** 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.

- 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

- Stability of the overdamped Langevin equation in double-well potential
- Ergodicity of Lévy-driven SDEs arising from multiclass many-server queues
- Uniform polynomial rates of convergence for a class of Lévy-driven controlled SDEs arising in multiclass many-server queues
- On sub-geometric ergodicity of diffusion processes
- CLT for the capacity of the range of stable random walks
- Functional CLT for the range of stable random walks
- Periodic homogenization of a Lévy-type process with small jumps
- Limit theorems for a stable sausage
- A CLT for degenerate diffusions with periodic coefficients, and application to homogenization of linear PDEs
- Subexponential upper and lower bounds in Wasserstein distance for Markov processes
- Capacity of the range of random walks on groups
- Nonlocal quadratic forms with visibility constraint
- Learning from non-irreducible Markov chains
- Hoeffding's inequality for non-irreducible Markov model
- On subgeometric ergodicity of regime switching diffusion processes
- Convex hulls of stable random walks
- Periodic homogenization of a class of weakly coupled systems of linear PDEs
- Skew-product decomposition of Brownian motion on ellipsoid
- Invariance principle for the capacity and the cardinality of the range of stable random walks

- (P. Lazić) 48
^{th }Probability Summer School Saint-Flour, France, July 8-20, 2018 - (I. Valentić) 48
^{th }Probability Summer School Saint-Flour, France, July 8-20, 2018 - (P. Lazić) Warm-Up Week - Bielefeld Stochastic Summer, Bielefeld, Germany, August 20-23, 2018
- (I. Valentić) Warm-Up Week - Bielefeld Stochastic Summer, Bielefeld, Germany, August 20-23, 2018
- (V. Wagner) Warm-Up Week - Bielefeld Stochastic Summer, Bielefeld, Germany, August 20-23, 2018
- (P. Lazić) Sums & Products of Random Matrices, Bielefeld, Germany, August 27-31, 2018
- (I. Valentić) Sums & Products of Random Matrices, Bielefeld, Germany, August 27-31, 2018
- (V. Wagner) Sums & Products of Random Matrices, Bielefeld, Germany, August 27-31, 2018
- (Petra Lazić) 9
^{th}International Conference on Stochastic Analysis and Its Applications*,*Bielefeld, Germany, September 3-7, 2018 - (N. Sandrić)
*Ergodicity of piecewise Ornstein-Uhlenbeck processes with jumps,*9^{th}International Conference on Stochastic Analysis and Its Applications*,*Bielefeld, Germany, September 3-7, 2018 - (I. Valentić) 9
^{th}International Conference on Stochastic Analysis and Its Applications*,*Bielefeld, Germany, September 3-7, 2018 - (V. Wagner) 9
^{th}International Conference on Stochastic Analysis and Its Applications*,*Bielefeld, Germany, September 3-7, 2018 - (V. Wagner)
*Boundary behaviour of pure-jump Markov processes on sets*, Inverse and Spectral Problems for (Non)-Local Operators, Leipzig, Germany, September 10-14, 2018 - (P. Lazić)
*Ergodicity of Diffusion Processes*, 6^{th}Fractional Calculus, Probability and Non-Local Operators: Applications and Recent Developments, Bilbao, Spain, September 26-28, 2018 - (N. Sandrić)
*Ergodicity of piecewise Ornstein-Uhlenbeck processes with jumps**,*Analysis and Stochastics Seminar, Technische Universität Dresden, Dresden, Germany, November 4-10, 2018 - (N. Sandrić)
*Wasserstein distances*, ERASMUS Week, Technische Universität Dresden, Dresden, Germany, December 10-14, 2018 - (V. Wagner) Universität Bielefeld, Bielefeld, Germany, December 16-23, 2018
- (V. Wagner)
*An introduction to the theory of (non-local) Dirichlet forms*, ERASMUS Seminar, Universität Bielefeld, Bielefeld, Germany, March 18-22, 2019 - (V. Wagner)
, Analysis of nonlocal and nonsmooth models, Universität Bielefeld, Bielefeld, Germany, March 25-29, 2019*Nonlocal quadratic forms with visibility constraint* - (P. Lazić)
*Sub-geometric ergodicity of diffusion processes,*Analysis and Stochastics Seminar, Technische Universität Dresden, Dresden, Germany, April 28 - May 11, 2019 - (I. Valentić)
*Central limit theorem for periodic singular diffusions,*Analysis and Stochastics Seminar, Technische Universität Dresden, Dresden, Germany, April 28 - May 11, 2019 - (N. Sandrić) Technische Universität Dresden, Dresden, Germany, March 30 - April 06, 2019
- (N. Sandrić)
*Ergodicity of piecewise Ornstein-Uhlenbeck processes with jumps*, Fujian Normal University, Fuzhou, China, May 31 - June 07, 2019 - (P. Lazić) Probability, Analysis, and Applications, University of Zagreb, Zagreb, Croatia, June 12-14, 2019
- (N. Sandrić) Probability, Analysis, and Applications, University of Zagreb, Zagreb, Croatia, June 12-14, 2019
- (I. Valentić) Probability, Analysis, and Applications, University of Zagreb, Zagreb, Croatia, June 12-14, 2019
- (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 - (P. Lazić) 2
^{nd }Korean Croatian Summer Probability Camp, University of Dubrovnik, Dubrovnik, Croatia, June 17-19, 2019 - (N. Sandrić) 2
^{nd }Korean Croatian Summer Probability Camp, University of Dubrovnik, Dubrovnik, Croatia, June 17-19, 2019 - (V. Wagner) 2
^{nd }Korean Croatian Summer Probability Camp, University of Dubrovnik, Dubrovnik, Croatia, June 17-19, 2019 - (P. Lazić) 11
^{th }International Conference on Extreme Value Analysis, University of Zagreb, Zagreb, Croatia, July 1-5, 2019 - (N. Sandrić) 11
^{th }International Conference on Extreme Value Analysis, University of Zagreb, Zagreb, Croatia, July 1-5, 2019 - (I. Valentić) 11
^{th }International Conference on Extreme Value Analysis, University of Zagreb, Zagreb, Croatia, July 1-5, 2019 - (V. Wagner) 11
^{th }International Conference on Extreme Value Analysis, University of Zagreb, Zagreb, Croatia, July 1-5, 2019 - (P. Lazić)
*Sub-geometric ergodicity of diffusion processes and Lévy-driven SDEs*, A satellite Summer School to the 9^{th}International Conference on Lévy Processes, Athens, Greece, July 8-12, 2019 - (I. Valentić)
*Central limit theorem for periodic singular diffusions*, A satellite Summer School to the 9^{th}International Conference on Lévy Processes, Athens, Greece, July 8-12, 2019 - (P. Lazić)
*Sub-geometric ergodicity of diffusion processes and Lévy-driven SDEs*, 9^{th}International Conference on Lévy Processes, Samos, Greece, July 15-19, 2019 - (I. Valentić)
*Central limit theorem for periodic singular diffusions*, 9^{th}International Conference on Lévy Processes, Samos, Greece, July 15-19, 2019 - (P. Lazić)
*Sub-geometric ergodicity of diffusion processes*, IRTG Summer School on Probability Theory and Stochastic Dynamics, Bielefeld, Germany, August 19-23, 2019 - (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 - (N. Sandrić)
*Periodic homogenization of a Lévy-type process with small jumps*, Stochastics Seminar, Friedrich-Schiller Universität Jena, Jena, Germay, October 24, 2019 - (N. Sandrić)
*Stochastic stability of Markov processes*, Mathematical Colloquium, Friedrich-Schiller Universität Jena, Jena, Germay, November 14, 2019 - (V. Wagner)
*Boundary Harnack inequality for regional operators*, Arbeitsgruppe Angewandte Analysis Seminar, Universität Bielefeld, Bielefeld, Germany, November 9-17, 2019 - (N. Sandrić)
*Periodic homogenization of a Lévy-type process with small jumps*, Analysis and Stochastics Seminar, Technische Universität Dresden, Dresden, Germany, December 12, 2019 - (P. Lazić)
*Ergodičnost procesa difuzija,*Simpozij studenata doktorskih studija PMF-a, University of Zagreb, Zagreb, Croatia, February 28, 2020 - (I. Valentić)
*Periodička homogenizacija za procese L*Simpozij studenata doktorskih studija PMF-a, University of Zagreb, Zagreb, Croatia, February 28, 2020*é*vyjevog tipa, - (P. Lazić) Technische Universität Dresden, Dresden, Germany, August 23 - September 5, 2020
- (N. Sandrić)
*Limit theorems for a stable sausage*, Probability and Financial Mathematics Seminar, Pennsylvania State University, USA, February 26, 2021 - (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 - (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 - (I. Valentić)
*A CLT for degenerate diffusions with periodic coefficients, and application to homogenization of linear PDE's,*8^{th }European Congress of Mathematics, Portorož, Slovenia*,*June 21-25, 2021 - (N. Sandrić)
*Periodic homogenization of linear degenerate PDEs,*Non-local operators, probability and singularities (on-line seminar), June 29, 2021 - (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 - (P. Lazić)
*On subgeometric ergodicity of regime-switching diffusion processes*, Analysis and Stochastics Seminar, Technische Universität Dresden, Dresden, Germany, August 21 - September 4, 2021 - (N. Sandrić) Technische Universität Dresden, Dresden, Germany, August 23 - September 17, 2021
- (I. Valentić)
*A CLT for degenerate diffusions with periodic coefficients, and application to homogenization of linear PDE's,*22^{nd}European Young Statisticians Meeting*,*Athens, Greece*,*September 6-10, 2021 - (P. Lazić) Technische Universität Dresden, Dresden, Germany, November 23 - December 6, 2021
- (N. Sandrić) Technische Universität Dresden, Dresden, Germany, November 27 - December 4, 2021
- (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 - (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 - (P. Lazić)
*Subgeometric ergodicity of regime-switching diffusion processes,*7^{th}Croatian Mathematical Congress, Split, Croatia, June 17-20, 2022 - (I. Valentić)
*Geometric approach to construction of Wright-Fisher diffusions*Applied Probability Seminar, University of Warwick, Warwick, UK, June 24, 2022*,* - (P. Lazić)
*Subgeometric ergodicity of regime-switching diffusion processes,*Summer School of the 10^{th}International Conference on L*é*vy Processes, Mannheim, Germany, July 16-17, 2022 - (I. Valentić)
Summer School of the 10*Skew-product decomposition of Brownian motion on ellipsoid,*^{th}International Conference on L*é*vy Processes, Mannheim, Germany, July 16-17, 2022 - (I. Valentić)
*Geometric approach to construction of Wright-Fisher diffusions*, 10^{th}International Conference on Lévy Processes, Mannheim, Germany, July 18-22, 2022 - (N. Sandrić)
*Periodic homogenization of a class of weakly coupled systems of linear PDEs,*Analysis and Stochastics Seminar, Technische Universität Dresden, Dresden, Germany, July 3-30, 2022 - (N. Sandrić) Technische Universität Dresden, Dresden, Germany, August 21 - September 17, 2022
- (N. Sandrić)
*Subexponential upper and lower bounds in Wasserstein distance for Markov processes*, Stochastics Seminar, Friedrich-Schiller Universität Jena, Jena, Germay, November 17, 2022 - (V. Wagner)
*Semilinear equations for nonlocal operators*, Nonlocal operators and Markov processes III, Bedlewo, Poland, March 20-24, 2023