The Strong Law of Large Numbers for random semigroups on uniformly smooth Banach spaces
S. V. Dzhenzher, V. Zh. Sakbaev
TL;DR
The work extends the Strong Law of Large Numbers for random operator products from Hilbert spaces to general Banach spaces by studying random semigroups generated by i.i.d. bounded generators. It proves SLLN in the Strong Operator Topology on uniformly smooth Banach spaces, showing $e^{A_1 t/n}\cdots e^{A_n t/n}\to e^{(\mathbb{E}A)t}$ uniformly in $t$ on compact intervals, and provides a second approach yielding SLLN in the Weak Operator Topology for all Banach spaces via a seminorm induced by a positive operator. The proofs combine Chernoff-type equivalence, independence of random generators, and Burkholder-type martingale inequalities in $p$-smooth spaces, delivering a robust framework for random semigroup LLNs with potential applications to open quantum dynamics beyond Hilbert spaces. The results broaden the applicability of LLN for random operator products and offer tools for analyzing random quantum channels in Banach-space settings. Overall, the paper supplies two complementary LLN results for random semigroups: a SOT-based theorem in uniformly smooth spaces and a WOT-based method valid for general Banach spaces, with explicit probabilistic bounds and uniform-in-$t$ convergence on finite intervals.
Abstract
We consider random linear continuous operators $Ω\to \mathcal{L}(\mathcal{X}, \mathcal{X})$ on a Banach space $\mathcal{X}$. For example, such random operators may be random quantum channels. The Law of Large Numbers is known when $\mathcal{X}$ is a Hilbert space, in the form of the usual Law of Large Numbers for random operators, and in some other particular cases. Instead of the sum of i.i.d. variables, there may be considered the composition of random semigroups $e^{A_it/n}$. We obtain the Strong Law of Large Numbers in Strong Operator Topology for random semigroups of bounded linear operators on a uniformly smooth Banach space. We also develop another approach giving the SLLN in Weak Operator Topology for all Banach spaces.