  Ultralight axion-like particles are well-motivated dark matter candidates
that are the target of numerous direct detection efforts. In the vicinity of
the Solar System, such particles can be treated as oscillating scalar fields.
The velocity dispersion of the Milky Way determines a coherence time of about
10^6 oscillations, beyond which the amplitude of the axion field fluctuates
stochastically. Any analysis of data from an axion direct detection experiment
must carefully account for this stochastic behavior to properly interpret the
results. This is especially true for experiments sensitive to the gradient of
the axion field that are unable to collect data for many coherence times.
Indeed, the direction, in addition to the amplitude, of the axion field
gradient fluctuates stochastically. We present the first complete stochastic
treatment for the gradient of the axion field, including multiple
computationally efficient methods for performing likelihood-based data
analysis, which can be applied to any axion signal, regardless of coherence
time. Additionally, we demonstrate that ignoring the stochastic behavior of the
gradient of the axion field can potentially result in failure to discover a
true axion signal.
