| Russian Mathematical Surveys |
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This article is cited in 1 scientific paper (total in 1 paper) Brief communications Supercriticality conditions for branching walks in a random killing environment with a single reproduction centre V. A. Kutsenkoa, S. A. Molchanovb, E. B. Yarovayaa a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow b HSE University
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     A branching random walk on the one-dimensional lattice $\mathbb{Z}$ with continuous time is considered. A field of independent identically distributed random variables $\mathcal{M}=\{\mu(x,\cdot),x\in\mathbb{Z}\setminus\{0\}\}$ defined on some probability space $(\Omega,\mathcal{F},\mathsf{P})$ is defined on the lattice outside zero. It is assumed that each random variable $\mu(x,\cdot\,)$ takes values in the interval $[0,c]$, $c\geqslant0$, and has positive density on it. The field $\mathcal{M}$ forms a ‘random killing environment’ on $\mathbb{Z}$, which specifies the intensity of particle disappearance in the branching random walk. A realization of the field $\mathcal{M}$ is denoted by $\mathcal{M}(\omega)=\{\mu(x,\omega),x\in\mathbb{Z}\setminus\{0\}\}$, $\omega\in\Omega$. We introduce additionally a parameter $\Lambda\geqslant 0$ responsible for the intensity with which particles are reproduced at zero and a parameter $\varkappa>0$ which controls the intensity which which particles walk on the lattice. Assume that there is a single particle on $\mathbb{Z}$ at the moment of time $t=0$. The further evolution proceeds as follows. If a particle is at zero, then, within time $h\to0$, it splits into two particles with probability $\Lambda h+o(h)$, moves with probability $\varkappa h+o(h)$ to one of the equiprobable adjacent points, or remains at its place with probability $1-\Lambda h-\varkappa h+o(h)$. If the particle is at a point $x\neq 0$, then, within time $h\to0$, it disappears with probability $\mu(x,\omega) h+o(h)$, moves with probability $\varkappa h+o(h)$ to one of the equiprobable adjacent points, or remains at its place with probability $1-\mu(x,\omega)-\varkappa h+o(h)$. New particles evolve independently of one another and of the whole prehistory, in accordance with the same law. The process of branching of particles at $x\in\mathbb{Z}$ is described by the potential $V(x,\omega):=\Lambda\delta_0(x)-\mu(x,\omega)(1-\delta_0(x))$, where $\delta_y(x)$ is the Kronecker delta. The behaviour of the branching random walk at a moment of time $t$ is specified by the set of numbers of particles $N_{t}(y,\omega)$ at the points $y\in\mathbb{Z}$. Normally, the mean numbers of particles are considered [1], [2], namely, $m_1(t,x,y,\omega)=\mathsf{E}_{x} N_{t}(y,\omega)$, where $\mathsf{E}_{x}$ is the mean under the condition $N_0(y,\omega)=\delta_y(x)$. We are going to study the probability $P(\Lambda,\varkappa,c)$ of the realization of an environment in which the quantity $m_1(t,x,y,\omega)$ shows an exponential growth (supercriticality) for fixed $\Lambda$, $\varkappa$, and $c$. Formally, we have
$$
\begin{equation*}
P(\Lambda,\varkappa,c)=\mathsf{P}\biggl\{\omega\in\Omega\colon\lim_{t\to\infty} \frac{m_1(t,x,y,\omega)}{C(x,y)e^{\lambda(\omega) t}}=1\ \forall\, x,y \in\mathbb{Z}\biggr\},
\end{equation*}
\notag
$$
where the functions $C$, $\lambda$, and $m_1$ are positive and depend additionally on $\Lambda$, $\varkappa$, and $c$.
This study is aimed at estimating $P(\Lambda,\varkappa,c)$ as a function of $\Lambda,\varkappa$, and $c$. We use the approach described, for example, in [2] and [3] and write the Cauchy problem for $m_1(t,x,y,\omega)$ as follows: $\partial m_1(t,x,y,\omega)/\partial t=(\varkappa\Delta m_1)(t,x,y,\omega)+V(x,\omega)m_1(t,x,y,\omega)$ with the initial condition $m_1(0,x,y,\omega)=\delta_y(x)$, where $\varkappa\Delta f(x)=\varkappa[f(x+1)/2+f(x-1)/2-f(x)]$ is the difference Laplacian on $\mathbb{Z}$. Throughout, all operators are defined on $l_2(\mathbb{Z})$. Introducing the random self-adjoint operator $H(\omega)=\varkappa\Delta+V(x,\omega)$, we can represent the above Cauchy problem in the form $\partial m_1(t,x, y, \omega)/\partial t= H(\omega)m_1(t,x,y,\omega)$, $m_1(0,x,y,\omega)=\delta_y(x)$. The behaviour of $m_1$ in problems of this kind depends on the structure of the spectrum of the operator $H(\omega)$. Lemma 1. The spectrum of $H(\omega)$ consists almost surely of the non-random essential part $[-2\varkappa-c;0]$ and also maybe a random eigenvalue $\lambda(\Lambda,\varkappa,c,\omega)>0$. The proof follows the scheme of reasoning in [4]. In particular, for each $\lambda_0\in[-2\varkappa-c;0]$ and almost any realization of the environment $\omega_0\in\Omega$, we manage to construct a sequence of orthonormal functions $\{f_i(x,\lambda_0,\omega_0)\}\in l_2(\mathbb{Z})$, $i\in\mathbb{N}$, such that $\|H(\omega)f_n-\lambda_0f_n\|\to0$ as $n\to\infty$. It follows that any $\lambda_0\in[-2\varkappa-c;0]$ is almost surely in the essential spectrum. The random eigenvalue $\lambda(\Lambda,\varkappa,c,\omega)$ can appear due to a one-point perturbation of the self-adjoint operator at zero. The following lemma provides an explicit form of the eigenfunction $u_\lambda(x)$ corresponding to $\lambda>0$. This assertion is proved by substituting the expression for $u_\lambda(x)$ into the eigenvalue problem. Lemma 2. If there exists a positive eigenvalue $\lambda(\Lambda,\varkappa,c,\omega)$, then the corresponding eigenfunction is representable by an absolutely convergent series: where $\gamma\colon a \rightsquigarrow b$ is a path $\{a=x_1,x_2,\dots,x_n\not=b\}$ from $a$ to $b$ with transitions to adjacent lattice points that (i) does not go through the point $0$ and (ii) is assumed not to contain the point $b$. The value $u_\lambda(0)$ is $1$. For the the realization of the ‘worst’ environment $\mu(x,\omega)\equiv c$, the series for $u_\lambda(x)$ in Lemma 2 can be expressed in terms of elementary functions. The same is true for the ‘best’ environment $\mu(x,\omega)\equiv 0$. This implies the following result. Theorem 1. If $\Lambda\geqslant\sqrt{2\varkappa c+c^2}-c$, then the eigenvalue $\lambda(\omega)$ lies in the interval $[\sqrt{(\Lambda+c)^2+\varkappa^2}-(\varkappa+c); \sqrt{\Lambda^2+\varkappa^2}-\varkappa]$ and, accordingly, $P(\Lambda,\varkappa,c)=1$. Now we consider a non-random environment in which particles are killed at the points $x=-1$ and $x=1$ with intensities $\mu_{-1}>0$ and $\mu_{1}>0$, respectively. Then the Cauchy problem takes the form $\partial m_1(t,x, y)/\partial t=H_{1}m_1(t,x,y)$, $m_1(0,x,y)=\delta_y(x)$, where $H_1:=\varkappa \Delta+\delta_0(x)\Lambda-\delta_1(x)\mu_1-\delta_{-1}(x)\mu_{-1}$. Lemma 3. The operator $H_1$ has a positive eigenvalue if and only if $\Lambda>(\mu_1+\mu_{-1}+2\sigma\mu_1\mu_{-1})/[(1+\sigma\mu_1)(1+\sigma\mu_{-1})]$, where $\sigma=2/\varkappa$. Consider the set of environments $\Omega_1=\{\omega\in\Omega\colon \mu(1,\omega)=\mu_{1}, \mu(-1,\omega)=\mu_{-1}\}$. The mean number of particles in the non-random environment is almost surely larger than their mean number in any environment in $\Omega_1$. This remark and Lemma 3 yield the following result. Theorem 2. $P(\Lambda,\varkappa,c)\leqslant\mathsf{P}\{\Lambda> (\xi_1+\xi_2+2\sigma\xi_1\xi_{2})/[(1+\sigma\xi_1)(1+\sigma\xi_{2})]\}$, where the $\xi_i$ are independent copies of $\mu(x,\omega)$. Thus, new approaches to study branching random walks in random environments are proposed. 
Citation in format AMSBIB
\Bibitem{KutMolYar23} |
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