Spectral methods and their applications in analysis of branching random walks

Spectral methods are widely used in stochastic analysis. We establish a link between branching random walks with signed branching sources of various intensities and the structure of the spectrum of the evolutionary operator for the mean population size of particles at the lattice points. In this connection, we solve the problem of the number of eigenvalues of a finite-dimensional non-self-adjoint perturbation $A+B$ of a non-self-adjoint operator $A$ in a real Hilbert space. We prove that the number of eigenvalues of the operator $A+B$ exceeding the upper bound of the spectrum of the operator $A$ is majorized by the number of positive eigenvalues of the operator $B$, and the number of eigenvalues of the operator $A+B$ smaller than the lower bound of the spectrum of the operator $A$ is majorized by the number of negative eigenvalues of the operator $B$.