  The nonlocal bosonic theory obtained from integrating out all anticommuting
and auxiliary variables in a globally supersymmetric theory is characterized by
the Nicolai map. We present a universal formula for the latter in terms of an
ordered exponential of the integrated coupling flow operator, which can be
canonically constructed. Also for supersymmetric gauge theories, this allows us
to perturbatively construct the Nicolai map explicitly in terms of tree
diagrams. For off-shell supersymmetry this works in any gauge, in the on-shell
case the Landau gauge is required. The dimensional reduction of $D{=}10$ super
Yang-Mills to maximally supersymmetric SU($N$) matrix mechanics (the BFSS
model) is known to provide a regularization of the $D{=}11$ supermembrane in
light-cone gauge via its incarnation as a one-dimensional gauge theory of
area-preserving membrane diffeomorphisms. We show how a well-defined
corresponding Nicolai map perturbatively linearizes the supermembrane in the
small-tension regime and points the way for a computation of vertex-operator
correlators.
