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We consider a noncooperative $n$-player principal eigenvalue game which isassociated with an infinitesimal generator of a stochastically perturbedmulti-channel dynamical system -- where, in the course of such a game, eachplayer attempts to minimize the asymptotic rate with which the controlled statetrajectory of the system exits from a given bounded open domain. In particular,we show the existence of a Nash-equilibrium point (i.e., an $n$-tuple ofequilibrium linear feedback operators) that is distinctly related to a uniquemaximum closed invariant set of the corresponding deterministic multi-channeldynamical system, when the latter is composed with this $n$-tuple ofequilibrium linear feedback operators.

On noncooperative $n$-player principal eigenvalue games