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The dimension of the boundary of super-Brownian motion

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We show that the Hausdorff dimension of the boundary of d-dimensional super-Brownian motion is 0, if \(d=1\), \(4-2\sqrt{2}\), if \(d=2\), and \((9-\sqrt{17})/2\), if \(d=3\).

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References

  1. Abraham, C., Le Gall, J.-F.: Excursion Theory for Brownian motion indexed by the Brownian tree. arXiv:1509.06616v2 (2017)

  2. Abraham, R., Werner, W.: Avoiding probabilities for Brownian snakes and super-Brownian motion. Elect. J. Probab. 2(3), 27 (1997)

    MathSciNet  MATH  Google Scholar 

  3. Bertoin, J.: Lévy Processes. Cambridge University Press, Cambridge (1996)

    MATH  Google Scholar 

  4. Brezis, H., Oswald, L.: Singular solutions for some semi-linear elliptic equations. Arch. Rat. Mech. Anal. 99, 249–259 (1987)

    Article  MATH  Google Scholar 

  5. Brezis, H., Peletier, L.A., Terman, D.: A very singular solution of the heat equation with absorption. Arch. Rat. Mech. Anal. 95, 185–209 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  6. Caballero, M.E., Lambert, A., Bravo, G.U.: Proof(s) of the Lamperti representation of continuous-state branching processes. Probab. Surv. 6, 62–89 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cox, J.T., Durrett, R., Perkins, E.: Rescaled voter models converge to super-Brownian motion. Ann. Probab. 28, 185–234 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cox, J.T., Perkins, E.: Rescaled Lotka–Volterra models converge to super-Brownian motion. Ann. Probab. 33, 904–947 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dawson, D., Iscoe, I., Perkins, E.: Super-Brownian motion: path properties and hitting probabilities. Probab. Theory Relat. Fields 83, 135–205 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dawson, D., Perkins, E.: Historical processes. Mem. Am. Math. Soc. 93(454), 179 (1991)

    MathSciNet  MATH  Google Scholar 

  11. Durrett, R., Perkins, E.: Rescaled contact processes converge to super-Browian motion in two or more dimensions. Probab. Theory Relat. Fields 114, 309–399 (1999)

    Article  MATH  Google Scholar 

  12. Dynkin, E.B.: Diffusions, Superdiffusions and Partial Differential Equations. AMS Colloquium Publications, Providence (2002)

    Book  MATH  Google Scholar 

  13. de Haan, L., Stadtmüller, U.: Dominated variation and related concepts and Tauberian theorems for Laplace transforms. J. Math. Anal. Appl. 108, 344–365 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  14. Harris, S.C.: Travelling-waves for the F-K-P-P equation via probabilistic arguments. Proc. R. Soc. Edinburgh Sect. A 125, 503–517 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hawkes, J.: Potential theory of Lévy processes. Proc. Lond. Math. Soc. 3(38), 335–352 (1979)

    Article  MATH  Google Scholar 

  16. Hong, J.: Renormalization of local times of super-Brownian motion. arXiv:1711.06447 (2017)

  17. Hong, J., Mytnik, L., Perkins, E.: On the topological boundary of the range of super-Brownian motion-extended version. arXiv:1809.04238 (2018)

  18. Iscoe, I.: A weighted occupation time for a class of measure-valued branching processes. Probab. Theory Relat. Fields 71, 85–116 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  19. Iscoe, I.: On the supports of measure-valued critical branching Brownian motion. Ann. Probab. 16, 200–221 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. North-Holland Mathematical Library (1989)

  21. Kyprianou, A.E.: Travelling wave solutions to the K-P-P equation: alternatives to Simon Harris’ probabilistic analysis. Ann. I. H. Poincaré 40, 53–72 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lalley, S., Zheng, X.: Spatial epidemics and local times for critical branching random walks in dimensions 2 and 3. Probab. Theory Relat. Fields 148, 527–566 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Le Gall, J.-F.: The Brownian snake and solutions of \({\varDelta } u=u^2\) in a domain. Probab. Theory Relat. Fields 102, 393–432 (1995)

    Article  MATH  Google Scholar 

  24. Le Gall, J.-F.: Spatial Branching Processes, Random Snakes and Partial Differential Equations. Lectures in Mathematics, ETH Zurich, Birkhäuser, Basel (1999)

  25. Le Gall, J.-F., Weill, M.: Conditioned Brownian trees. Ann. Inst. J. Poincaré Probab. Statist. 42, 455–489 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Mueller, C., Mytnik, L., Perkins, E.: On the boundary of the support of super-Brownian motion. Ann. Probab. 45, 3481–3534 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  27. Mueller, C., Tribe, R.: Stochastic p.d.e.’s arising from the long range contact and long range voter processes. Probab. Theory Relat. Fields 102, 519–545 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  28. Mytnik, L.: Stochastic partial differential equation driven by stable noise. Probab. Theory Relat. Fields 123, 157–201 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  29. Mytnik, L., Perkins, E.: Pathwise uniqueness for stochastic heat equations with Hölder continuous coefficients: the white noise case. Probab. Theory Relat. Fields 149, 1–96 (2011)

    Article  MATH  Google Scholar 

  30. Perkins, E.: Continuity of measure-valued processes. In: Seminar on Stochastic Processes, 1990, Birkhäuser, Boston, pp. 261–268 (1991)

  31. Perkins, E.: Dawson-Watanabe Superprocesses and Measure-valued Diffusions. In: Ecole d’Eté de Probabilités de Saint Flour 1999, Lecture Notes in Math. 1781, Springer-Verlag (2002)

  32. Rogers, L.C.G., Williams, D.: Diffusions, Markov Processes and Martingales, vol. 2. Cambridge University Press, Cambridge (1994)

    MATH  Google Scholar 

  33. Stroock, D.W., Varadhan, S.R.S.: Multidimensional Diffusion Processes. Springer-Verlag, Berlin (1979)

    MATH  Google Scholar 

  34. Sugitani, S.: Some properties for the measure-valued branching diffusion processes. J. Math. Soc. Jpn. 41, 437–462 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  35. Takeuchi, J.: On the sample paths of the symmetric stable processes. J. Math. Soc. Jpn. 16, 109–126 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  36. Taliaferro, S.: Asymptotic behavior of solutions of \(y^{\prime \prime }=\phi (t)y^\lambda \). J. Math. Anal. Appl. 66, 95–134 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  37. Veron, L.: Singular solutions of some nonlinear elliptic equations. Nonlinear Anal. Theory Methods Appl. 5, 225–242 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  38. Yor, M.: On some exponential functionals of Brownian motion. Adv. Appl. Probab. 24, 509–531 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  39. Yor, M.: Generalized meanders as limits of weighted Bessel processes, and an elementary proof of Spitzer’s asymptotic result on Brownian windings. Studia Sci. Math. Hungar. 33, 339–343 (1997)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

LM thanks Paul Balança for enjoyable and very helpful discussions on this problem. We thank L. Ryzhik for showing us the proof of Proposition 3.3, and I. Benjamini whose comments about the boundary of the range of a scaling limit of tree-indexed random walks gave us a strong motivation to carry out this work. We thank two anonymous referees and Jieliang Hong for carefully reading the manuscript, making suggestions to improve readability and spotting a number of misprints and inconsistencies. EP thanks the Technion for hosting him during a visit where some of this research was carried out. LM thanks UBC and EP for hosting him during his visits.

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Authors and Affiliations

  1. Faculty of Industrial Engineering and Management, Technion – Israel Institute of Technology, 32000, Haifa, Israel

    Leonid Mytnik

  2. Department of Mathematics, The University of British Columbia, 1984 Mathematics Road, Vancouver, B.C., V6T 1Z2, Canada

    Edwin Perkins

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  1. Leonid Mytnik
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  2. Edwin Perkins
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Correspondence to Leonid Mytnik.

Additional information

L. Mytnik is supported in part by the Israel Science Foundation (Grant No. 1325/14).

E. Perkins supported by an NSERC Discovery Grant.

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Mytnik, L., Perkins, E. The dimension of the boundary of super-Brownian motion. Probab. Theory Relat. Fields 174, 821–885 (2019). https://doi.org/10.1007/s00440-018-0866-5

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  • DOI: https://doi.org/10.1007/s00440-018-0866-5

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