The Annals of Probability

On the boundary of the support of super-Brownian motion

Carl Mueller, Leonid Mytnik, and Edwin Perkins

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We study the density $X(t,x)$ of one-dimensional super-Brownian motion and find the asymptotic behaviour of $P(0<X(t,x)\le a)$ as $a\downarrow0$ as well as the Hausdorff dimension of the boundary of the support of $X(t,\cdot)$. The answers are in terms of the leading eigenvalue of the Ornstein–Uhlenbeck generator with a particular killing term. This work is motivated in part by questions of pathwise uniqueness for associated stochastic partial differential equations.

Article information

Source
Ann. Probab., Volume 45, Number 6A (2017), 3481-3534.

Dates
Received: December 2015
Revised: August 2016
First available in Project Euclid: 27 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1511773657

Digital Object Identifier
doi:10.1214/16-AOP1141

Mathematical Reviews number (MathSciNet)
MR3729608

Zentralblatt MATH identifier
06838100

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60] 60J68: Superprocesses
Secondary: 35K15: Initial value problems for second-order parabolic equations

Keywords
Super-Brownian motion Hausdorff dimension stochastic partial differential equations

Citation

Mueller, Carl; Mytnik, Leonid; Perkins, Edwin. On the boundary of the support of super-Brownian motion. Ann. Probab. 45 (2017), no. 6A, 3481--3534. doi:10.1214/16-AOP1141. https://projecteuclid.org/euclid.aop/1511773657


Export citation

References

  • [1] Brezis, H., Peletier, L. A. and Terman, D. (1986). A very singular solution of the heat equation with absorption. Arch. Ration. Mech. Anal. 95 185–209.
  • [2] Chen, Y. (2015). Pathwise uniqueness for the SPDE’s of some super-Brownian motion with immigration. Ann. Probab. 43 3359–3467.
  • [3] Coddington, E. A. and Levinson, N. (1955). Theory of Ordinary Differential Equations. McGraw-Hill, New York.
  • [4] Dawson, D. A. and Perkins, E. A. (1991). Historical processes. Mem. Amer. Math. Soc. 93 iv+179.
  • [5] de Haan, L. and Stadtmüller, U. (1985). Dominated variation and related concepts and Tauberian theorems for Laplace transforms. J. Math. Anal. Appl. 108 344–365.
  • [6] Fristedt, B. E. and Pruitt, W. E. (1971). Lower functions for increasing random walks and subordinators. Z. Wahrsch. Verw. Gebiete 18 167–182.
  • [7] Hawkes, J. (1975/76). On the potential theory of subordinators. Z. Wahrsch. Verw. Gebiete 33 113–132.
  • [8] Hawkes, J. (1979). Potential theory of Lévy processes. Proc. Lond. Math. Soc. (3) 38 335–352.
  • [9] Kamin, S. and Peletier, L. A. (1985). Singular solutions of the heat equation with absorption. Proc. Amer. Math. Soc. 95 205–210.
  • [10] Karatzas, I. and Shreve, S. E. (1979). Brownian Motion and Stochastic Calculus. Springer, New York.
  • [11] Mueller, C., Mytnik, L. and Perkins, E. (2014). Nonuniqueness for a parabolic SPDE with $\frac{3}{4}-\varepsilon$-Hölder diffusion coefficients. Ann. Probab. 42 2032–2112.
  • [12] Mueller, C., Mytnik, L. and Perkins, E. (2015). On the boundary of the support of super-Brownian motion. Preprint. Available at arXiv:1512.08610.
  • [13] Mueller, C. and Perkins, E. A. (1992). The compact support property for solutions to the heat equation with noise. Probab. Theory Related Fields 93 325–358.
  • [14] Mytnik, L. (1998). Weak uniqueness for the heat equation with noise. Ann. Probab. 26 968–984.
  • [15] Mytnik, L. and Perkins, E. (2003). Regularity and irregularity of $(1+\beta)$-stable super-Brownian motion. Ann. Probab. 31 1413–1440.
  • [16] Mytnik, L. and Perkins, E. (2011). Pathwise uniqueness for stochastic heat equations with Hölder continuous coefficients: The white noise case. Probab. Theory Related Fields 149 1–96.
  • [17] Mytnik, L., Perkins, E. and Sturm, A. (2006). On pathwise uniqueness for stochastic heat equations with non-Lipschitz coefficients. Ann. Probab. 34 1910–1959.
  • [18] Perkins, E. (2002). Dawson–Watanabe superprocesses and measure-valued diffusions. In Lectures on Probability Theory and Statistics (Saint-Flour, 1999). Lecture Notes in Math. 1781 125–324. Springer, Berlin.
  • [19] Riesz, F. and Sz.-Nagy, B. (1955). Functional Analysis. Frederick Ungar, New York.
  • [20] Taylor, S. J. (1961). On the connexion between Hausdorff measures and generalized capacity. Math. Proc. Cambridge Philos. Soc. 57 524–531.
  • [21] Uchiyama, K. (1980). Two contrasting properties of solutions for one-dimensional stochastic partial differential equations. Z. Wahrsch. Verw. Gebiete 54 75–116.