A Brownian snake is a stochastic Markov process on the space of stopped paths. It has been extensively studied.,[1] 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.
The Brownian snake approach was originally developed by Jean-François Le Gall.[2] It has since been applied in fragmentation theory,[3] partial differential equation[4] or planar map[5] [6]
Let
D(\R+,\R)
\R+
\R
d
(w,z)
w\inD(\R+,\R)
z\in\R+
w(t)=w(t\wedgez)
w
z
N) | |
(J | |
s\geq0 |
\{+1,-1\}
N
N | |
J | |
0 |
=+1
N | |
\beta | |
s |
:=
N| | |
|\hat{\beta} | |
s |
In words,
N | |
\beta | |
s |
N | |
J | |
s |
\sigmaN
N
\betaN
N | |
(η | |
s) |
s\in\R+ |
N | |
η | |
0 |
=0
N | |
J | |
s |
=+1
s\in[s1,s2]
η | |
s1 |
(t)=η | |
s2 |
(t)
t\leq
\beta | |
s1 |
(η | |
s2 |
(t-\beta | |
s1 |
)-η | |
s1 |
(\beta | |
s1 |
)) | |||||||||||
|
η | |
s1 |
N | |
J | |
s |
=-1
s\in[s1,s2]
η | |
s1 |
(t)=η | |
s2 |
(t)
t\leq
\beta | |
s2 |
See animation for an illustration. We call this process a snake and
N | |
\beta | |
s |
N | |
(X | |
t) |
t\geq
N
N
1/2
On the other hand, we may define from our process
N) | |
(η | |
0\leqs\leq\sigmaN |
\hat{X}t
t\in\R+
s1,s2,...,sn\in[0,\sigmaN]
\beta | |
si |
=t
f
N | |
\hat{X} | |
t(f):= |
nf(η | |
\sum\limits | |
i=1 |
N | |
s(t)) |
Then
X
\hat{X}
We take the limit of the previous system as
N\toinfty
N | |
\beta | |
s |
\betas
\betas
However, we may define a probability
Ra,b((u,y),d(w,z))
Ra,b
z=b
w(t)=u(t)
0\leqt\leqa
(w(a+t))0\leq
We may also define
y(da,db) | |
\gamma | |
s |
(inf0\leq\betar,\betas)
\beta0=y
Qs((u,y),d(w,z))=\int
y(da,db)R | |
\gamma | |
a,b |
((u,y),d(w,z))
A stochastic process with this semigroup is called a Brownian snake.
(Xt)
t\in\R+ |
\phi(z)=z2
However, unlike the previous case, we must be more careful in the definition of the process
\hat{X}
t\in\R+
s1,s2,...
\betas=t
ls(t)
\betas
\sigma=inf\{s\geq0,ls(0)\gequ\}
f
X
\hat{X}
The previous example can be generalized in many ways:
D(\R+,E)
(E,d)
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. 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.