Team:
Project leader: Dr. Iulian Cîmpean
Mentor: Acad. Viorel Barbu
Budget: 246.950 ron
Abstract: The theme of this project is at the interplay between analysis, probability and differential equations, with focus on qualitative and quantitative aspects of diffusion processes with possibly singular coefficients or in irregular environments. The aim is to explore in three (interconnected) directions: The first one deals with the study of local operators and associated diffusion Markov processes, the second one concerns with ergodic properties for singular stochastic differential equations, while the third one focuses on probabilistic solutions to boundary value problems under low regularity.
Objectives:
Analysis of local operators and the corresponding diffusion processes.
The study of ergodic properties for stochastic differential equations with singular coefficients on Hilbert spaces.
Probabilistic representations for boundary value problems.
Results:
I. Scientific papers:
L.Beznea, I. Cimpean, M. Roeckner, Path continuity of Markov processes and locality of Kolmogorov operators, Stoch PDE: Anal Comp (2023). DOI:10.1007/s40072-023-00304-x arXiv:2208.06036
L.Beznea, I. Cimpean, O. Lupascu-Stamate, I. Popescu, A. D. Zarnescu, From Monte Carlo to neural networks approximations of boundary value problems, submitted (2022). arxiv:2209.01432
M. Bucataru, I. Cimpean, L. Marin, Stable reconstruction of discontinuous solutions to the Cauchy problem in steady-state anisotropic heat conduction with non-smooth coefficients, ESAIM: Mathematical Modelling and Numerical Analysis, Vol 57, Nr 2, 1029 - 1062 (2023).
M. Bucataru, I. Cimpean, L. Marin, A Gradient-Based Regularization Algorithm for the Cauchy Problem in Steady-State Anisotropic Heat Conduction, Computers and Mathematics with Applications 119, 220-240, (2022).
I. Cimpean, Ergodicity of Markov processes by Girsanov transformation and applications, in progress.
L. Beznea, M. Bezzarga, I. Cimpean, Markovian flows driving Markov processes and multiplicative Lp- semigroups, in progress.
L. Beznea, I. Cimpean, M. Roeckner, From resolvents to Hunt processes in a nutshell and applications to generalized Mehler semigroups, in progress.
II. Talks at workshops/conferences/seminars:
Iulian Cimpean, On the construction of Hunt processes with applications to Mehler semigroups, Atelier de travail en Stochastique et EDP, Bucharest, October 20-21, 2020 (online).
Iulian Cimpean, Ergodicity of Markov semigroups and applications to singular SDEs, TOTA 2021 (online) Instituto Superior Tecnico, Universidade de Lisboa, 25 iunie 2021.
Iulian Cimpean, Ergodicity of Markov semigroups and applications to singular SDEs on Hilbert spaces, CSH Workshop (online): “Stochastic Dynamics for Complex Systems” Complexity Science Hub Vienna, 30 septembrie - 01 octombrie 2021.
Iulian Cimpean, On the path continuity of Markov processes, Bielefeld Stochastic Afternoon at Bielefeld University (online), 15 decembrie 2021.
Iulian Cimpean, On the path continuity of Markov processes, Stochastic Analysis Seminar - Imperial College London, 1 februarie 2022.
Iulian Cimpean, On the path continuity of Markov processes, CSH WORKSHOP: “STOCHASTIC DYNAMICS FOR COMPLEX SYSTEMS”, Complexity Science Hub, Vienna 1-3 iunie 2022.
Iulian Cimpean, From Monte Carlo to neural networks approximations of boundary value problems, Bielefeld Stochastic Afternoon at Bielefeld University, 10 august 2022.
Iulian Cimpean, Ergodicity of Markov semigroups and application to singular SDEs on Hilbert spaces, XV`eme Colloque Franco-Roumain de Mathematiques Appliqu ́ees, 29 august-2 septembrie 2022.
III. Research stages:
BCAM Bilbao (invited by Prof. Arghir Zarnescu), 5-14 iunie 2022.
Bielefeld University (invited by Prof. Michael R ̈ockner), 7-28 august 2022.