Research Videos

In this video Miha describes our new results on the ergodicity and the rate of decay of the tail of the stationary distribution for a broad class of storage models, encompassing constant, linear, and power-type release rates with both finite and infinite activity input process. Our results are expressed in terms of the asymptotics of the release rate, the tail-decay rate  of the L\'evy measure of the input process and its (possibly infinite) first moment. Our framework unifies and significantly extends classical results on the stability of storage models. 

This presentation is based on the paper ``Stability of storage processes with general release rates'' M. Brešar, A. Mijatović and N. Sandrić, available on arXiv

In this video I describe our new results on the equivalence of the finiteness of the second moment of a d-dimensional Levy process and the integrability at infinity of the convex distance (with respect to the scale-invariant measure $t^{-1}dt$) between the appropriately rescaled process at time $t$ and a standard Gaussian vector in $\R^d$.

Our integrability criterion demonstrates why Berry-Esseen bounds have to deteriorate in the limit as $\delta\to0$, when the Levy process is assumed to have $(2+\delta)$ moments. Moreover our criterion shows that in the Non-Normal Gaussian Domain of Attraction, the corresponding convex distance cannot be bounded above by any function integrable at infinity with respect to $t^{-1}dt$.

This presentation is based on the paper ``Multivariate CLT for L\'evy processes: convergence rates without moment assumptions'' J. Gonzalez Cazares, D. Kramer-Bang and A. Mijatovic, available at arXiv:2510.06891 

In this video I discuss a recently established central limit theorem for superdiffusive reflected Brownian motion (RBM) in generalised parabolic domains. The main result describes Gaussian fluctuations of the unbounded component of the reflected Brownian motion around its superdiffusive power function to which it is almost surly proportional. A joint limit law with the appropriately scaled d-dimensional bounded component of the  RBM is given. Interestingly, the CLT fails when the domain narrows too rapidly.  This presentatin is based on the paper "Central limit theorem for superdiffusive reflected Brownian motion",  A. Mijatović, I. Sauzedde and A. Wade, available at arXiv:2412.14267.

A short video describing the superdiffusive almost sure behaviour of RBM in generalised parabolic domains, providing the background for this limit theorem, can be found on my YouTube channel Prob-AM at the link here.

The presentation used in the video is above. For details of the proofs see the paper "Central limit theorem for superdiffusive reflected Brownian motion",  A. Mijatović, I. Sauzedde and A. Wade, available on arXiv at  arXiv:2412.14267

The video describes upper and lower bounds on the rate of convergence in the short-time stable domain of attraction.  Two new couplings of pure-jump Levy processes are developed for obtaining the upper bounds in the normal and non-normal domains of stable attraction. The lower bounds on the rates hold for the entire stable domain of attraction without any additional structural assumptions on the Levy process. Interestingly, the rate of convergence in the non-normal domain cannot be faster than logarithmic, while the normal domain of attraction (i.e. when the slowly varying function can be taken constant) is always polynomial. The above couplings turn out to be rate optimal. This is joint work with J. González Cázares and D. Kramer-Bang. 

The presentation used in the video is above. For details see the "Asymptotically optimal Wasserstein couplings for the small-time stable domain of attraction", J. González Cázares, D. Kramer-Bang and A. Mijatović, on arXiv:2411.03609

In this presentation I explain why a Bessel-driven kinetic model exhibits anomalous diffusive behaviour. The model can be viewed as a Bessel process with a stochastic dimension. Unlike the Bessel process, at large times, this model exhibits superdiffusive growth at rates greater than square root. 

The first half of the presentation sets out the model and the results. The second half discusses the proofs. 

This presentation, used in the YouTube video, is based on the paper "Superdiffusive limits for Bessel-driven stochastic kinetics", M. Brešar, C. da Costa, A. Mijatović, A. Wade

This presentation describes our exact simulation algorithm for the first-passage event of a general subordinator. It is based on the paper "Fast exact simulation of the first-passage event of a subordinator", J. González Cázares, F. Lin, A. Mijatović arXiv:2306.06927

The slides used, in used in the YouTube presentation, explain our new algorithm for the exact simulation of the first-passage event for general subordinators and disucss our complexity results. For elements of the proof and applications of this algorithm, please see Part II of this presentation. 

GitHub repository with implementation of exact simulation algorithm in Python and Julia  

This presentation describes our exact simulation algorithm for the first-passage event of a general subordinator. It is based on the paper "Fast exact simulation of the first-passage event of a subordinator", J. González Cázares, F. Lin, A. Mijatović arXiv:2306.06927

The slides used, in used in the YouTube presentation, explain the ideas behind the proof of the complexity theorem for the expected running time of our new algorithm for the exact simulation of the first-passage event for general subordinators. A numerical example is also discussed. 

GitHub repository with implementation of exact simulation algorithm in Python and Julia  

This presentation describes our exact simulation algorithm for the first passage event. It is based on the recent paper "Fast exact simulation of the first passage of a tempered stable subordinator across a non-increasing function", J. González Cázares, F. Lin, A. Mijatović arXiv:2303.11964

The slides used in the presentation explaining our new algorithm for the exact simulation of the first passage event for tempered stable subordinators. For applications of this algorithm, please see Part 2 of this presentation. 

GitHub repository with implementation of exact simulation algorithm in Python and Julia  

Jorge describes Monte Carlo estimation, using our exact simulation algorithm for the first-passage event of a tempered stable subordinator,  of a solution of a fractional PDE.  See Part 1 of this presentation for the description of the exact simulation algorithm.

The presentation used in this video is based on our recent paper "Fast exact simulation of the first passage of a tempered stable subordinator across a non-increasing function", J. González Cázares, F. Lin, A. Mijatović arXiv:2303.11964

GitHub repository with implementation of exact simulation algorithm in Python and Julia  

In this video I discuss how a normally reflecting Brownian particle in an unbounded generalized  parabolic domain can be positive recurrent. The main result described in the presentation states sharp polynomial rates of decay of the tails of the invariant distribution and polynomial convergence rates of the total variation distance of the reflected diffusion to stationarity.

For an intuitive description of the proofs in this paper see Part 2 of this video.

This presentation is based on the recent paper by M. Bresar, A. Mijatovic, and A.R. Wade, "Brownian motion with asymptotically normal reflection in unbounded domains: from stability to transience", (March 2023). arXiv link is arXiv:2303.06916

Part 2 of the presentation on the paper "Brownian motion with asymptotically normal reflection in unbounded domains: from stability to transience", M. Bresar, A. Mijatovic, and A.R. Wade (March 2023), discusses the ideas behind the proof. arXiv link is arXiv:2303.06916

The presentation used in part 2 video on the paper "Brownian motion with asymptotically normal reflection in unbounded domains: from stability to transience", M. Bresar, A. Mijatovic, and A.R. Wade (March 2023), arXiv link is arXiv:2303.06916

In this video I describe what happens with an obliquely reflecting Brownian particle in a generalized  parabolic domain. 

This is a based on our recent paper by M. Menshikov, A. Mijatovic & A.R. Wade, "Reflecting Brownian motion in generalized parabolic domains: explosion and superdiffusivity", (2022), to appear in  Ann. Inst. Henri Poincar\'e Probab. Stat.;  arXiv link is here

Based on paper by M. Menshikov, A. Mijatovic & A.R. Wade, "Reflecting Brownian motion in generalized parabolic domains: explosion and superdiffusivity", (2022), to appear in  Ann. Inst. Henri Poincar\'e Probab. Stat.;  arXiv link is here

Based on paper by M. Menshikov, A. Mijatovic & A.R. Wade, "Reflecting Brownian motion in generalized parabolic domains: explosion and superdiffusivity", (2022), to appear in  Ann. Inst. Henri Poincar\'e Probab. Stat.;  arXiv link is here

Based on paper by M. Menshikov, A. Mijatovic & A.R. Wade, "Reflecting Brownian motion in generalized parabolic domains: explosion and superdiffusivity", (2022), to appear in  Ann. Inst. Henri Poincar\'e Probab. Stat.;  arXiv link is here

In this video David gives a brief description of the characterisation of the  Hölder continuity of the convex minorant of any Levy process. Our characterisation covers all Hölder exponents and nearly all Levy processes. This talk is based on the paper ``Hölder continuity of the convex minorant of a Levy process'',  coauthored by David Bang (my PhD student at Warwick), Jorge Gonzalez Cazares (research fellow at Warwick) and AM.

The paper "Hölder continuity of the convex minorant of a Lévy process?'' is on arXiv at: https://arxiv.org/abs/2207.12433

 The related Stick-Breaking Representation Theorem video is here https://youtu.be/hEg4YmxOgXA

In this video I describes our recent results on the growth rate of the derivative of boundary of the the convex hull of a path of a Levy process. The paper "How smooth can the convex hull of a Levy path be?" is coauthored by David Bang (my PhD student at Warwick), Jorge Gonzalez Cazares (research fellow at Warwick) and AM.

The paper "How smooth can the convex hull of a Lévy path be?'' is on arXiv at: https://arxiv.org/abs/2206.09928

 Related Stick-Breaking Representation Theorem video https://youtu.be/hEg4YmxOgXA

The stick-breaking representation theorem (SBRT) for convex minorants is stated at 8'38''. First half of the video discusses the derivation of the fluctuation theory using convex minorants. The second half is about the proof of SBRT.

The paper "Convex minorants and the fluctuation theory of Lévy processes'' is on arXiv at: http://arxiv.org/abs/2105.15060

My PhD student David Bang gives a short presentation of the results in the paper "Asymptotic shape of the concave majorant of a Lévy process", coauthored by Jorge González Cázares and myself.

The paper is available on arXiv: https://arxiv.org/pdf/2106.09066.pdf