Asymptotic behavior of survival probability for a branching random walk with a barrier

1.   Introduction and results

We consider the branching random walk (BRW) on ${{\mathbb{R}}}.$ At time $0,$ an initial ancestor (denoted by $\phi$) is located at the origin. At time $1,$ the ancestor dies and reproduces (including the number and displacement of its children) according to the distribution of a point process $L,$ i.e., $\phi$ gives birth to $N(\phi)$ children who are located at $\zeta_i(\phi), 1\leq i\leq N(\phi)$ ($N(\phi)$ can be $0$) and the law of the random vector $(N(\phi), \zeta_i(\phi), 1\leq i\leq N(\phi))$ is $L.$ These children (also called particles) consist the first generation. Each of the particles in the first generation reproduces its own children who are thus in the second generation and are positioned (with respect to their parent) according to the same distribution of $L.$ All particles reproduce independently according to the same law $L$ as time goes on. The particle system formed in this way is called a (time-homogeneous) branching random walk. Hence BRW can be viewed as that we attach a displacement information to each particle in a Galton-Watson tree $\mathbf{T}.$ For a given particle $u\in\mathbf{T}$ we write $V(u)\in{{\mathbb{R}}}$ for the position of $u$ and $|u|$ for the generation at which $u$ is alive. In the present paper, we focus on the barrier problem of BRW and a more general time-inhomogeneous model. The so-called barrier is in fact a function $f: {{\mathbb{N}}} {\rightarrow} {{\mathbb{R}}}.$ For any realization of the BRW, if a particle $u$ satisfies $V(u) > f(|u|),$ then we remove $u$ and all its descendants. The surviving particles (i.e., which have not been removed) form a new system, which is called a BRW with barrier. For any $i\leq |u|,$ we conventionally write $u_i$ for the ancestor of $u$ in generation $i.$ It is evident to see that $u$ is survival if and only if $V(u_i)\leq f(i), \forall i\leq |u|.$ Let $\kappa$ be the $\log-$Laplace transform of $L,$ that is to say

Obviously, another equivalent expression of $\kappa(\theta)$ is $\kappa(\theta) = \log{{\mathbb{E}}}(\sum_{|u| = 1}e^{-\theta V(u)}).$ We always assume that

which means that the underlying Galton-Watson process is supercritical, i.e., the survival probability of the particle system (BRW) is positive. Under the assumption that there exists $\vartheta > 0$ such that

where $\kappa'$ presents the derivative of $\kappa.$ Hammersley ^[1], Kingman ^[2] and Biggins ^[3] showed that

The above result enlightens the approach for the barrier problem, which is a topic motivated by the parallel simulation, see Lubachevsky et al. ^[4,5]. We first introduce some notations before we recall some achieved results on the barrier problem of BRW. On the Galton-Watson tree $\mathbf{T}$ we define a partial order $>$ such that $u > v$ if $v$ is the ancestor of $u$. We write $u\geq v$ if $u > v$ or $u = v$ (i.e., the particle $u$ is exactly the particle $v$). Define an infinite path $u_{\infty}$ through the tree $\mathbf{T}$ as a sequence of particles $(u_i)_{i\in N}$ such that

We write $\mathcal{T_{\infty}}$ the collection of the infinite path. Let

Hence one see that $\rho(\varepsilon, \alpha)$ presents the survival probability for the BRW with a barrier

The first result on the the barrier problem of BRW can be found in Biggins et al. ^[6]. Under Assumptions (1.1) and (1.2) they claimed that

From the view of (1.3), we can have a better understanding on this conclusion. That is to say, when critical slope of the barrier is determined by the first order of $\min_{u\in\mathbf{T}, \; |u| = n}V(u).$ Under a slightly stronger assumption, Jaffuel ^[7] refine the result (1.4). ^[7] showed that under (1.1), (1.2) and the assumption

It is true that

where the explicit form of the positive constant $a_{\kappa''(\vartheta)}$ (depending on $\kappa''(\vartheta)$) is obtained. Combining (1.6) with (1.4), we can prove the following statement.

Proposition 1.1. If (1.1), (1.2) and (1.5) hold, then $\rho(\varepsilon, \alpha) > 0$ when $\varepsilon > 0, \alpha > 1/3.$

Proof. It is obvious when $\alpha > 1$ since the definition domain of the barrier $f$ is ${{\mathbb{N}}}.$ Now we deal with the case $\alpha\in(1/3, 1).$ Let $\bar{a}$ be a constant such that $\bar{a}\in(a_{\kappa''(\vartheta)}, \varepsilon).$ Define

We see $j$ is finite since $\alpha > 1/3.$ Choose $k$ large enough such that $k > j\varepsilon^{-\frac{1}{\alpha}},$ which ensures that $\varepsilon(n+k)^{\alpha} > \bar{a}n^{1/3}, \forall n\in{{\mathbb{N}}}^+.$ Note that

we can find $a_- > 0$ small enough such that

Hence it is true that $\varepsilon i^{\alpha} > a_-i$ for $1\leq i\leq k$ and $\varepsilon i^{\alpha} > a_-k+\bar{a}(i-k)^{1/3}$ for $i > k.$ By Markov property we see

(1.4) tells us that $P_1 > 0$ and (1.6) means that $P_2 > 0,$ hence we have $\rho(\varepsilon, \alpha) > 0.$

The decay rate of $\rho(\varepsilon, 1)$ had been obtained in Gantert et al. ^[8]. When (1.1) and (1.2) hold, ^[8] obtained the explicit negative constant $c$ such that

(We remind that under (1.1), (1.2), (1.5) and some extra assumptions, the lower bound of $\rho(\varepsilon, 1)$ had also been obtained in ^[8].) In the present paper, we want to extend the upper bound of the rate to some non-linear barrier. It is evident to see that $\rho(\varepsilon, \alpha) = 0, \forall \alpha > 0$ when $\varepsilon = 0$ and $\rho(\varepsilon, \alpha)$ is non-decreasing on $\varepsilon$ when the positive constant $\alpha$ is fixed. Combining these two facts with Proposition 1.1, we see that for any given $\alpha > 1/3,$ it is reasonable and meaningful to ask the question about the decay rate of $\rho(\varepsilon, \alpha)$ as $\varepsilon\downarrow 0$. In the present paper, we wonder whether the decay rate of $\rho(\varepsilon, \alpha)$ (when $\alpha > 1/3$) as $\varepsilon\downarrow 0$ will be the same as the one of $\rho(\varepsilon, 1)$ (as $\varepsilon\downarrow 0$). Furthermore, if they are different, will the order be different? In other word, we want to investigate the impact of $\alpha$ on the decay rate. Now we give the first result in the present paper.

Theorem 1.1. If (1.1) and (1.2) hold, then for $\alpha > 1,$ we have

and for $\alpha\in(1/3, 1),$ we have

Remark 1.1. We remind that the limit of the right-hand-side of (1.7) as $\alpha\downarrow 1$ is the exact value of the corresponding one in ^[8], hence our result can be viewed as an extension for the upper bound part in ^[8].

In fact, this asymptotic behavior can be shown for a more general model called a branching random walk with varying environment (BRWve). Let us describe the model as follows. For a sequence of time-inhomogeneous branching random walks $\{(\mathbf{T}^{(n)}, V^{(n)})\}_{n\in{{\mathbb{N}}}},$ we only consider the generations from $0$ to $n$ in $(\mathbf{T}^{(n)}, V^{(n)}),$ where $\mathbf{T}^{(n)}$ presents the (time-inhomogeneous) Galton CWatson tree of the genealogy of this process and $V^{(n)}$ the displacements of the particles in $\mathbf{T}^{(n)}.$ Let $\{L_t, t\in[0, 1]\}$ be a family of laws of point processes. All particles reproduce independently but the law of reproduce is determined in the following way. For particle $u\in\mathbf{T}^{(n)}, |u| = i < n,$ the reproduce law of $u$ is $L_{\frac{i+1}{n}}.$ This model has been studied in several papers. Fang and Zeitouni ^[9] showed that the asymptotic behavior of the maximal displacement $\max_{u\in\mathbf{T}^{(n)}, |u| = n}V(u)$ under some special settings (two time intervals) on the reproduction law $\{L_t, t\in[0, 1]\}.$ Mallein ^[10] has generalized the result in ^[9] to more general reproduction law (a sequence of macroscopic time intervals). For a smoothly varying environment, Mallein ^[11] obtained a new asymptotic behavior of the maximal displacement. However, there is no result on the barrier problem of the BRWve. In the present paper, we want to extend Theorem 1.1 to some BRWve with special settings on the varying environment. Define

Assume that there exists $\vartheta, v > 0$ such that for any $s, t\in[0, 1],$

and

Furthermore, we assume that $\kappa''_t(\vartheta)$ satisfies that

Obviously, BRW is a special case of BRWve when the family $\{L_t, t\in[0, 1]\}$ is a constant one. In order to deal with the new model (BRWve), from now on we redefine the survival probability $\rho(\varepsilon, \alpha)$ as

Now we give a generalized version of Theorem 1.1.

Theorem 1.2. Denote $\sigma^2_{-}: = \min_{t\in[0, 1]}\vartheta^2\kappa''_t(\vartheta),$ $\gamma_{\sigma}: = \frac{\pi^2\sigma^2_{-}}{2}$. If (1.8), (1.9) and (1.10) hold, then for $\alpha > 1,$ we have

and for $\alpha\in(1/3, 1),$ we have

2.   Preliminary for the proof

Let us give a sketch of the proof. First we give a decomposition of the survival probability of the BRWve with barrier. Secondly, we transfer BRWve to a triangular array of independent centered random variables by the version of time-inhomogeneous many-to-one formula which has been introduced in ^[10]. Then the survival probability will be dominated by a series of small deviation probabilities of the triangular array random variables. At last, applying a time-inhomogeneous version of small deviation principle which has been given in ^[11], the estimate for the upper bound will becomes a extremal problem of some continuous functions.

The many-to-one formula, which is essentially a kind of measure transformation, is a basic tool in the study of the branching random walks, It can be traced down to the early works of Peyrière ^[12] and Kahane and Peyrière ^[13]. We refer to Biggins and Kyprianou ^[14] for more variations of this result. Let $\tau_{n, k}$ be a random measure on ${{\mathbb{R}}}$ such that for any $x\in{{\mathbb{R}}}$ we have

For any given $n$, we introduce a series of independent random variables $\{X_{n, k}\}_{k\in{{\mathbb{N}}}^+, k\leq n}$ whose distributions are $\{\tau_{n, k}\}_{n, k\in{{\mathbb{N}}}^+}$ and define

The following theorem shows the relationship between $S^{(n)}_k$ and the BRWve.

Theorem 2.1. (Mallein ^[10]) For any $n, k\in{{\mathbb{N}}}^+, \; k\leq n,$ and a measurable function $f:{{\mathbb{R}}}^n\rightarrow [0, +\infty),$ we have

By many-to-one formula, the barrier problem of a BRWve becomes equivalently to the small deviation problem for a time-inhomogeneous random walk.

The small deviation problem is a classic topic which attracts intensive attention for many years. We refer to Mogul'skiĭ^[15], Borovkov & Mogul'skiĭ ^[16], Shao ^[17] and Lv & Hong ^[18] as the small deviation principle for sums of independent random variables. In our proof, a time-inhomogeneous version of a small deviation principle which has been given in ^[11] will be used. We state it as follows.

Theorem 2.2. (Mallein ^[11]) Let $\{\tilde{X}_{n, k}\}_{n, k\in{{\mathbb{N}}}, k\leq n}$ be a triangular array of independent centered random variables. We assume that there exists$\; \sigma\in\mathcal{C}[0, 1]$ with $\sigma_{-}: = \min_{s\in[0, 1]}\sigma(s) > 0$ and $u > 0$ such that for any $n, k\in{{\mathbb{N}}}, k\leq n,$

and

Set $g, h\in \mathcal{C}[0, 1]$ and $g(0) < 0 < h(0).$ Denote $\tilde{S}^{(n)}_k: = \tilde{S}_0+\sum_{i = 1}^k\tilde{X}_{n, i}.$ Then we have

This conclusion extends the main result in ^[15] to the time-inhomogeneous case.

3.   Proof of Theorem 1.2

Recall the barrier function $f(i): = \varepsilon i^{\alpha}-\kappa_1'(\vartheta)i$, hence $f(i) = \varepsilon i^{\alpha}-\frac{i\kappa_1(\vartheta)}{\vartheta}$ from (1.8). We define

where the exact value of positive constants $a, b$ will be given later. From the definition of $\rho(\varepsilon, \alpha)$, we see for any $n\in{{\mathbb{N}}},$ it is true that

Define $T^{(n)}_i = \vartheta S^{(n)}_i+i\kappa_1(\vartheta).$ By Markov inequality and Theorem 2.2, it is true that

By the same way we get

Note that for any $n\in[Nk, (N+1)k],$ it is true that

Hence we have

Taking $N{\rightarrow}\infty,$ from (3.1) we get

We observe that

To apply Theorem 2.2, we need to verify that the sequence $\{T^{(n)}_i\}$ satisfies all conditions in Theorem 2.2. According to Theorem 2.1, we see

We observe that (1.8) and the above equality imply that

thus we see ${{\mathbb{E}}}(T^{(n)}_i) = 0, \forall i, n\in{{\mathbb{N}}}.$ Moreover, Theorem 2.1 tells that

Hence we have

where $Var$ presents the variation, that is to say, (1.10) ensures that $\{T^{(n)}_i\}$ meets (2.1). Next we check (2.2). Note that

and

Therefore, (1.9) ensures that $\{T^{(n)}_i\}$ meets (2.2). We rewrite the probability in (3.4) as

Let $(l-1)k$ play the role as $n$ in Theorem 2.2, from (3.3) and (3.4) we get

Recall the definition of $\sigma_{-}$ and define $\gamma_{\sigma}: = \frac{\pi^2\sigma^2_{-}}{2},$ we get

where $\varphi(x): = \vartheta a x^{\alpha}+\left(\frac{3\gamma_{\sigma}}{\vartheta^2b^2}-\vartheta b\right)\sqrt[3]{1-x}-\frac{3\gamma_{\sigma}}{\vartheta^2b^2}, x\in[0, 1].$ Because of the monotonicity of $\rho(\varepsilon, \alpha)$ on $\varepsilon$, by a similar argument as (3.2)–(3.3) we can see that for any $a > 0,$

By the light of (3.6) and (3.7), next we need to consider how to take the value of $a, b$ to get the minmum of $\sup_{x\in[0, 1]}\varphi(x).$

(ⅰ) For the case $\alpha > 1,$ we let positive constants $a, b$ satisfy that

Note that

(3.8) implies that $\max_{x\in[0, 1]}\varphi'(x)\leq 0.$ That is to say,

Combining (3.5), (3.6) with (3.8) we get

Noting that

hence we choose $b = \left(\frac{3\gamma_{\sigma}(\alpha-1)}{\alpha\vartheta^3}\right)^{1/3},$ which satisfies the second condition in (3.8) and the last line in (3.9) will take its maximum. Finally, from (3.7) we complete the proof of (1.11).

(ⅱ) Now we consider the case $\alpha\in(\frac{1}{3}, 1)$. Recall that

Hence it is true that

From this point we choose $b = (3\gamma_{\sigma})^{1/3}/\vartheta$ such that $\vartheta b-\frac{3\gamma_{\sigma}}{\vartheta^2b^2} = 0,$ hence it is true that

By direct caculation we see

thus the best choice of $a$ is $a: = \frac{b}{3\alpha}.$ Finally we get

which completes the proof of (1.12).

Acknowledgments

The author thanks the editor and the referees for the coming valuable comments and suggestions, which improves the quality of this paper greatly.

This research is supported by the Fundamental Research Funds for the Central Universities (NO.2232021D-30).

Conflict of interest

The author declares no conflict of interest.