Local convex path action ($I(a)={\int }_{-\infty }^{0}g({\dot{a}}_s)ds$)

A local path action integrates a nonnegative convex cost g of the path's instantaneous input rate, assigning infinity to paths that are not absolutely continuous. When $g(\mu )=0$, the typical-rate path has zero cost. On any interval with prescribed total input, Jensen inequality makes a constant derivative minimize its cost. With a queue workload constraint this leads to constant-rate burst paths and the linear workload rate for a local convex action. Such a formula alone does not establish goodness in an unspecified path topology; goodness must be checked or assumed as part of the input large-deviation principle.