Brownian snake

The Brownian snake approach was originally developed by Jean-François Le Gall.{{Cite journal |last=Le Gall |first=Jean-Francois |date=1991 |title=Brownian Excursions, Trees and Measure-Valued Branching Processes |url=https://www.jstor.org/stable/2244522 |journal=The Annals of Probability |volume=19 |issue=4 |pages=1399–1439 |jstor=2244522 |issn=0091-1798}} It has since been applied in fragmentation theory,{{Cite journal |last1=Abraham |first1=Romain |last2=Serlet |first2=Laurent |date=2002-07-01 |title=Poisson Snake and Fragmentation |url=https://projecteuclid.org/journals/electronic-journal-of-probability/volume-7/issue-none/Poisson-Snake-and-Fragmentation/10.1214/EJP.v7-116.full |journal=Electronic Journal of Probability |volume=7 |issue=none |doi=10.1214/EJP.v7-116 |s2cid=12003800 |issn=1083-6489|doi-access=free }} partial differential equation{{Cite journal |last=Abraham |first=Romain |date=2000-10-01 |title=Reflecting Brownian snake and a Neumann–Dirichlet problem |url=https://www.sciencedirect.com/science/article/pii/S0304414900000272 |journal=Stochastic Processes and Their Applications |language=en |volume=89 |issue=2 |pages=239–260 |doi=10.1016/S0304-4149(00)00027-2 |issn=0304-4149|doi-access=free }} or planar map{{Cite journal |last=Le Gall |first=Jean-François |date=2019-09-01 |title=Brownian geometry |url=https://doi.org/10.1007/s11537-019-1821-7 |journal=Japanese Journal of Mathematics |language=en |volume=14 |issue=2 |pages=135–174 |doi=10.1007/s11537-019-1821-7 |s2cid=255314865 |issn=1861-3624}}{{Cite journal |last=Miermont |first=Grégory |date=2013 |title=The Brownian map is the scaling limit of uniform random plane quadrangulations |url=https://projecteuclid.org/journals/acta-mathematica/volume-210/issue-2/The-Brownian-map-is-the-scaling-limit-of-uniform-random/10.1007/s11511-013-0096-8.full |journal=Acta Mathematica |volume=210 |issue=2 |pages=319–401 |doi=10.1007/s11511-013-0096-8 |s2cid=119140342 |issn=0001-5962|doi-access=free |arxiv=1104.1606 }}

Let $D(\backslash R\_+,\backslash R)$ be the space of càdlàg functions from $\backslash R\_+$ to $\backslash R$, equipped with a metric $d$ compatible with the Skorokhod topology. We define a stopped path as a couple $(w,z)$ where $w\backslash in\; D(\backslash R\_+,\backslash R)$ and $z\backslash in\; \backslash R\_+$ are such that $w(t)=w(t\backslash wedge\; z)$. In other words, $w$ is constant after $z$.

Now, we consider a jump process $(J\_s^N)\_\{s\backslash geq\; 0\}$ with states $\backslash \{+1,-1\backslash \}$ and jump rate $N$, such that $J^N\_0\; =\; +1$. We set:$$\backslash hat\{\backslash beta\}^N\_s\; :=\; \backslash int\_0^s\; J^N\_\{s\text{'}\}ds\text{'}$$and then $\backslash beta^N\_s\; :=\; |\backslash hat\{\backslash beta\}\_s^N|$ to be the process reflected on 0.

In words, $\backslash beta^N\_s$ increases with speed 1, until $J^N\_s$ jumps, in which case it decreases with speed 1, and so on. We define the stopping time $\backslash sigma\_N$ to be the $N$-th hitting time of 0 by $\backslash beta^N$.

We now define a stochastic process $(\backslash eta^N\_s,\backslash beta^N\_s)\_\{s\backslash in\; \backslash R\_+\}$ on the set of stopped paths as follows:

• $\backslash eta\_0^N\; =\; 0$
• if $J^N\_s\; =\; +1$ for $s\backslash in\; [s\_1,s\_2]$ then:
• $\backslash eta\_\{s\_1\}(t)=\backslash eta\_\{s\_2\}(t)$ for $t\backslash leq\; \backslash beta\_\{s\_1\}$
• $\backslash Big(\backslash eta\_\{s\_2\}(t-\backslash beta\_\{s\_1\})-\backslash eta\_\{s\_1\}(\backslash beta\_\{s\_1\})\backslash Big)\_\{0\backslash leq\; t\backslash leq\; \backslash beta\_\{s\_2\}-\backslash beta\_\{s\_1\}\}$ is distributed as a Brownian motion independent from $\backslash eta\_\{s\_1\}$
• if $J\_s^N\; =\; -1$ for $s\backslash in\; [s\_1,s\_2]$ then $\backslash eta\_\{s\_1\}(t)=\backslash eta\_\{s\_2\}(t)$ for $t\backslash leq\; \backslash beta\_\{s\_2\}$

See animation for an illustration. We call this process a snake and $\backslash beta\_s^N$ 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.

{{Hide|Show/hide animation|

File:Gif discrete-time snake.gif}}

We now consider a measure-valued branching process $(X^N\_t)\_\{t\backslash geq\; 0\}$ starting with $N$ particles, such that each particle dies with rate $N$, and upon its death gives birth to two offspring with probability $1/2$.

On the other hand, we may define from our process $(\backslash eta\_s^N,\backslash beta\_s^N)\_\{0\backslash leq\; s\backslash leq\; \backslash sigma^N\}$ a measure-valued random process $\backslash hat\{X\}\_t$ as follows: note that for any $t\backslash in\; \backslash R\_+$, there will almost surely be finitely many times $s\_1,s\_2,\backslash dots,s\_n\backslash in\; [0,\backslash sigma^N]$ such that $\backslash beta\_\{s\_i\}=t$. We then set for any measurable function $f$:

$\backslash hat\{X\}^N\_t(f):=\; \backslash sum\backslash limits\_\{i=1\}^nf(\backslash eta^N\_s(t))$

Then $X$ and $\backslash hat\{X\}$ are equal in distribution.

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

However, we may define a probability $R\_\{a,b\}((u,y),d(w,z))$ on stopped paths such that:

• $R\_\{a,b\}$-almost surely $z=b$ and $w(t)=u(t)$ for $0\backslash leq\; t\backslash leq\; a$
• The law of $(w(a+t))\_\{0\backslash leq\; t\backslash leq\; b-a\}$ is the law of a standard Brownian motion.

We may also define $\backslash gamma\_s^y(da,db)$ to be the distribution of $(\backslash inf\_\{0\backslash leq\; r\backslash leq\; s\}\backslash beta\_r,\backslash beta\_s)$ if $\backslash beta\_0=y$. Finally, define the transition semigroup on the set of stopped paths:

$Q\_s((u,y),d(w,z))\; =\; \backslash int\; \backslash gamma\_s^y(da,db)R\_\{a,b\}((u,y),d(w,z))$

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 $(X\_t)\_\{t\backslash in\; \backslash R\_+\}$ with branching mechanism $\backslash phi(z)=z^2$, started on a Dirac in 0.

However, unlike the previous case, we must be more careful in the definition of the process $\backslash hat\{X\}$. Indeed, for $t\backslash in\; \backslash R\_+$ we cannot just list the times $s\_1,s\_2,\backslash dots$ such that $\backslash beta\_s=t$. Instead we use the local time $l\_s(t)$ associated with $\backslash beta\_s$: we first define the stopping time $\backslash sigma\; =\; \backslash inf\backslash \{s\backslash geq\; 0,\; l\_s(0)\backslash geq\; u\backslash \}$. Then we define for any measurable $f$:$$\backslash hat\{X\}\_t(f):=\; \backslash int\_0^\backslash sigma\; f(\backslash eta\_s(t))dl\_s(t)$$ Then, as before, we obtain that $X$ and $\backslash hat\{X\}$ are equal in distribution. See the animation for the construction of the branching process from the Brownian snake.

{{Hide|Animation for the branching process associated with the Brownian snake|File:Construction of a Branching process associated with the Brownian snake.gif}}

The previous example can be generalized in many ways:

• We may consider $D(\backslash R\_+,E)$ where $(E,d)$ 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).

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.

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Category:Markov processes