Probability Theory and Related Fields

, Volume 93, Issue 3, pp 325–358 | Cite as

The compact support property for solutions to the heat equation with noise

  • Carl Mueller
  • Edwin A. Perkins


We consider all solutions of a martingale problem associated with the stochastic pde\(u_t = \tfrac{1}{2}u_{xx} + u^\gamma \dot W\) and show thatu(t,·) has compact support for allt≧0 ifu(0,·) does and if γ<1. This extends a result of T. Shiga who derived this compact support property for γ≦1/2 and complements a result of C. Mueller who proved this property fails if γ≧1.


Stochastic Process Probability Theory Mathematical Biology Compact Support Heat Equation 
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. Albeverio, S., Fenstad, J.E., Hoegh-Krohn, R., Lindstrom, T.: Nonstandard methods in stochastic analysis and mathematical physics. New York: Academic Press 1986Google Scholar
  2. Anderson, R.M.: Star-finite representations of measure spaces. Trans. Am. Math. Soc.271, 667–687 (1982)Google Scholar
  3. Cutland, N.: Nonstandard measure theory and its applications. Bull. Lond. Math. Soc.15, 529–589 (1983)Google Scholar
  4. Dawson, D.A., Iscoe, I., Perkins, E.A.: Super-Brownian motion: path properties and hitting probabilities. Probab. Theory Relat. Fields83, 135–205 (1989)Google Scholar
  5. Dawson, D.A., Perkins, E.A.: Historical processes. Mem. Am. Math. Soc.93 (1991)Google Scholar
  6. Dellacherie, C., Meyer, P.-A.: Probabilities and potential, I–IV. (North-Holland Math. Stud., vol. 29) Amsterdam. North Holland 1978Google Scholar
  7. Dellacherie, C., Meyer, P.-A.: Probabilités et potential, Chaps. V–VIII. Paris: Hermann 1980Google Scholar
  8. Fitzsimmons, P.J.: Construction and regularity of measure-valued branching processes. Isr. J. Math.64, 337–361 (1988)Google Scholar
  9. Fitzsimmons, P.J.: Correction and addendum to “Construction and regularity of measure-valued branching processes”. (Preprint 1990)Google Scholar
  10. Itô, K., McKean, H.P. Jr.: Diffusion processes and their sample paths. Berlin Heidelberg New York: 1974Google Scholar
  11. Iscoe, I.: On the supports of measure-valued critical branching Brownian motion Ann. Probab.16, 200–221 (1988)Google Scholar
  12. Knight, F.B.: Essentials of Brownian motion and diffusion. (Am. Math. Soc. Surv., vol. 18) Providence: Am. Math. Soc. 1981Google Scholar
  13. Konno, N., Shiga, T.: Stochastic partial differential equations for some measure-valued diffusions. Probab. Theory Relat. Fields79, 201–225 (1988)Google Scholar
  14. Mueller, C.: On the support of solutions to the heat equation with noise. Stochastics37, 225–246 (1991)Google Scholar
  15. Perkins, E.A.: A space-time property of a class of measure-valued branching diffusions Trans. Am. Math. Soc.305, 743–795 (1988)Google Scholar
  16. Perkins, E.A.: On the continuity of measure-valued processes. In: Seminar on stochastic processes 1990, pp. 261–268. Boston: Birkhäuser 1991Google Scholar
  17. Reimers, M.: One dimensional stochastic partial differential equations and the branching measure diffusion. Probab. Theory Relat. Fields81, 319–340 (1989)Google Scholar
  18. Roelly-Coppoletta, S.: A criterion of convergence of measure-valued processes: application to measure branching processes. Stochastics17, 43–65 (1986)Google Scholar
  19. Rogers, L.C.G., Williams, D.: Diffusions, Markov processes and martingales, vol. 2: Itô calculus. Chichester: Wiley 1987Google Scholar
  20. Sharpe, M.J.: General theory of Markov processes. New York: Academic Press 1988Google Scholar
  21. Shiga, T.: Two contrastive properties of solutions for one-dimensional stochastic partial differential equations (Preprint 1990)Google Scholar
  22. Walsh, J.B.: An introduction to stochastic partial differential equations. (Lect. Notes Math., vol. 1180) Berlin Heidelberg New York: Springer 1986Google Scholar
  23. Watanabe, S.: A limit theory of branching processes and continuous state branching processes. J. Math. Kyoto Univ.8, 141–167 (1968)Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Carl Mueller
    • 1
  • Edwin A. Perkins
    • 2
  1. 1.Department of MathematicsUniversity of RochesterRochesterUSA
  2. 2.Department of MathematicsUniversity of British ColumbiaVancouverCanada

Personalised recommendations