We prove the Poincaré inequality for vector fields on the balls of the control distance by integrating along subunit paths. Our method requires that the balls are representable by means of suitable “controllable almost exponential maps”.

The present paper is devoted to the theory of discontinuous Markoff processes, that is processes where the transitions between states take place either by “jumps” of some specified kind, or by other means. States are taken as point$x$in an abstract space;$phases$are points ($x, t$) in the product state×time space; sets of states are denoted by$X$, sets of phases by$S$.

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