# Analysis-Stochastik-Seminar (Archive)

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# Program SoSe 2016

## Minicourses

###
Jump Process Convergence

Robert Patterson
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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

### Dates:

14.07.16,

10:00 AM,

Room 406
07.07.16,

10:00 AM,

Room 406
30.06.16,

10:00 AM,

Room 406
23.06.16,

10:00 AM,

Room 406
###
Uncertainty Quantification for hysteresis operators

Olaf Klein
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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.

### Dates:

16.06.16,

10:00 AM,

Room 406
02.06.16,

10:00 AM,

Room 406
26.05.16,

10:00 AM,

Room 406
19.05.16,

10:00 AM,

Room 406
###
An overview on classical physical methods (Lagrange, Hamilton, Hamilton-Jacobi)

Holger Stephan
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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.

### Dates:

12.05.16,

10:00 AM,

Room 406 (Solving the Hamilton-Jacobi equation by the Hopf-Lax method)

21.04.16,

10:00 AM,

Room 406 (From Lagrange to Hamilton and Hamilton-Jacobi)

18.02.16,

09:00 AM,

Room 406
11.02.16,

10:00 AM,

Room 406
# Program SoSe 2015 and WiSe 2015/16

## Winter term

### Part II: Applications

###
04.02.16, 10:00 AM, Room 406

Michiel Renger
Large deviations of stochastic processes (formal) Part 3

###
21.01.16, 10:00 AM, Room 406

Michiel Renger
Large deviations of stochastic processes (formal) Part 2
###
14.01.16, 10:00 AM, Room 406

Michiel Renger
Large deviations of stochastic processes (formal)
We like to dive a bit deeper into large deviatioLarge deviations of stochastic processes (formal) ns of stochastic processes. With this in mind, I will discuss the general ideas from the book of Feng and Kurtz on a formal level.

###
10.12.15, 10:00 AM, Room 406

Christian Hirsch
Large deviations in loss networks
A loss network can be described by a family of links in a discrete space
appearing and vanishing according to a Poisson point process. Capacity
constraints give rise to an interacting particle system whose
large-deviation behavior is considered. This talk is based on a paper of
C. Graham and S. Méléard (MPRF '96).

###
03.12.15, 10:00 AM, Room 406

Thomas Frenzel
On the Relation of Large Deviations and Gradient Flows
We consider a sequence of stochastic processes that satisfy a LDP. Since we consider continuous time processes the rate functional is acting on paths.
We draw the connection to gradient flows by observing that the deterministic limit path obeys a gradient flow.
The talk is based on the paper "On the Relation between Gradient Flows and the Large-Deviation Principle,
with Applications to Markov Chains and Diffusion" by Mielke, Peletier and Renger.

###
26.11.15, 10:00 AM, Room 406

Franziska Flegel
Large deviations for normalized local times of jump processes
We use the Gärter-Ellis theorem to find the rate function for the normalized local times of jump processes restricted to a finite domain. As we will see, the rate function relates to the Dirichlet form of the underlying Markov generator.

###
19.11.15, 10:00 AM, Room 406

Holger Stephan
Inequalities for Markov operators and the direction of time (Part 2)
###
12.11.15, 10:00 AM, Room 405

Holger Stephan
Inequalities for Markov operators and the direction of time
Im diesem Übersichtsvortrag wird der Zusammenhang einiger grundlegender Begriffe aus Analysis (stetige Funktionen, Radonmaße) Stochastik (Momente, W-Maße) und statistischer Physik (Beobachtungen, Zustände) besprochen. Des weiteren wird untersucht, in welcher zeitlichen Richtung Markowoperatoren (rückwärts) und ihre adjungierten (vorwärts) wirken. Das führt auf Ungleichungen für Markowoperatoren (Entropieungleichungen), die als Audruck der zeitlichen Irreversibilität der zugrundeliegenden physikalischen Prozesse interpretiert werden können.

###
05.11.15, 10:00 AM, Room 405

Michiel Renger
An alternative derivation of Schilder's theorem using abstract white noise
Analogous to what we discussed last time, we use Gärtner-Ellis to derive Schilder's Theorem. This time however, we will not work with Brownian motion via Itô, but rather with its time derivative (white noise) directly. We will see that the abstract definition of white noise is a very natural formulation to apply Gärtner-Ellis to.

###
22.10.15, 10:00 AM, Room 405

Robert Patterson
Schilder's Theorem as a Special Case of Gärtner-Ellis
## Sommer term

###
08.07.15, 10:00 AM, Room 405

Sebastian Jachalski
Law of the iterated logarithm and stochastic Itô-equation

###
01.07.15, 10:00 AM, Room 405

Robert Patterson
Schilder's theorem
### Part I: Basics

###
24.06.15, 10:00 AM, Room 405

Christian Hirsch
Gärtner-Ellis (Chapter V of Frank den Hollander's ''Large Deviations'')
###
17.06.15, 10:00 AM, Room 405

Christian Hirsch
Gärtner-Ellis (Chapter V of Frank den Hollander's ''Large Deviations'')
We consider LDPs for dependent random vectors. In particular, we prove the Gärtner-Ellis Theorem, which illustrates the strong connection between LDPs and the existence of certain asymptotic Laplace functionals.

###
10.06.15, 10:00 AM, Room 405

Chiranjib Mukherjee
Non-compact spaces
We extend classical Donsker-Varadhan theory of large deviations to non-compact spaces. The main idea is to compactify the quotient space of probability measures on R^{d} under the action of the additive group of translations.

###
03.06.15, 10:00 AM, Room 405

Chiranjib Mukherjee
Occupation Measures, Weak LDP
###
27.05.15, 10:00 AM, Room 405

Chiranjib Mukherjee
Occupation Measures, Weak LDP
Large deviations for occupation measures for a class of Markov processes taking values in a compact space (weak LDP, Donsker-Varadhan)

###
20.05.15, 10:00 AM, Room 405

Markus Mittnenzweig
Large Deviations for Markov sequences

###
13.05.15, 10:00 AM, Room 405

Markus Mittnenzweig
Large Deviations for Markov sequences
###
06.05.15, 10:00 AM, Room 405

Michiel Renger
[den Hollander] Chapter III.5, III.6, and [Dembo and Zeitouni] Chapter I.2 and Lemma 4.1.23
We consider two very essential tools: the contraction principle and exponential tightness, and possibly the connection with `good' rate functions.

###
29.04.15, 10:00 AM, Room 405

Michiel Renger
Sanov's large-deviation principle and Entropy
To offer some background, I want to discuss the relation between Sanov's large-deviation principle, yielding a relative entropy, and the physical notions of Boltzmann entropy and Helmholtz free energy.

###
22.04.15, 10:00 AM, Room 405

Renato Soares dos Santos
General Theory (Chapter III of Frank den Hollander's ''Large Deviations'')

###
15.04.15, 10:00 AM, Room 405

Thomas Frenzel
The Empirical Measure, Countable State Space (Chapter II)
###
08.04.15, 10:00 AM, Room 405

Thomas Frenzel
The Pair Empirical Measure (Chapter II of Frank den Hollander's ''Large Deviations'')
In a first step we deal with large deviations for the pair empirical measure with a finite state space. It is noteworthy that the sequence of the pair empirical measure considered here does not satisfy the strong i.i.d. assumption. The extension to a countable state space is done with the standard emprirical measure.

###
18.03.15, 10:00 AM, Room 405

Franziska Flegel
Proof of Cramér's Theorem (Chapter I of Frank den Hollander's ''Large Deviations'')
###
11.03.15, 10:00 AM, Room 405

Franziska Flegel
Introduction, Cramér's Theorem (Chapter I of Frank den Hollander's ''Large Deviations'')
We define what we understand by ''Large Deviation'' and find a result for the case of i.i.d. sequences under the assumption that all exponential moments are finite (Cramér's Theorem).