
  • [13] S. Mizohata, The theory of partial differential equations, Cambridge Univ. Press, London, 1973. MR 0599580 (58:29033)
  • [14] O. A. Oleinik and E. D. Radkevich, Second order equations with nonnegative characteristic form, Plenum Press, New York and London, 1973. MR 0457908 (56:16112)
  • [15] M. Tomisaki, A construction of diffusion processes with singular product measures, Z. Wahrsch. Verw. Gebiete 53 (1980), 51-70. MR 576897 (83h:60084)
  • [16] -, Dirichlet forms associated with direct product diffusion processes, Lecture Notes in Math., vol. 923, Springer-Verlag, Berlin and New York, 1981.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 60J60

Retrieve articles in all journals with MSC: 60J60

Additional Information

Article copyright: © Copyright 1986 American Mathematical Society

American Mathematical Society