  The millihertz gravitational-wave frequency band is expected to contain a
rich symphony of signals with sources ranging from galactic white dwarf
binaries to extreme mass ratio inspirals. Many of these gravitational-wave
signals will not be individually resolvable. Instead, they will incoherently
add to produce stochastic gravitational-wave confusion noise whose frequency
content will be governed by the dynamics of the sources. The angular structure
of the power of the confusion noise will be modulated by the distribution of
the sources across the sky. Measurement of this structure can yield important
information about the distribution of sources on galactic and extra-galactic
scales, their astrophysics and their evolution over cosmic timescales.
Moreover, since the confusion noise is part of the noise budget of LISA,
mapping it will also be essential for studying resolvable signals. In this
paper, we present a Bayesian algorithm to probe the angular distribution of the
stochastic gravitational-wave confusion noise with LISA using a spherical
harmonic basis. We develop a technique based on Clebsch-Gordan coefficients to
mathematically constrain the spherical harmonics to yield a non-negative
distribution, making them optimal for expanding the gravitational-wave power
and amenable to Bayesian inference. We demonstrate these techniques using a
series of simulations and analyses, including recovery of simulated distributed
and localized sources of gravitational-wave power. We also apply this method to
map the gravitational-wave foreground from galactic white-dwarfs using a
simplified model of the galactic white dwarf distribution.
