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May 11, 2022, 07:49

Stochastic Stability and Potential Theory of Markov Processes

 

Stochastic Stability and Potential Theory of Markov Processes is an installation research project financed by the Croatian Science Foundation (grant no. UIP-2017-05-8958), and co-financed by the University of Zagreb, Faculty of Science. Project duration is 60 months, from May 1 2018 to April 30 2023.

 

Aims of the Project 

The focus of research of this project are two fundamental problems in the theory and applications of Markov processes: (i) stochastic stability, and (ii) potential theory of Markov processes. Both proposed topics belong to the classical direction in the analysis of Markov processes and are motivated by the current direction of research in the theory and applications of stochastic processes. Stochastically stable Markov processes naturally arise as mathematical models in a number of phenomena arising in nature and engineering, such as problems related to population dynamics and homogenization of heterogeneous structures. Potential theory originates from mathematical physics as a basic tool in studying electrostatic and gravitational problems. The goal of this project is to improve and advance existing methods, models and approaches and to develop new ones with aim of better understanding the nature of Markov processes through their stochastic stability and potential theoretical properties. For this purpose, approaches and tools from various mathematical disciplines, such as probability theory, theory of stochastic processes, mathematical analysis, and theory of differential equations will be combined.

 

Team Members
  • Nikola Sandrić (PI), Associate Professor, Department of Mathematics, Faculty of Science, University of Zagreb
  • Vanja Wagner, Assistant Professor, Department of Mathematics, Faculty of Science, University of Zagreb
  • Petra Lazić, Ph.D. student, Department of Mathematics, Faculty of Science, University of Zagreb
  • Ivana Valentić, Ph.D. student, Department of Mathematics, Faculty of Science, University of Zagreb 

 

Publications
  1. Stability of the overdamped Langevin equation in double-well potential
  2. Ergodicity of Lévy-driven SDEs arising from multiclass many-server queues
  3. Uniform polynomial rates of convergence for a class of Lévy-driven controlled SDEs arising in multiclass many-server queues
  4. On sub-geometric ergodicity of diffusion processes
  5. CLT for the capacity of the range of stable random walks
  6. Functional CLT for the range of stable random walks
  7. Periodic homogenization of a Lévy-type process with small jumps
  8. Limit theorems for a stable sausage
  9. A CLT for degenerate diffusions with periodic coefficients, and application to homogenization of linear PDEs
  10. Subexponential upper and lower bounds in Wasserstein distance for Markov processes
  11. Capacity of the range of random walks on groups
  12. Learning from non-irreducible Markov chains
  13. Hoeffding's inequality for non-irreducible Markov model
  14. On subgeometric ergodicity of regime switching diffusion processes
  15. Convex hulls of stable random walks

 

Conferences, Workshops, Schools, and Visits
  1. (P. Lazić) 48th Probability Summer School Saint-Flour, France, July 8-20, 2018
  2. (I. Valentić) 48th Probability Summer School Saint-Flour, France, July 8-20, 2018
  3. (P. Lazić) Warm-Up Week - Bielefeld Stochastic Summer, Bielefeld, Germany, August 20-23, 2018
  4. (I. Valentić) Warm-Up Week - Bielefeld Stochastic Summer, Bielefeld, Germany, August 20-23, 2018
  5. (V. Wagner) Warm-Up Week - Bielefeld Stochastic Summer, Bielefeld, Germany, August 20-23, 2018
  6. (P. Lazić) Sums & Products of Random Matrices, Bielefeld, Germany, August 27-31, 2018
  7. (I. Valentić) Sums & Products of Random Matrices, Bielefeld, Germany, August 27-31, 2018
  8. (V. Wagner) Sums & Products of Random Matrices, Bielefeld, Germany, August 27-31, 2018
  9. (Petra Lazić) 9th International Conference on Stochastic Analysis and Its ApplicationsBielefeld, Germany, September 3-7, 2018
  10. (N. Sandrić) Ergodicity of piecewise Ornstein-Uhlenbeck processes with jumps, 9th International Conference on Stochastic Analysis and Its ApplicationsBielefeld, Germany, September 3-7, 2018
  11. (I. Valentić) 9th International Conference on Stochastic Analysis and Its ApplicationsBielefeld, Germany, September 3-7, 2018
  12. (V. Wagner) 9th International Conference on Stochastic Analysis and Its ApplicationsBielefeld, Germany, September 3-7, 2018
  13. (V. Wagner) Boundary behaviour of pure-jump Markov processes on setsInverse and Spectral Problems for (Non)-Local Operators, Leipzig, Germany, September 10-14, 2018
  14. (P. Lazić) Ergodicity of Diffusion Processes, 6th Fractional Calculus, Probability and Non-Local Operators: Applications and Recent Developments, Bilbao, Spain, September 26-28, 2018
  15. (N. Sandrić) Ergodicity of piecewise Ornstein-Uhlenbeck processes with jumpsAnalysis and Stochastics Seminar, Technische Universität Dresden, Dresden, Germany, November 4-10, 2018
  16. (N. Sandrić) Wasserstein distances, ERASMUS Week, Technische Universität Dresden, Dresden, Germany, December 10-14, 2018
  17. (V. Wagner) Universität Bielefeld, Bielefeld, Germany, December 16-23, 2018
  18. (V. Wagner) An introduction to the theory of (non-local) Dirichlet forms, ERASMUS Seminar, Universität Bielefeld, Bielefeld, Germany, March 18-22, 2019
  19. (V. Wagner) Nonlocal quadratic forms with visibility constraint, Analysis of nonlocal and nonsmooth models, Universität Bielefeld, Bielefeld, Germany, March 25-29, 2019
  20. (P. Lazić) Sub-geometric ergodicity of diffusion processes, Analysis and Stochastics Seminar, Technische Universität Dresden, Dresden, Germany, April 28 - May 11, 2019
  21. (I. Valentić) Central limit theorem for periodic singular diffusions, Analysis and Stochastics Seminar, Technische Universität Dresden, Dresden, Germany, April 28 - May 11, 2019
  22. (N. Sandrić) Technische Universität Dresden, Dresden, Germany, March 30 - April 06, 2019
  23. (N. Sandrić) Ergodicity of piecewise Ornstein-Uhlenbeck processes with jumpsFujian Normal University, Fuzhou, China, May 31 - June 07, 2019
  24. (P. Lazić) Probability, Analysis, and Applications, University of Zagreb, Zagreb, Croatia, June 12-14, 2019
  25. (N. Sandrić) Probability, Analysis, and Applications, University of Zagreb, Zagreb, Croatia, June 12-14, 2019
  26. (I. Valentić) Probability, Analysis, and Applications, University of Zagreb, Zagreb, Croatia, June 12-14, 2019
  27. (V. Wagner) Boundary behaviour of Markov processes corresponding to regional Dirichlet forms, Probability, Analysis, and Applications, University of Zagreb, Zagreb, Croatia, June 12-14, 2019
  28. (P. Lazić) 2nd Korean Croatian Summer Probability Camp, University of Dubrovnik, Dubrovnik, Croatia, June 17-19, 2019
  29. (N. Sandrić) 2nd Korean Croatian Summer Probability Camp, University of Dubrovnik, Dubrovnik, Croatia, June 17-19, 2019
  30. (V. Wagner) 2nd Korean Croatian Summer Probability Camp, University of Dubrovnik, Dubrovnik, Croatia, June 17-19, 2019
  31. (P. Lazić) 11th International Conference on Extreme Value Analysis, University of Zagreb, Zagreb, Croatia, July 1-5, 2019
  32. (N. Sandrić) 11th International Conference on Extreme Value Analysis, University of Zagreb, Zagreb, Croatia, July 1-5, 2019
  33. (I. Valentić) 11th International Conference on Extreme Value Analysis, University of Zagreb, Zagreb, Croatia, July 1-5, 2019
  34. (V. Wagner) 11th International Conference on Extreme Value Analysis, University of Zagreb, Zagreb, Croatia, July 1-5, 2019
  35. (P. Lazić) Sub-geometric ergodicity of diffusion processes and Lévy-driven SDEs, A satellite Summer School to the 9th International Conference on Lévy Processes, Athens, Greece, July 8-12, 2019
  36. (I. Valentić) Central limit theorem for periodic singular diffusions, A satellite Summer School to the 9th International Conference on Lévy Processes, Athens, Greece, July 8-12, 2019
  37. (P. Lazić) Sub-geometric ergodicity of diffusion processes and Lévy-driven SDEs, 9th International Conference on Lévy Processes, Samos, Greece, July 15-19, 2019
  38. (I. Valentić) Central limit theorem for periodic singular diffusions, 9th International Conference on Lévy Processes, Samos, Greece, July 15-19, 2019
  39. (P. Lazić) Sub-geometric ergodicity of diffusion processes, IRTG Summer School on Probability Theory and Stochastic Dynamics, Bielefeld, Germany, August 19-23, 2019
  40. (P. Lazić) General criteria for exponential convergence towards the minimal quasi-stationary distribution, IRTG Summer School on Probability Theory and Stochastic Dynamics, Bielefeld, Germany, August 19-23, 2019
  41. (N. Sandrić) Periodic homogenization of a Lévy-type process with small jumps, Stochastics Seminar, Friedrich-Schiller Universität Jena, Jena, Germay, October 24, 2019
  42. (N. Sandrić) Stochastic stability of Markov processes, Mathematical Colloquium, Friedrich-Schiller Universität Jena, Jena, Germay, November 14, 2019
  43. (V. Wagner) Boundary Harnack inequality for regional operators, Arbeitsgruppe Angewandte Analysis Seminar, Universität Bielefeld, Bielefeld, Germany, November 9-17, 2019
  44. (N. Sandrić) Periodic homogenization of a Lévy-type process with small jumps, Analysis and Stochastics Seminar, Technische Universität Dresden, Dresden, Germany, December 12, 2019
  45. (P. Lazić) Ergodičnost procesa difuzija, Simpozij studenata doktorskih studija PMF-a, University of Zagreb, Zagreb, Croatia, February 28, 2020
  46. (I. Valentić) Periodička homogenizacija za procese Lévyjevog tipa, Simpozij studenata doktorskih studija PMF-a, University of Zagreb, Zagreb, Croatia, February 28, 2020
  47. (P. Lazić) Technische Universität Dresden, Dresden, Germany, August 23 - September 5, 2020
  48. (N. Sandrić) Limit theorems for a stable sausage, Probability and Financial Mathematics Seminar, Pennsylvania State University, USA, February 26, 2021
  49. (I. Valentić) Periodic homogenization of a Lévy-type process with small jumps, LMS Women in Mathematics Day 2021, University of Plymouth, UK, March 26, 2021
  50. (I. Valentić) A CLT for degenerate diffusions with periodic coefficients, and application to homogenization of linear PDE's, Modern Stochastics: Theory and Applications V, Kyiv, Ukraine, June 1-4, 2021
  51. (I. Valentić) A CLT for degenerate diffusions with periodic coefficients, and application to homogenization of linear PDE's, 8th European Congress of Mathematics, Portorož, SloveniaJune 21-25, 2021
  52. (N. Sandrić) Periodic homogenization of linear degenerate PDEs, Non-local operators, probability and singularities (on-line seminar), June 29, 2021
  53. (N. Sandrić) Subexponential upper and lower bounds in Wasserstein distance for Markov processes, Analysis and Stochastics Seminar, Technische Universität Dresden, Dresden, Germany, July 8-30, 2021
  54. (P. Lazić) On subgeometric ergodicity of regime-switching diffusion processes, Analysis and Stochastics Seminar, Technische Universität Dresden, Dresden, Germany, August 21 - September 4, 2021
  55. (N. Sandrić) Technische Universität Dresden, Dresden, Germany, August 23 - September 17, 2021
  56. (I. Valentić) A CLT for degenerate diffusions with periodic coefficients, and application to homogenization of linear PDE's, 22nd European Young Statisticians Meeting, Athens, GreeceSeptember 6-10, 2021
  57. (P. Lazić) Technische Universität Dresden, Dresden, Germany, November 23 - December 6, 2021
  58. (N. Sandrić) Technische Universität Dresden, Dresden, Germany, November 27 - December 4, 2021
  59. (V. Wagner) Semilinear equations for nonlocal operators: beyond the fractional Laplacian & Regional nonlocal operators, Fractional differential equations, Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, January 4 - April 29, 2022
  60. (N. Sandrić) Limit theorems for a stable sausage, Fractional kinetics, hydrodynamic limits and fractals, Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, March 21-25, 2022

 

Organized meetings