A Brownian snake is a stochastic Markov process on the space of stopped paths. It has been extensively studied.,[1][2] and was in particular successfully used as a representation of superprocesses.

Informally, superprocesses are the scaling limit of branching processes, except each particle splits and dies at infinite rates. The Brownian snake is a stochastic object that enables the representation of the genealogy of a superprocess, providing a link between super-Brownian motion and Brownian trees. In other words, even though infinitely many particles are constantly born, we can still keep track of individual trajectories in space, or of when two given present-day particles have split from a common ancestor in the past.

History

edit

The Brownian snake approach was originally developed by Jean-François Le Gall.[2][3] It has since been applied in fragmentation theory,[4] partial differential equation[5] or planar map[6][7]

The simplest setting

edit

Let   be the space of càdlàg functions from   to  , equipped with a metric   compatible with the Skorokhod topology. We define a stopped path as a couple   where   and   are such that  . In other words,   is constant after  .

Now, we consider a jump process   with states   and jump rate  , such that  . We set: and then   to be the process reflected on 0.

In words,   increases with speed 1, until   jumps, in which case it decreases with speed 1, and so on. We define the stopping time   to be the  -th hitting time of 0 by  . We now define a stochastic process   on the set of stopped paths as follows:

  •  
  • if   for   then:
    •   for  
    •   is distributed as a Brownian motion independent from  
  • if   for   then   for  

See animation for an illustration. We call this process a snake and   the head of the snake. This process is not yet the Brownian snake, but a good introduction. The path is erased when the snake head moves backwards, and is created anew when it moves forward.

Show/hide animation
 
The left panel is the "health bar", which goes to 0 as the number of times the snake hits 0 increases. The panel below just shows how the head of the snake moves. The large panel represents the non-constant part of the snake in black, with the head as a red dot.

Duality with a branching Brownian motion

edit

We now consider a measure-valued branching process   starting with   particles, such that each particle dies with rate  , and upon its death gives birth to two offspring with probability  .

On the other hand, we may define from our process   a measure-valued random process   as follows: note that for any  , there will almost surely be finitely many times   such that  . We then set for any measurable function  :

 

Then   and   are equal in distribution.

The Brownian snake

edit

We take the limit of the previous system as  . In this setting, the head of the snake keeps jittering. In fact, the process   tends towards a reflected Brownian motion  . The definitions are no longer valid for a number of reasons, in particular because   is almost surely never monotonous on any interval.

However, we may define a probability   on stopped paths such that:

  •  -almost surely   and   for  
  • The law of   is the law of a standard Brownian motion.

We may also define   to be the distribution of   if  . Finally, define the transition semigroup on the set of stopped paths:

 

A stochastic process with this semigroup is called a Brownian snake.

We may again find a duality between this process and a branching process. Here the branching process will be a super-Brownian motion   with branching mechanism  , started on a Dirac in 0.

However, unlike the previous case, we must be more careful in the definition of the process  . Indeed, for   we cannot just list the times   such that  . Instead we use the local time   associated with  : we first define the stopping time  . Then we define for any measurable  :  Then, as before, we obtain that   and   are equal in distribution. See the animation for the construction of the branching process from the Brownian snake.

Animation for the branching process associated with the Brownian snake
 
The left panel shows the "health bar" of the snake, which decreases with the local time the head spends on 0. The panel below shows the movement of the snake head according to a Brownian motion reflected on 0. The central panel shows: in red the head of the current snake, in black the current snake, in green the past snakes. The branching superprocess   is obtained once the health bar reaches 0, by taking all of the green paths.

Generalisation

edit

The previous example can be generalized in many ways:

  • We may consider   where   is a complete separable metric space.
  • Instead of a Brownian motion, the underlying movement of the snake can be very general class of Markov processes (see Superprocess).
edit

The Brownian snake can be seen as a way to represent the genealogy of a superprocess, the same way a Galton-Watson tree may encode the hidden genealogy of a Galton–Watson process.[2] Indeed, for two points of the Brownian snake, their common ancestor will be the infimum of the snake's head position between them.

If we take a Brownian snake and construct a real tree from it, we obtain a Brownian tree.[2]

References

edit
  1. ^ Li, Zenghu (2011), Li, Zenghu (ed.), "Measure-Valued Branching Processes", Measure-Valued Branching Markov Processes, Probability and Its Applications, Berlin, Heidelberg: Springer, pp. 29–56, doi:10.1007/978-3-642-15004-3_2, ISBN 978-3-642-15004-3, retrieved 2022-12-20
  2. ^ a b c d Le Gall, Jean-Francois (1999-07-01). Spatial Branching Processes, Random Snakes and Partial Differential Equations. Springer Science & Business Media. ISBN 978-3-7643-6126-6.
  3. ^ Le Gall, Jean-Francois (1991). "Brownian Excursions, Trees and Measure-Valued Branching Processes". The Annals of Probability. 19 (4): 1399–1439. ISSN 0091-1798. JSTOR 2244522.
  4. ^ Abraham, Romain; Serlet, Laurent (2002-07-01). "Poisson Snake and Fragmentation". Electronic Journal of Probability. 7 (none). doi:10.1214/EJP.v7-116. ISSN 1083-6489. S2CID 12003800.
  5. ^ Abraham, Romain (2000-10-01). "Reflecting Brownian snake and a Neumann–Dirichlet problem". Stochastic Processes and Their Applications. 89 (2): 239–260. doi:10.1016/S0304-4149(00)00027-2. ISSN 0304-4149.
  6. ^ Le Gall, Jean-François (2019-09-01). "Brownian geometry". Japanese Journal of Mathematics. 14 (2): 135–174. doi:10.1007/s11537-019-1821-7. ISSN 1861-3624. S2CID 255314865.
  7. ^ Miermont, Grégory (2013). "The Brownian map is the scaling limit of uniform random plane quadrangulations". Acta Mathematica. 210 (2): 319–401. arXiv:1104.1606. doi:10.1007/s11511-013-0096-8. ISSN 0001-5962. S2CID 119140342.