scholarly journals On the martingale problem for super-Brownian motion

2001 ◽  
pp. 195-201
Author(s):  
Richard F. Bass ◽  
Edwin A. Perkins
Keyword(s):  
Brownian Motion ◽  
Martingale Problem ◽  
Super Brownian Motion

Singular Spacetime Itô's Integral and a Class of Singular Interacting Branching Particle Systems

2003 ◽  
Vol 06 (02) ◽  
pp. 321-335 ◽  
Author(s):  
Hao Wang
Keyword(s):  
Brownian Motion ◽  
Unique Solution ◽  
Martingale Problem ◽  
Particle Systems ◽  
Dual Process ◽  
Singular Function ◽  
Singular Coefficient ◽  
Duality Method ◽  
Super Brownian Motion ◽  
Branching Diffusions

In Wang,8 a class of interacting measure-valued branching diffusions [Formula: see text] with singular coefficient were constructed and characterized as a unique solution to ℒε-martingale problem by a limiting duality method since in this case the dual process does not exist. In this paper, we prove that for any ε ≠ 0 the superprocess with singular motion coefficient is just the super-Brownian motion. The singular motion coefficient is handled as a sequential limit motivated by Antosik et al.1 Thus, the limiting superprocess is investigated and identified as the motion coefficient converges to a singular function. The representation of the singular spacetime Itô's integral is derived.

A Stochastic Calculus Approach for the Brownian Snake

2000 ◽  
Vol 52 (1) ◽  
pp. 92-118 ◽  
Author(s):  
Jean-Stéphane Dhersin ◽  
Laurent Serlet
Keyword(s):  
Brownian Motion ◽  
Wide Class ◽  
Stochastic Equation ◽  
Martingale Problem ◽  
Stochastic Calculus ◽  
Brownian Snake ◽  
Super Brownian Motion ◽  
Path Dependent ◽  
Definition Of ◽  
Well Posed

AbstractWe study the “Brownian snake” introduced by Le Gall, and also studied by Dynkin, Kuznetsov, Watanabe. We prove that Itô’s formula holds for a wide class of functionals. As a consequence, we give a new proof of the connections between the Brownian snake and super-Brownian motion. We also give a new definition of the Brownian snake as the solution of a well-posed martingale problem. Finally, we construct a modified Brownian snake whose lifetime is driven by a path-dependent stochastic equation. This process gives a representation of some super-processes.

scholarly journals An Application of the Backbone Decomposition to Supercritical Super-Brownian Motion with a Barrier

2012 ◽  
Vol 49 (03) ◽  
pp. 671-684
Author(s):  
A. E. Kyprianou ◽  
A. Murillo-Salas ◽  
J. L. Pérez
Keyword(s):  
Brownian Motion ◽  
Wave Equation ◽  
Relative Ease ◽  
Branching Brownian Motion ◽  
Analytical Properties ◽  
Super Brownian Motion ◽  
Exit Measure ◽  
The Right

We analyse the behaviour of supercritical super-Brownian motion with a barrier through the pathwise backbone embedding of Berestycki, Kyprianou and Murillo-Salas (2011). In particular, by considering existing results for branching Brownian motion due to Harris and Kyprianou (2006) and Maillard (2011), we obtain, with relative ease, conclusions regarding the growth in the right-most point in the support, analytical properties of the associated one-sided Fisher-Kolmogorov-Petrovskii-Piscounov wave equation, as well as the distribution of mass on the exit measure associated with the barrier.

scholarly journals Kolmogorov's test for super-Brownian motion

1998 ◽  
Vol 26 (3) ◽  
pp. 1041-1056 ◽  
Author(s):  
Jean-Stéphane Dhersin ◽  
Jean-François Le Gall
Keyword(s):  
Brownian Motion ◽  
Super Brownian Motion
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