Propagation of the Front of Random Walk with Periodic Branching Sources

Abstract

We consider the model of branching random walk on an integer lattice $\mathbb{Z}^{d}$ with periodic sources of branching. It is supposed that the regime of branching is supercritical and the Cramér condition is satisfied for a jump of the random walk. The theorem established describes the rate of front propagation for particles population over the lattice as the time increases unboundedly. The proofs are based on fundamental results related to the spatial spread of general branching random walk.

Study of the branching random walk (BRW) with periodic sources of branching and death of particles was initiated in papers [1, 2], where the asymptotic expansion for mean local populations of particles at the time instance $t$ as $t\to\infty$ was established. In this work we first managed to show that with probability one a properly transformed cloud of studied particles approaches a certain limit set in the Hausdorff metric as time goes to infinity, namely, we prove that the population of particles in the supercritical BRW with periodically located sources of branching and death of particles propagates asymptotically linear in the case of light tails of distribution of the walk jump, that is, when the Cramér condition is satisfied. It is worth noting that, unlike works [1, 2], we do not assume that the random walk is symmetric, that is, we do not exclude that it can eventually have a drift. Our results are obtained both for a one-dimensional problem formulation (studying the maximum or minimum among the positions of particles on $\mathbb{Z}$) and for a multidimensional formulation (studying the propagation of particle cloud in the space), that is, for the lattice $\mathbb{Z}^{d}$ for $d\in\mathbb{N}$, $d>1$.

We assume that all considered values are defined on the same probabilistic space $\left(\Omega,\mathcal{F},{\mathsf{P}}\right)$, where the space of elementary outcomes $\Omega$ consists of elements $\omega$. The studied model combines the random walk of particles over the lattice $\mathbb{Z}^{d}$ and their possible branching (and death) in the set $\Gamma\subset\mathbb{Z}^{d}$ with a certain structure. Firstly, we describe the process of random walk of a particle over the lattice $\mathbb{Z}^{d}$, $d\in\mathbb{N}$. Random walk is given by the Markov chain $S=\{S(t),t\geqslant 0\}$ with continuous time and infinitesimal matrix $A=(a(x,y))_{x,y\in\mathbb{Z}^{d}}$ (see, e.g., [3, Chapter 8, Section 2]). Here, $a(x,y)$ is the intensity of transfer of the Markov chain $S$ from the state $x$ to the state $y$. Assume that the matrix $A$ is conservative, that is, $\sum_{y\in\mathbb{Z}^{d}}a(x,y)=0$, $a(x,y)\geqslant 0$ for $x\neq y$ and $a(x,x)\in(-\infty,0)$, and the Markov chain $S$ is irreducible, that is, a random walk $S$ with a positive probability can occur from any vertex $\mathbb{Z}^{d}$ to any other. We consider the one-dimensional random walk $S$, that is,

$$a(x,y)=a(x-y,0)=a(0,y-x)\quad\textrm{and}\quad a:=-a(0,0).$$

(1)

Now, we describe the set of sources of branching and death of particles. Let $g_{1}$, $\ldots$, $g_{d}$ be a set of linearly independent (not necessarily orthogonal) vectors in $\mathbb{Z}^{d}$. We call a lattice an (infinite) set

$$\Gamma:=\left\{g\in\mathbb{Z}^{d}:g=\sum_{j=1}^{d}n_{j}g_{j},\ n_{j}\in\mathbb{Z},\ j=1,\ldots,d\right\},$$

(2)

and the set $\{g_{j}\}_{j=1}^{d}$ is a basis of the lattice $\Gamma$. Note that different bases may generate the same lattice. Assume that only in the nodes of the lattice $\Gamma$ the particles can yield descendants and then immediately die. The mechanism of branching is given by the infinitesimal generating function

$$b(s)=\sum_{k=0}^{\infty}b_{k}s^{k},\quad s\in[0,1],$$

$$\textrm{where}\quad b_{k}\geqslant 0\quad\textrm{for}\quad k\neq 1,\quad b:=-b_{1}\in(0,\infty)\quad\textrm{and}\quad\sum_{k=0}^{\infty}b_{k}=0.$$

(3)

This means that the probability $p_{k}(t)$ of a particle to transform to $k$ particles for time $t$ is

$$p_{k}(t)=b_{k}t+o(t),\quad k\neq 1,\quad\textrm{and}\quad p_{1}(t)=1+b_{1}t+o(t),\quad t\to 0+.$$

(4)

At the initial time instance $t=0$, the lattice has a single particle appearing at the point $x{\in}\mathbb{Z}^{d}$. New particles evolutionize according to the same probabilistic laws as the parent particle, independently of each other and of the process prehistory.

Let us summarize this model description. If the starting point $x$ of an initial particle does not belong to a lattice $\Gamma$, then the parent particle performs random walk according to a Markov chain $S$ before falling into the set $\Gamma$, that is, to some point $y\in\Gamma$. At the point $y$ the particle may, with given intensities, either leave it, or continue to perform random walk until the next arrival at $\Gamma$, or leave $k$, $k=0,1,2,\ldots$, descendants in place of itself at the point $y$ and instantaneously die. New particles behave in the same manner, independently of each other and of the parent particle.

We consider the supercritical regime (see, e.g., [4, Chapter 1, Section 6]), that is, the population of particles survives with a positive probability, which, in terms of generating function, is written as the relation

$$\beta:=b\,^{\prime}(1)=\sum_{k=1}^{\infty}kb_{k}\in(0,\infty).$$

(5)

Other regimes are not interesting from the point of view of the formulated problem, because the critical and subcritical BRW degenerate with probability one and in such cases the studied problem about propagation of a population makes no sense. By $N(t)$ we denote the random set of particles in the BRW existing on a lattice at the time instance $t\geqslant 0$, and by $X^{v}(t)=\left(X^{v}_{1}(t),\ldots,X^{v}_{d}(t)\right)$ we denote the position of a particle $v$ from the set $N(t)$ at the time instance $t$. We introduce an event $\mathcal{S}\in\mathcal{F}$ at which the population of particles does not degenerate. Due to condition (5) we have ${\mathsf{P}}\left(\mathcal{S}\right)\in(0,1]$.

We think that the jumps of random walk have light tails. This means that the Cramér condition is fulfilled. More accurately, let the following mathematical expectation hold for all $\theta$ from some neighborhood of the point $0\in\mathbb{R}^{d}$:

$${\mathsf{E}}e^{\langle\theta,Y\rangle}<\infty,$$

(6)

where the vector $Y=\left(Y_{1},\ldots,Y_{d}\right)$ describes a jump of random walk $S$ and $\langle\cdot,\cdot\rangle$ denotes the Euclidean dot product in $\mathbb{R}^{d}$.

We introduce a function

$$m(\theta,\phi):=\frac{\left(\beta+b\right){\mathsf{E}}e^{-\left\langle\theta,S\left(\tau_{\Gamma}^{(0)}\right)-S(0)\right\rangle-\phi\tau_{\Gamma}^{(0)}}}{\phi+a+b-(\phi+a){\mathsf{E}}e^{-\left\langle\theta,S\left(\tau_{\Gamma}^{(1)}\right)\right\rangle-\phi\tau_{\Gamma}^{(1)}}}$$

(7)

for those values $\theta\in\mathbb{R}^{d}$ and $\phi\in\mathbb{R}$ for which it is correctly defined and nonnegative, that is, the mathematical expectations in formula (7) are finite, and

$$(\phi+a){\mathsf{E}}e^{-\left\langle\theta,S\left(\tau_{\Gamma}^{(1)}\right)\right\rangle-\phi\tau_{\Gamma}^{(1)}}<\phi+a+b,\quad\phi>-a,$$

(8)

where $a$ and $b$ are defined, respectively, in formulas (1) and (3). Here, $\tau_{\Gamma}^{(0)}$ is the time of first reaching the set $\Gamma$ by a particle performing random walk $S$ and $\tau_{\Gamma}^{(1)}$ is the time of its first returning to $\Gamma$. In Lemma 2 given below, we establish that the domain of a nonnegative function $m(\cdot,\cdot)$ is different from the empty set.

Let $\mathcal{P}_{t}:=\left\{X^{v}(t)/t:v\in N(t)\right\}\subset\mathbb{R}^{d}$ be a random set of normalized positions of all particles existing in the BRW at the time instance $t$. Put

$$\mathcal{P}:=\left\{z\in\mathbb{R}^{d}:\Theta_{1}(z)\supset\Theta_{2}(z)\right\},$$

(9)

where

$$\Theta_{1}(z):=\left\{\theta\in\mathbb{R}^{d}:m(\theta,-\left\langle z,\theta\right\rangle)\geqslant 1\right\},$$

$$\Theta_{2}(z):=\left\{\theta\in\mathbb{R}^{d}:\left(a-\left\langle z,\theta\right\rangle\right){\mathsf{E}}e^{-\left\langle\theta,S\left(\tau_{\Gamma}^{(1)}\right)\right\rangle+\left\langle\theta,z\right\rangle\tau_{\Gamma}^{(1)}}<a+b-\left\langle z,\theta\right\rangle,\;a-\left\langle z,\theta\right\rangle>0\right\}.$$

It is easy to see that by formula (8) the set $\Theta_{2}(z)$ is a domain of all possible values of the variable $\theta$ at which the nonnegative function $m(\theta,-\left\langle z,\theta\right\rangle)$ is defined for a fixed variable $z$.

The main result of the current work shows that $\mathcal{P}$ is a convex set which is an asymptotic shape of the considered BRW. Recall that, according to paper [5], the asymptotic shape of a BRW is a convex set to which a convex hull of all positions of particles in the BRW at the instance $t$, normalized by the multiplier $t$, tends in the sense of the Hausdorff metric as $t\to\infty$. It is well-known that for the sets $D$, $F\subset\mathbb{R}^{d}$ the Hausdorff distance between them is given by the formula

$$\Delta\left(D,F\right):=\inf\{\varepsilon\geqslant 0:D\subset F_{\varepsilon},F\subset D_{\varepsilon}\},$$

where $D_{\varepsilon}:=\cup_{x\in D}\left\{z\in\mathbb{R}^{d}:||x-z||\leqslant\varepsilon\right\}$ and $||\cdot||$ is the Euclidean norm in $\mathbb{R}^{d}$. It is natural to call the boundary of the set $\mathcal{P}$ in $\mathbb{R}^{d}$ the limit shape of the propagation front of the BRW.

Theorem. Let conditions (1), (5), and (6) be met. Then the set $\mathcal{P}$ is compact and convex, and the relation holds:

$$\Delta\left(\mathcal{P}_{t},\mathcal{P}\right)\to 0\quad\textit{a.s. on event}\quad\mathcal{S}\quad\textit{as}\quad t\to\infty.$$

(10)

This theorem implies an important corollary at lattice dimension of $d=1$. In this case the entire population of particles at time instance $t$ is within the maximum and minimum among the coordinates of all particles existing on the integer-valued axis at the time instance $t$. We write out, for instance, the statement for the maximum defined as $M_{t}:=\max\{X^{v}(t):v\in N(t)\}$, $t\geqslant 0$.

Corollary. If at $d=1$ relations (1), (5), and (6) take place, then

$$\frac{M_{t}}{t}\to r\quad\text{a.s. on event}\quad\mathcal{S}\quad\textrm{as}\quad t\to\infty,$$

where the limit $r:=\sup\left\{z\in\mathbb{R}:\Theta_{1}(z)\supset\Theta_{2}(z)\right\}$ is finite.

Thus, for BRWs with periodic sources of branching, we have established a theorem that is an analog of the main results of papers [6–8] demonstrating an asymptotically linear propagation of the population of particles in a supercritical BRW with a finite number of sources of branching and death of particles in the case when the tails of distribution of the random walk jump are light. One of the methods for studying the BRW with a finite number of catalyzers in papers [6–8] is in analyzing the system of integral equations of renewal equations type, in which the number of equations is equal to the number of sources. For a BRW with periodic catalyzers this approach seems to be futile, because there arises a complex system of infinite number of equations. However, due to the periodic position of branching sources, in the current work we managed to consider the BRWs with periodic catalyzers as a BRW being in some sense homogeneous and apply the well-known results established for the general homogeneous BRW to it. We first prove the results describing the spatial propagation of a population of particles for BRWs with an infinite number of branching sources in the case when these sources are located periodically.

Before we proceed to proving the main result, we introduce several auxiliary designations and establish two lemmas. Recall that due to [3, Chapter 8, Section 2] a random walk $S$ with continuous time satisfying condition (1) is a generalized Poisson process, that is, the formula holds:

$$S(t)=S(0)+\sum\limits_{i=1}^{\Pi(t)}Y^{(i)},\quad t\geqslant 0,$$

(11)

where $\{\Pi(t),t\geqslant 0\}$ is a Poisson process of intensity $a$ and $Y^{(1)},Y^{(2)},\ldots$ are independent equally distributed random vectors having the same distribution as $Y$. By $\tau^{(i)}$ we denote the length of time interval between the $(i-1)$th and $i$th jumps of the Poisson process $\Pi$, where $i\in\mathbb{N}$. It is well-known that $\tau^{(1)},\tau^{(2)},\ldots$ are independent equally distributed random variables and $\tau^{(i)}\sim\textrm{Exp}(a)$, $i\in\mathbb{N}$.

We introduce an array of events

$$B(n,m):=\left\{\sum_{j=n}^{m}Y^{(j)}\in\Gamma,\sum_{j=n}^{l}Y^{(j)}\notin\Gamma,l=1,\ldots,m-1\!\right\},\quad n,m\in\mathbb{N},\quad m\geqslant n,$$

(12)

where the set $\Gamma$ was introduced in (2).

Lemma 1. There exist constant $c>0$ and $q\in(0,1)$ at which the inequality holds:

$${\mathsf{P}}\left(B(1,k)\right)\leqslant cq^{k},\quad k\in\mathbb{N}.$$

(13)

Proof. To prove an auxiliary Markov chain, we recall several necessary denotations from paper [9]. Define on $\mathbb{Z}^{d}$ the following equivalence relation: the points $y,z\in\mathbb{Z}^{d}$ are called equivalent if $y-z\in\Gamma$. The corresponding factorspace is denoted by $\Upsilon:=\mathbb{Z}^{d}/\Gamma$ and is referred to as the fundamental set of vertices. This set can always be put in correspondence to some set $\{z^{(1)},\ldots,z^{(p)}\}$ of pairwise nonequivalent elements from $\mathbb{Z}^{d}$. For a fixed choice $\Upsilon=\{z^{(1)},\ldots,z^{(p)}\}$ any point $y\in\mathbb{Z}^{d}$ can be uniquely represented as

$$y=u_{y}+\gamma_{y},\quad\textrm{where}\quad u_{y}\in\Upsilon\quad\textrm{and}\quad\gamma_{y}\in\Gamma.$$

(14)

The number of vertices $p$ in the fundamental set of vertices $\Upsilon$ can be found as follows. Consider the set $\mathcal{C}:=\left\{y\in\mathbb{R}^{d}:y=\sum_{j=1}^{d}x_{j}g_{j},\ 0\leqslant x_{j}<1,\ j=1,\ldots,d\right\}.$ Then $p$ is the number of points in the intersection $\mathcal{C}\cap\mathbb{Z}^{d}$. In what follows, we use the choice $\Upsilon=\mathcal{C}\cap\mathbb{Z}^{d}=\{z^{(1)},\ldots,z^{(p)}\}$, where, without loss of generality, we put $z^{(1)}=0$.

Now, on the basis of the random walk $S=\left\{S(t),t\geqslant 0\right\}$, we construct an auxiliary Markov chain $V=\{V(t),t\geqslant 0\}$ with values in $\Upsilon$, putting $V(t)=u_{S(t)}$ for each $t\geqslant 0$ (the values $u_{y}$ have been defined in (14) for $y\in\mathbb{Z}^{d}$). By construction of the Markov chain $V$, the time of first reaching the set $\Gamma$ by the random walk $S$ is equal to the time of first reaching the state $z^{(1)}=0$ by the Markov chain $V$. Therefore, by Theorem 13.4.2 from [10, Chapter 13] applied to the auxiliary Markov chain, there exist constants $c>0$ and $q\in(0,1)$ such that the sought relation (13) takes place. Note that in this case a significant role is played by periodicity of the set $\Gamma$ which allows reducing the study of a Markov chain $S$ with a countable space of states to considering a finite Markov chain $V$. Lemma 1 is proved.

Lemma 2. If relations $(1)$, $(5)$, and $(6)$ hold, then for each unit vector ${U\in\mathbb{R}^{d}}$, that is, $||U||=1$, the function $m(\theta U,\phi)$ takes on finite values for all $\theta$ from some neighborhood of the point $0\in\mathbb{R}$ and all sufficiently large $\phi$.

Proof. Due to the Cauchy–Bunyakovsky–Schwartz inequality and relation (13) established in Lemma 1, for $\phi>-a$ we have

$${\mathsf{E}}e^{-\theta\left\langle U,\,S\left(\tau_{\Gamma}^{(1)}\right)\right\rangle-\phi\tau_{\Gamma}^{(1)}}=\sum_{k=1}^{\infty}\left({\mathsf{E}}e^{-\phi\tau^{(1)}}\right)^{k}{\mathsf{E}}\exp\!\left\{\!-\theta\left\langle\!U,\sum_{j=1}^{k}Y^{(j)}\!\right\rangle\!\right\}\mathbb{I}\!\left(B(1,k)\right)$$

$${}\leqslant\sum_{k=1}^{\infty}\left(\frac{a}{\phi+a}\right)^{k}\left({\mathsf{E}}\exp\left\{-2\theta\left\langle U,\sum_{j=1}^{k}Y^{(j)}\right\rangle\right\}\right)^{\frac{1}{2}}\left({\mathsf{P}}\left(B(1,k)\right)\right)^{\frac{1}{2}}$$

$${}\leqslant\sqrt{c}\sum_{k=1}^{\infty}\left(\frac{a}{\phi+a}\right)^{k}\left({\mathsf{E}}e^{-2\theta\left\langle U,Y\right\rangle}\right)^{k/2}q^{k/2}=\frac{a\sqrt{c}\sqrt{q{\mathsf{E}}e^{-2\theta\left\langle U,Y\right\rangle}}}{\phi+a-a\sqrt{q{\mathsf{E}}e^{-2\theta\left\langle U,Y\right\rangle}}}<\infty$$

(15)

under condition $\dfrac{a}{\phi+a}\sqrt{q{\mathsf{E}}e^{-2\theta\langle U,Y\rangle}}<1$. The last inequality holds when we choose $\phi$ sufficiently large and $\theta$ close to zero, because in this case the expression ${\mathsf{E}}e^{-2\theta\langle U,Y\rangle}$ is finite due to (6). Moreover, relation (15) implies that such choice of $\theta$ and, perhaps, a further increase in $\phi$ lead to validity of the inequality $(\phi+a){\mathsf{E}}e^{-\theta\left\langle U,\,S\left(\tau_{\Gamma}^{(1)}\right)\right\rangle-\phi\tau_{\Gamma}^{(1)}}<\phi+a+b$. Similarly to the derivation of relation (15), we establish the finiteness of the mean value ${\mathsf{E}}e^{-\theta\left\langle U,S\left(\tau_{\Gamma}^{(0)}\right)-S(0)\right\rangle-\phi\tau_{\Gamma}^{(0)}}$ for the same values of $\theta$ and $\phi$. The last results in the finiteness of the function $m(\theta U,\phi)$ for all $\theta$ from some neighborhood $0\in\mathbb{R}$ and all sufficiently large $\phi$. Lemma 2 is proved.

Thus, according to (11), if in our BRW a particle falls into the source of branching, then it waits for time $\tau_{w}\sim\textrm{Exp}(a)$ until the possible withdrawal from it. Similarly, due to (4) (see, e.g., [11, Chapter 1, Section 1]) the time of waiting for the possible branching is $\tau_{br}\sim\textrm{Exp}(b)$. If $\tau_{w}<\tau_{br}$, which occurs with a probability $a/(a+b)$, then the particle leaves the branching source without descendants. Otherwise, if $\tau_{w}>\tau_{br}$ (with a probability $b/(a+b)$), then in this branching source the particle produces a random number of descendants and dies immediately.

Recall that $\tau_{\Gamma}^{(0)}$ is the time of first reaching the set $\Gamma$ and $\tau_{\Gamma}^{(1)}$ is the time of first returning to $\Gamma$ of a particle performing random walk $S$. Assume that, if a random walk $S$ starts at a point from the set $\Gamma$, then $\tau_{\Gamma}^{(0)}=0$ and $\tau_{\Gamma}^{(1)}\geqslant\tau^{(1)}$. If it starts at a point not in the set $\Gamma$, then $\tau_{\Gamma}^{(0)}\geqslant\tau^{(1)}$ and counting $\tau_{\Gamma}^{(1)}$ begins at the time instance $\tau_{\Gamma}^{(0)}$. By denoting the time intervals between the next returns to $\Gamma$ by $\tau_{\Gamma}^{(2)},\tau_{\Gamma}^{(3)},\ldots$, we obtain the fact that a random walk $S$ falls into $\Gamma$ at time instances $\tau_{\Gamma}^{(0)}$, $\tau_{\Gamma}^{(0)}+\tau_{\Gamma}^{(1)}$, $\ldots$. Unlike the enumeration $\tau^{(1)},\tau^{(2)},\ldots$ of times of waiting at points of the lattice at which the random walk $S$ performs jumps, we introduce a separate enumeration for times of waiting only at points from the set $\Gamma$ to which the random walk $S$ arrives and denote them by $\tau_{w}^{(1)},\tau_{w}^{(2)},\ldots$. In our BRW these random variables correspond to a sequence (independent of them) of independent equally distributed random variables $\tau_{br}^{(1)},\tau_{br}^{(2)},\ldots$. Obviously, the trajectories of random walk $S$ and the trajectories of BRW, if they started at the same point, coincide while $\tau_{w}^{(k)}<\tau_{br}^{(k)}$ for all $k=1,2,\ldots,\nu-1$. As we have $\tau_{w}^{(\nu)}>\tau_{br}^{(\nu)}$ for some $\nu$, the behavior of the BRW is different from the behavior of random walk. In the BRW the parent particle dies, leaving after it a random number of descendants, and in the random walk the particle continues to move over the lattice $\mathbb{Z}^{d}$. Put $L$ to be equal to the lifetime of a parent particle in BRW. Then it is clear that

$$L=\sum_{k=0}^{\nu-1}\tau_{\Gamma}^{(k)}+\tau_{br}^{(\nu)},$$

(16)

where $\nu:=\min\{k\in\mathbb{N}:\tau_{w}^{(k)}>\tau_{br}^{(k)}\}$. We can easily see that $\nu<\infty$ with probability 1. Indeed, the probability of the opposite event, that is, when $\tau_{w}^{(k)}\leqslant\tau_{br}^{(k)}$ for all $k\in\mathbb{N}$, is equal to zero, because $\tau_{w}^{(k)}$ and $\tau_{br}^{(k)}$ are independent for any $k\in\mathbb{N}$ and have an exponential distribution, respectively, with parameters $a$ and $b$.

Proof of Theorem. The proof is based on applying Theorem A and its corollary from paper [12], as well as their generalizations in [5, Subsection 4.2]. These statements are formulated for the so called general BRW in $\mathbb{R}^{d}$ the description of which we recall.

The general BRW is given by a triple $(Z,M,\chi)$ in which for each particle the process $Z$ is responsible for branching, the process $M$ is responsible for motion in space, and the process $\chi$ is responsible for its value depending on the age of this particle at counting the size of population or subpopulation. For a point process $Z$ in $\mathbb{R}^{d}\times\mathbb{R}_{+}$ each point corresponds to a descendant. The first $d$ coordinates of a point is the deviation of the descendant from the initial position of the parent particle, and the last coordinate is the age of the parent particle at the time instance of birth of this descendant. Motion of the parent particle in the space is described by the random process $M$. A particle born at a point $z$, will be at the point $z+M(t)$ after time $t$. The random characteristic $\chi$ is a nonnegative random process assigning the value to a particle in the population depending on the age. In other words, $\chi(t)$ is the ‘‘weight’’ of a particle having the age $t$. For instance, if we want to know the size of population at the current time instance $t$, then we should use the standard random characteristic $\chi(t)=\mathbb{I}(0\leqslant t<\ell)$, where $\ell$ is the lifetime of the particle and then sum random characteristics at points $t-\sigma_{v}$ over all particles $v$, where $\sigma_{v}$ is the time instance of birth of the particle $v$. Thus, we obtain the number of particles existing at the time instance $t$.

Let us show that our BRW with periodically positioned branching sources can be considered within the described general BRW. For instance, we put $M=S$, that is, motion of a particle occurs following a random walk with an infinitesimal matrix $A$. We prescribe a point process $Z$. Each its point has coordinates $(S(L)-S(0),L)$, where the lifetime $L$ of a parent particle in our BRW satisfies Eq. (16). The number $\xi$ of such points is random and has a probabilistic generating function $b_{0}(s):={\mathsf{E}}s^{\xi}=\sum_{k=0,k\neq 1}^{\infty}\dfrac{b_{k}}{b}s^{k}$, $s\in[0,1]$. It remains to determine the random characteristic $\chi$, which in our case is given in the standard way as $\chi(t)=\mathbb{I}(0\leqslant t<L)$, $t\geqslant 0$.

We apply the arguments from [5, Subsection 4.2] that are generalizations of Theorem A and its corollaries from paper [12] to the thus prescribed general BRW. If we make sure that the equality

$$m(\theta,\phi)={\mathsf{E}}\int e^{-\langle\theta,z\rangle}e^{-\phi t}Z(dz,dt)$$

(17)

holds for all $\theta\in\mathbb{R}^{d}$ and $\phi\in\mathbb{R}$, at which both parts of the equality are defined, then the mentioned arguments imply the conclusion: the asymptotic shape of the thus prescribed general BRW is determined by the formula

$$\left\{z\in\mathbb{R}^{d}:\inf\limits_{\theta}\left\{\ln\,m\!\left(\theta,-\langle z,\theta\rangle\right)\right\}\geqslant 0\right\}.$$

However, it is easy to see that the last set is the set $\mathcal{P}$ introduced above in (9). Thus, we obtain result (10) of our theorem if we check Eq. (17). Note that, due to Lemma 2, for each unit vector $U$ there exist some $\theta>0$ and $\phi\in\mathbb{R}$ such that $m(\theta U,\phi)<\infty$. The rest of the proof of the theorem is devoted to establishing formula (17).

As was shown above, the studied BRW with periodic sources of branching can be considered within the general BRW if we prescribe a triple $(Z,M,\chi)$ properly. Taking into account the explicit formulas for a given point process $Z$, we arrive at the relation

$${\mathsf{E}}\int e^{-\langle\theta,z\rangle}e^{-\phi t}Z(dz,dt)={\mathsf{E}}\xi{\mathsf{E}}e^{-\langle\theta,S(L)-S(0)\rangle}e^{-\phi L},$$

(18)

where ${{\mathsf{E}}\xi=1+\beta/b}$. According to definition of time $L$, we have

$$S(L)=S\left(\sum_{k=0}^{\nu-1}\tau_{\Gamma}^{(k)}+\tau_{br}^{(\nu)}\right)=S\left(\sum_{k=0}^{\nu-1}\tau_{\Gamma}^{(k)}\right).$$

To reduce the notation, we assume that the starting point of BRW $x$ belongs to the set $\Gamma$. Otherwise, in the below formulas there arises an additional indicator

$$\mathbb{I}\left(x+\sum_{j=1}^{k_{0}}Y^{(j)}\in\Gamma,x+\sum_{j=1}^{l_{0}}Y^{(j)}\notin\Gamma,l_{0}=1,\ldots,k_{0}-1\right).$$

With this indicator the calculations are performed in the same manner as with the indicator $\mathbb{I}\left(B(1,k_{1})\right)$, where the set $B(1,k_{1})$ is introduced in (12). In addition, without loss of generality, we assume $x=0$ (recall that $0\in\Gamma$). Considering the indices $k_{1},\ldots,k_{n}\in\mathbb{N}$, for $u=1,\ldots,n$ we put $K(u):=k_{1}+\ldots+k_{u}$. Consequently, due to (11) we have

$${\mathsf{E}}e^{-\langle\theta,S(L)-S(0)\rangle-\phi L}=\sum_{n=0}^{\infty}{\mathsf{E}}\exp\left\{-\left\langle\theta,S\left(\sum\limits_{k=0}^{n}\tau_{\Gamma}^{(k)}\right)\right\rangle-\phi\left(\sum_{k=0}^{n}\tau_{\Gamma}^{(k)}+\tau_{br}^{(n+1)}\right)\right\}\mathbb{I}\left(\nu=n+1\right)$$

$${}=\sum_{n=0}^{\infty}\sum_{k_{1},\ldots,k_{n}=1}^{\infty}\!{\mathsf{E}}\exp\!\left\{-\sum_{u=1}^{n}\left\langle\theta,\!\!\sum_{j=K(u-1)+1}^{K(u)}\!\!Y^{(j)}\right\rangle\!-\phi\left(\sum_{u=1}^{n}\sum_{j=K(u-1)+1}^{K(u)}\tau^{(j)}+\tau_{br}^{(n+1)}\right)\right\}$$

$${}\times\prod_{u=1}^{n}\mathbb{I}\left(B\left(K(u-1)+1,K(u)\right)\right)\prod_{u=1}^{n}\mathbb{I}\left(\tau^{(K(u-1)+1)}<\tau_{br}^{(u)}\right)\mathbb{I}\left(\tau^{(K(n)+1)}>\tau_{br}^{(n+1)}\right)$$

$${}=\sum_{n=0}^{\infty}\sum_{k_{1},\ldots,k_{n}=1}^{\infty}\left({\mathsf{E}}e^{-\phi\tau^{(1)}}\right)^{K(n)-n}{\mathsf{E}}e^{-\phi\tau_{br}^{(n+1)}}\mathbb{I}\left(\tau^{(K(n)+1)}>\tau_{br}^{(n+1)}\right)$$

$${}\times\prod_{u=1}^{n}{\mathsf{E}}\exp\!\left\{-\left\langle\theta,\!\sum_{j=K(u-1)+1}^{K(u)}\!\!Y^{(j)}\!\right\rangle\right\}$$

$${}\times\mathbb{I}\left(B\left(K(u-1)+1,K(u)\right)\right)\prod_{u=1}^{n}{\mathsf{E}}e^{-\phi\tau^{(K(u-1)+1)}}\mathbb{I}\left(\tau^{(K(u-1)+1)}<\tau_{br}^{(u)}\right)$$

$${}=\sum_{n=0}^{\infty}\frac{b}{\phi+a+b}\left(\frac{\phi+a}{\phi+a+b}\right)^{n}\sum_{k_{1},\ldots,k_{n}=1}^{\infty}\left({\mathsf{E}}e^{-\phi\tau^{(1)}}\right)^{K(n)}\prod_{u=1}^{n}{\mathsf{E}}\exp\!\left\{-\left\langle\theta,\!\sum_{j=1}^{k_{u}}Y^{(j)}\!\right\rangle\!\right\}\mathbb{I}\!\left(B\left(1,k_{u}\right)\right)$$

$${}=\frac{b}{\phi+a+b}\sum_{n=0}^{\infty}\!\left(\frac{\phi+a}{\phi+a+b}\!\right)^{n}\prod_{u=1}^{n}\left(\sum_{k_{u}=1}^{\infty}\left({\mathsf{E}}e^{-\phi\tau^{(1)}}\right)^{k_{u}}{\mathsf{E}}\exp\left\{-\left\langle\theta,\sum_{j=1}^{k_{u}}Y^{(j)}\right\rangle\right\}\mathbb{I}\left(B\left(1,k_{u}\right)\right)\right)$$

$${}=\frac{b}{\phi+a+b}\sum_{n=0}^{\infty}\left(\frac{\phi+a}{\phi+a+b}\right)^{n}\left({\mathsf{E}}e^{-\left\langle\theta,\,S\left(\tau_{\Gamma}^{(1)}\right)\right\rangle-\phi\tau_{\Gamma}^{(1)}}\right)^{n}$$

$${}=b\left(\phi+a+b-\left(\phi+a\right){\mathsf{E}}e^{-\left\langle\theta,\,S\left(\tau_{\Gamma}^{(1)}\right)\right\rangle-\phi\tau_{\Gamma}^{(1)}}\right)^{-1},$$

(19)

where the events $B(n,m)$, $n,m\in\mathbb{N}$, $n\leqslant m$, have been introduced in (12) and $\phi>-a$. Here, we have essentially relied upon the fact that the set $\Gamma$ is closed with respect to addition and subtraction of its elements. Indeed, it follows from these properties $\Gamma$ that $\sum_{j=1}^{K(u)}Y^{(j)}\in\Gamma$, $u=1,\ldots,n$, iff $\sum_{j=K(u-1)+1}^{K(u)}Y^{(j)}\in\Gamma$, $u=1,\ldots,n$. Moreover, in establishing formula (19), we have taken into account that the jumps $Y^{(j)}$, $j\in\mathbb{N}$, are independent, and therefore, the nonintersecting sets of such random jumps are also independent (see, e.g., the grouping lemma in [13, Chapter 3, Corollary 3.7]). Finally, we use the independence of random vectors $Y^{(j)}$, $j\in\mathbb{N}$, and times of waiting $\tau^{(i)}$, $i\in\mathbb{N}$. The last equality in (19) holds under the condition ${\dfrac{\phi+a}{\phi+a+b}{\mathsf{E}}e^{-\left\langle\theta,\,S\left(\tau_{\Gamma}^{(1)}\right)\right\rangle-\phi\tau_{\Gamma}^{(1)}}<1}$. However, this inequality has been already established in Lemma 2. Now, the validity of relation (17) follows from formulas (18) and (19). This completes the proof of the theorem.

Example 1. Suppose that $d=1$, $g_{1}=2$, that is, $\Gamma=\{2k:k\in\mathbb{Z}\}$. Assume that $A$ is a discrete Laplace operator, that is, $a(x,y)/a=1/2$ if $|x-y|=1$ and $a(x,y)=0$ in all other cases. Let, for instance, a starting point $x$ be the origin of coordinates $0$. Then $\tau_{\Gamma}^{(1)}=\tau^{(1)}+\tau^{(2)}$, and the value $S\left(\tau_{\Gamma}^{(1)}\right)$ can be equal to $-2$, $0$, or $2$, respectively, with probabilities $1/4$, $1/2$, or $1/4$. Therefore,

$$m(\theta,\phi)=\frac{\beta+b}{\phi+a+b-(\phi+a)^{-1}a^{2}\cosh^{2}\theta},$$

$$\Theta_{2}(z)=\left\{\theta\in\mathbb{R}:a^{2}\cosh^{2}\theta<\left(a-z\theta\right)^{2}+b\left(a-z\theta\right),\;a-z\theta>0\right\},$$

where, as usual, $\cosh$ denotes the hyperbolic cosine. Put

$$\Theta^{\ast}_{1}(z):=\left\{\theta\in\mathbb{R}:a^{2}\cosh^{2}\theta\geqslant(a-z\theta)^{2}-\beta(a-z\theta)\right\}.$$

It is easy to see that the point $z\in\mathbb{R}$ is contained in the set $\mathcal{P}$ being the asymptotic shape of the BRW due to the established theorem iff the inclusion $\Theta^{\ast}_{1}(z)\supset\Theta_{2}(z)$ holds. Note that $0\in\mathcal{P}$, because $\Theta^{\ast}_{1}(0)=\mathbb{R}$. Suppose, for instance, that $a=b=\beta=1$. There exist several ways to find the left and right ends of the segment $\mathcal{P}$ using Wolfram Mathematica. One of the ways is in considering the discrete values of $z$ with a step, for instance, of 0.001 and testing the inclusion $\Theta^{\ast}_{1}(z)\supset\Theta_{2}(z)$ for each of them. Then we determine the left and right boundaries of the set $\mathcal{P}$ with an accuracy up to $0.001$. Another way is in graphic representation on the plane of the set $\left\{(z,\theta):z\in\mathbb{R},\theta\in\Theta^{\ast}_{1}(z)\right\}$, $\left\{(z,\theta):z\in\mathbb{R},\theta\in\Theta_{2}(z)\right\}$ and revealing how their cross sections correspond at various fixed $z$. As a result, we obtain $\mathcal{P}\approx[-1.122,1.122]$. In terms of corollary of the theorem we proved, we have $r\approx 1.122$ (with an accuracy up to $0.001$).

Example 2. Consider an analog of Example 1 in the two-dimensional space ($d=2$). Let $g_{1}=(2,0)$ and $g_{2}=(0,2)$, that is, $\Gamma=\left\{(2k,2m):k,m\in\mathbb{Z}\right\}$. Assume that the coordinates of the vector of jump $Y=(Y_{1},Y_{2})$ are independent and ${\mathsf{P}}(Y_{i}=1)={\mathsf{P}}(Y_{i}=-1)=1/2$, $i=1,2$. The starting point of the BRW is located at the origin of coordinates. Then, again, $\tau_{\Gamma}^{(1)}=\tau^{(1)}+\tau^{(2)}$. The random variable $S\left(\tau^{(1)}_{\Gamma}\right)$ takes on the values $(2,2)$, $(-2,2)$, $(2,-2)$, and $(-2,-2)$ with probability $1/16$, the values $(0,2)$, $(2,0)$, $(-2,0)$, and $(0,-2)$ with probability $1/8$, and the value $(0,0)$ with probability $1/4$. Hence,

$${\mathsf{E}}e^{-\left\langle\theta,S\left(\tau_{\Gamma}^{(1)}\right)\right\rangle-\phi\tau_{\Gamma}^{(1)}}=\frac{a^{2}\cosh^{2}\theta_{1}\cosh^{2}\theta_{2}}{(\phi+a)^{2}}\quad\textrm{and}$$

$$\Theta_{2}(z)=\left\{\theta=(\theta_{1},\theta_{2})\in\mathbb{R}^{2}:a^{2}\cosh^{2}\theta_{1}\cosh^{2}\theta_{2}<\left(a-\langle z,\theta\rangle\right)^{2}+b\left(a-\langle z,\theta\rangle\right),\;a-\langle z,\theta\rangle>0\right\}.$$

Suppose that

$$\Theta^{\ast}_{1}(z):=\left\{\theta=(\theta_{1},\theta_{2})\in\mathbb{R}^{2}:a^{2}\cosh^{2}\theta_{1}\cosh^{2}\theta_{2}\geqslant(a-\langle z,\theta\rangle)^{2}-\beta(a-\langle z,\theta\rangle)\right\}.$$

Then, it follows from the proved theorem that for our BRW the asymptotic shape $\mathcal{P}$ includes a point ${z=(z_{1},z_{2})\in\mathbb{R}^{2}}$ iff for it the inclusion $\Theta^{\ast}_{1}(z)\supset\Theta_{2}(z)$ holds. It is clear that $0\in\mathcal{P}$, because $\Theta^{\ast}_{1}(0)=\mathbb{R}^{2}$. Put $a=b=\beta=1$. Further, using Wolfram Mathematica, we can discretize the values $z_{1}$ and $z_{2}$ with a step, for instance, of $0.1$ and test the inclusion $\Theta^{\ast}_{1}(z)\supset\Theta_{2}(z)$ for each pair of values $z_{1}$ and $z_{2}$. If it is true, then we put the point $z$ with such coordinates to the set $\mathcal{P}$. Otherwise, we should proceed to the next pair of values $z_{1}$ and $z_{2}$. Using such algorithm, we succeeded to plot the set $\mathcal{P}$ graphically on the plane (see Fig. 1).

Fig. 1

Limit shape of front $\partial\mathcal{P}$ in Example 2.

Fig. 2

Set $\mathcal{P}\subset\mathbb{R}^{3}$ in Example 3.

Example 3. Consider the case when at each point of the lattice the particles may leave descendants, that is, $\Gamma=\mathbb{Z}^{d}$, $d\in\mathbb{N}$. Then $\tau_{\Gamma}^{(1)}=\tau^{(1)}$ and $S\left(\tau_{\Gamma}^{(1)}\right)=Y^{(1)}$. Therefore, the representation holds:

$$m(\theta,\phi)=\frac{\beta+b}{\phi+b-H(-\theta)},\quad\textrm{where}\quad H(\theta):=\sum_{y\in\mathbb{Z}^{d}}e^{\left\langle\theta,y\right\rangle}a(0,y)=a\left({\mathsf{E}}e^{\left\langle\theta,Y^{(1)}\right\rangle}-1\right),\;\theta\in\mathbb{R}^{d}.$$

Note that, using Eq. (11), we can easily prove that ${\mathsf{E}}e^{\left\langle\theta,S(t)\right\rangle}=e^{tH(\theta)}$, $\theta\in\mathbb{R}^{d}$, $t\geqslant 0$. That is why the function $H$ is called the logarithmic generating function of moments for the random variable $S(1)$. The function $\Lambda(z):=\sup_{\theta\in\mathbb{R}^{d}}\left(\left\langle z,\theta\right\rangle-H(\theta)\right)$, $z\in\mathbb{R}^{d}$, is always nonnegative and is called the function of deviations of the random variable $S(1)$ (see [10, Chapter 9, Subsection 1.2]). Thus, it follows from our theorem that within the considered example the asymptotic shape of the population has the form $\mathcal{P}=\left\{z\in\mathbb{R}^{d}:\Lambda(z)\leqslant\beta\right\}$. Consider a special case when $d=3$ and the coordinates of the vector $Y=(Y_{1},Y_{2},Y_{3})$ are independent and have the following distributions:

$${\mathsf{P}}\left(Y_{1}=n\right)={\mathsf{P}}\left(Y_{1}=-n\right)=\frac{\rho_{1}^{n-1}e^{-\rho_{1}}}{2(n-1)!},$$

$${\mathsf{P}}\left(Y_{2}=n\right)={\mathsf{P}}\left(Y_{2}=-n\right)=\frac{\rho_{2}^{n-1}e^{-\rho_{2}}}{2(n-1)!},\quad n\in\mathbb{N},$$

$${\mathsf{P}}\left(Y_{3}=1\right)={\mathsf{P}}\left(Y_{3}=-1\right)=1/2.$$

Then, the formula holds:

$$H(\theta)=a\frac{R(\rho_{1},\theta_{1})}{2}\cdot\frac{R(\rho_{2},\theta_{2})}{2}\cdot\cosh\theta_{3}-a,\quad\theta\in\mathbb{R}^{3},$$

where $R(\rho,s):=e^{\rho\left(e^{s}-1\right)+s}+e^{\rho\left(e^{-s}-1\right)-s}$, $\rho>0$, and $s\in\mathbb{R}$. According to [10, Chapter 9, Subsection 1.2], the upper bound in the definition of the function $\Lambda$ is reached at the point $\theta^{\ast}\in\mathbb{R}^{d}$ (that is, the relation $\Lambda(z)=\langle z,\theta^{\ast}\rangle-H\left(\theta^{\ast}\right)$ holds) iff $z=\nabla H(\theta^{\ast})$. Put $a=\beta=1$, $\rho_{1}=0.7$, and $\rho_{2}=0.8$. We construct the graph of the set $\mathcal{P}\subset\mathbb{R}^{3}$ (see Fig. 2) on the basis of Wolfram Mathematica and using the description of the set $\mathcal{P}$ by the parameters $\theta^{\ast}=\left(\theta^{\ast}_{1},\theta^{\ast}_{2},\theta^{\ast}_{3}\right)\in\mathbb{R}^{3}$.

Thus, in this work we for the first time studied the propagation of a population of particles in the supercritical BRW with periodic branching sources in the case when the tails of the walk jump satisfy the Cramér condition. Due to introduction of an auxiliary process, it became possible to consider the mentioned BRW as in some sense space-homogeneous process. This allowed reducing the formulated problem to the problem already solved for the general BRW. As a result, we proved the theorem establishing the asymptotically linear (in time) propagation of a population of particles in the space and found the limit form of the cloud of particles with linearly normalized coordinates. This limit set $\mathcal{P}$ is called the asymptotic shape of the BRW and is expressed, in particular, through the combined Laplace transform of time and place of the first returning of a particle to the periodic set $\Gamma$. In Examples 1–3 we demonstrated how we can find $\mathcal{P}$ in practice for $d=1$, $d=2$, and $d=3$. The corresponding results are given in Figs. 1 and 2.