Calculating the stochastic work done on a quantum system while strongly
coupled to a reservoir is a formidable task, requiring the calculation of the
full eigenspectrum of the combined system and reservoir. Here we show that this
issue can be circumvented by using a polaron transformation that maps the
system into a new frame where weak-coupling theory can be applied. It is shown
that the work probability distribution is invariant under this transformation,
allowing one to compute the full counting statistics of work at strong
reservoir coupling. Crucially this polaron approach reproduces the Jarzynski
fluctuation theorem, thus ensuring consistency with the laws of stochastic
thermodynamics. We apply our formalism to a system driven across the
Landau-Zener transition, where we identify clear signatures in the work
distribution arising from a non-negligible coupling to the environment. Our
results provide a new method for studying the stochastic thermodynamics of
driven quantum systems beyond Markovian, weak-coupling regimes.Comment: 15 pages, 3 figures, comments welcom