| Russian Mathematical Surveys |
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This article is cited in 9 scientific papers (total in 10 papers) Brief Communications Martingale method for studying branching random walks N. V. Smorodinaa, E. B. Yarovayab a St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences b Lomonosov Moscow State University
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     We consider a continuous-time branching random walk (BRW) on $\mathbb{Z}^{d}$, $d\in \mathbb{N}$. The underlying random walk of the BRW is given by the transition intensity matrix $A=(a(x-y))_{x,y\in \mathbb{Z}^{d}}$, where the even function $a(x)$ satisfies the conditions $\sum_{x\in\mathbb{Z}^{d}}a(x)= 0$, $a(x)\geqslant 0$ for $x\ne 0$, and $a(0)<0$. The random walk is also assumed to be irreducible, that is, for every $z\in\mathbb{Z}^{d}$ there exist $z_{1},\dots,z_{k}\in\mathbb{Z}^{d}$ such that $z=\sum_{i=1}^{k}z_{i}$ and $a(z_{i})\ne 0$ for $i=1,\dots,k$. The branching process at a point $x \in \mathbb{Z}^{d}$ is determined by an infinitesimal generating function $f(u,x)=\sum_{k=0}^\infty b_k(x) u^k$ defined for $0\leqslant u \leqslant1$, where $b_k(x)\geqslant0$ for $k\ne 1$, $b_1(x)\leqslant 0$, and $\sum_{k\ne 1} b_k(x)={|b_1|<\infty}$. It is assumed that $f^{(r)}(1,x)<\infty$ for every $x \in \mathbb{Z}^{d}$ and $r=1,2$. The quantity $\beta(x)=f'(1,x)=\sum_{k}kb_{k}(x)$ is called the intensity of branching at the point $x \in \mathbb{Z}^{d}$. We assume that $\beta(x) \to 0$ as $\|x\| \to\infty$, where $\|\,{\cdot}\,\|$ is the norm in $L_{2}(\mathbb{Z}^{d})$, and that the function $f^{(2)}(1,x)=\sum_{k}k(k-1)b_{k}(x)$ is bounded on $\mathbb{Z}^{d}$. The functions $a(x)$ and $f(u,x)$ define a branching Markov process. Namely, a particle located at a point $x$ jumps to a point $ y\ne x$ in short time $h$ with probability $p(h,x,y)=a(x-y)h+o(h)$, or it does not make such a transition but at the same time produces offspring of $k\ne 1$ particles remaining at $x$ with probability $p_{*}(h,x,k)=b_{k} h+o(h)$ (we assume that the particle itself is included in this number; for $k=0$ we say that the particle dies), or, with probability $1-\sum_{ y\ne x}a(x-y)h-\sum_{k\ne 1}b_{k}(x)h+o(h)$, no changes occur to the particle. Individual particles evolve independently of one another. Now, the main object of our investigation is the number of particles $\mu_{t,x}(y)$ at time $t$ at an arbitrary fixed point $y\in \mathbb{Z}^{d}$ under the assumption that at the initial moment of time we have only one particle, which is located at a point $x\in \mathbb{Z}^{d}$. As shown, for example, in [1], the first moment $m_{1}(t,x,y)=\mathsf{E}\mu_{t,x}(y)$ of the random variable $\mu_{t,x}(y)$ is a solution $u(t,x)$ of the Cauchy problem $\partial_{t}u=\mathcal{A}u+\mathcal{B}u$ with initial condition $u(0,x)=\delta(y-x)$, where the symmetric operator of convolution type $\mathcal{A}$ generated by the matrix $A$ and the diagonal operator of coordinatewise multiplication $\mathcal{B}$ act on the function $\varphi\in L_{2}(\mathbb{Z}^d)$ as follows: $(\mathcal{A} \varphi)(x)=\sum_{y\in\mathbb{Z}^d}a(x- y)\varphi(y)$ and $(\mathcal{B} \varphi)(x)=\beta(x)\varphi(x)$ for any $x\in\mathbb{Z}^d$. In most publications on this topic the proofs of limit theorems for the number of particles $\mu_{t,x}(y)$ are united by a common research method based on an analysis of the asymptotics of the moments of $\mu_{t,x}(y)$ (see, for example, [2]). We use a method based on the martingale technique, which was originally used by Doob to study branching processes and further developed by Biggins [3] and Ioffe [4] for BRWs. We introduce a completely different martingale, whose construction is based on the use of methods of the spectral theory of self-adjoint operators in a Hilbert space. Let us proceed directly to the construction of the martingale. Let $X_{x}(t)$ denote a BRW defined, as above, by the functions $a(x)$ and $f(u,x)$ and satisfying the condition $X_{x}(0)=\delta_{x}$, so that at the initial moment of time we have exactly one particle, which is located at the point $x$. In what follows the process $X_{x}(t)$ is considered as a Markov process with values in the space $\mathcal{M}$ of all discrete finite integer measures on $\mathbb{Z}^d$. Each element of $M\in\mathcal{M}$ has the form $M=\sum_{j=1}^{k} \delta_{y_j}$, where $k\in\mathbb{N}\cup\{0\}$ and $y_j \in \mathbb{Z}^d$. The points $y_j$ in this representation are not necessarily different, which reflects the fact that several particles can be located at one lattice point at the same time, and the particles located at the same point differ only by their numbers in the list of particles $\{y_1,y_2,\dots,y_k\}$. In other words, each $y_j$ corresponds to a separate particle, which we encode by its lattice site $y_j$ and its number $j$ in the list. Since only symmetric functions of $X_{x}(t)$ are considered below, the particular choice of the numbering of particles does not play any role. For $M\in\mathcal{M}$ we denote the set of all particles by the symbol $\{M\}$. We will represent this set as $\{M\}=\{y_1,y_2,\dots,y_k\}$, and any lattice point can occur several times in this representation, which corresponds to the fact that this point can contain several particles. Thus, we treat BRW $X_x(t)$ as an $\mathcal{M}$-valued Markov random process. We denote by $\mathcal{F}_t$ the filtration corresponding to this process. Now for all $t\geqslant 0$, $x\in\mathbb{Z}^d$, and $\varphi\in L_{2}(\mathbb{Z}^d)$ we consider the random variable
$$
\begin{equation*}
I_{t,x}(\varphi)=\displaystyle\sum_{y \in \{X_{x}(t)\}}\varphi(y)=\int_{\mathbb{Z}^d}\varphi \,dX_{x}(t).
\end{equation*}
\notag
$$
One can show that the operator $\mathcal{A}+\mathcal{B}$ is bounded and self-adjoint in $L_{2}(\mathbb{Z}^d)$, and its positive spectrum can only consist of eigenvalues of finite multiplicity (possibly condensing to zero). Assume also that
$$
\begin{equation*}
\lambda_0=\sup_{\|h\|=1}\biggl\{(\mathcal{A}h,h)+\sum_{x\in\mathbb{Z}^d}\beta(x)h^2(x)\biggr\}>0.
\end{equation*}
\notag
$$
It follows from the Krein–Rutman theorem [5] that $\lambda_0$ is a simple eigenvalue of the operator $\mathcal{A}$, which corresponds to a strictly positive eigenfunction $\varphi_0\in L_{2}(\mathbb{Z}^d)$.
Theorem 1. For any $x\in\mathbb{Z}^d$ the process $\eta(t,x)=e^{-\lambda_0 t}I_{t,x}(\varphi_0)$, where $\eta(0 ,x)=\varphi_0(x)$, is a non-negative $(\mathcal{F}_t)$-martingale. Since the martingale $\eta(t,x)$ is non-negative, it follows from Doob’s theorem ([6], Chap. 3) that the limit $\eta(\infty,x)=\lim_{t\to\infty}\eta(t,x)$ exists almost surely. The following theorem shows that this limit also exists in the sense of mean square convergence. Theorem 2. The relation $\lim_{t\to\infty}\sup_{x\in\mathbb{Z}^d}\mathsf{E}(\eta(t,x)- \eta(\infty,x))^2=0$ holds. Using Theorems 1 and 2 we obtain the following. Theorem 3. The relation $\lim_{t\to\infty}\sup_{x\in\mathbb{Z}^d}\mathsf{E} \big(e^{- \lambda_0t}\mu_{t,x}(y)-\varphi_0(y)\eta(x,\infty)\big)^2=0$ holds. Theorems 1–3 show that the martingale approach makes it possible to obtain a limit theorem on the number of particles $\mu_{t,x}(y)$ by studying its first two moments, and also to generalize results obtained for BRWs with a finite number of particle branching sources [7] to models with an infinite number of sources of positive intensity. In contrast to the method of moments, with the help of which the convergence in the distribution of random variables $e^{-\lambda_0t}\mu_{t,x}(y)$ was proved in [7] and [2], the martingale approach makes it possible to prove a stronger statement on the convergence of these random variables in mean square. 
Citation in format AMSBIB
\Bibitem{SmoYar22} |
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