Past

We discuss estimating the growth exponents for positive solutions to the

random parabolic Anderson's model with small parameter k. We show that

behaviour for the case where the spatial variable is continuous differs

markedly from that for the discrete case.

The probabilistic approach to homogenization can be adapted to fully

degenerate situations, where irreducibility is insured from a Doeblin type

condition. Using recent results on weak sense Poisson equations in a

similar framework, obtained jointly with A. Veretennikov, together with a

regularization procedure, we prove the homogenization result. A similar

approach can also handle degenerate random homogenization.

It is known for elliptic problems with smooth coefficients

that the solution is smooth in the interior of the domain;

low regularity is only possible near the boundary.

The $hp$-version of the FEM allows us to exploit this

property if we use meshes where the element size grows

porportionally to the element's distance to the boundary

and the approximation order is suitably linked to the

element size. In this way most degrees of freedom are

concentrated near the boundary.

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In this talk, we will discuss convergence and complexity

issues of the boundary concentrated FEM. We will show

that it is comparable to the classical boundary element

method (BEM) in that it leads to the same convergence rate

(error versus degrees of freedom). Additionally, it

generalizes the classical FEM since it does not require

explicit knowledge of the fundamental solution so that

it is also applicable to problems with (smooth) variable

coefficients.

Consider a spectrally one-sided Levy process X and reflect it at

its past infimum I. Call this process Y. We determine the law of the

first crossing time of Y of a positive level a in terms of its

'scale' functions. Next we study the exponential decay of the

transition probabilities of Y killed upon leaving [0,a]. Restricting

ourselves to the case where X has absolutely continuous transition

probabilities, we also find the quasi-stationary distribution of

this killed process. We construct then the process Y confined in

[0,a] and prove some properties of this process.