A central limit problem in random evolutions
In this article the following model of random evolutions is considered: let \(A,A_ 1,A_ 2,..\). be a sequence of i.i.d. random generators of strongly continuous operator semigroups defined densely in a Banach space B and let \(Y_ n(t)=\exp (n^{-1}A_{[n^ 2t]})...\exp (n^{-1}A_ 2)\exp (n^{-1}A_ 1)Y_ n(0)\) be a sequence of stochastic processes with sample paths in \(D_ B[0,\infty)\), where \(Y_ n(0)\) converges weakly to Y(0). The main topic is a study of the limiting behaviour of \(Y_ n(t)\) as \(n\to\infty \). It is shown that under suitable assumptions of measurability, boundedness of operators, moments restrictions and some other conditions, \(\{Y_ n\}\) is relatively compact in the topology of weak convergence and any limit process Y lies in \(C_ B[0,\infty).\) Furthermore, for each \(\theta\) in a separable subspace of the dual space \(B^*\), the martingale \[ M_ n^{\theta}(t)=(\theta,Y_ n(t)-Y_ n(0)-\sum^{[n^ 2t]}_{i=1}\int_{\Omega}[\exp (n^{-1}A(\omega))- I]Y_ n((i-1)/n^ 2)P(d\omega)) \] converges weakly to \((\theta,\quad Y(t)-Y(0)-1/2\int^{t}_{0}\int_{\Omega}A^ 2(\omega)Y(s)P(d\omega)ds),\) and the similar result is true for the martingale \(M_ n^{\theta}(t)^ 2-V_ n^{\theta}(t)\), where \(V_ n^{\theta}(t)\) is an appropriate centering sequence converging weakly to the quadratic variation process \(\int^{t}_{0}\int_{\Omega}(\theta,A(\omega)Y(s))^ 2P(d\omega)ds\). For the proof an equivalent form of the so-called compact containment criterion is derived that reduces investigations to real valued processes. The mirror approach in the dual space \(B^*\) allows to construct a dual sequence of processes having a weak limit, which implies the uniqueness in law of Y. Three special cases illustrating typical areas of applications are discussed: bounded operators, transport phenomena and random classical mechanics.
scientific article; zbMATH DE number 3872395