Week 1:

• Sigma-algebras and filtrations
• Weak convergence theory going beyond Polish spaces

Week 2:

The continuous and cadlag path spaces

• their filtrations and sigma-algebras
• compact sub sets (Arzela-Ascoli and generalisations)

Week 3:

• Paths of bounded variation
• Topologies on this path space

Week 4:

• Properties of the bounded variation space needed for stochastic analysis
• Worked example

Many models for hysteresis effects like magnetization or plasticity are influenced by uncertainties since there are derived from measurements. Moreover, the observable initial macroscopic state may be generated by several initial microscopic states that can not (easily) be identified, and the corresponding different evolutions of the microscopic states can generate different observable macroscopic evolutions.

To quantify the influence of these uncertainties to the models the methods of uncertainty quantification are be applied to the hysteresis operators in these models.

Talk 1
Uncertainty quantification for hysteresis operators: Introduction and relay operator

In the first talk, some fundamental aspects of uncertainty quantification are presented and the notion of hysteresis operators is introduced. As first example, the relay operator is presented and relay operators with stochastic thresholds and stochastic initial states will be considered. Moreover,the output of the relay operator for piece monotone inputs function with stochastic values for the local extrema will be discussed.

Talk 2
Uncertainty quantification for hysteresis operators: the play operator

The play operator is introduced. This operator can be considered a model for a mechanical play, and is used as building block for further hysteresis operators.

Play operators with stochastic initial states, stochastic yield limits and stochastic values for the local extrema of the input function will be will be investigated. Also the sum of play operators with stochastic initial states will be considered.

Many modern concepts in mathematics like gradient flows, large deviation principle or entropy methods have their origins in classical mechanics. Here the motion of a system of particles can be described in an equivalent way by different equations (Lagrange, Hamilton or Hamilton-Jacobi). In the mini-course we discuss the connection of these deterministic equations from an (convex) analytical and physical point of view. Disturbing the deterministic motion with some noise, we arrive at stochastic equations, the deterministic trajectory becomes a stochastic process.
Contrary, the investigation of the deterministic limit of a stochastic process is connected with Laplace's method, vanishing viscosity or large deviation principle. Using this, stochastic equations can be a tool for solving the mechanical equations. Thus one can obtain the Hopf-Lax solution of the potential free Hamilton-Jacobi equation. Using some ideas of Vlassov (WKB-method), this method can be understood as a general nonlinear Fourier transform.