Abstract: We analyze temperature measurements in the pacific ozean from the KSST database. Using a numerical embedding technique -the isomap algorithm- it is possible to show that the system lives basically on a three dimensional manifold. From the embedding transform, three time series are obtained which are further processed using a backfitting algorithm, eventually yielding a system of three coupled difference equations. The methods are discussed in detail and the final result is interpreted in terms of an oscillatory system coupled to a delayed feedback equation. Physically, one can argue that this third component represents the thermocline depth, which might relate to the El nino events.

Abstract: Geometric singular perturbation theory offers an efficient framework for the study of ordinary differential equations with well-separated time scales, so-called slow-fast systems. It combines the construction of invariant manifolds, which allow a low-dimensional effective description of the dynamics reduced to slow variables, with a local analysis near bifurcation points. We present extensions of this theory to systems of slow-fast stochastic differential equations, constructing, in particular, neighbourhoods of invariant manifolds in which sample paths concentrate. We also discuss the effect of noise on some generic bifurcations, and their links with phenomena such as relaxation oscillations, hysteresis and stochastic resonance. Joint work with Barbara Gentz (WIAS, Berlin). Reference: N. Berglund, B. Gentz, Geometric singular perturbation theory for stochastic differential equations, J. Differential Equations 191:1-54 (2003)

Abstract: As an example we consider the Swift-Hohenberg equation on a large (but still bounded) domain near its change of stability. This equation is a toy model for the convective instability in the Rayleigh-B\'enard model. It is well known that sufficiently close to the bifurcation, solutions are approximated by a dominating periodic pattern. But on sufficiently large domains this pattern is modulated by solutions of a Ginzburg-Landau equation. We discuss the influence of noise given by thermal fluctuations on the approximation, and give rigorous error estimates in the stochastic case. This approximation also extends to long time behaviour given by invariant measures, and has applications in pattern formation below the threshold of instability. Joint work with M. Hairer (Warwick) and G.A. Pavliotis (Imperial).

Abstract: Starting from the example of the Dansgaard--Oeschger warming events, we discuss the importance of residence-time distributions as a measure for stochastic resonance. For the paradigm of the motion of a periodically forced Brownian particle in a bistable potential, we explain the relation between first-passage-time and residence-time distributions. Going beyond the exponential asymptotics known from the Wentzell--Freidlin theory, we are able to give rigorous expressions for the densities of these distributions. In a broad range of forcing frequencies and amplitudes, the distributions are found to be close to periodically modulated exponential ones, where the periodic modulations are governed by a universal function, depending on a single parameter related to the forcing period. Joint work with Nils Berglund (CPT-CNRS Luminy, France).

Abstract: It is shown that the synchronization of dissipative systems persists when they are disturbed byadditive noise no matter how large the intensity of the noise provided asymptotically stable stationary stochastic solutions are used instead of asymptotically stable equilibria. This is joint work with Tomas Caraballo (University of Sevilla) and will appear in the Proceedings of the Royal Society of London

Abstract: We discuss an asymptotic estimation of a nonlinear Laplace transform of stochastic line integrals on a compact Riemannian manifold. With the aid of the Large deviation principle, the scheme of classical Laplace method works well. We express the limit of a proportionally re-scaled expectation and compare it with the case for mean empirical measures.

Abstract: In this talk I will introduce a time discretization for the solution of stochastic PDEs based on approximating the integrals directly. This is similar to the Lawson scheme or an exponential integrator for the deterministic case. I will introduce post-processing for both the deterministic and stochastic case - and present a Theorem on strong convergence. Depending on the regularity of the noise, solution and error norm we examine the rates of convergence. We examine the efficiency and errors numerically for the Allen-Cahn and Burgers equation.

Abstract: We give a short review on the dynamics of the Navier-Stokes equations and of possible relations with the phenomenology of turbulence, mostly related to the classical theory of Kolmogorov. Some old results from the author and new results (from a joint work in progress with F. Flandoli, M. Gubinelli and M. Hairer) are also presented.

Abstract: Ito integrals of random Banach valued functions w.r.t. compensated Poisson random measures (cPrm) are discussed in [Ru]. The Levy - Ito decomposition Theorem claims that any additive process is uniquely decomposed in a continuous semimartingale driven by a Brownian motion and a pure jump semimartingale driven by a jump martingale obtained by an Ito integral w.r.t. a cPrm. In [ARu] we prove that the Levy -Ito decomposition theorem holds also on separable Banach spaces of type 2. (It holds also on general separable Banach spaces (see Dettweiler [Dett]) but in this case the integral w.r.t. a cPrm is not defined in terms of an Ito integral) . These results [ARu], [Ru] permit us [MRu2] to define non Gaussian additive noise and to study [MRu] stochastic differential equations (SDE)with non Gaussian additive noise on separable Banach spaces. Existence and uniqueness is proven under local Lipshitz conditions for the drift and noise coefficients. The Ito formel for Banach valued functions applied to the solutions of such SDEs is proven in [RuZ]. This permits us to analyze solutions of others SDEs [MRu] .

[ARu] Albeverio, Sergio; Rüdiger, Barbara, Stochastic integration with respect to compensated Poisson random measures and the Lévy ^ÖIto decomposition theorem on separable Banach spaces, Stoch. An. and Appl. to appear 2005.

[Dett] E. Dettweiler, Banach space valued processes with independent increments and stochastic integrals, pp. 54 -83 in Probability in Banach spaces IV, Proc., Oberwolfach 1982, Springer, Berlin (1982).

[MRu] A.Mandrekar, B. Rüdiger, Existence and uniqueness of path wise solutions for stochastic integral equations driven by non Gaussian noise on separable Banach spaces, preprint SFB 611 February 2003, extended version preprint SFB 611 October 2004.

[MRu2]V. Mandrekar, B. Rüdiger, Lévy noise and stochastic integrals w.r.t. martingales on Banach spaces , preprint SFB 611 march 2005. [Ru] B. Rüdiger, Stochastic integration w.r.t. compensated Poisson random measures on separable Banach spaces, Stoch. Stoch. Reports, Vol. 76 No. 3 (June 2004) .

[RuZ] B. Rüdiger, G. Ziglio The Ito formula for Banach valued stochastic integrals obtained by integration with respect to compensated Poisson random measures, preprint SFB 611 October 2004

Abstract: The averaging method has been used to study random or non-autonomous systems on a fast time scale. We apply this method to a random abstract evolution equation on a fast time scale whose long time behavior can be characterized by a random attractor or a random inertial manifold. The main purpose is to show that the long-time behavior of such a system can be described by a deterministic evolution equation with averaged coefficients. Our first result provides an averaging result on finite time intervals which we use to show that under a dissipativity assumption the attractors of the fast time scale systems are upper semi-continuous when the scaling parameter goes to zero. Our main result deals with a global averaging procedure. Under some spectral gap condition we show that inertial manifolds of the fast time scale system tend to an inertial manifold of the averaged system when the scaling parameter goes to zero. These general results can be applied to semilinear parabolic differential equations containing a scaled ergodic noise on a fast time scale which includes scaled almost periodic motions.

Abstract: The phenomenology of tsunami waves is described and a theory of surface waves is derived from the main set of the fluidodynamic equations using the pseudo-differential operators approach. A formula for the expression of the front and the amplitude of the tsunami waves is derived using the saddle point equation. An equivalent description in terms of Hamilton-Jacobi equation is shown. The solution is reconducted to the integration over the characteristic given by the usual Hamilton equations associated to the Hamilton-Jacobi equation. An interesting remark is about the speed of the integration along the charecteristic which is many order of magnitudes faster than the usual integration over the lattice since the domain of tsunami propagation are very large. The case of non-uniform bottom is also described and connected to the WKB method. Some numerical results are shown.