  A long-standing puzzle in the study of black-hole accretion concerns the
presence or not of thermal instability. Classical theory predicts the
encircling accretion disk is unstable, as do some self-consistent MHD
simulations of the flow. Yet observations of strongly accreting sources
generally fail to exhibit cyclic or unstable dynamics on the expected
timescales. This paper checks whether turbulent fluctuations impede thermal
instability. It also asks if it makes sense to conduct linear stability
analyses on a turbulent background. These issues are explored with a set of MRI
simulations in thermally unstable local boxes in combination with stochastic
equations that approximate the disk energetics. These models show that the
disk's thermal behaviour deviates significantly from laminar theory, though
ultimately a thermal runaway does occur. We find that the disk temperature
evolves as a biased random walk, rather than increasing exponentially, and thus
generates a broad spread of outcomes, with instability often delayed for
several thermal times. We construct a statistical theory that describes some of
this behaviour, emphasising the importance of the `escape time' and its
associated probability distribution. In conclusion, turbulent fluctuations on
there own cannot stabilise a disk, but they can weaken and delay thermal
instability.
