CONTENTS -- Preface -- Participants -- Stochastic parallel transport on the d-dimensional torus Ana Bela Cruzeiro and Paul Malliavin -- 1. Basis of Lie algebra of vector elds with vanishing divergence -- 2. Constants of structure of the Lie algebra and Christofell symbols -- 3. Stochastic parallel transport, symmetries of the noise -- 4. Transfer energy matrix of the stochastic parallel transport -- 5. Exact computation of the energy transfer matrix for d = 2 -- 6. Qualitative estimation of the energy transfer matrix for d > 2 -- References -- Riemannian geometry of Di (S1)=S1 revisited Maria Gordina -- 1. Introduction -- 2. Virasoro algebra -- 3. Diff(S1)=S1 as a K ahler manifold -- Acknowledgments -- References -- Ergodic theory of SDE's with degenerate noise Antti Kupiainen -- 1. Stationary states for SDE's -- 2. A model of heat conduction -- 3. Closure equations -- 4. Conclusions -- References -- Dynkin's isomorphism without symmetry Yves Le Jan -- 1. Introduction -- 2. The nite case -- 2.1. Dual processes -- 2.2. A twisted Gaussian measure -- 2.3. Positivity -- 3. The in nite case -- 3.1. Some calculations in Gaussian space -- 3.2. A class of Markov processes in duality -- 3.3. An extension of Dynkin's isomorphism -- References -- Large deviations for the two-dimensional Yang-Mills measure Thierry L evy -- 1. The Yang-Mills measure as a Gibbs measure -- 1.1. The Brownian motion -- 1.2. The Yang-Mills action -- 1.3. The necessity to consider holonomy -- 2. The Yang-Mills measure as a stochastic process -- 2.1. Finite-dimensional distributions -- 2.2. The Yang-Mills process -- 3. The large deviations principle -- 4. Disintegration of YMT according to the topology -- 4.1. The fundamental group of U(N) -- 4.2. A homotopic splitting of the Yang-Mills process -- 4.3. Topology of principal bundles -- References.

Laplace operator in networks of thin bers: Spectrum near the threshold S. Molchanov and B. Vainberg -- 1. Introduction -- 2. Historical remarks -- 3. Scattering solutions -- 4. Spectrum of the problem on the limiting graph -- 5. Resolvent convergence of the operator H" -- 6. The GC at near the threshold -- 7. Effective potential -- References -- Adiabatic limits and quantum decoherence Rolando Rebolledo and Dominique Spehner -- 1. Introduction -- 2. Adiabatic perturbations -- 3. Adiabatic limits -- 4. Invariant states and adiabatic perturbations. Decoherence -- 5. The quantum exclusion semigroup -- References -- Gauge theory in two dimensions: Topological, geometric and probabilistic aspects Ambar N. Sengupta -- 1. Introduction -- 2. Yang-Mills Gauge Theory -- 3. Wilson loop integrals in two dimensions -- 4. Yang-Mills on surfaces and Chern-Simons: the symplectic limit -- 4.1. From four dimensions to three: the Chern-Simons form -- 4.2. From three dimensions to two: the U(1) bundle over the space of connections on a surface -- 4.3. Connection on the U(1) bundle over the space of connections -- 4.4. From Chern-Simons to Yang-Mills on a surface -- 4.5. The symplectic limit -- 5. Concluding Remarks -- Acknowledgments -- References -- Near extinction of solution caused by strong absorption on a ne-grained set V. V. Yurinsky and A. L. Piatnitski -- 1. Introduction -- 2. Near Extinction of Solution -- 2.1. Weak solutions -- 2.2. The main result -- 2.3. Proof of Theorem 2.1 -- 3. A theorem on homogenization -- Acknowlegments -- Appendix A. -- A.1. Proof of Lemma 2.1. -- A.2. An embedding inequality -- A.3. Inequalities for random chessboard -- References.