  The thermodynamics of a quantum system interacting with an environment that
can be assimilated to a harmonic oscillator bath has been extensively
investigated theoretically. In recent experiments, the system under study
however does not interact directly with the bath, but though a cavity or a
transmission line. The influence on the system from the bath is therefore seen
through an intermediate system, which modifies the characteristics of this
influence. Here we first show that this problem is elegantly solved by a
transform, which we call the Vernon transform, mapping influence action kernels
on influence action kernels. We also show that the Vernon transform takes a
particularly simple form in the Fourier domain, though it then must be
interpreted with some care. Second, leveraging results in quantum
thermodynamics we show how the Vernon transform can also be used to compute the
generating function of energy changes in the environment. We work out the
example of a system interacting with two baths of the Caldeira-Leggett type,
each of them seen through a cavity.
