Supercritical Branching Random Walks on Nested Lattices

Introduction

Traditionally, models of branching random walks (BRW) on the lattice $\mathbb{Z}^d$ are considered for a fixed lattice dimension $d$. At the same time, “dimension-independent” random walk models defined on lattices of arbitrary dimension are well known—for instance, the random walk model defined by the Laplace operator. For this model, the dependence of the random walk operators’ eigenvalues on the lattice dimension was investigated in [6]. The research method proposed in [6] relied heavily on the specific structure of the Laplace operator.

This raises the question: is it possible to describe—in “reasonable terms” and in a “sufficiently constructive and simple” manner—a family of random walk models on the lattices $\mathbb{Z}^d$ ($d \in\mathbb{N}$) such that the model on $\mathbb{Z}^{d+1}$ inherits, in some natural sense, the properties of the model on the lower-dimensional lattice $\mathbb{Z}^d$, and such that all these models can be viewed as realizations of a single “dimension-independent” model?

The article is structured as follows. Section 2 introduces key definitions regarding families of random walks consistent across nested lattices, enabling the study of the “inheritance” of process properties when transitioning to a lattice of higher dimensionality. Section 3 reviews the properties of the Green’s function, which represents the expected time a random walk spends at a point in $\mathbb{Z}^d$. Section 4 outlines the description of a supercritical branching random walk (BRW) with a single particle generation source located at a point in $\mathbb{Z}^d$ (an inhomogeneous branching medium). Section 5 presents proofs of the main results: Theorem 4, concerning the monotonic decrease of the supercritical BRW evolution operator’s eigenvalues with respect to the dimension $d$, and Theorem 5, regarding the asymptotic expansion of the principal eigenvalue for a supercritical BRW with a high-intensity particle

2020 Mathematics Subject Classification. Primary 60J27; Secondary 60J80, 05C81, 60J85. Key words and phrases. Branching random walks, Nested lattices, Evolution operator, Exponential growth of particle numbers. reproduction source. The concluding section, 6, summarizes the main results and discusses the impact of the spatial dimension $d$ on particle population growth.

On the Hereditary Properties of Random Walks

Let us recall the necessary definitions and properties of a continuous-time symmetric random walk on $\mathbb{Z}^{d}$ ($d \in\mathbb{N}$) without particle generation at lattice sites (i.e., without branching); see, for example, [4]. We are interested in random walks on lattices $\mathbb{Z}^{d}$ of various dimensions $d$. Therefore, we reformulate the axiomatic framework for random walks from [4] to account for this factor: a random walk on the lattice $\mathbb{Z}^{d}$ is defined by a transition rate matrix $A_d = (a_d(x,y))_{x,y\in\mathbb{Z}^{d}}$, the elements of which satisfy the following conditions:

$1^{(d)}$. $a_d(x,y) \ge0$ for $x \ne y$ and $a_d(x,x) < 0$,

$2^{(d)}$. $\sum_{y} a_d(x,y) = 0$,

$3^{(d)}$. $a_d(x,y) = a_d(y,x)$, i.e., the random walk is symmetric,

$4^{(d)}$. the random walk is irreducible, i.e., all points $y \in\mathbb{Z}^{d}$ are reachable,

$5^{(d)}$. $\sigma^2 = -\frac{1}{a_d(0,0)}\sum_{x\in\mathbb{Z}^{d}} |x|^2 a_d(0,x) < \infty$, i.e., the jumps have finite variance,

$6^{(d)}$. $a_d(x,y) = a_d(0,y-x)$, i.e., the random walk is spatially homogeneous.

Remark 1. It will be convenient to denote the function $a_d(0,x)$ by omitting the first zero argument:

and in certain cases, it will be convenient to use coordinate notation for this function:

Definition 1. We say that random walks with intensity matrices

satisfying axioms $1^{(d)}$–$6^{(d)}$, form a nested lattice-consistent set of random walks (or, more briefly, random walks on nested lattices) if, for each $d=1,2,\ldots$, the relation

holds.

In the case where, for each $d=1,2,\ldots$, the more general relation

holds, the consistent set of random walks is called increasing.

Clearly, the concept of a consistent set of random walks is a special case of the concept of an increasing consistent set of random walks. We have chosen to retain the concept of a consistent set of random walks as a separate category, since random walks defined by the Laplace operator possess precisely this property.

Let $\phi_d(\theta)$ denote the Fourier transform of a random walk on $\mathbb{Z}^{d}$ satisfying conditions $1^{(d)}$–$6^{(d)}$.

As is easily seen, $\phi_d(\theta)\le0$ for all $\theta\in[-\pi,\pi]^{d}$. However, as shown in [4 Lemma 2.1.2], for a transient random walk (i.e., when axiom $4^{(d)}$ holds), a stronger property applies:

Proof. By virtue of axioms $1^{(d+1)}$ and $2^{(d+1)}$,

whence, using the coordinate representation of the vector $x=(x_1,\ldots,x_d,x_{d+1})$ on the right-hand side of the resulting equality, we obtain

Let us discard the terms in the last sum for which $x_{d+1}\ne0$. Then the corresponding sum does not increase, and for the remaining terms in it (in which $x_{d+1}=0$), the condition $x=(x_1,\ldots,x_d,0)\ne0$ implies the condition $x=(x_1,\ldots,x_d)\ne0$, and therefore

whence, by virtue of condition (2) and axioms $1^{(d)}$ and $2^{(d)}$, we obtain:

Lemma 1 is proved. □

Proof. By virtue of axiom $2^{(d+1)}$,

and, therefore,

whence, using the coordinate representations of the vectors

on the right-hand side of the resulting equality, we obtain

@page 3/12 starts_mid=0 ends_mid=1

Proof. By virtue of axioms $1^{(d+1)}$ and $2^{(d+1)}$,

whence, using the coordinate representation of the vector $x=(x_1,\ldots,x_d,x_{d+1})$ on the right-hand side of the resulting equality, we obtain

Let us discard the terms in the last sum for which $x_{d+1}\ne0$. Then the corresponding sum does not increase, and for the remaining terms in it (in which $x_{d+1}=0$), the condition $x=(x_1,\ldots,x_d,0)\ne0$ implies the condition $x=(x_1,\ldots,x_d)\ne0$, and therefore

whence, by virtue of condition (2) and axioms $1^{(d)}$ and $2^{(d)}$, we obtain:

Lemma 1 is proved. □

Proof. By virtue of axiom $2^{(d+1)}$,

and, therefore,

whence, using the coordinate representations of the vectors

on the right-hand side of the resulting equality, we obtain

Let us now discard the terms in the last sum for which $x_{d+1} \ne0$. Then the corresponding sum will not decrease, and for the remaining terms in it (in which $x_{d+1}=0$), the condition $x=(x_1,\ldots,x_d,0)\ne0$ will imply the condition $x=(x_1,\ldots,x_d)\ne0$, and therefore

Hence, by virtue of (2) we obtain

Lemma 2 is proven. $\square$

As shown, for example, in [4], for each $d=1,2,\ldots$ and the model described above, the probabilities of a particle transitioning over time $t$ from point $x$ to point $y$ on $\mathbb{Z}^d$ can be represented as $p_d(t,x,y)=p_d(t,0,y-x)=p_d(t,y-x)$, and as $h\downarrow0$, they satisfy the relations

from which it follows that the probabilities $p_d(t,x)$ satisfy the Kolmogorov backward equations

where $\delta(\cdot)$ is the Kronecker delta function on $\mathbb{Z}^d$.

In the sequel, we consider a random walk either with finite jump variance (axiom $5^d$)

or with heavy tails, where the function $a_d(x)$ depends on $\alpha\in(0,2)$—i.e., $a_d(x):=a_{d,\alpha}(x)$—and exhibits regular behavior at infinity:

where $|\cdot|$ is the Euclidean norm on $\mathbb{R}^d$ and $H(\cdot)$ is a positive, symmetric, continuous function on $S^{d-1}=\{x\in\mathbb{R}^d:|x|=1\}$. Under this assumption, the jump variance becomes infinite [3]

When condition (6) holds for $a_{d,\alpha}(x)$, we replace axiom $5^d$ with axiom $5^{d,\alpha}$:

i.e., the jumps have infinite variance, and will assume that all other axioms hold for each fixed $\alpha$.

When using axiom (7), we emphasize the dependence on $\alpha$ in the notation for the transition intensities $a_{d,\alpha}(0,x)$ and write them—as in Remark 1—by omitting the zero argument:

Remark 2. It is important to note that the proof of Lemmas 1 and 2 did not actually rely on axiom $5^{d}$, and these lemmas remain valid under axiom $5^{d,\alpha}$ for any fixed $\alpha\in(0,2)$.

Green's function of a random walk

Applying the Fourier transform to equation (4) (see, e.g., [4, 5]), we obtain a representation for the Green’s function of the random walk:

where $\langle\cdot,\cdot\rangle$ denotes the scalar product in $\mathbb{R}^{d}$.

An alternative representation for the Green’s function also holds (see, e.g., [4]); it expresses the function via the Laplace transform of the transition probabilities $p_{d}(t,x)$ and takes the form:

This representation clarifies the probabilistic meaning of the function $G_{0}(x)$: it represents the mean residence time of the random walk at point $x\in\mathbb{Z}^{d}$ as $t\to\infty$.

The asymptotic properties of the Green’s function $G_{\lambda}(x)$ for an arbitrary $\lambda$ were investigated in [2].

As is evident from (8) and (9), the Green’s function $G_{\lambda}(x)$ actually depends on the dimension $d$ as well; therefore, to emphasize this dependence, we will henceforth use the notation $G_{\lambda}(x):=G_{\lambda}(d,x)$:

According to (3), $\phi_{d}(\theta)\leq0$, and consequently, at $x=0$, the Green’s function

decreases monotonically with respect to $\lambda$.

Recall the main properties of the Green function that play a key role in analyzing the asymptotic properties of a random walk; for details, see [5].

Suppose condition (5) holds. As $\lambda\to0$, the function $G_{\lambda}(d,0)$ tends to infinity in dimensions $d=1$ and $d=2$, whereas for $d\geq3$ it tends to a finite value $G_{0}(d,0)<\infty$.

Suppose condition (6) holds. As $\lambda\to0$, the function $G_{\lambda}(d,0)$ tends to infinity in dimension $d=1$ for any fixed $\alpha\in[1,2)$, whereas it tends to a finite value $G_{0}(d,0)<\infty$ in dimension $d=1$ for any fixed $\alpha\in(0,1)$ and in dimensions $d\geq2$ for any fixed $\alpha\in(0,2)$.

Since $\lvert G_{\lambda}(0)\rvert< \frac{1}{\lambda}$, the function $G_{\lambda}(0)$ tends to zero as $\lambda\to\infty$ for any lattice dimension $d\in\mathbb{N}$, under both condition (5) and condition (6).

Following, for instance, [3], we call a random walk recurrent if $G_0 := G_0(0,0) = \infty$, and transient if $G_0 < \infty$. By virtue of (8), establishing the recurrence of a random walk requires upper and lower bounds on the Fourier transform $\phi(\theta)$.

Supercritical BRW with a single source

The question of the growth rate of particle numbers, $\lambda_0(d) > 0$, arises not only in the analysis of simple random walks; the properties of branching random walks also depend heavily on it. Therefore, in this section, we recall the fundamental facts regarding the theory of branching random walks that are necessary for our purposes.

We assume the presence of a single particle at the initial time $t = 0$, which begins a random walk on $\mathbb{Z}^d$ according to the rules described in § 2. At a specific point in $\mathbb{Z}^d$ (the branching source), the initial particle may produce an arbitrary number of offspring—including itself—or die. Without loss of generality, we assume the branching source is located at the origin, $x = 0$. We now turn to the description of the branching mechanism. Branching at the source $x = 0$ is described by a Markov branching process defined by the infinitesimal generating function $f(u) = \sum_{n=0}^{\infty} b_n u^n$ ($0 \le u \le1$), where $b_i$ represents the intensity of a particle dividing into $i$ offspring (see, e.g., [4] for details). We assume that $b_n \ge0$ for $n \ne1$, $-\infty< b_1 < 0$, and $\sum_{n=0}^{\infty} b_n = 0$. Let us denote by

the source intensity, where the last sum characterizes the average number of particles produced. Let $\beta^{(k)} = f^{(k)}(1) < \infty$ for $k = 1$ or $k = 2$.

To define the BRW process, we combine a random walk and a branching mechanism in the manner described, for example, in [4]. Then (see, e.g., [4]), the random time until branching at the origin follows an exponential distribution with parameter $-a_d(0) - b_1$. Subsequently, the particle transforms into a random number of particles or moves to another lattice site. The new particles evolve according to the same rules, independently of one another and of the entire history of the process.

Of primary interest in the study of the described model is the behavior of the local particle numbers $\mu_t(y)$ at an arbitrary point $y \in\mathbb{Z}^d$, the total population size $\mu_t = \sum_{y \in\mathbb{Z}^d} \mu_t(y)$, and their moments $m_n(t,x,y) = \mathbb{E}_x\mu_t^n(y)$ and $m_n(t,x) = \mathbb{E}_x\mu_t^n$, where $\mathbb{E}_x$ denotes the expectation given that, at the initial time, the system contained a single particle located at point $x$. Using the derivation scheme from [4], we can proceed to linear differential equations for $m_1(t,x)$ and $m_1(t,x,y)$ in the space $\ell^p(\mathbb{Z}^d)$, $1 \le p \le\infty$:

where the operator $\mathcal{H}_{\beta}$ is defined by the equality

in which $(\mathcal{V}_0u)(x) = u(0)\delta_0(x)$, and the operator $\mathcal{A}_d : \ell^p(\mathbb{Z}^d) \to\ell^p(\mathbb{Z}^d)$, $1 \le p \le\infty$, is defined by the equality

In the space $\ell^2(\mathbb{Z}^d)$, the operator $\mathcal{A}_d$ is self-adjoint.

The behavior of the branching random walk depends significantly on the recurrence property. Let $\beta_c$ denote the minimum source intensity such that, for $\beta> \beta_c$, the spectrum of the operator $\mathcal{H}_\beta$ contains an eigenvalue $\lambda_0(d) > 0$ (see, e.g., [4]); this eigenvalue is a solution to the equation

where $\beta> 0$. We call a BRW subcritical if $\beta< \beta_c$, critical if $\beta= \beta_c$, and supercritical if $\beta> \beta_c$. For a BRW with a single branching source and $\beta> \beta_c$, limit theorems have been proven (see, e.g., [4, 5]) establishing the exponential growth of particle numbers with parameter $\lambda_0(d) > 0$, both at individual lattice sites and across the entire lattice. It is shown in [5] that for the single-source case, the following relations hold: if $G_0 = \infty$, then $\beta_c = 0$, and if $G_0 < \infty$, then $\beta_c = G_0^{-1} = 1/G_0$.

For the BRW models under consideration, the transition from finite to infinite jump variance gives rise to new effects. Thus, although the fundamental differential and integral equations for the generating functions and the equations for the particle count moments remain unchanged when condition (6) is satisfied, the properties and asymptotic behavior of the functions $\phi(\theta)$ and $p(t,x,y)$—and, consequently, of the functions $G_\lambda(x,y)$ and $m_1(t,x,y)$—undergo qualitative changes. For instance, in the case of infinite jump variance, the condition $G_0 < \infty$ can hold in dimension $d = 1$ for $\alpha\in(0,1)$ and in dimensions $d \ge2$ for $\alpha\in(0,2)$, whereas in the case of finite jump variance, $G_0 < \infty$ holds only in dimensions $d \ge3$. This implies that for the lattices $\mathbb{Z}^1$ and $\mathbb{Z}^2$, the value of $\beta_c$ can be strictly positive; that is, for $d = 1$ or $d = 2$, a heavy-tailed branching random walk (unlike the finite-variance case) can be subcritical even when the branching process at the source is supercritical.

Main Results

In this section, we prove the main assertions of the paper: Theorem 4, concerning the monotonic decrease with respect to the dimension $d$ of the eigenvalues of the evolution operator for a supercritical branching random walk, and Theorem 5, regarding the asymptotic expansion of the principal eigenvalue for a supercritical branching random walk in the regime of high particle-multiplication source intensity.

We begin by analyzing the growth of the particle numbers $\lambda_0(d) > 0$ as a function of $d$, which is equivalent to analyzing the growth of the solution to the equation $G_\lambda(d,0) = \beta^{-1}$.

Proof. For an arbitrary $d$, we estimate the denominator of the integrand in (10) using Lemma 2 as follows:

From this, we obtain

Let us express the right-hand side of the resulting inequality as an iterated integral:

Here, the expression in parentheses is simply $(2\pi)^dG_{\lambda}(d,0)$; therefore,

Since the integrand does not depend on $\theta_{d+1}$, we obtain

Lemma 3 is proved. $\square$

We are now ready to prove the first of the main assertions of this section.

Proof. By virtue of (10), for each $d=1,2,\ldots$, the function $G_{\lambda}(d,0)$ is monotonically decreasing and tends to zero as $\lambda\to\infty$. Therefore, for a given $\beta>0$, there exists an integer $d(\beta)$ such that for $d>d(\beta)$, equation (12) no longer has a positive solution $\lambda_0(d)$, whereas it does have a solution at $d=d(\beta)$. Then, by Lemma 3, equation (12) has a solution $\lambda_0(d)>0$ for each $d=1,2,\ldots,d(\beta)$; that is, for these values of $d$, the inequality $\beta>G_0^{-1}(0,d)$ holds [4]. Moreover, due to the monotonicity of $G_{\lambda}(d,0)$ with respect to $\lambda$, the solution $\lambda_0(d)$ is unique.

By virtue of (3), the function $\phi_d(\theta)$ is strictly negative for almost all $\theta\in[-\pi,\pi]^d$. Then it follows from (10) that $G_{\lambda}(d,0)<\lambda^{-1}$, and therefore,

To prove the remaining inequalities, it suffices to show that the inequality

holds for each $d=1,2,\ldots,n-1$.

Finally, we note that, by Lemma 3, for any value $\lambda>0$, the function $G_\lambda(d,0)$ is monotonically decreasing in $d$. This, combined with the obvious fact that $G_\lambda(d,0)$ monotonically decreases with respect to $\lambda$, implies relation (13), thereby completing the proof of Theorem 4. □

Proof. The constructions presented below hold true whether condition (5) or condition (6) is satisfied. Accordingly, when condition (6) holds, we will omit the index $\alpha$. In both cases, by virtue of (10), equation (12) takes the form

Our goal is to find a solution $\lambda$ to this equation as a function of the variable $\beta$; that is, to express $\lambda$ in the form $\lambda=\lambda(d,\beta)$. Unfortunately, determining the asymptotic behavior of $\lambda(d,\beta)$ as $\beta\to\infty$ is difficult, because in this case, the functions involved cannot be expanded into Taylor series. Therefore, we perform a change of variables in equation (14) by setting

In terms of the new variables $\tau$ and $\gamma$, equation (14) takes the form

Let $\gamma=\gamma(d,\tau)$ be the solution to this equation, expressed as a function of $\tau$. Then, by virtue of (15), the functions $\lambda(d,\beta)$ and $\gamma(d,\tau)$ are related by the following simple expressions:

Consequently, the problem of finding the asymptotic behavior of $\lambda(d,\beta)$ as $\beta\to\infty$ reduces to the analogous problem of finding the asymptotic behavior of $\gamma(d,\tau)$ as $\tau\to0$. And the final problem, as shown below, reduces to the standard problem of finding an implicit function.

Indeed, since

equation (16) obtained above has a unique solution $\gamma=1$ at $\tau=0$. Moreover, the function $\widetilde{F}(\gamma,\tau)$ satisfies the conditions of the classical implicit function theorem [1]: $\widetilde{F}(\gamma,\tau)$ is infinitely differentiable with respect to both variables, and

which is crucial for applying the implicit function theorem. In this case, by the implicit function theorem, equation (16) defines $\gamma$ as an infinitely continuously differentiable function of $\tau$ in some neighborhood of the point $\tau=0$:

To find $\gamma'(0)$ we differentiate identity (17):

whence for $\tau=0$ we get

Therefore,

which gives us the following representation of the function $\gamma(\tau)$:

To find $\gamma'(0)$, it remains for us to calculate two quantities: $\widetilde{F}'_{\gamma}(1,0)$ and $\widetilde{F}'_{\tau}(1,0)$. Based on definition (16) of the function $\widetilde{F}(\gamma,\tau)$, we have

and, consequently, $\widetilde{F}'_{\gamma}(1,0)=-1$. Again, based on definition (16), we have

where

Here, each integral under the summation sign vanishes, and the last sum equals $-a_d(0)$ by virtue of axiom $2^{(d)}$; consequently, (19) takes the form:

Thus, from (18), we obtain

where, we recall, $a_d(0)<0$. From this, reverting from the variables $\gamma$ and $\tau$ to the variables $\lambda$ and $\beta$, we obtain

Theorem 5 is proved. □

Corollary 3. Suppose condition (5) holds; then, as $\beta\to\infty$, for each $d\in\mathbb{N}$, the following relation holds:

If, however, condition (6) holds, then as $\beta\to\infty$, for each $d\in\mathbb{N}$ and each $\alpha\in(0,2)$, the following relation holds:

Proof. The assertion of Corollary 3 follows directly from Theorem 5. □

Conclusion

This paper proposes a new approach to describing families of random walks on lattices of varying dimensions, based on the concept of a set of random walks that is consistent with respect to nested lattices. Within this framework, it is proven that when transitioning from $\mathbb{Z}^d$ to $\mathbb{Z}^{d+1}$, the intensity $\lvert a_d(0)\rvert$ does not decrease, whereas the Fourier transform $\phi_d(\theta)$ decreases monotonically with $d$. These properties hold true regardless of whether the jump variance is finite or infinite under condition (6). The main result of the paper is an asymptotic formula for the positive eigenvalue of the mean-population operator in a supercritical branching random walk with a single branching source:

which holds for a wide class of transition intensities, including models where the random walk jump variance may be infinite.

References

Department of Probability Theory, Lomonosov Moscow State University, Moscow, 119234, Russia

Email address: [email protected]