Probability Theory and Related Fields

, Volume 89, Issue 1, pp 89–115 | Cite as

A probabilistic approach to one class of nonlinear differential equations

  • E. B. Dynkin


We establish connections between positive solutions of one class of nonlinear partial differential equations and hitting probabilities and additive functionals of superdiffusion processes. As an application, we improve results on superprocesses by using the recent progress in the theory of removable singularities for differential equations.


Differential Equation Partial Differential Equation Stochastic Process Probability Theory Mathematical Biology 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [A] Adams, R.A.: Sobolev Spaces. New York San Francisco London: Academic Press 1975Google Scholar
  2. [ADN] Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Commun. Pure Appl. Math.12, 623–727 (1959)Google Scholar
  3. [AM] Adams, D.R., Meyers, N.G.: Bessel potentials. Inclusion relation among classes of exceptional sets. Indiana Univ. Math. J.22, 873–905 (1973)Google Scholar
  4. [BP] Baras, P., Pierre, M.: Singularités éliminables pour des équations semi-linéares. Ann. Inst. Fourier34 185–206 (1984)Google Scholar
  5. [BS] Brezis, H., Strauss, W.A.: Semi-linear second-order elliptic equations inL 1. J. Math. Soc. Japan25, 565–590 (1973)Google Scholar
  6. [BV] Brezis, H., Veron, L.: Removable singularities of some nonlinear equations. Arch. Ration. Mech. Anal.75, 1–6 (1980)Google Scholar
  7. [C] Chung, K.L.: Probabilistic approach in potential theory to the equilibrium problem. Ann. Institut Fourier23, 313–322 (1973)Google Scholar
  8. [DIP] Dawson, D.A., Iscoe, I., Perkins, E.A.: Super-Brownian motion: path properties and hitting probabilities. Probab. Th. Rel. Fields83, 135–206 (1989)Google Scholar
  9. [DP] Dawson, D.A., Perkins, E.A.: Historical processes 1990. (preprint)Google Scholar
  10. [DM] Dellacherie, C., Meyer, P.-A.: Probabilités et potentiel: Théorie discrète du potential. Paris, Hermann 1983Google Scholar
  11. [Do] Dynkin, E.B.: Theory of Markov processes Oxford London New York Paris: Pergamon Press 1960Google Scholar
  12. [D] Dynkin, E.B.: Markov processes. Berlin Göttingen Heidelberg: Springer 1965Google Scholar
  13. [D1] Dynkin, E.B.: Representation for functionals of superprocesses by multiple stochastic integrals, with applications to self-intersection local times. Astérisque157–158, 147–171 (1988)Google Scholar
  14. [D2] Dynkin, E.B.: Regular transition functions and regular superprocesses. Trans. Am. Math. Soc.316, 623–634 (1989)Google Scholar
  15. [D3] Dynkin, E.B.: Branching particle systems and superprocesses. Ann. Probab. (in press)Google Scholar
  16. [D4] Dynkin, E.B.: Path processes and historical processes. Probab. Th. Rel. Fields (in press)Google Scholar
  17. [F] Fitzsimmons, P.J.: Construction and regularity of measure-valued Markov branching processes. Isr. J. Math.64, 337–361 (1988)Google Scholar
  18. [GT] Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Berlin Heidelberg New York: Springer 1977Google Scholar
  19. [I] Iscoe, I.: On the support of measure-valued critical branching Brownian motion. Ann. Probab.16, 200–221 (1988)Google Scholar
  20. [Ko] Koshelev, A.I.: A priori estimates inL p and generalized solutions of elliptic equations and systems. AMS Trans. Ser.2, 2 D, 105–171 (1962)Google Scholar
  21. [KN] Kondrat'yev, V.A., Nikishkin, V.A.: On asymptotic boundary behavior of a solution of a singular boundary value problem for a semilinear elliptic equation. Differ. Uravn.26, 465–468 (1990)Google Scholar
  22. [L] Lions, P.L.: Isolated singularities in semilinear problems. J. Differ. Equations38, 441–450 (1980)Google Scholar
  23. [LN] Loewner, C., Nirenberg, L.: Partial differential equations invariant under conformal or projective transformations. In: Ahlfors, L. et al. (eds.) Contributions to analysis, pp. 245–272. New York: Academic Press 1974Google Scholar
  24. [M] Meyer, P.-A.: Probability and potential, Waltham: Blaisdell 1966Google Scholar
  25. [Mey] Meyers, N.G.: A theory of capacities for potentials of functions in Lebesgue classes. Math. Scand.26, 255–292 (1970)Google Scholar
  26. [Mi] Miranda, C.: Partial differential equations of elliptic type, vol. 2, 2nd edn. Berlin Heidelberg New York: Springer 1970Google Scholar
  27. [MT] Matheron, G.: Random sets and integral geometry. New York: Wiley 1975Google Scholar
  28. [P] Perkins, E.A.: Polar sets and multiple points for super-Brownian motion. Ann. Probab.18, 453–491 (1990)Google Scholar
  29. [RV] Richard, Y., Véron, L.: Isotropic singularities of solutions of nonlinear elliptic inequalities. Ann. Inst. Henri Poincare6, 37–72 (1989)Google Scholar
  30. [V1] Véron, L.: Singularités éliminables d'équations elliptiques non linéaires. J. Differ. Equations41, 87–95 (1981)Google Scholar
  31. [V2] Véron, L.: Singular solutions of some nonlinear elliptic equations. Nonlinear Anal. Theory Methods Appl.5, 225–242 (1981)Google Scholar
  32. [V3] Véron, L.: Weak and strong singularities of nonlinear elliptic equations. Proc. Symp. Pure Math.45, 477–495 (1986)Google Scholar
  33. [V4] Véron, L.: Semilinear elliptic equations with uniform blow-up on the boundary. Université F. Rablais, Tour (preprint 1990)Google Scholar
  34. [VV] Vàsquez, J.L., Véron, L.: Isolated singularities of some semilinear elliptic equations. J. Differ. Equations60, 301–321 (1985)Google Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • E. B. Dynkin
    • 1
  1. 1.Department of MathematicsCornell UniversityIthacaUSA

Personalised recommendations