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.