Paul Dobson 

My main interest area is applications of stochastic analysis and probability theory to statistics and data science. More precisely, my interests include:

• Markov Chain Monte Carlo algorithms

• Bayesian Inverse Problems

• Degenerate Diffusion processes

• Irreversible Markov processes, in particular Piecewise Deterministic Markov Processes

• SDE theory

• Malliavin calculus

• Ergodic Theory

• Accelerated optimization algorithms and ordinary differential equations: the convex non Euclidean case (with J. M. Sanz-Serna and K. Zygalakis) preprint available at (https://arxiv.org/abs/2410.19380)

• Splitting schemes for second order approximations of piecewise-deterministic Markov processes (with A. Bertazzi and P. Monmarché) preprint available at https://arxiv.org/abs/2301.02537

• Poisson Equations with locally-Lipschitz coefficients and Uniform in Time Averaging for Stochastic Differential Equations via Strong Exponential Stability (with D. Crisan, B. Goddard, M. Ottobre and I. Souttar) accepted to Annales de l’Institut Henri Poincaré and available at (https://arxiv.org/abs/2204.02679) 

• Accelerated Bayesian imaging by relaxed proximal-point Langevin sampling (with T. Klatzer, Y. Altmann, M. Pereyra, J.M. Sanz-Serna and K. Zygalakis) published in SIAM Journal on Imaging Sciences (https://epubs.siam.org/doi/abs/10.1137/23M1594832) also available at https://arxiv.org/abs/2308.09460

• Infinite Dimensional Piecewise Deterministic Markov Processes (with J. Bierkens) published in Stochastic Processes and their applications (https://doi.org/10.1016/j.spa.2023.08.010) also available at https://arxiv.org/abs/2205.11452

• On the connections between optimization algorithms, Lyapunov functions, and differential equations: theory and insights (with J.M. Sanz-Serna and K. Zygalakis) published in SIAM Journal on Optimization (https://doi.org/10.1137/23M1625287) also available at https://arxiv.org/abs/2305.08658

• Efficient Bayesian computation for low-photon imaging problems (with S. Melidonis, Y. Altmann, M. Pereyra and K. Zygalakis) published in SIAM journal on Imaging Sciences (https://epubs.siam.org/doi/10.1137/22M1502240) at also available on arXiv at https://arxiv.org/pdf/2206.05350.pdf

• Fast non mean-field networks: uniform in time averaging (with J. Barré, M. Ottobre, E. Zatorska) published in SIAM Journal on Mathematical Analysis (https://epubs.siam.org/doi/abs/10.1137/20M1328646) and also available on arXiv at https://arxiv.org/abs/2003.14230 

• Subgeometric Hypocoercivity for piecewise-deterministic Monte Carlo methods (with C. Andrieu, A. Wang) published in SIAM Journal on Mathematical Analysis (https://projecteuclid.org/journals/electronic-journal-of-probability/volume-26/issue-none/Subgeometric-hypocoercivity-for-piecewise-deterministic-Markov-process-Monte-Carlo-methods/10.1214/21-EJP643.full) and also available on arXiv at https://arxiv.org/abs/2011.09341

• Approximations of Piecewise Determinstic Markov Processes and their convergence properties (with A. Bertazzi and J. Bierkens) published in Stochastic Processes and their applications (https://www.sciencedirect.com/science/article/pii/S0304414922001958) also available on arXiv at (https://arxiv.org/abs/2109.11827) 

•  Uniform in time convergence for the Euler method and a Pathwise Approach to Derivative Estimates for Diffusion Semigroups (with D. Crisan and M. Ottobre)  published in transactions of the AMS (https://www.ams.org/journals/tran/2021-374-05/S0002-9947-2021-08301-4/) also available on arXiv at https://arxiv.org/abs/1905.03524

• Long-time behaviour of Degenerate diffusions: UFG-type SDE and time-inhomogeneous hypoelliptic processes (with T. Cass, D. Crisan, M. Ottobre) – published in Electronic Journal of Probability (https://projecteuclid.org/journals/electronic-journal-of-probability/volume-26/issue-none/Long-time-behaviour-of-degenerate-diffusions--UFG-type-SDEs/10.1214/20-EJP577.full) also available on arXiv at https://arxiv.org/abs/1805.01350. 

• Comparison of some reversible and non-reversible sampling schemes (with I. Fursov, G. Lordand M. Ottobre) published in Computational Geosciences https://link.springer.com/article/10.1007/s10596-020-09947-4 .

The code for approximations of Piecewise Deterministic Markov Processes is available on Andrea Bertazzi's github page (https://github.com/andreabertazzi). In particular for the SPA paper the code is https://github.com/andreabertazzi/Euler_PDMC_algorithms and for the preprint the code is available at https://github.com/andreabertazzi/splittingschemes_PDMP.

I gave talks at the following occasions:

• 20th Internet Seminar on Evolution Equations, Baronissi, Italy, on "Feller theory''.

• Intercontinental conference in analysis, London, on the "Geometric, analytic and Probabilistic implications of the UFG condition''.

• North British Probability Seminar, University of Edinburgh on using techniques for non-autonomous SDEs to understand the convergence of degenerate diffusion processes.

• Probability working seminar, University of Edinburgh, two seminars on "Kolmogorov equations'', two seminars on "Small Random Perturbations of Peano Phenomena'' and one seminar on the "Stability of Stochastic Differential Equations''.

• At the Postgraduate Colloquium at both the University of Edinburgh and Heriot-Watt University on "The Moment Problem''.

•  Joint CDT colloquium in Oxford University and in Edinburgh on "Geometric, analytic and Probabilistic implications of the UFG condition''.

• Mathematical Encounters (March 2019), Buenos Aires, on "Degenerate Diffusion Processes: Beyond Hörmander's condition".

•  Hausdorff Institute Workshop in Bonn, Germany: Problems of roughness, geometry and random fluctuations on "Degenerate Diffusion Processes: Beyond Hörmander's condition".

• MCQMC conference in Oxford, "Infinite dimensional Piecewise Deterministic Markov Processes". Youtube link 

2023 to present - Assistant professor at Heriot-Watt University

2021 to 2023- Post doc at the University of Edinburgh

2020 to 2021 - Post doc at TU Delft

2019 to 2020 - Post doc position at the Vrije Universiteit.

2015 to 2019 - PhD Maxwell Institute Graduate School for Analysis and its Applications, Edinburgh. Based at Heriot-Watt University.

2011-2015 - MMath degree from University of Oxford

A copy of my PhD thesis is available at https://drive.google.com/file/d/18OocmRSc-Q00YbI30g-LjAgVNQJYGizk/view?usp=sharing.