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Spectral properties of random nonselfadjoint matrices and operators
, 2001
"... We describe some numerical experiments which determine the degree of spectral instability of medium size randomly generated matrices which are far from selfadjoint. The conclusion is that the eigenvalues are likely to be intrinsically uncomputable for similar matrices of a larger size. We also desc ..."
Abstract

Cited by 7 (4 self)
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We describe some numerical experiments which determine the degree of spectral instability of medium size randomly generated matrices which are far from selfadjoint. The conclusion is that the eigenvalues are likely to be intrinsically uncomputable for similar matrices of a larger size. We also describe a stochastic family of bounded operators in infinite dimensions for almost all of which the eigenvectors generate a dense linear subspace, but the eigenvalues do not determine the spectrum. Our results imply that the spectrum of the nonselfadjoint Anderson model changes suddenly as one passes to the infinite volume limit.
Maslowski B.: Lower estimates of transition densities and bounds on exponential ergodicity for stochastic PDE’s
"... A formula for the transition density of a Markov process defined by an infinitedimensional stochastic equation is given in terms of the Ornstein–Uhlenbeck bridge and a useful lower estimate on the density is provided. As a consequence, uniform exponential ergodicity and Vergodicity are proved for ..."
Abstract

Cited by 5 (2 self)
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A formula for the transition density of a Markov process defined by an infinitedimensional stochastic equation is given in terms of the Ornstein–Uhlenbeck bridge and a useful lower estimate on the density is provided. As a consequence, uniform exponential ergodicity and Vergodicity are proved for a large class of equations. We also provide computable bounds on the convergence rates and the spectral gap for the Markov semigroups defined by the equations. The bounds turn out to be uniform with respect to a large family of nonlinear drift coefficients. Examples of finitedimensional stochastic equations and semilinear parabolic equations are given. 1. Introduction. The