### Malliavin calculus

Head of the seminar: Prof. A.A.Dorogovtsev

Secretary of the seminar: Ia.Korenovska

E-mail address: yaroslavaka@mail.ru

17.00, room 208.

- Семинар 26.05.2020
Докладчик:

**Xia Chen**(University of Tennessee)Тема:

**Parabolic Anderson models – Large scale asymptotics**Abstract. The model of the parabolic Anderson equation is relevant to some problems arising from physics such as the particle movement in disorder media, population dynamics, and to the KPZ equations through a suitable transformation.

In the name of intermittency, broadly speaking, there has been increasing interest in the asymptotic behaviors of the system, over a large scale of the time or space, formulated in a quench or annealed form. By the multiplicative structure of the equation, the model is expected to grow geometrically. Hence, the ideas and methods developed from the area of large deviations become relevant and effective to some problems on intermittency.

The talk is to provide some general view on the recent development over the topic of intermittency of this model.

- Семинар 19.05.2020
Докладчик:

**Мария Белозерова**(Одесский национальный университет имени И.И. Мечникова)Тема:

**Асимптотическое поведение решений систем стохастических дифференциальных уравнений со взаимодействием** - Семинар 12.05.2020
Докладчик:

**Victor Marx**(Leipzig University)Тема:

**Smoothing properties of a diffusion on the Wasserstein space**Abstract:

*We study in this talk diffusion processes defined on the L_2-Wasserstein space of probability measures on the real line. We will introduce the construction of a diffusion inspired by (but slightly different from) the modified massive Arratia flow, studied by Konarovskyi and von Renesse. Then, our aim is to show that this diffusion has smoothing properties, similar to those of the standard Euclidean Brownian motion. Namely, we will first show that this process restores uniqueness of McKean-Vlasov equations with a drift coefficient which is not Lipschitz-continuous in its measure argument, extending the standard results obtained by Jourdain and foll. Secondly, we will present a Bismut-Elworthy-Li integration by parts formula for the semi-group associated to this diffusion.* - Семинар 28.04.2020
Докладчик:

**Max von Renesse**Тема:

**Molecules as metric measure spaces with lower Kato Ricci curvature**Joint work with Batu Güneysu (Humboldt University Berlin)

Abstract. In this talk we shall present a new result which connects the analysis of the Schrödinger semigroup associated to a molecule to the theory of metric measure spaces with lower Ricci curvature bounds. We show that the ground state transformation associated to this molecule creates naturally a metric measure space which has lower Ricci curvature bounds in terms of a Kato class function. This has numerous applications, for instance we show stochastic completeness of the corresponding metric measure space, and we also demonstrate that this setting is good enough to drive it semigroup gradient estimates using a variant of the Bismut derivative formula. - Семинар 21.04.2020
Докладчик:

**Vitalii Konarovskyi**Тема:

**On the existence and uniqueness of solutions to the Dean-Kawasaki equation**Abstract. We consider the Dean-Kawasaki equation with smooth drift interaction potential and show that measure-valued martingale solutions exist only in certain parameter regimes in which case they are given by finite Langevin particle systems with mean-field interaction. The proof is based on the Girsanov transform and log-Laplace duality. This is joint work with Max von Renesse and Tobias Lehmann. - Семинар 14.04.2020
Докладчик:

**Е.В. Глиняная**Тема:

**Предельные теоремы для числа кластеров в потоке Арратья** - Семинар 07.04.2020
Докладчик:

**PD Dr. Yana Kinderknecht (Butko)**(Technical University of Braunschweig)Тема:

**Chernoff approximation of operator semigroups generated by Markov processes**Abstract: We present a method to approximate operator semigroups generated by Markov processes and, therefore, transition probabilities of these processes. This method is based on the Chernoff theorem. In some cases, Chernoff approximations provide also discrete time Markov processes approximating the considered (continuous time) processes (in particular, Euler-Maruyama Schemes for the related SDEs). In some cases, Chernoff approximations have the form of limits of n iterated integrals of elementary functions as n→∞ (in this case, they are called Feynman formulae) and can be used for direct computations and simulations of Markov processes. The limits in Feynman formulae sometimes coincide with (or give rise to) path integrals with respect to probability measures (such path integrals are usually called Feynman-Kac formulae). Therefore, Feynman formulae can be used to approximate the corresponding path integrals and to establish relations between different path integrals.

In this talk, we discuss Chernoff approximations for (semigroups generated by) Feller processes in ℝ^d. We are also interested in constructing Chernoff approximations for Markov processes which are obtained by different operations from some original Markov processes, assuming that Chernoff approximations for the original processes are already known. In this talk, we present Chernoff approximations for such operations as: a random time change via an additive functional of a process, a subordination (i.e., a random time change via an independent a.s. nondecreasing 1-dim. Lévy process), killing of a process upon leaving a given domain, reflecting of a process. These results allow, in particular, to obtain Chernoff approximations for subordinate diffusions on star graphs and compact Riemannian manifolds. Moreover, Chernoff approximations can be further used to approximate solutions of some time-fractional evolution equations and hence to approximate marginal densities of the corresponding non-Markovian stochastic processes. - Семинар 31.03.2020Докладчик:
**Г. В. Рябов**Тема:**Преобразования винеровской меры и обобщение теоремы Гирсанова** - Семинар 17.03.2020Докладчик:
**Н. Б. Вовчанский**Тема:**Каплинг в методе дробных шагов для броуновской сети** - Семинар 10.03.2020Докладчик:
**А. Ю. Пилипенко**Тема:**О возмущении малым шумом дифференциальных уравнений с нелипшицевыми коэффициентами**