## LECTURES

The proposed lecturers for this school, and suggested topics of their lectures, are the following:

1. Hakima BESSAIH (University of Wyoming Laramie, WY, USA): *An introduction to stochastic fluid dynamics. *

Abstract: We will give an overview of some models motivated by Hydrodynamics. These models are described by the Navier-Stokes equations and related models driven by a noise. We will study existence and uniqueness of solutions, the continuous dependence with respect to initial conditions and the longtime behavior of solutions through their invariant measures and/or random attractors. Moreover, we will tackle some results about regularity of solutions. We will mainly deal with the 2D case. The 3D case will be studied thought some approximations like the alpha models. \\

2. María J. GARRIDO-ATIENZA (University of Sevilla, Spain): *Fractional Calculus and Stochastic Differential/Partial Equations driven by fractional noise.*

Abstract: The first goal of this course will be to study the existence and uniqueness of equations driven by Hölder continuous functions, for which we will make use of tools of Fractional Calculus. Remarkable differences will appear depending than the Hölder exponent is greater or smaller than 1/2. More precisely, we will pay special attention to the study of pathwise solutions of stochastic equations driven by a fractional Brownian motion, when the Hurst parameter H>1/2 as well as when 1/3<H<1/2, emphasizing the different methods needed to handle both situations. In a second part of the course, and thanks to the fact that the solutions of the equations will generate a cocycle, we will investigate the longtime behaviour of the solutions by analyzing the random attractor associate to these equations.

Annie MILLET (Université Paris 1, France): *An introduction to Large Deviations for SDEs and SPDEs.*

Abstract: The aim of the course is to give an introduction to usual techniques to prove Large Deviations Principles (LDP) either from scratch (for empirical means, Markov chains and Gaussian processes) or to deduce them from already known LDP results by means of "continuous maps except on an exponentially small set" or the Varadhan Lemma. The course will cover the following 6 topics: 1. The log Laplace transform and its Legendre transform. 2. The Ellis Görtner theorem and applications to Markov chains. 3. Schilder's theorem for Brownian motion and some general Gaussian processes. 4. The contraction principle; the Freidlin-Wentzell inequality. 5. Varadhan's Lemma and the inverse Bryc Lemma; the weak convergence approach to LDP. 6. Applications to some SPDEs (such as the stochastic 2D Navier Stokes equations).

David NUALART (University of Kansas, USA): *Rough Path Analysis.*

Abstract: This course is an introduction to the theory of rough paths analysis, which aims to solve multidimensional differential equations controlled by a deterministic input forcing, which has finite p--variaton for some p>1. First we will treat the case p<2, when the equation can be formulated using Young's integral, thus one can establish the existence and uniqueness of a solution to the equation. For the case p between 2 and 3, we will define integrals with respect to rough paths, applying Gubinelli's approach of controlled paths. Using rough paths integrals, we will construct and show the continuity of the so-called Ito- Lyons map that defines the solution to the differential equation as a function of the enhanced input signal. As an application we will discuss the classical Ito and Stratonovich SDE driven by a standard Brownian and SDE driven by a fractional Brownian motion.

Björn SCHMALFUSS, (Universität Jena, Germany): *Random dynamical systems.*

Abstract: The content of these lectures is to consider dynamical systems under the influence of noise. At first we give a description of a general (ergodic) noise. Examples for this kind of noise are the white noise, the Ornstein-Uhlenbeck noise, fractional noise and Lévy noise. Considering systems under the influence of noise we have to study instead of a semigroup a so-called cocycle. Under appropriate measurability assumptions a random dynamical system is a cocycle. We will introduce steady states, bifurcations of steady states, stable, unstable, inertial manifolds, and attractors for random dynamical systems. We then will consider several examples from the theory of stochastic random pde's generating random dynamical systems and discuss their dynamical behavior by means of these objects.

Michael SCHEUTZOW (TU Berlin, Germany): *Stochastic Delay Equations.*

Abstract: Unlike ordinary stochastic differential equations in d-dimensional Euclidean space driven by multidimensional Brownian motion, the drift and/or diffusion coefficients of a stochastic delay differential equation (SDDE) do not only depend on the current state but also on the state in the past. Therefore, solutions are not Markov processes when considering in the Euclidean phase space, but typically generate Markov processes taking values on some function space, for example the space of continuous functions on [-1,0] in case the maximal delay is 1. We will briefly discuss existence and uniqueness of solutions and then turn to the question of existence and uniqueness of invariant probability measures for the solution Markov process. Methods developed in joint work with Martin Hairer and Jonathan Mattingly and further refined in joint work with Alex Kulik will be discussed.