2021, Spring

Abstract: We will discuss the problem of uniqueness of probability solutions to the Cauchy problem for the Fokker--Planck--Kolmogorov equation with an unbounded drift coefficient and the unit diffusion coefficient. It will be demonstrated that in the one-dimensional case there holds uniqueness and in all other dimensions it fails. We will present an example of an equation with the unit diffusion matrix and a smooth drift coefficient for which the Cauchy problem with every probability initial condition has an infinite-dimensional simplex of probability solutions. Finally we will discuss the generalization of the known Ambrosio-Figalli-Trevisan superposition principle for probability solutions, according to which such a solution is generated by a solution to the corresponding martingale problem.

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Abstract: We consider recurrence times for non-negative AR(1)-sequences X(n + 1) = a X(n) + b(n + 1), where 0 < a < 1 and for some related processes. If the tails of i.i.d. innovations b(n) decrease logarithmically then the chain X(n) may be either positive or null recurrent. We determine the tail behavior of recurrence times in both cases. In the null recurrent case we also construct a positive harmonic function for X(n) killed at entering [0,c], where c > 0.

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Abstract: For a multivariate random walk with i.i.d. jumps satisfying the Cramér moment condition and having mean vector with at least one negative component, we derive the exact asymptotics of the probability of ever hitting the positive orthant that is being translated to infinity along a fixed vector with positive components. This problem is motivated by and extends results from a paper by F. Avram et al. (2008) on a two-dimensional risk process. Our approach combines the large deviation techniques from a series of papers by A. Borovkov and A. Mogulskii from around 2000 with new auxiliary constructions, which enable us to extend their results on hitting remote sets with smooth boundaries to the case of boundaries with a “corner” at the “most probable hitting point”. We also discuss how our results can be extended to the case of more general target sets. [Joint work with Yuqing Pan.]

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Abstract: The talk is mainly based on the joint work with Michael Scheutzow  https://projecteuclid.org/euclid.aop/1603180872. We develop a new  generalized coupling  approach to the study of stochastic delay equations with Hölder continuous coefficients, for which analytical PDE-based methods are not available. We prove that such equations possess unique weak solutions, and establish weak ergodic rates for the corresponding segment processes. We also prove, under additional smoothness assumptions on the coefficients, stabilization rates for the sensitivities in the initial value of the corresponding semigroups. Some recent extensions to stochastic equations with jump noise will be discussed, as well.

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Abstract: Various large deviations principles for birth-death processes will be considered, depending on the rate of tending to infinity of the normalizing function. Some analogue for solutions of Ito stochastic equations will also be considered.

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Abstract: We consider an approximation of a Wiener process by a sequence of compound Poisson processes and prove the convergence of some functionals of this compound Poisson processes to the local time of the Wiener process.

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Abstract: We consider a random walk on a multidimensional integer lattice with random bounds on local times, conditioned on the event that it hits a high level before its death. We introduce an auxiliary ``core'' process that has a regenerative structure and plays a key role in our analysis. We obtain a number of representations for the distribution of the random walk in terms of the similar distribution of the ``core'' process.  Based on that, we prove a number of limiting results by letting the high level to tend to infinity. In particular, we generalise results for a simple symmetric one-dimensional random walk obtained earlier in the paper by Benjamini and Berestycki (2010).

The talk is based on a joint work with Sergey Foss: [1] Foss S. and Sakhanenko A.   Properties of  Conditioned Random Walks  on  Integer Lattices with Random Local Constraints. arXiv:2007.05165 [math.PR]; (accepted to Memorial volume for Vladas Sidoravicius in the Springer series ``Progress in Probability“)

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Abstract: Positive recurrence of a diffusion with switching and with recurrent and transient regimes is established under suitable conditions on the drift in all regimes and on the intensities of switching. The approach is based on a "new old idea" of an embedded Markov chain with alternating jumps: one jump increases the average of the square norm of the process, while the next jump decreases it. Under suitable balance conditions positive recurrence follows. In the simplest case of a one-dimensional diffusion with an additive Wiener process and two regimes, the result is briefly presented on pp. 224-228 at https://publications.hse.ru/mirror/pubs/share/direct/438402359.pdf

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Abstract: Is it possible to confine a Markov process with values in R^d to a given domain D? That is, to construct a new process, which on one hand has as many properties of the initial one as possible, but on the other is forced to remain in D?

For diffusions, intuition offers at least two such confined versions: the absorbing and the reflecting at/off the boundary. Both notions lose their meaning in the context of jump processes.

Nevertheless, turns out there is some universal way to talk about absorbing, reflecting and other "versions" of processes: these operations can be thought of as imposing various conditions on the free process generator. For example, consider d-dimensional Brownian motion. Its generator is the Laplace operator. In the lecture I shall explain why it is natural to define the reflecting Brownian motion in D as a process with generator being the Laplace-Neumann operator.

I shall start with a brief overview of the reflecting processes theory and its history and after that give a construction of reflecting Levy processes in the above sense.

Video, Slides