We observe that Y^(H)Y^{H} and widetilde(Y)^(K)\widetilde{Y}^{K} are constant processes until time RR, that R > 0R>0 a.s. and that T_(a)!=T_(b)T_{a} \neq T_{b} a.s. 我们观察到 Y^(H)Y^{H} 和 widetilde(Y)^(K)\widetilde{Y}^{K} 是恒定的过程,直到时间 RR ,那个 R > 0R>0 a.s. 和那个 T_(a)!=T_(b)T_{a} \neq T_{b} a.s.
One of three things can happen at time RR. Firstly, if R=T_(a)R=T_{a}, there is a reproduction event in widetilde(Y)^(K)\widetilde{Y}^{K} but not in Y^(H)Y^{H}. If we rank in a non-increasing order these new particles, they again satisfy the partial ordering. Moreover, as H_(R) <= K_(R)H_{R} \leq K_{R}, applying the selection procedure to both models preserves this partial ordering, therefore 以下三种情况之一可能发生在 time RR .首先,如果 ,则 R=T_(a)R=T_{a} 中存在 widetilde(Y)^(K)\widetilde{Y}^{K} 繁殖事件,但在 中没有 Y^(H)Y^{H} 。如果我们以非递增顺序对这些新粒子进行排序,它们将再次满足部分排序。此外,由于 H_(R) <= K_(R)H_{R} \leq K_{R} ,将选择过程应用于两个模型会保留此部分排序,因此
AA x inR,#{j <= H_(R):Y_(R)^(H)(j) >= x} <= #{j <= K_(R): widetilde(Y)_(R)^(K)(j) >= x}\forall x \in \mathbb{R}, \#\left\{j \leq H_{R}: Y_{R}^{H}(j) \geq x\right\} \leq \#\left\{j \leq K_{R}: \widetilde{Y}_{R}^{K}(j) \geq x\right\}
If R=T_(b)R=T_{b}, then there is a reproduction event in Y^(H)Y^{H} and widetilde(Y)^(K)\widetilde{Y}^{K}. We use the same point process to construct the child of the particle that reproduces in each process. Once again, ranking in a non-increasing order these new particles, then applying the selection, we have 如果 ,则 R=T_(b)R=T_{b} 中存在 和 widetilde(Y)^(K)\widetilde{Y}^{K} 中的 Y^(H)Y^{H} 重现事件。我们使用相同的 point 过程来构建在每个过程中再现的粒子的子对象。再一次,以非递增的顺序对这些新粒子进行排序,然后应用选择,我们有
AA x inR,#{j <= H_(R):Y_(R)^(H)(j) >= x} <= #{j <= K_(R): widetilde(Y)_(R)^(K)(j) >= x}\forall x \in \mathbb{R}, \#\left\{j \leq H_{R}: Y_{R}^{H}(j) \geq x\right\} \leq \#\left\{j \leq K_{R}: \widetilde{Y}_{R}^{K}(j) \geq x\right\}
Finally, if R=S!in{T_(a),T_(b)}R=S \notin\left\{T_{a}, T_{b}\right\}, the maximal size of at least one of the populations is modified. Even if this implies the death of some particles in Y^(H)Y^{H} and/or widetilde(Y)^(K)\widetilde{Y}^{K}, the property (7.3) is preserved at time RR. 最后,如果 ,则 R=S!in{T_(a),T_(b)}R=S \notin\left\{T_{a}, T_{b}\right\} 至少修改一个总体的最大大小。即使这意味着 and/or widetilde(Y)^(K)\widetilde{Y}^{K} 中的 Y^(H)Y^{H} 某些粒子死亡,属性 (7.3) 也会在 time RR 保留。
Now fix t > 0t>0 and assume that H_(s) <= K_(s)H_{s} \leq K_{s} for every 0 <= s <= t0 \leq s \leq t. As HH and KK are integer-valued càdlàg processes, they attain their maxima on compact sets. Therefore, they are both a.s. finite on the interval [0,t][0, t], so the number of particles is a.s. finite in both processes Y^(H)Y^{H} and widetilde(Y)^(K)\widetilde{Y}^{K}. Thus there is a.s. a finite sequence of times (R_(k))\left(R_{k}\right) smaller than tt such that Y^(H)Y^{H} or widetilde(Y)^(K)\widetilde{Y}^{K} is modified at each time R_(k)R_{k}. Using this coupling on each time interval of the form [R_(k),R_(k+1)]\left[R_{k}, R_{k+1}\right] yields (7.3). 现在修复 t > 0t>0 并假设 H_(s) <= K_(s)H_{s} \leq K_{s} 对于每个 0 <= s <= t0 \leq s \leq t .由于 HH 和 KK 是整数值 càdlàg 进程,它们在紧集上达到最大值。因此,它们在区间 [0,t][0, t] 上都是 a.s. 有限的,因此粒子的数量在过程 Y^(H)Y^{H} 和 widetilde(Y)^(K)\widetilde{Y}^{K} 中都是 a.s. 有限的。因此,存在一个小于 tt 或 Y^(H)Y^{H}widetilde(Y)^(K)\widetilde{Y}^{K} 在每次 时被修改的有限时间 (R_(k))\left(R_{k}\right) 序列 R_(k)R_{k} 。在形式的 [R_(k),R_(k+1)]\left[R_{k}, R_{k+1}\right] 每个时间间隔上使用此耦合可生成 (7.3)。
Using this lemma, we can prove that the cloud of particles in a kk-BRW drifts at linear speed c_(k)c_{k}. Note that by the coupling described in the proof of Lemma 7.1, this result can be obtained as a consequence of Theorem 1.1. However, we believe the following proof to be of independent interest, as it can be generalized to more diverse continuous-time branching random walks with selection. 使用这个引理,我们可以证明 kk -BRW 中的粒子云以线性速度 c_(k)c_{k} 漂移。请注意,通过引理 7.1 证明中描述的耦合,这个结果可以作为定理 1.1 的结果得到。然而,我们认为以下证明具有独立的兴趣,因为它可以推广到更多样化的连续时间分支随机游走。
Lemma 7.5. For any k inNk \in \mathbb{N}, there exists c_(k)inRc_{k} \in \mathbb{R} such that 引理 7.5。对于任何 k inNk \in \mathbb{N} ,存在 c_(k)inRc_{k} \in \mathbb{R} 这样的
Moreover, if Y_(0)^(k)(1)=Y_(0)^(k)(2)=dots=Y_(0)^(k)(k)=0Y_{0}^{k}(1)=Y_{0}^{k}(2)=\ldots=Y_{0}^{k}(k)=0, we have 此外,如果 Y_(0)^(k)(1)=Y_(0)^(k)(2)=dots=Y_(0)^(k)(k)=0Y_{0}^{k}(1)=Y_{0}^{k}(2)=\ldots=Y_{0}^{k}(k)=0 ,我们有
c_(k)=i n f_(t > 0)(E[Y_(t)(1)])/(t)=s u p_(t > 0)(E[Y_(t)(k)])/(t).c_{k}=\inf _{t>0} \frac{\mathbf{E}\left[Y_{t}(1)\right]}{t}=\sup _{t>0} \frac{\mathbf{E}\left[Y_{t}(k)\right]}{t} .
The proof of this lemma is adapted from [5, Proposition 2]. 这个引理的证明改编自 [5, 命题 2]。
Proof. We prove that (Y_(t)^(k)(1))\left(Y_{t}^{k}(1)\right) is a sub-additive process. We then use Kingman’s sub-additive ergodic theorem (see [19, Theorem 4] and [18, Theorem 9.14]), 证明。我们证明这是一个 (Y_(t)^(k)(1))\left(Y_{t}^{k}(1)\right) 子加法过程。然后,我们使用 Kingman 的子加性遍历定理(参见 [19,定理 4] 和 [18,定理 9.14]),
stating that if ( {:X_(s,t),0 <= s <= t)\left.X_{s, t}, 0 \leq s \leq t\right) is a càdlàg family of random variables satisfying 表示 if ( {:X_(s,t),0 <= s <= t)\left.X_{s, t}, 0 \leq s \leq t\right) 是满足
{:[AA0 <= s <= t <= u","X_(s,u) <= X_(s,t)+X_(t,u)" a.s "],[AA h >= 0","(X_(s+h,t+h),0 <= s <= t)=^((d))(X_(s,t),0 <= s <= t)],[AA h >= 0","(X_(s+h,t+h),0 <= s <= t)" is independent of "(X_(s,t),0 <= s <= t <= h)],[EE A > 0","AA t >= 0","-At <= E(X_(0,t)) < oo],[E(|s u p_(0 <= s <= t <= 1)X_(s,t)|) < oo]:}\begin{aligned}
& \forall 0 \leq s \leq t \leq u, X_{s, u} \leq X_{s, t}+X_{t, u} \text { a.s } \\
& \forall h \geq 0,\left(X_{s+h, t+h}, 0 \leq s \leq t\right) \stackrel{(d)}{=}\left(X_{s, t}, 0 \leq s \leq t\right) \\
& \forall h \geq 0,\left(X_{s+h, t+h}, 0 \leq s \leq t\right) \text { is independent of }\left(X_{s, t}, 0 \leq s \leq t \leq h\right) \\
& \exists A>0, \forall t \geq 0,-A t \leq \mathbf{E}\left(X_{0, t}\right)<\infty \\
& \mathbf{E}\left(\left|\sup _{0 \leq s \leq t \leq 1} X_{s, t}\right|\right)<\infty
\end{aligned}
then gamma:=lim_(t rarr oo)(1)/(t)E(X_(0,t))\gamma:=\lim _{t \rightarrow \infty} \frac{1}{t} \mathbf{E}\left(X_{0, t}\right) exists, is finite and is equal to i n f_(t >= 0)(E(X_(0,t)))/(t)\inf _{t \geq 0} \frac{\mathbf{E}\left(X_{0, t}\right)}{t} (by sub-additivity), and 然后 gamma:=lim_(t rarr oo)(1)/(t)E(X_(0,t))\gamma:=\lim _{t \rightarrow \infty} \frac{1}{t} \mathbf{E}\left(X_{0, t}\right) 存在,是有限的并且等于 i n f_(t >= 0)(E(X_(0,t)))/(t)\inf _{t \geq 0} \frac{\mathbf{E}\left(X_{0, t}\right)}{t} (通过子加法),并且
lim_(t rarr oo)(1)/(t)X_(0,t)=gammaquad" a.s. and in "L^(1)\lim _{t \rightarrow \infty} \frac{1}{t} X_{0, t}=\gamma \quad \text { a.s. and in } L^{1}
We construct on the same probability space a family (Y_(s,t)^(k)(j),0 <= s <= t,j <= k)\left(Y_{s, t}^{k}(j), 0 \leq s \leq t, j \leq k\right), such that for all s >= 0,(Y_(s,s+t)^(k),t >= 0)s \geq 0,\left(Y_{s, s+t}^{k}, t \geq 0\right) is a kk-branching random walk, and (Y_(s,t)^(k)(1),0 <= s <= t)\left(Y_{s, t}^{k}(1), 0 \leq s \leq t\right) is sub-additive. 我们在相同的概率空间上构造一个族 (Y_(s,t)^(k)(j),0 <= s <= t,j <= k)\left(Y_{s, t}^{k}(j), 0 \leq s \leq t, j \leq k\right) ,使得 for all s >= 0,(Y_(s,s+t)^(k),t >= 0)s \geq 0,\left(Y_{s, s+t}^{k}, t \geq 0\right) 是一个 kk 分支随机游走,并且 (Y_(s,t)^(k)(1),0 <= s <= t)\left(Y_{s, t}^{k}(1), 0 \leq s \leq t\right) 是 sub-addive 的。
Let N^(1),dots,N^(k)N^{1}, \ldots, N^{k} be kk i.i.d. Poisson processes with unit intensity. For all s >= 0s \geq 0, we set Y_(s,s)^(k)(1)=Y_(s,s)^(k)(2)=cdots=Y_(s,s)^(k)(k)=0Y_{s, s}^{k}(1)=Y_{s, s}^{k}(2)=\cdots=Y_{s, s}^{k}(k)=0, i.e. particles start at position 0 at time ss. Then the process evolves as follows: at each time tt such that N_(t)^(j)!=N_(t-)^(j)N_{t}^{j} \neq N_{t-}^{j}, the jj th largest particle alive at time t-t- in Y_(s,∙)^(k)Y_{s, \bullet}^{k} creates a new child, and the leftmost particle is erased. 设 N^(1),dots,N^(k)N^{1}, \ldots, N^{k} i.i.d kk .具有单位强度的 Poisson 过程。对于所有 s >= 0s \geq 0 ,我们设置 Y_(s,s)^(k)(1)=Y_(s,s)^(k)(2)=cdots=Y_(s,s)^(k)(k)=0Y_{s, s}^{k}(1)=Y_{s, s}^{k}(2)=\cdots=Y_{s, s}^{k}(k)=0 ,即粒子在时间 ss 上从位置 0 开始。然后,该过程将按如下方式发展:在每次 tt , N_(t)^(j)!=N_(t-)^(j)N_{t}^{j} \neq N_{t-}^{j} 在 t-t- 时间存活的 jj 第 th 个最大粒子 会 Y_(s,∙)^(k)Y_{s, \bullet}^{k} 创建一个新的子粒子,而最左边的粒子将被擦除。
By definition, we observe that for all s >= 0,(Y_(s,s+t)^(k),t >= 0)s \geq 0,\left(Y_{s, s+t}^{k}, t \geq 0\right) is a kk-branching random walk starting with kk particles at position 0 at time 0 . In particular, (7.6) is satisfied. Moreover, Y_(s,∙)^(k)Y_{s, \bullet}^{k} is measurable with respect to the Poisson processes (N_(t+s)^(j)-N_(s)^(j),t >= s,j <= k)\left(N_{t+s}^{j}-N_{s}^{j}, t \geq s, j \leq k\right), therefore is independent of (Y_(u,v)^(k),0 <= u <= v <= s)\left(Y_{u, v}^{k}, 0 \leq u \leq v \leq s\right). This shows (7.7), i.e. that this process is ergodic. 根据定义,我们观察到 for all s >= 0,(Y_(s,s+t)^(k),t >= 0)s \geq 0,\left(Y_{s, s+t}^{k}, t \geq 0\right) 是一个 kk 分支随机游走,从 kk 时间 0 位置 0 的粒子开始。特别是 (7.6) 得到满足。此外, Y_(s,∙)^(k)Y_{s, \bullet}^{k} 是 Poisson 过程 (N_(t+s)^(j)-N_(s)^(j),t >= s,j <= k)\left(N_{t+s}^{j}-N_{s}^{j}, t \geq s, j \leq k\right) 可测量的,因此 独立于 (Y_(u,v)^(k),0 <= u <= v <= s)\left(Y_{u, v}^{k}, 0 \leq u \leq v \leq s\right) 。这表明 (7.7),即这个过程是遍历的。
Moreover, one can observe that the construction described here is the same as the one given in the proof of Lemma 7.4. Therefore, for all s <= ts \leq t this process couples the kk-branching random walks (Y_(s,t+h)^(k),h >= 0)\left(Y_{s, t+h}^{k}, h \geq 0\right) and (Y_(t,t+h)^(k),h >= 0)\left(Y_{t, t+h}^{k}, h \geq 0\right) in such a way that for all j <= kj \leq k and h >= 0h \geq 0, one has 此外,可以观察到这里描述的结构与引理 7.4 证明中给出的结构相同。因此,对于所有这些 s <= ts \leq t 过程,耦合 kk -分支随机游走 (Y_(s,t+h)^(k),h >= 0)\left(Y_{s, t+h}^{k}, h \geq 0\right) , (Y_(t,t+h)^(k),h >= 0)\left(Y_{t, t+h}^{k}, h \geq 0\right) 并且以这种方式,对于所有 j <= kj \leq k 和 h >= 0h \geq 0 ,都有
Indeed, the kk-branching random walk (Y_(s,t+h)^(k),h >= 0)\left(Y_{s, t+h}^{k}, h \geq 0\right) is coupled with the kk branching random walk (Y_(t,t+h)^(k)+Y_(s,t)(1),h >= 0)\left(Y_{t, t+h}^{k}+Y_{s, t}(1), h \geq 0\right) which starts with kk particles at position Y_(s,t)(1)Y_{s, t}(1). In particular, we have Y_(s,u)^(k)(1) <= Y_(s,t)^(k)(1)+Y_(t,u)^(k)(1)Y_{s, u}^{k}(1) \leq Y_{s, t}^{k}(1)+Y_{t, u}^{k}(1) a.s., proving (7.5). 事实上, kk 分支随机游走 (Y_(s,t+h)^(k),h >= 0)\left(Y_{s, t+h}^{k}, h \geq 0\right) 与 kk 分支随机游走 (Y_(t,t+h)^(k)+Y_(s,t)(1),h >= 0)\left(Y_{t, t+h}^{k}+Y_{s, t}(1), h \geq 0\right) 相耦合,后者从 kk 位置 Y_(s,t)(1)Y_{s, t}(1) 的粒子开始。特别是,我们有 Y_(s,u)^(k)(1) <= Y_(s,t)^(k)(1)+Y_(t,u)^(k)(1)Y_{s, u}^{k}(1) \leq Y_{s, t}^{k}(1)+Y_{t, u}^{k}(1) a.s.,证明 (7.5)。
To prove the last two conditions, we observe that Y_(s,∙)(1)Y_{s, \bullet}(1) increases by at most 1 at each time one of the Poisson processes jumps. Moreover, t|->Y_(s,t)t \mapsto Y_{s, t} is non-decreasing, thus, for all t >= 0t \geq 0, 为了证明最后两个条件,我们观察到每次其中一个泊松进程跳跃时, Y_(s,∙)(1)Y_{s, \bullet}(1) 最多增加 1。此外, t|->Y_(s,t)t \mapsto Y_{s, t} 是非递减的,因此,对于所有 t >= 0t \geq 0 ,
E(|s u p_(0 <= s <= t <= 1)Y_(s,t)(1)|) <= k quad" and "quad0 <= E(Y_(0,t)(1)) < oo\mathbf{E}\left(\left|\sup _{0 \leq s \leq t \leq 1} Y_{s, t}(1)\right|\right) \leq k \quad \text { and } \quad 0 \leq \mathbf{E}\left(Y_{0, t}(1)\right)<\infty
proving both (7.8) and (7.9). 证明 (7.8) 和 (7.9)。
As a result, by Kingman’s sub-additive ergodic theorem, setting 因此,根据 Kingman 的子加性遍历定理,将
we have lim_(t rarr oo)(Y_(0,t)(1))/(t)=c_(k)\lim _{t \rightarrow \infty} \frac{Y_{0, t}(1)}{t}=c_{k} a.s. 我们有 lim_(t rarr oo)(Y_(0,t)(1))/(t)=c_(k)\lim _{t \rightarrow \infty} \frac{Y_{0, t}(1)}{t}=c_{k} A.S.
With the same construction, one can observe that (Y_(s,t)^(k)(k),0 <= s <= t)\left(Y_{s, t}^{k}(k), 0 \leq s \leq t\right) is a super-additive sequence, satisfying similar integrability assumptions as Y_(s,t)(1)Y_{s, t}(1). Therefore, setting 使用相同的构造,可以观察到 是一个 (Y_(s,t)^(k)(k),0 <= s <= t)\left(Y_{s, t}^{k}(k), 0 \leq s \leq t\right) 超级加法序列,满足与 Y_(s,t)(1)Y_{s, t}(1) 类似的可积性假设。因此,设置
we have lim_(t rarr oo)(Y_(0,t)(k))/(t)=d_(k)\lim _{t \rightarrow \infty} \frac{Y_{0, t}(k)}{t}=d_{k} a.s. As Y_(s,t)(k) <= Y_(s,t)(1)Y_{s, t}(k) \leq Y_{s, t}(1), we have d_(k) <= c_(k)d_{k} \leq c_{k}. We now prove these two quantities to be equal. 我们有 lim_(t rarr oo)(Y_(0,t)(k))/(t)=d_(k)\lim _{t \rightarrow \infty} \frac{Y_{0, t}(k)}{t}=d_{k} A.S.作为 Y_(s,t)(k) <= Y_(s,t)(1)Y_{s, t}(k) \leq Y_{s, t}(1) ,我们有 d_(k) <= c_(k)d_{k} \leq c_{k} 。我们现在证明这两个量是相等的。
We define a sequence of hitting times (T_(n),n >= 0)\left(T_{n}, n \geq 0\right) by setting T_(0)=0T_{0}=0, and T_(n+1)T_{n+1} is the first time after time T_(n)T_{n} where the last kk children are born from the same particle, and that this particle was the rightmost particle before the series of branching events. The probability that the next kk branching events are as such is 1//k^(k) > 01 / k^{k}>0, therefore T_(n) < ooT_{n}<\infty a.s. Moreover, by definition, we have Y_(0,T_(n))(1)=Y_(0,T_(n))(k)Y_{0, T_{n}}(1)=Y_{0, T_{n}}(k) a.s., all particles being at the same position at that time As a result, we have 我们通过设置 T_(0)=0T_{0}=0 来定义一系列撞击时间 (T_(n),n >= 0)\left(T_{n}, n \geq 0\right) ,并且 T_(n+1)T_{n+1} 是最后一次 kk 从同一粒子中诞生的最后一个 T_(n)T_{n} 子粒子,并且该粒子是一系列分支事件之前最右边的粒子。下一个 kk 分支事件的概率是 1//k^(k) > 01 / k^{k}>0 ,因此 T_(n) < ooT_{n}<\infty a.s.此外,根据定义,我们有 Y_(0,T_(n))(1)=Y_(0,T_(n))(k)Y_{0, T_{n}}(1)=Y_{0, T_{n}}(k) a.s.,所有粒子在那个时候都处于同一位置,因此,我们有
l i m i n f_(t rarr oo)(Y_(0,t)(1)-Y_(0,t)(k))/(t)=0quad" a.s. "\liminf _{t \rightarrow \infty} \frac{Y_{0, t}(1)-Y_{0, t}(k)}{t}=0 \quad \text { a.s. }
proving that c_(k)=d_(k)c_{k}=d_{k}, and that (Y_(0,t)(1))/(t)\frac{Y_{0, t}(1)}{t} and (Y_(0,t)(k))/(t)\frac{Y_{0, t}(k)}{t} have the same limit. 证明 c_(k)=d_(k)c_{k}=d_{k} , 和 that (Y_(0,t)(1))/(t)\frac{Y_{0, t}(1)}{t} 和 具有相同的 (Y_(0,t)(k))/(t)\frac{Y_{0, t}(k)}{t} 限制。
Finally, we consider a kk-branching random walk Y^(k)Y^{k} starting from an arbitrary initial configuration. After a finite amount of time tt, the process contains kk particles. From that point on, the process can be bounded from above and from below by kk-branching random walks starting with kk particles at position Y_(t)^(k)(1)Y_{t}^{k}(1) and Y_(t)^(k)(k)Y_{t}^{k}(k) respectively. Therefore, by the previous results, we also obtain 最后,我们考虑从任意初始配置开始的 kk -branching 随机游走 Y^(k)Y^{k} 。在有限的时间 tt 之后,该过程包含 kk 粒子。从那时起,该过程可以通过 kk -分支随机游走从上方和下方界定,分别从 kk 位置 Y_(t)^(k)(1)Y_{t}^{k}(1) 处的 Y_(t)^(k)(k)Y_{t}^{k}(k) 粒子开始。因此,通过前面的结果,我们还得到了
In this section, we use Lemma 7.4 to compare the asymptotic behaviour of the continuous-time branching random walk with selection Y^(k)Y^{k} with a discrete-time branching random walk with selection. This latter model being well-studied, we are able to deduce Lemma 7.3 from it. The discrete-time branching random walk with selection of the rightmost kk individuals was introduced by Brunet and Derrida in [8] to study noisy FKPP equations. In that article, they conjecture that the cloud of particles drifts at speed v_(k)v_{k}, that satisfies 在本节中,我们使用 Lemma 7.4 来比较带选择的连续时间分支随机游走与带选择的 Y^(k)Y^{k} 离散时间分支随机游走的渐近行为。后一个模型经过充分研究,我们能够从中推断出引理 7.3。Brunet 和 Derrida 在 [8] 中引入了选择最 kk 右边个体的离散时间分支随机游走,用于研究嘈杂的 FKPP 方程。在那篇文章中,他们推测粒子云以 的速度 v_(k)v_{k} 漂移,这满足
v_(k)-v=-(chi)/((log k+3log log k+o(log log k))^(2))," as "k rarr oo,v_{k}-v=-\frac{\chi}{(\log k+3 \log \log k+o(\log \log k))^{2}}, \text { as } k \rightarrow \infty,
for some explicit constants v inRv \in \mathbb{R} and chi > 0\chi>0. 对于某些显式常量 v inRv \in \mathbb{R} 和 chi > 0\chi>0 .
We now describe more precisely the discrete-time kk-branching random walk. Let k inNk \in \mathbb{N} and M\mathcal{M} be the law of a point process on Z\mathbb{Z}. The system starts with kk particles on Z\mathbb{Z}. At each integer time nn, every particle dies while giving birth to offspring. The children of a given individual are positioned around their parent according to an i.i.d. point process with law M\mathcal{M}. Among all the children of 我们现在更精确地描述了离散时间 kk 分支随机游走。设 k inNk \in \mathbb{N} 和 M\mathcal{M} 是 上的点过程的 Z\mathbb{Z} 定律。系统从 kk 上的 Z\mathbb{Z} 粒子开始。在每个整数时间 nn ,每个粒子在生下后代时都会死亡。给定个体的子代根据 i.i.d. 点过程与定律 M\mathcal{M} .在所有的子代中