Desgaste de las cuchillas

The usual birth-death process is a continuous-time Markov jump process on the non-negative inte- gers. If the current state isnthen the next transitions can only be made to staten+ 1(abirth) or state

n−1(adeath); furthermore the transition rates depend only on the number of individuals alive. For many models it is reasonable to expect that the transition rates depend on the number of individuals

as well astheir locations. Preston (1977) describes a process which takes into account thepositions

of the individuals; hence the name ‘spatial birth-death process’. The relevance of these processes lies in the close relationship to Gibbs processes and especially in the way they provide a means to simulate such processes, as suggested by Ripley (1977). When the process is time-reversible

(Definition 1.24) then one can findequilibrium distributions(Definition 1.22) for the process; these actually turn out to be various kinds of Gibbs states or distributions (Preston 1977).

Suppose the individuals {ξ} live in some space X and recall that Xe denotes the Exponential

space (cf. Section 1.1.1) ofX. A spatial birth-death processΦis a continuous-time Markov jump (cf. Definition 1.29) process taking values inXe. The process has a Markov property: the probability

of the next transition depends only on the current configuration. Following Preston (1977) we now describe the construction of a spatial birth-death processΦ, which evolves in time with individuals dying or being born, given by the following recipe:

1. At any timetthere are only a finite number of individuals alive.

2. If at timetthe configuration of alive individuals isx ={ξ1, . . . , ξn}then there exists a finite

measureB(x,·)on(X,B(X))such that the probability of an individual being born in some

W ∈ B(X)in the interval[t, t+δ]isB(x, W)δ+o(δ).

3. If at time t the configuration of alive individuals is x = {ξ1, . . . , ξn, ξ} then there exists a

B(X) measurable function d(x,·) : X → _R+ such that the probability of the individual

ξ dying in the interval[t, t+δ]isd(x\ {ξ}, ξ)δ+o(δ).

4. The probability that there is more than one transition in[t, t+δ]iso(δ).

IfB(x;·) is absolutely continuous with respect to some measure µ on (X,B(X))then there ex- ists a positive measurable functionb : Xe × X → R+ such that B(x, W) =

R

W b(x, ξ)µ(dξ),

W ∈ B(X). Refer tob as the birth rate and d the (per capita) death rate; then B(x,·) denotes the total birth rate and D(x) = P

ξ∈xd(x\ {ξ}, ξ) the total death rate. In the terminology of

Fernándezet al.(2002) a spatial birth-death process whose birth and death rates,b(x,·)andd(x,·)

respectively, do not depend on the current configurationxis called afree processsince then there is no interactions between the individuals. Conversely if the birth/death rates do depend on the current configuration, then such a process is referred to as an interacting spatial birth-death process since there is some kind of interaction.

1.3.1

Simulation of Spatial Birth-Death Processes

SupposeΦis a spatial birth-death process on some bounded W ∈ B(X)with birth rate b(·) and death rated(·), which may depend on the current state of the process. Therefore in order to simulate

Φ the state of the birth death process outside W must be fixed, and a realization of Φ simulated

conditionalon this fixed state outside W (cf. Section 1.1.10). IfΦ (t) = xthen the time till the next ‘jump’ or transition is exponentially distributed with rateα(x) = B(x, W) +D(x), where

B(x, W) = R_W b(x, ξ)µ(dξ)andD(x) = P

ξ∈xd(x\ {ξ}, ξ). With probability

B(x,W)

α(x) the next

transition is a birth, else it is a death. Births are drawn from density _Bb₍(_x,Wx,·)₎; deaths are drawn from density d_D(x₍\·_x,₎·). The following describes how to obtain a sample path ofΦon[0, T].

Algorithm 1.1 (Simulation ofΦ).

InitializeΦ(0)(this can be at any arbitrary configuration); sett= 0. whilet ≤T:

drawτ ∼Exponential with rateα(Φ (t)) =B(Φ (t), W) +D(Φ (t)). drawU ∼Uniform(0,1).

ifU ≤ B_α(Φ(_(Φ(t)_t,W₎₎): drawξ∼ _B(Φ(b(_t·₎)_,W); setΦ (t) = Φ (t−)∪ {ξ}. else: drawξ∼ _D(Φ(d(·)_t)); setΦ (t) = Φ (t−)\ {ξ}.

sett=t+τ. returnΦ (T).

1.3.2

Simulation of Point Processes via Spatial Birth-Death Processes

In this section we look at how one can employ spatial birth-death processes in order to sample point processes defined by a density, such as Gibbs processes. If a spatial birth-death process is statistically indistinguishable from its time-reversal then it is time-reversible (Definition 1.24). A time-reversible process has an equilibrium distribution (Preston 1977, Proposition 8.3), ie. if the process is started with this distribution then its distribution is preserved for all time. The equilibrium distributions of spatial birth-death processes are of particular importance, since they represent the so-called Gibbs states or distributions from statistical physics. Under suitable conditions the spatial birth-death process will converge to a unique equilibrium distribution (Ripley 1977; Preston 1977, Theorem7.1). Iff is someµ-integrable function such that

b(x, ξ)f(x) = d(x\ {ξ};ξ)f(x∪ {ξ}), for allξ /∈x (1.25) thenbanddsatisfy the equations ofdetailed balance(Definition 1.25) with respect tof. In this case the process with birth and death ratesbanddrespectively is time-reversible. Moreoverπ = R

f dµ

is the unique equilibrium distribution of the spatial birth-death process (Preston 1977, Lemma 8.2). Example 1.3. For constant birth rateb and unit death rate per point, detailed balance calculations show that the process converges to a Poisson(b)process on some bounded W ⊂ _Rd_{. The density}

e(1−bmd[W])_bn(x)_{(cf. Example 1.1). Then for anyξ /}∈_x

bf(x) =be(1−bmd[W])_bn(x)_=e(1−bmd[W])_bn(x∪{ξ}) _{=d(x;ξ)f(x∪ {ξ}).}

Now consider a point processX with densityf. Suppose that one can define a birth-death process

Ψso that its birth and death rates satisfy the detailed balance condition in Eq.(1.25)with respect to

f. Then, provided thatΨis irreducible, Ψwill be ergodic and hence converge to the point process

X. It is easily shown that if the birth rate ofΨis equal to the Papangelou conditional intensity ofX

(Eq. 1.9) and the death rate per point is one then detailed balance is satisfied.

For most point processes of interest the Papangelou conditional intensity`(·;x)depends on the current configurationx. In this case simulatingΨwith birth rateb=`may pose practical problems since calculating the total birth rateB(x) = R_W`(ξ;x)µ(dξ)may be difficult, if not impossible. However if`is bounded inxso that`(·;x)≤`∗(·)for allxthen it is possible to simulateΨwithout

having to computeB(·). The idea is to define another processΦ with birth rateb∗ = `∗ and unit death rate per point, so that simulation ofΦis straight forward. A realization ofΨis obtained by

couplingits evolution to that ofΦ: transitions ofΦare considered as ‘proposed’ transitions inΨand accepted so as to ensure the correct transition rates forΨ. Specifically, since the per capita death rate for both processes is one, deaths inΦare always accepted. The birth rate of Φis higher, so births need to be censored and accepted with probability <sub>`</sub>`∗ (cf. Example 1.2).

Remark 1.2. Suppose X, Y are two point processes with respective distributions πX, πY and Pa-

pangelou conditional intensities`X, `Y. If `X ≥ `Y then it is possible to construct coupled spatial

birth-death processesΦX,ΦY with equilibriaπX, πY respectively such thatΦX(t) ⊇ΦY (t)for all

t. Existence of suchΦX andΦY shows thatXis stochastically larger thanY; cf. Theorem 1.3.