Bessel Processes, the Brownian Snake and Super-Brownian Motion

2015 ◽  
pp. 89-105 ◽  
Author(s):  
Jean-François Le Gall
Keyword(s):  
Brownian Motion ◽  
Bessel Processes ◽  
Brownian Snake ◽  
Super Brownian Motion

Wiener's test for super-Brownian motion and the Brownian snake

1997 ◽  
Vol 108 (1) ◽  
pp. 103-129 ◽  
Author(s):  
Jean-Stéphane Dhersin ◽  
Jean-François Le Gall
Keyword(s):  
Brownian Motion ◽  
Brownian Snake ◽  
Super Brownian Motion

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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