## The Annals of Probability

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

### On the boundary of the support of super-Brownian motion

Carl Mueller, Leonid Mytnik, and Edwin Perkins

#### 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