  Spectral line-shapes provide a window into the local environment coupled to a
quantum transition in the condensed phase. In this paper, we build upon a
stochastic model to account for non-stationary background processes produced by
broad-band pulsed laser stimulation. In particular, we consider the
contribution of pair-fluctuations arising from the full bosonic many-body
Hamiltonian within a mean-field approximation, treating the coupling to the
system as a stochastic noise term. Using the It{\^o} transformation, we
consider two limiting cases for our model which lead to a connection between
the observed spectral fluctuations and the spectral density of the environment.
In the first case, we consider a Brownian environment and show that this
produces spectral dynamics that relax to form dressed excitonic states and
recover an Anderson-Kubo-like form for the spectral correlations. In the second
case, we assume that the spectrum is Anderson-Kubo like, and invert to
determine the corresponding background. Using the Jensen inequality, we obtain
an upper limit for the spectral density for the background. The results
presented here provide the technical tools for applying the stochastic model to
a broad range of problems.
