Electronic Journal of Probability

A boundary local time for one-dimensional super-Brownian motion and applications

Thomas Hughes

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For a one-dimensional super-Brownian motion with density $X(t,x)$, we construct a random measure $L_{t}$ called the boundary local time which is supported on $BZ_{t} := \partial \{x:X(t,x) = 0\}$, thus confirming a conjecture of Mueller, Mytnik and Perkins [13]. $L_{t}$ is analogous to the local time at $0$ of solutions to an SDE. We establish first and second moment formulas for $L_{t}$, some basic properties, and a representation in terms of a cluster decomposition. Via the moment measures and the energy method we give a more direct proof that $\text{dim} (BZ_{t}) = 2-2\lambda _{0}> 0$ with positive probability, a recent result of Mueller, Mytnik and Perkins [13], where $-\lambda _{0}$ is the lead eigenvalue of a killed Ornstein-Uhlenbeck operator that characterizes the left tail of $X(t,x)$. In a companion work [6], the author and Perkins use the boundary local time and some of its properties proved here to show that $\text{dim} (BZ_{t}) = 2-2\lambda _{0}$ a.s. on $\{X_{t}(\mathbb{R} ) > 0 \}$.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 54, 58 pp.

Received: 27 April 2018
Accepted: 1 April 2019
First available in Project Euclid: 5 June 2019

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Zentralblatt MATH identifier

Primary: 60J68: Superprocesses
Secondary: 60J55: Local time and additive functionals 60H15: Stochastic partial differential equations [See also 35R60] 28A78: Hausdorff and packing measures

super-Brownian motion local time stochastic pde Hausdorff dimension

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Hughes, Thomas. A boundary local time for one-dimensional super-Brownian motion and applications. Electron. J. Probab. 24 (2019), paper no. 54, 58 pp. doi:10.1214/19-EJP303. https://projecteuclid.org/euclid.ejp/1559700304

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  • [1] Brezis, H. and Friedman, A. (1983) Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures et Appl. 62, 73–97.
  • [2] Brezis, H., Peletier, L.A., and Terman, D. (1986) A very singular solution of the heat equation with absorption. Arch. Rat. Mech. Anal. 95, 185–209.
  • [3] Dawson, D., and Perkins, E. (1991) Historical Processes, Memoirs of the AMS, 93, no. 454, 179pp.
  • [4] Durrett, R. (2011) Probability: Theory and examples, Cambridge University Press, Cambridge.
  • [5] Ethier, S., and Kurtz, T. (1986) Markov Processes: Characterization and convergence, John Wiley & Sons Inc., New York.
  • [6] Hughes, T., and Perkins, E. (2018) On the boundary of the zero set of one-dimensional super-Brownian motion and its local time, To Appear in Ann. Henri Poincaré.
  • [7] Kamin, S., Peletier, L.A. (1985) Singular solutions of the heat equation with absorption, Proc. Am. Math. Soc. 95 205–210.
  • [8] Karatzas, I., Shreve, S. (1988) Brownian motion and stochastic calculus, Springer-Verlag, New York.
  • [9] Konno, N., Shiga, T. (1988) Stochastic partial differential equations for some measure-valued diffusions, Probab. Th. Rel. Fields 79, 201–225.
  • [10] Le Gall, J.-F. (1999). Spatial Branching Processes, Random Snakes and Partial Differential Equations, Lectures in Mathematics, ETH Zurich, Birkhäuser, Basel.
  • [11] Marcus, M. and Veron, L. (1999) Initial trace of positive solutions of some nonlinear parabolic equations, Comm. in PDE 24, 1445–1499.
  • [12] Mörters, P. and Peres, Y. (2010) Brownian Motion., Cambridge University Press, Cambridge.
  • [13] Mueller, C., Mytnik, L., and Perkins, E. (2017) On the boundary of the support of super-Brownian motion, Ann. Probability 45, 3481–3543.
  • [14] Mytnik L. (2002) Stochastic partial differential equation driven by stable noise, Probab. Theory Relat. Fields 123, 157–201.
  • [15] Mytnik, L. and Perkins, E. (2011) Pathwise uniqueness for stochastic heat equations with Hölder continuous coefficients: the white noise case, Probab. Theory Relat. Fields 149, 1–96.
  • [16] Perkins, E. (1995) On the martingale problem for interactive measure-valued branching diffusions, Memoirs of the American Math. Soc. 115, No. 549, 1–89.
  • [17] Perkins, E. (2001) Dawson-Watanabe Superprocesses and Measure-valued Diffusions, in Lectures on Probability Theory and Statistics, Ecole d’Eté de probabilités de Saint-Flour XXIX-1999, Ed. P. Bernard, Lecture Notes in Mathematics 1781, 132–335, Springer, Berlin.
  • [18] Reimers, M. (1989) One dimensional stochastic partial differential equations and the branching measure diffusion Probab. Th. Rel. Fields 81, 319–340.
  • [19] Rogers, L.C.G., and Williams, D. (2000) Diffusions, Markov processes and martingales, Volume 2: Itô calculus, Cambridge University Press, Cambridge.
  • [20] Walsh, J. (1986) An introduction to stochastic partial differential equations, Lecture Notes in Math., 1180, pp. 265–439.
  • [21] Watanabe, S. (1968) A limit theorem of branching processes and continuous state branching processes J. Math. Kyoto Univ. 8-1, 141–167.
  • [22] Zhu, P. (2017) On the Hausdorff dimension of the boundary of the $1+\beta $ branching super-Brownian motion. Undergraduate Summer Research Report, UBC.