  This article introduces the degenerate special Lagrangian equation (DSL) and
develops the basic analytic tools to construct and study its solutions. The DSL
governs geodesics in the space of positive graph Lagrangians in $\mathbb{C}^n.$
Existence of geodesics in the space of positive Lagrangians is an important
step in a program for proving existence and uniqueness of special Lagrangians.
Moreover, it would imply certain cases of the strong Arnold conjecture from
Hamiltonian dynamics.
  We show the DSL is degenerate elliptic. We introduce a space-time Lagrangian
angle for one-parameter families of graph Lagrangians, and construct its
regularized lift. The superlevel sets of the regularized lift define
subequations for the DSL in the sense of Harvey--Lawson. We extend the
existence theory of Harvey--Lawson for subequations to the setting of domains
with corners, and thus obtain solutions to the Dirichlet problem for the DSL in
all branches. Moreover, we introduce the calibration measure, which plays a
r\^ole similar to that of the Monge--Amp\`ere measure in convex and complex
geometry. The existence of this measure and regularity estimates allow us to
prove that the solutions we obtain in the outer branches of the DSL have a
well-defined length in the space of positive Lagrangians.
