  Sawin recently gave an axiomatic characterization of multiple Dirichlet
series over the function field $\mathbb{F}_{q}(T)$ and proved their existence
by exhibiting the coefficients as trace functions of specific perverse sheaves.
However, he did not prove that these series actually converge anywhere, instead
treating them as formal power series.
  In this paper, we prove that these series do converge in a certain region,
and moreover that the functions obtained by analytically continuing them
satisfy functional equations.
  For convergence, it suffices to obtain bounds on the coefficients, for which
we use the decomposition theorem for perverse sheaves, in combination with the
Kontsevich moduli space of stable maps to construct a suitable
compactification.
  For the functional equations, the key identity is a multi-variable
generalization of the relationship between a Dirichlet character and its
Fourier transform; in the multiple Dirichlet series setting, this uses a
density trick for simple perverse sheaves and an explicit formula for
intermediate extensions from the complement of a normal crossings divisor.
