Generalized principal eigenvalues for parabolic operators in bounded domains

We introduce here new generalized principal eigenvalues for linear parabolic operators withheterogeneous coefficients in space and time. We consider a bounded spatial domain and anunbounded time intervalI:I=R,R+orR−, and operators with coefficients having a fairlygeneral dependence on space and time. The notions we introduce rely on the parabolic maximumprinciple and extend some earlier definitions introduced for elliptic operators [3, 5].We first show that these eigenvalues hold the key to understanding the large time behaviorand entire solutions of heterogeneous Fisher-KPP type equations. We then describe the relationof these quantities with principal Floquet bundles for parabolic operators which provides furthercharacterizations of the principal eigenvalues. These allow us to derive monotonicity proper-ties and comparisons between generalized principal eigenvalues, as well as perturbation resultsand further properties involving limit operators. We show that the sign of these eigenvaluesencodes different versions of the maximum principle for parabolic operators. Lastly, we ex-plicitly compute the generalized principal eigenvalues for several classes of operators such asspatial-independent, periodic, almost periodic, uniquely ergodic or random stationary ergodiccoefficients.