Workshops

• Women in Applied Mathematics (online workshop) February 19 2021

• Stochastic Differential Equations: Analysis & Modelling,  June 7-8 2023, Heraklion, Crete

Group working seminars

The Group working seminars are taking place online via the Zoom platform. For any further information please contact Georgia Karali. 

February 2020 – May 2020 (The seminars  were taking place  on  Tuesdays, 3 pm)

•  A basic introduction to Stochastic Analysis (Speaker: Dimitra Antonopoulou):. In this seminar, basic concepts of stochastic analysis are discussed in order to describe the derivative of stochastic functions  (i.e., functions belonging to a probability space (Ω, F, P)). This includes explanation of the measurable and adapted functions, Martingales, stopping time, Brownian motion and canonical Brownian motion (Wiener process).

• The action of Malliavin operator on the Itô-stochastic integrals (Speaker: Dimitra Antonopoulou):  Here we examine the behavior of  stochastic functions (square integrable random variables)  in combination with the action of Malliavin operator on them. This is actually the application of Malliavin operator on the Itô stochastic integrals.

• Symmetric square integrable functions and the n-fold iterated Itô integral (Part I) (Speaker: Dimitris Farazakis):  The specific topic of this lecture is basic for the study of the Wiener- Itô chaos expansion and generally of the Malliavin calculus. Symmetric square integrable functions, Itô isometry and the Wiener process form the basic structure of the n-fold iterated Itô integral.

• Symmetric square integrable functions and the n-fold iterated Itô integral (Part II) (Speaker: Dimitris Farazakis):  By using the previous lecture we complete the multiple Itô integral representation. The action of the expectation on the square form of the n-fold iterated Itô integrals studied. In addition,we discuss some useful examples combining in this way the tensor product with the n-fold iterated Itô integral as well as the Hermite polynomial. 

• Hermite polynomials (Speaker: Georgia Karali):  In this seminar, we exploit Hermite’s polynomial and its properties according with some examples. Moreover, we combine and discuss how this particular polynomial blends with the multiple stochastic Itô-integrals.

October 2020 – December 2020 (The seminars  were taking place  on  Tuesdays, 4 pm)

•  The Wiener-Itô chaos expansion (Speaker: Dimitra Antonopoulou):  We represent the Wiener- Itô chaos expansion on the probability space (Ω, F, P). We also analyze some basic keys of its proof in combination with Itô  representation theorem on the Ft-measurable random variables, convergence in L2(Ω), Hermite polynomials, symmetrization and Itô isometry.

• The Wiener-Itô chaos expansion & the Skorohod integral (Speaker: Dimitris Farazakis):  Firstly we introduce the basic directions of the Skorohod integral. After that an extension of this type of integral in terms of Wiener chaos expansion is completed. Finally we mention the Skorohod-integrable random variable as well as some corresponding examples.                                              .

• The Hilbert topological vector space (Speaker: Dimitris Farazakis):  In this lecture we adapt the topological structure of the Hilbert space to interfaces and free boundary problems. That is an introduction to the L^p-spaces especially when p=2, i.e., when we have the case where H is an-space over a measure space of Borel subsets (white noise case).

• The isonormal Gaussian process and its association with the Brownian motion (Speaker: Georgia Karali):  This lecture is a step before the action of Malliavin operator on the Itô-stochastic integrals. Inner product, isonormal Gaussian process, Brownian motion and the Malliavin operator (and its properties) compose the major topics of this session.

• Generation and propagation of reaction-diffusion equations (Speaker: Kostas Tzirakis):  The main topic is related to the asymptotic behavior of the solution  of a reaction-diffusion equation as ε tends to zero. Here ε is a small positive parameter. Then a study about the propagation of the interface with respect to the normal velocity follows.

• Metastable patterns for the Cahn-Hilliard equation (Part I) (Speaker: Kostas Tzirakis):  We introduce the integrated Cahn-Hilliard equation and discuss the spectrum of the linearized integrated Cahn-Hilliard operator. Then, we examine the dynamics of this model in one-dimensional in a neighborhood of an equilibrium having N+1 transition layers.                                                         

• Metastable patterns for the Cahn-Hilliard equation (Part ΙΙ) (Speaker: Kostas Tzirakis):  Based on the previous lecture we extend the study of the Cahn-Hilliard equation (in one-dimension) in a finite interval, in a neighborhood of an equilibrium having N+1 transition layers.  

February 2021 – May 2021  (The seminars  were taking place  on  Wednesdays, 3 pm)

• An introduction to reaction-diffusion equations (Speaker: Georgia Karali):  This seminar aims to provide an introduction concerning the physical and mathematical background of  some deterministic and well known in the literature reaction-diffusion equations..

• Surface dynamics (Speaker: Georgia Karali):  Based on the previous lecture,we analyze the surface evolution equations. That is a discussion about mean field partial differential equation, macroscopic cluster evolution laws and transport structure. 

•  A deterministic modified Cahn-Hilliard/Allen-Cahn model (Speaker: Georgia Karali):  Here, we discuss the deterministic modified Cahn-Hilliard/Allen-Cahn model and it’s physical background, i.e, surface diffusion including particle/particle interactions (Cahn-Hilliard) and adsorption to and desorption from the surface (Allen-Cahn).                                                        .

•  Space-time white noise in the sense of Walsh (Part I) (Speaker: Dimitris Farazakis):  The purpose of this seminar is to give a path in order to reach the space-time white noise in the sense of Walsh. This demands the introduction of the Brownian sheet by using the Schwartz distribution (the topology of the Schwartz space and its metric). Then, an extensive discussion about the tempered distributions is made.

•  Space-time white noise in the sense of Walsh (Part II) (Speaker: Dimitra Antonopoulou):  Here, we describe topics such as Fourier transform, Radon measures, measurable functions and martingale measure (worthy martingale measure). Then we express and analyze the space-time white noise in the sense of Walsh.

• Stochastic reaction-diffusion equations (Part I) (Speaker: Dimitris Farazakis):  In this lecture, we first discuss some basic notions of the deterministic version of  popular and well known reaction-diffusion deterministic equations. Then we discuss their stochastic versions with space-time white noise in the sense of Walsh as it was describe in the previous lectures.

• Stochastic reaction diffusion equations (Part II) (Speaker: Dimitra Antonopoulou):  We continue the study of particular stochastic reaction diffusion equations and expoloit them further with the presence of space-time noise.

October 2021 – December 2021  (The seminars  were taking place  on  Wednesdays, 4 pm)

•  Frechet differential functions and their relation with Gateaux derivative (Speaker: Georgia Karali): The purpose of this lecture is to describe the Frechet  and Gateaux derivative as a basic tools of the next section, i.e.,  Malliavin calculus.

• The Malliavin derivative (Part I) (Speaker: Dimitris Farazakis): At first, a brief representation was developed into Wiener space. The main goal is to extend the idea of the derivative operator to the stochastic functions, i.e., functions defined to the complete probability space (Ω, F, P). This is the main idea of the Malliavin derivative.                                                   .

• The Malliavin derivative (Part II) (Speaker: Dimitra Antonopoulou): By using an isonormal Gaussian process  associated with Hilbert space, we defined the smooth random variables in order to proceed to their ‘Malliavin’ derivation, through Malliavin operator. After-that some basic properties of the Malliavin operator follows as well as some useful examples.                                                          

• The absolute continuity criterion (Speaker: Dimitra Antonopoulou): In this presentation we study the absolute continuity criterion through the existence and uniqueness of the modified model of Cahn-Hilliard/Allen-Cahn. This proof is established by using Malliavin calculus as we developed in the two previous lectures.                                                            

• The existence of probability density  (Speaker: Dimitris Farazakis): Through the absolute continuity criterion we prove that the density in the case of the stochastic model Cahn-Hilliard/Allen-Cahn exists. This result requires a localization argument, i.e., the study of the regularity spaces related with Malliavin calculus (non-degenerate properties and requirements of Malliavin derivative with respect to the stochastic functions).

• Stochastic Cahn-Hilliard equation and the Landau-Ginzburg free energy functional (Speaker: Kostas Tzirakis): Here we develop the basic keys of the theory of  Landau-Ginzburg free energy functional  and also we combine this theory with stochastic Cahn-Hilliard equation. More precisely, chemical potential, gradient flow and variational derivative are described.

• Stochastic Allen-Cahn equation and the Landau-Ginzburg theory (Speaker: Kostas Tzirakis): Keeping the knowledge from the previous lecture,  we express the combination of stochastic Allen-Cahn equation and the  free energy functional.

February-May 2022 (The seminars were taking place on Fridays, 4 pm)

• Basic inequalities (Speaker: Georgia Karali): The specific topic of this lecture is to analyze some basic inequalities in the field of the deterministic partial differential equations, as Holder inequality, Jensen inequality and Minkowski inequality.

• Martingales (Speaker: Dimitris Farazakis):  We have spoken in the previous seminar about this topic of stochastic analysis, but here we immerse in the section of Martingales as a path for the next lecture, i.e., the quadratic variation. 

• Hormander’s theorem through Mallliavin calculus (Speaker: Dimitris Cheliotis): The aim of this lecture is to introduce basic field of Malliavin calculus and then to present  the smoothness of transition probabilities for a diffusion under Hormander’s condition.

• Quadratic variation and covariation processes (Part I) (Speaker: Kostas Tzirakis): Firstly, the quadratic variation process is a useful part of Burkholder-Davis-Gundy inequality, which it is the next seminar. Thus, topics as partition, mesh, quadratic variation of Brownian motion have been spoken. 

• Quadratic variation and covariation processes (Part II)  (Speaker: Dimitra Antonopoulou): We represent variation and covariation processes for martingales and their properties. 

• Doob’s inequality (Speaker: Georgia Karali): Here we present the Doob’s inequality and its contribution to other fundamental stochastic inequalities. 

October 2022 – December 2022  (The seminars  were taking place  on  Mondays, 5 pm)

• Integration by parts and Girsanov theorem (Speaker: Dimitris Farazakis): Here we study the behavior of a stochastic process with respect to the change of original measure. The new measure is an equivalent probability measure.

• The Clark-Ocone formula (Speaker: Kostas Tzirakis): We use the previous seminar in order to study the Clark-Ocone formula as a cornerstone of Malliavin calculus on finance. In addition, this is an introduction for the next seminar related to the generalized Clark-Ocone formula.

• The generalized Clark-Ocone formula (Speaker: Kostas Tzirakis): This type of formula represents a measurable random variable as a stochastic integral with respect to the Wiener process derived from the Girsanov theorem. This formula has a major financial impact.

• Applications of Malliavin calculus in finance (Part I) (Speaker: Dimitris Farazakis): A discussion about the Black & Scholes formula, i.e., firstly we analyze the action of the generalized Clark-Ocone formula on a portfolio analysis (safe investment, risky investment, self-financing portfolio).

• Applications of Malliavin calculus in finance (Part II) (Speaker: Dimitris Farazakis): In this lecture it is proven a Black & Scholes formula. In addition, we represent some relevant examples.

• Applications of Malliavin calculus in finance (Part III) (Speaker: Dimitris Farazakis): By using the integration by parts formula, we focus on the computation of Greeks, i.e., the quantities expressing the sensitivity of the price of derivatives.

Janouary-June 2023 (The seminars were taking place on Fridays, 7 pm)

• Sharp interface limit (Part I) (Speaker: Georgia Karali): We study in a theoretical framework diffuse interface models and their sharp interface limits. We refer to the physical background and some basic properties.

• Sharp interface limit (Part II) (Speaker: Georgia Karali): We discuss the sharp interface limit for the stochastic Cahn-Hilliard equation. Specifically, we estimate the convergence of stochastic Cahn-Hilliard equation to a Hele-Shaw problem with stochastic forcing on the curvature equation.

•  Front propagation phenomena (Part I) (Speaker: Dimitris Farazakis): We discuss the general category of reaction-diffusion equations (in exterior domains), that have major impact to the propagation phenomena in the presence of obstacles. Then, we refer the three phases of the wave propagation problem and the respectively mathematical statement (the front has not reached the obstacle, the wave front is reaching the obstacle, the shape of the wave front after hitting the obstacle).

• Front propagation phenomena (Part II) (Speaker: Dimitris Farazakis): We are interested in the study of the Nagumo reaction-diffusion equation as a set of coupled partial differential equations. Then, we discuss what is happening in higher space dimensions.

• Spectral estimates for stochastic partial differential equations (Part I) (Speaker: Kostas Tzirakis): We discuss the general framework of spectral estimates for initial-boundary value problems for stochastic parabolic differential equations.

• Spectral estimates for stochastic partial differential equations (Part II) (Speaker: Georgia Karali): We represent the dynamics of the Cahn-Hilliard / Allen-Cahn equation in a neighborhood of a layered equilibrium parameterized by a small positive constant ε. Then, the spectrum of the linearized Cahn-Hilliard / Allen-Cahn operator is investigated.