Past

Ergodic Markov processes possess invariant measures. In the case if transition probabilities or SDE coefficients depend on a parameter, it is important to know whether these measures depend regularly on this parameter. Results of this kind will be discussed. Another close topic is whether approximations to Markov diffusions possess ergodic properties similar to those of the limiting processes. Some partial answer to this question will be presented.

An interior-point method for solving mathematical programs with

equilibrium constraints (MPECs) is proposed. At each iteration of the

algorithm, a single primal-dual step is computed from each subproblem of

a sequence. Each subproblem is defined as a relaxation of the MPEC with

a nonempty strictly feasible region. In contrast to previous

approaches, the proposed relaxation scheme preserves the nonempty strict

feasibility of each subproblem even in the limit. Local and superlinear

convergence of the algorithm is proved even with a less restrictive

strict complementarity condition than the standard one. Moreover,

mechanisms for inducing global convergence in practice are proposed.

Numerical results on the MacMPEC test problem set demonstrate the

fast-local convergence properties of the algorithm.

I shall talk on recent results on behaviour of solutions of

2D Navier-Stokes Equation (and some other related equations), perturbed by a random force, proportional to the square root of the viscosity. I shall discuss some properties of the solutions, uniform in the viscosity, as well as the inviscid limit.

Hamiltonian Feynman path integrals, or Feynman (path) integrals over

trajectories in the phase space, are values, which some

pseudomeasures, usually called Feynman (pseudo)measures (they are

distributions, in the sense of the Sobolev-Schwartz theory), take on

functions defined on trajectories in the phase space; so such

functions are integrands in the Feynman path integrals. Hamiltonian

Feynman path integrals (and also Feynman path integrals over

trajectories in the configuration space) are used to get some

representations of solutions for Schroedinger type equations. In the

talk one plans to discuss the following problems.

The trapezoidal rule for numerical integration is remarkably accurate when

the integrand under consideration is smooth and periodic. In this

situation it is superior to more sophisticated methods like Simpson's rule

and even the Gauss-Legendre rule. In the first part of the talk we

discuss this phenomenon and give a few elementary examples. In the second

part of the talk we discuss the application of this idea to the numerical

evaluation of contour integrals in the complex plane.

Demonstrations involving numerical differentiation, the computation

of special functions, and the inversion of the Laplace transform will be

presented.

Marstrand's Theorem is a one of the classic results of Geometric Measure Theory, amongst other things it says that fractal measures do not have density. All methods of proof have used symmetry properties of Euclidean space in an essential way. We will present an elementary history of the subject and state a version of Marstrand's theorem which holds for spaces whose unit ball is a polytope.

We present a large deviation result for the behaviour of the

end-point of a diffusion under the influence of a strong drift. The rate

function can be explicitely determined for both attracting and repelling

drift. It transpires that this problem cannot be solved using

Freidlin-Wentzel theory alone. We present the main ideas of a proof which

is based on the Girsanov-Formula and Tauberian theorems of exponential type.

Let {X_n} be a sequence of bounded, real-valued random variables.

Assume that the partial-sums processes {S_n}, where S_n=X_1+...+X_n,

satisfies the large deviation principle with a convex rate-function, I().

Given an observation of the process {X_n}, how would you estimate I()? This

talk will introduce an estimator that was proposed to tackle a problem in

telecommunications and discuss it's properties. In particular, recent

results regarding the large deviations of estimating I() will be presented.

The significance of these results for the problem which originally motivated

the estimator, estimating the tails of queue-length distributions, will be

demonstrated. Open problems will be mentioned and a tenuous link to Oxford's

Mathematical Institute revealed.