Optimal stationary markings

Many specific problems ranging from theoretical probability to applications in statistical physics, combinatorial optimization and communications can be formulated as an optimal tuning of local parameters large systems interacting particles. Using the framework stationary point processes Euclidean space, we pose it a problem marking given process. The quality is evaluated terms scores calculated covariant manner for all points function proposed marked configuration. In absence total order configurations scores, identify intensity-optimality optimality two natural ways defining marking. We derive tightness integrability conditions under which intensity-optimal markings exist further stabilization making them equivalent locally ones. present examples motivating proposed, general framework. Finally, discuss various possible approaches leading uniqueness results.