  This work considers stochastic operators in general inner-product spaces, and
in particular, systems with stochastically time-varying input delays of a known
probability distribution. Stochastic dissipativity and stability are defined
from an operator-theoretic perspective, and the well-known open-loop
dissipativity conditions for closed-loop/network stability are extended to the
stochastic case. Criteria are derived to identify dissipative nonlinear systems
with stochastic input delays, and this result is used to find
delay-distribution-dependent linear matrix inequality conditions for stochastic
dissipativity of a linear system with input delays of a known probability
distribution. A numerical experiment demonstrates the utility of the resulting
criteria for robust plant analysis and controller design, highlighting
significantly reduced conservatism compared to deterministic methods.
