×

zbMATH — the first resource for mathematics

A boundary local time for one-dimensional super-Brownian motion and applications. (English) Zbl 1467.60063
Summary: 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 C. Mueller et al. [Ann. Probab. 45, No. 6A, 3481–3534 (2017; Zbl 1412.60123)]. \(L_{t}\) 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 et al. [loc. cit.], 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 [T. Hughes and E. Perkins, Ann. Inst. Henri Poincaré, Probab. Stat. 55, No. 4, 2395–2422 (2019; Zbl 1434.60234)], 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 \(\{X_{t}(\mathbb{R} ) > 0 \}\).

MSC:
60J68 Superprocesses
60J55 Local time and additive functionals
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
28A78 Hausdorff and packing measures
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] Brezis, H. and Friedman, A. (1983) Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures et Appl.62, 73–97. · Zbl 0527.35043
[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. · Zbl 0627.35046
[3] Dawson, D., and Perkins, E. (1991) Historical Processes, Memoirs of the AMS, 93, no. 454, 179pp. · Zbl 0754.60062
[4] Durrett, R. (2011) Probability: Theory and examples, Cambridge University Press, Cambridge. · Zbl 1202.60002
[5] Ethier, S., and Kurtz, T. (1986) Markov Processes: Characterization and convergence, John Wiley & Sons Inc., New York. · Zbl 0592.60049
[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. · Zbl 0607.35046
[8] Karatzas, I., Shreve, S. (1988) Brownian motion and stochastic calculus, Springer-Verlag, New York. · Zbl 0638.60065
[9] Konno, N., Shiga, T. (1988) Stochastic partial differential equations for some measure-valued diffusions, Probab. Th. Rel. Fields79, 201–225. · Zbl 0631.60058
[10] Le Gall, J.-F. (1999). Spatial Branching Processes, Random Snakes and Partial Differential Equations, Lectures in Mathematics, ETH Zurich, Birkhäuser, Basel. · Zbl 0938.60003
[11] Marcus, M. and Veron, L. (1999) Initial trace of positive solutions of some nonlinear parabolic equations, Comm. in PDE24, 1445–1499. · Zbl 1059.35054
[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. Probability45, 3481–3543. · Zbl 1412.60123
[14] Mytnik L. (2002) Stochastic partial differential equation driven by stable noise, Probab. Theory Relat. Fields123, 157–201. · Zbl 1009.60053
[15] Mytnik, L. and Perkins, E. (2011) Pathwise uniqueness for stochastic heat equations with Hölder continuous coefficients: the white noise case, Probab. Theory Relat. Fields149, 1–96. · Zbl 1233.60039
[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. · Zbl 0823.60071
[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. Fields81, 319–340. · Zbl 0651.60069
[19] Rogers, L.C.G., and Williams, D. (2000) Diffusions, Markov processes and martingales, Volume 2: Itô calculus, Cambridge University Press, Cambridge. · Zbl 0949.60003
[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. · Zbl 0159.46201
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.