## WORKSHOP

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- Category: Conference
- Published on Monday, 30 October 2017 01:07
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There will be a mini-workshop organized after the school is finished. The mini-workshop, which is considered to be a separated activity from the school, aims to encourage potential collaborations in the future. It is scheduled on Thursday 15/03 afternoon and Friday 16/03 of the second week.

1. **Hakima Besaih - University of Wyoming, Laramie, USA**

Title: **Mean field limit of interacting filaments for 3D fluids***Abstract*: Families of N interacting curves are considered, with long range, mean field type, interaction. They generalize models based on classical interacting point particles to models based on curves. In this new set-up, a mean field result is proven, as N goes to infinity. The limit PDE is vector valued and, in the limit, each curve interacts with a mean field solution of the PDE. Some connection to the 3D Euler equation is established.

2. **The Anh Cung – University of Education, Hanoi, Vietnam**

Title: **On the existence and long-time behavior of solutions to stochastic three-dimensional Navier-Stokes-Voigt equations***Abstract*: We consider the 3D stochastic Navier-Stokes-Voigt equations in bounded domains with homogeneous Dirichlet boundary conditions. First, we prove the existence and uniqueness of solutions to the problem. Then we investigate the mean square exponential stability and the almost sure exponential stability of the stationary solutions.

3. **The Tuan Hoang - Institute of Mathematics, VAST, Hanoi, VietnamTitle: Asymptotic separation between solutions of Caputo fractional stochastic differential equations**

*Abstract : *In this talk, we formulate a generic condition on the coefficient of a stochastic phase oscillator for which the Lyapunov exponent is negative.

Consequently, the generated random dynamical system exhibits a synchronization

4. **Ghaus ur Rahman - University of Swat Khyber Pakhtunkhawa, Pakistan**.

Title: **Dynamical Aspects of Stochastic Childhood Diseases Model**

*Abstract*: In this talk, we look at the dynamics of an epidemic model of the infectious childhood diseases. We take a look at the asymptotic behavior of the stochastic model near the equilibrium points, and additionally study the model when the coefficients of the noise are small. Moreover we show that the model is ergodic.

5. **Javed Hussain - Sukkur IBA University, Pakistan**

Title: **Large deviation principle for stochastic heat equation on Hilbert manifold**

*Abstract*: In this talk our aim is to present Large Deviation property for the solution of stochastic heat equation on Hilbert manifold with stratonovich noise. For this we are going to employ the weak convergence method for studying large deviation principle . (Joint work with Prof. Zdzislaw Brzezniak, University of York).

6. **Max-Olivier Hongler - EPFL STI SMT-GE, Lausanne Switzerland**

Title: **Solvable nonlinear reaction-diffusion equations and their connection to optimal control problems.***Abstract*: Generalising a connection between nonlinear discrete velocities Boltzmann equation and solvable optimal control problem, we construct a new class of fully solvable nonlinear reaction-diffusion dynamics. In parallel, we show how the cost function of the control problem is naturally related to a large deviations principle.

7. **Viet Hung Pham, Institute of Mathematics Hanoi**

Title: **Persistence probability of random polynomials**

*Abstract*: We study the persistence probability of random polynomials where the polynomials stay positive or equivalently have no real roots. The motivation comes from some applications in statistical physics and evolutionary game theory. Using a method proposed by Dembo and Mukherjee, we can provide the persistence exponents. In joint work with Van Hao Can and Duong Manh Hong.

8. **Hoang Long Ngo – University of Education, Hanoi, Vietnam**

Title: **Geometric numerical integration for some classes of stochastic non-colliding particle systems.***Abstract*: We present a semi-implicit Euler-Maruyama approximation scheme for some classes of stochastic non-colliding particle systems such as the Dyson-Brownian motion. We study its rates of convergence in the strong sense and show that the scheme preserves some geometric properties of the systems.

This is a joint work with Dai Taguchi (Osaka University).

9. **Tien Dung Nguyen - FPT University, Hanoi, Vietnam**

Title: **Some sufficient conditions for Novikov's criterion**

Authors: Nguyen Tien Dung and Nguyen Van Tan*Abstract*: In this talk, we employ the techniques of Malliavin calculus to provide some sufficient conditions for a stochastic process to satisfy Novikov's criterion. In particular, we obtain an improvement for Buckdahn's results established in [1]. **References**

[1] R. Buckdahn, Anticipative Girsanov transformations. Probab. Theory Rel. Fields 89 (1991) 211-238.

[2] N.T. Dung, Some sufficient conditions for Novikov's criterion. To appear in Proceedings of the AMS, 2018.

10. **Toru Sera - Kyoto University, Japan**

Title: **Multiray generalization of the arcsine laws for interval maps***Abstract*: We present a distributional limit theorem for the occupation ratio measures of interval maps with indifferent fixed points. This limit theorem is a multiray extension of Thaler's generalized arcsine laws [1], and is also based on studies of occupation times of diffusion processes on multiray. This is a joint work with Kouji Yano (Kyoto University).

[1] M. Thaler, A limit theorem for sojourns near indifferent fixed points of one-dimensional maps, Ergodic Theory Dynam. Systems 22 (2002), no. 4, 1289--1312. MR1926288

11. **Yuzuru Sato - Hokkaido University, Japan**

Title: **Stochastic chaos in a turbulent swirling flow***Abstract*: We report the experimental evidence of the existence of a random attractor in a fully developed turbulent swirling flow. By defining a global observable which tracks the asymmetry in the flux of angular momentum imparted to the flow, we can first reconstruct the associated turbulent attractor and then follow its route towards chaos. We further show that the experimental attractor can be modeled by stochastic Duffing equations, that match the quantitative properties of the experimental flow, namely the number of quasi-stationary states and transition rates among them, the effective dimensions, and the continuity of the first Lyapunov exponents. Such properties can neither be recovered using deterministic models nor using stochastic differential equations based on effective potentials obtained by inverting the probability distributions of the experimental global observables. Our findings open the way to low dimensional modeling of systems featuring a large number of degrees of freedom and multiple quasi-stationary states.

12. **Michael Scheutzow - TU Berlin, Germany**

Title: *Minimal attractors.**Abstract*: It is well-known that a random attractor of a random dynamical system which attracts all compact sets is unique while this is not true for a random point attractor (which attracts all deterministic points). We show that if a random point attractor exists then there is always a smallest such point attractor (no matter whether the attraction is in the pullback sense or in probability).

We also provide generalizations to other families of attracted sets and provide an example showing that a smallest forward attractor may not exist. This is joint work with Hans Crauel (Frankfurt).

13. **Bjoern Schmalfuss - Friedrich-Schiller-Universitaet Jena, Germany**

Title: **Synchronization of a coupled system of stochastic parabolic differential equations.***Abstract*: We consider a system of two spde where the linear parts generate an analytic semigroup. The coupling operator is linear and positive. We formulate conditions for the existence of a random attractor for this system. This attractor has a particular shape which can be interpreted as synchronization.

14. **Hiroki Sumi - Kyoto University, Japan**

Title: **Weak mean stability in random holomorphic dynamical systems.***Abstract*: We consider random holomorphic dynamical systems generated by holomorphic families of rational maps on the Riemann sphere. We introduce the notion of “weak mean stability” and show several properties of weakly mean stable systems. Also, we show that generic systems are weakly mean stable. For the preprint, see H. Sumi, Negativity of Lyapunov Exponents and Convergence of Generic Random Polynomial Dynamical Systems and Random Relaxed Newton’s Methods, preprint, https://arxiv.org/abs/1608.05230.

15. **Thi Thanh Diu Tran - University of Luxembourg, Luxembourg**

Title: **Statistical inference for Vasicek-type model driven by Hermite processes***Abstract*: Let {Z_t^{q,H} ,t \ge {\rm{0}}} denote a Hermite process of order q\geq 1 and self-similarity parameter H\left( {\frac{{\rm{1}}}{{\rm{2}}},{\rm{1}}} \right) . This process is -self-similar, has stationary increments and exhibits long-range dependence. When q= 1 it corresponds to the fractional Brownian motion, whereas it is not Gaussian as soon as q\geq 2 .

In this talk, we deal with the following Vasicek-type model driven by Z^{q,H}:

X_{\rm{0}} = {\rm{0,}}dX_t = a\left( {b - X_t } \right)dt + dZ_t^{q,H} ,t \ge {\rm{0}}

where a>0 and b\in R are considered as unknown drift parameters. We provide estimators for a and b based on continuous-time observations. For all possible values of H and q, we prove strong consistency and we analyze the asymptotic fluctuations. Joint work with Ivan Nourdin

16. **Tat Dat Tran – MPI MIS, Germany**

Title: *Free energy functional method and its applications to Wright-Fisher model**Abstract*: In this talk, I will first introduce the most popular model in population genetics, the Wright-Fisher model, which can be approximated as a diffusion process in a Riemannian manifold with corners. The conditional density function of the process is known as the solution of a forward Kolmogorov equation (also known by physicists as a Fokker-Planck equation). The difficulty is that the generator is singular and the boundary of the domain is not smooth. I will then discuss about a concept from stochastic mechanism so-called free energy functional and how to apply it in this setting. In particular, I shall provide a necessary and sufficient condition in terms of mutation and selection coefficients under which the Wright-Fisher diffusion process will possess a free energy functional. This then also is a necessary and sufficient condition for the existence of a unique reversible probability density with respect to the Lebesgue measure. Moreover, by using Bakry-Emery techniques, I will show that under an additional condition, the flow of probability densities exponentially converges to the reversible one in some distance such as total variation. Some open problems will also be discussed.

17. **Ngoc Khue Tran - Pham Van Dong University, Quangngai, Vietnam**

Title: **Local asymptotic properties for the growth rate of a jump-type CIR process.***Abstract:* In this talk, we consider a one-dimensional jump-type Cox-Ingersoll-Ross process driven by a Brownian motion and a subordinator, whose growth rate is a unknown parameter. The Lévy measure of the subordinator is finite or infinite. Considering the process observed continuously or discretely at high frequency, we derive the local asymptotic properties for the growth rate. Three cases are distinguished: subcritical, critical and supercritical. Local asymptotic normality (LAN) is proved in the subcritical case, local asymptotic quadraticity (LAQ) is derived in the critical case, and local asymptotic mixed normality (LAMN) is shown in the supercritical case. Our approach is based on Girsanov's theorem and Malliavin calculus. This is a joint work with Mohamed Ben Alaya, Ahmed Kebaier and Gyula Pap.

18. **Guangqu Zheng - University of Luxembourg, Esch-sur-Alzette, Luxembourg**

Title: **Central limit theorem on Gaussian, Poisson Wiener chaos***Abstract*: This talkpresents a new/elementary strategy of proving CLT on Wiener chaos via Stein’s method of exchangeable pairs. We will begin with construction of Gaussian, Poisson Wiener chaos, then introduce the natural exchangeable pair couplings motivated by the classical Mehler formulae. Then a unified proof follows for CLT on Gaussian, Poisson Wiener chaos, together with a transfer principle.