Workshops

• Advances in partial differential equations in physics and materials science, IACM/FORTH, May 29-31, 2024.

• Stochastic Analysis in Mathematics and Natural Sciences: Theory and Applications, June 5-6 2025, IACM/FORTH, Crete 

 Group working seminars

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

December 2023 – May 2024 (The seminars were taking place on Thursdays, 4 pm)

• Asymptotic analysis methods for deterministic and stochastic reaction-diffusion equations (Speaker: Georgia Karali)

At first, a brief review of asymptotic methodologies on deterministic and stochastic reaction-diffusion equation is given. Then, we discuss limiting behavior of stochastic partial differential equations driven by multiplicative noise and deterministic non-autonomous terms. We also examine as a special case, reactions-diffusion systems driven by colored noise. This lecture introduces the basic mathematical tools of asymptotic analysis for the next lectures.

• Sharp Interface problems (Speaker: Georgia Karali)

A sharp interface problem for the Cahn-Hilliard equation is studied. More precisely and by using asymptotic analysis, we investigate that the sharp interface limit for the multidimensional homogeneous generalized Cahn-Hilliard equation is the homogeneous Hele-Shaw problem.  In addition, we analyze an analogous problem for the stochastic Cahn-Hilliard equation.

• Study of deterministic and stochastic free boundary problems (Speaker: Kostas Tzirakis)

We specialize in the study of the two-phase Stefan free boundary problems.

• Study of radial solutions (Speaker: Kostas Tzirakis)

We study radial solutions for the deterministic and stochastic combined Cahn-Hilliard/Allen-Cahn equation.

• Malliavin derivative with applications to finance I (Speaker: Dimitris Farazakis)

We introduce the canonical probability space and we define the Malliavin operator in this space. We apply this operator on the stochastic Ito integral by using the Malliavin formula. Then, we study existence and uniqueness theorem of stochastic reflected partial differential equation with only one reflecting barrier in combination with the definition of a localization argument (a localization of a random variable).

• Malliavin derivative with applications to finance II (Speaker: Dimitris Farazakis)

We investigate Malliavin calculus on stochastic reflected partial differential equations driven by multiplicative space-time white noise with Dirichlet conditions. The existence of absolute continuity of a random variable is examined, and as a consequence it implies that a density exists. In addition, we consider some applications in mathematical finance.

September 2024 – December 2024 (The seminars were taking place on Thursdays, 5 pm)

• Existence and uniqueness of the stochastic heat equation: (Speaker: Georgia Karali)  We provide a description of a weak formulation of the heat equation and its piecewise approximation as well, in order to derive existence and uniqueness results.
  
  

•  Localization spaces:  (Speaker: Kostas Tzirakis) This presentation is related to the spaces for which the main variable of the stochastic heat equation is localized, and then, convergence results are established.
  
  

• Malliavin derivative for the stochastic heat equation: (Speaker: Dimitris Farazakis) We first develop the process for applying the Malliavin’s formula for the stochastic heat equation, and then, we proceed with the proof of the corresponding existence and uniqueness problem.
  
  

• Regularity problem: (Speaker: Georgia Karali)  By using the Malliavin estimations of the previous lecture, we prove the L_{1,2}-regularity condition for u, where u represents the solution of the stochastic partial differential equation.
  
  

• Absolute continuity problem (Part I): (Speaker: Dimitris Farazakis) This presentation is an introduction about the sufficient condition in order to prove existence of density for the stochastic heat equation, and the connection between existence of density and the known results from Malliavin’s calculus.
  
  

• Absolute continuity problem (Part II): (Speaker: Dimitris Farazakis) In continuation to the previous lecture, we establish two main estimations concerning the upper and lower bound.

February 2025 – June 2025 (The seminars were taking place on Mondays, 4 pm)

• Density criteria for stochastic systems (Part I): We investigate generalized Hörmander-type conditions for the existence and smoothness of densities in degenerate stochastic differential equations.

• Density criteria for stochastic systems (Part II):  We extend classical results to non-Lipschitz and singular drift settings.

• Applications to spdes and non-semimartingale models (Part I)

We study non-semimartingale models to stochastic partial differential equations   with singularities.

• Applications to spdes and non-semimartingale models (Part II).

We focus on models that do not fit into the classical semimartingale framework and how stochastic calculus enables both quantitative and quantitative analysis of such systems.

•  Malliavin calculus for Lévy processes and analysis on Poisson spaces: A discussion of Malliavin calculus in the context of jump processes, including Poisson space differentiation, and applications.