53 research outputs found
On the point spectrum of some perturbed differential operators with periodic coefficients
Finiteness of the point spectrum of linear operators acting in a Banach space
is investigated from point of view of perturbation theory. In the first part of
the paper we present an abstract result based on analytical continuation of the
resolvent function through continuous spectrum. In the second part, the
abstract result is applied to differential operators which can be represented
as a differential operator with periodic coefficients perturbed by an arbitrary
subordinated differential operator
Parameter estimations for SPDEs with multiplicative fractional noise
We study parameter estimation problem for diagonalizable stochastic partial
differential equations driven by a multiplicative fractional noise with any
Hurst parameter . Two classes of estimators are investigated:
traditional maximum likelihood type estimators, and a new class called
closed-form exact estimators. Finally the general results are applied to
stochastic heat equation driven by a fractional Brownian motion
A note on error estimation for hypothesis testing problems for some linear SPDEs
The aim of the present paper is to estimate and control the Type I and Type
II errors of a simple hypothesis testing problem of the drift/viscosity
coefficient for stochastic fractional heat equation driven by additive noise.
Assuming that one path of the first Fourier modes of the solution is
observed continuously over a finite time interval , we propose a new
class of rejection regions and provide computable thresholds for , and ,
that guarantee that the statistical errors are smaller than a given upper
bound. The considered tests are of likelihood ratio type. The main ideas, and
the proofs, are based on sharp large deviation bounds. Finally, we illustrate
the theoretical results by numerical simulations.Comment: Forthcoming in Stochastic Partial Differential Equations: Analysis
and Computation
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