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 Free Access

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] William Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math. (2) 55 (1952), 468–519. MR 0047886
  • [2] Masatoshi Fukushima, Dirichlet forms and Markov processes, North-Holland Mathematical Library, vol. 23, North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1980. MR 569058
  • [3] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190
  • [4] R. Z. Has′minskiĭ, Ergodic properties of recurrent diffusion processes and stabilization of the solution of the Cauchy problem for parabolic equations, Teor. Verojatnost. i Primenen. 5 (1960), 196–214 (Russian, with English summary). MR 0133871
  • [5] G. A. Hunt, On positive Green’s functions, Proc. Nat. Acad. Sci. U. S. A. 40 (1954), 816–818. MR 0063538
  • [6] Kanji Ichihara, Some global properties of symmetric diffusion processes, Publ. Res. Inst. Math. Sci. 14 (1978), no. 2, 441–486. MR 509198, 10.2977/prims/1195189073
  • [7] Kanji Ichihara, Explosion problems for symmetric diffusion processes, Proc. Japan Acad. Ser. A Math. Sci. 60 (1984), no. 7, 243–245. MR 774562
  • [8] Kanji Ichihara and Hisao Watanabe, Double points for diffusion processes, Japan. J. Math. (N.S.) 6 (1980), no. 2, 267–281. MR 615172
  • [9] Kiyoshi Itô and Henry P. McKean Jr., Diffusion processes and their sample paths, Die Grundlehren der Mathematischen Wissenschaften, Band 125, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-New York, 1965. MR 0199891
  • [10] W. Littman, G. Stampacchia, and H. F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa (3) 17 (1963), 43–77. MR 0161019
  • [11] H. P. McKean Jr., Stochastic integrals, Probability and Mathematical Statistics, No. 5, Academic Press, New York-London, 1969. MR 0247684
  • [12] H. P. McKean Jr. and Hiroshi Tanaka, Additive functionals of the Brownian path, Mem. Coll. Sci. Univ. Kyoto Ser. A Math. 33 (1960/1961), 479–506. MR 0131295
  • [13] Sigeru Mizohata, The theory of partial differential equations, Cambridge University Press, New York, 1973. Translated from the Japanese by Katsumi Miyahara. MR 0599580
  • [14] O. A. Oleĭnik and E. V. Radkevič, Second order equations with nonnegative characteristic form, Plenum Press, New York-London, 1973. Translated from the Russian by Paul C. Fife. MR 0457908
  • [15] Matsuyo Tomisaki, A construction of diffusion processes with singular product measures, Z. Wahrsch. Verw. Gebiete 53 (1980), no. 1, 51–70. MR 576897, 10.1007/BF00531611
  • [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

DOI: http://dx.doi.org/10.1090/S0002-9947-1986-0860378-9
Article copyright: © Copyright 1986 American Mathematical Society