Solutions of a class of nonlinear master equations
The considered state space \(E\) is the set of non-negative integers, equipped with the discrete topology. Let \(Q=[q_{ij}]\) be a \(Q\)-matrix, totally stable and conservative, and \(Q_ \lambda=Q+\lambda[\delta_{i+1,j}-\delta_{i,j}]\) for \(\lambda\geq 0\). The operator acting on the space \(B(E)\) of bounded functions \(f\) is denoted by \(\Omega(\lambda)\), \[ \Omega(\lambda)f(i)=\sum_ j q_{i,j}(f(j)-f(i))+\lambda(f(i+1)-f(i)), \] \(\delta\) is Kronecker's. For \(s>0\), the set of all probability measures \(u\) on \(E\) with finite moment of order \(s\) is denoted by \({\mathcal P}_ s(E)\). The norm of \(u\in{\mathcal P}_ 1(E)\) is \(\| u\|=\sum u(y)y\). The main object of this article is the nonlinear master equation \[ (d/dt)u_ t(x)=\sum u_ t(y)\Omega(\| u_ t\|)\delta_ x(y) \] with initial condition \(u_ 0(y)=v(y)\), where \(v\in{\mathcal P}_ 1(E)\) is given. A solution of the above equation is by definition a probability distribution \(P\) on the canonical Skorokhod space of cadlag paths on \(E\), whose marginal distributions \(u_ t\) satisfy the above system. McKean introduced the Markov property for a solution: there is a transition function \(p(s,i;t,j)\) satisfying for all \(s,t\geq 0\) and \(i,j\in E\), \[ (d/ds)p(t,i;t+s,j)=\sum_ k p(t,i;t+s,k)\Omega(\| u_{t+s}\|)\delta_ j(k) \] and the conditional probability \[ P(X_{t+s}=j\mid F_ t)=p(t,X_ t;t+s,j). \] There is also a martingale formulation of the equation: for \(j\in E\), \(\delta_ j(X_ t)-\int^ t_ 0\Omega(\| u_ s\|)\delta_ j(X_ s)ds\) is a martingale, where \(u_ t=P\circ X^{-1}_ t\). In Theorem 1.1 the martingale problem is solved when for all \(k\in E\), (1.1) \(\sup(q_{i,k}: i\neq k)<\infty\), and (1.2) \(\sum_ j q_{k,j}(j^ \alpha-k^ \alpha)\leq M+\lambda k^ \alpha\), where \(\alpha>1\), \(M>0\), \(\lambda>1\) are fixed. With another condition (1.3), Theorem 1.2 shows that the martingale solution is the unique Markovian solution of the master equation. The birth-death model when \(a_ i=q_{i,i-1}\), \(b_ i=q_{i,i+1}\) is discussed in \(\S 4\), and uniqueness and existence of the stationary solutions are established in Theorem 4.2 under suitable conditions. A Markovian solution \(P\) is stationary if for all \(t\geq 0\), \(P_ t\circ X^{-1}=u\); \(u\) is called a stationary distribution. (Definition 4.1 given in the article is a little bit confusing!) Theorem 4.3 asserts the finiteness of the number of stationary distributions, whith some polynomial growth condition of \((a_ i)\) and \((b_ i)\) added. These are applied to the second Schlögl model, resulting in the change of the number of stationary distributions (Theorem 1.3). This is the phenomenon of phase transition.
scientific article; zbMATH DE number 98444
scientific article