I coorganised Warsaw Summer School in Probability.


I am an assistant professor in the Faculty of Mathematics, Informatics and Mechanics, University of Warsaw.  I was a post-doc at the Université de Genève working under the supervision of prof. Yvan Velenik. Further, I was a post-doc at the Prob-Lab working with dr Simon Harris and prof. Andreas Kyprianou.

Recently, I have been working in machine learning focusing in reinforcement learning. I am the PI of 500k research grant RL founded by the National Science Center. Contact me, if you are interested in Ph.D./post-doc positions.


My email address is: pmilos (at)


I am interested in probability. More particularly in stochastic models, branching processes, problems arising in the mathematical physics.

Recently I focus in research in machine learning and mostly in reinforcement learning.


You can read about it in my papers. Most of them are available on arXiv.

    Last update: Dec 18

  • Implementation of the PPO module in open source (Tensor2Tensor framework) Ongoing project with on model-based reinforcement learning (jointly with GoogleBrain)

  • The interchange process with reversals on the complete graph (with J. Björnberg, M. Kotowski and B. Lees) (pdf)

  • Phase transition for the interchange and quantum Heisenberg models on the Hamming graph (with R. Adamczak and M. Kotowski) (pdf)

  • Expert-augmented actor-critic for ViZDoom and Montezumas Revenge (with J. Michalewski, M. Gramulewicz) (pdf)

  • Learning to Run challenge solutions: Adapting reinforcement learning methods for neuromusculoskeletal environments, NIPS 2017 competition track, (pdf)

  • CLT for supercritical branching processes with heavy-tailed branching law (with R. Marks), 2018, submitted (pdf)

  • Hierarchical Reinforcement Learning with Parameters(with M. Klimek and H. Michalewski), Proceedings of the 1st Annual Conference on Robot Learning, PMLR 78:301-313, 2017 (pdf)

  • Existence of a phase transition of the interchange process on the Hamming graph (with B. Sengul), accepted to Elec. J. Prob. (pdf)

  • Maximal displacement of a supercritical branching random walk in a time-inhomogeneous random environment (with B. Mallein), accepted to Stoch. Proc. Appl. (pdf)

  • Brownian motion and Random Walk above Quenched Random Wall (with B. Mallein), Ann. Inst. H. Poincaré Probab. Statist. Vol. 54, No. 4 (2018), 1877-1916. (pdf)

  • Branching Brownian motion with absorption and the all-time minimum of branching Brownian motion with drift (with J. Berestycki, E. Brunet and S. Harris), J. Funct. Anal. Vol. 273, Iss 6, 2017, pp 2107-2143 (pdf)

  • Exact representation of truncated variation of Brownian motion, submitted, arXiv:1311.2415 (pdf)

  • Piracy as an ethical decision (with M. Kobus and M. Krawczyk), accepted to Inf. Econ. Pol. (with corrections)

  • The random interchange process on the hypercube (with R. Kotecky and D. Ueltschi), Elec. Comm. Prob. 21(2016), no. 4, 1-9. (pdf)

  • Delocalization of two-dimensional random surfaces with hard-core constraints (with R. Peled), Comm. Math. Phys 340(1) (2015), pp.1-46 (pdf)

  • CLT for Ornstein-Uhlenbeck branching particle system (with R. Adamczak), Elect. J. Probab. 20 (2015), pp no. 42, 1-35 (pdf)

  • Spatial CLT for the supercritical Ornstein-Uhlenbeck superprocess, accepted (with corrections) to J. Th. Prob, arXiv:1203.6661 (pdf)

  • U-statistics of Ornstein-Uhlenbeck branching particle system (with R. Adamczak), J. Th. Prob. 27 (2014), no. 4, pp. 1071-1111 (pdf)

  • On the discrete Gaussian Free Field with disordered pinning on $\mathbb{Z}^d$, $d\geq 2$ (with L. Coquille), Stoch. Proc. Appl. 123(9) (2013), pp. 3542-3559 (pdf)

  • On truncated variation, upward truncated variation and downward truncated variation for diffusions (with R. Lochowski), Stoch. Proc. Appl. 123 (2013), pp. 446-474 (pdf)

  • Inequality decomposition by population subgroups for ordinal data (with M. Kobus), J. Health Econ. (2012), 31:15-21 (pdf)

  • Occupation times of subcritical branching immigration systems with Markov motion, CLT and deviations principles, Inf. Dim. Anal. Quant. Prob. Rel. Top. (2012) Vol. 15, No. 1 (pdf)

  • Fluctuations of the occupation times for branching system starting from infinitely divisible point processes, 2011, arXiv:1001.5142 (pdf)

  • Occupation time fluctuation limits of infinite variance equilibrium branching systems, Inf. Dim. Anal. and Quant. Probab. 2009, Vol. 12. No.4, pp. 593-612 (pdf)

  • Occupation times of subcritical branching immigration system with Markov motions, Stoch. Proc. Appl. 119 (2009), pp. 3211-3237 (pdf)

  • A note on small branching fluctuation limits of catalytic superprocesses with immigration, IMPAN Preprint 689, 2008 (pdf)

  • Limit theorems for fluctuations of occupation times of branching systems, (in Polish), 2008, Ph.D. dissertation, in polish (pdf)

  • On Meritocratic Inequality Indices (with M. Kobus), Working Paper. RePEc Archive no. 10532, 2008 (pdf)

  • Occupation time fluctuations of Poisson and equilibrium branching systems in critical and large dimensions, Prob. Math. Stat. 2008, Vol. 28, Fasc. 2, pp. 235-256 (pdf)

  • Occupation time fluctuations of Poisson and equilibrium finite variance branching systems, Prob. Math. Stat. 2007, Vol. 27, Fasc. 2, pp. 181-203 (pdf)

With whom?

I have had a pleasure to collaborate with: Radek Adamczak, Julien Berestycki, Jakob Björnberg, Éric Brunet, Loren Coquille, Michał Gramulewicz, Simon Harris, Benjamin Lees, Rafał Łochowski, Roman Kotecký, Maciej Klimek, Martyna Kobus, Michał Kotowski, Rafał Marks, Henryk Michalewski, Bastien Mallein, Ron Peled, Batı Şengül, Daniel Ueltschi.



Math movies

A wonderful movie depicting the famous "sphere eversion".

Do you think that the Möbius transform is difficult? Not at all!

Short talk

Surprisingly many extraordinarily intelligent people give crappy talks on conferences.
Here a few hints by Richard Durrett

And some others by Agelos Georgakopoulos Clik

Tao blog

If you've never heard of Terrence Tao go here. Later read his amazing blog.


Do you know who your mathematical ancestor was. Or you would like to know that your grand-grand-...-grand supervisor was David Hilbert ...


Well, I am not a Ph.D. student any more but the Ph.D. comics still amuses me.