Singular diffusion: analytic and stochastic approaches, I

Potsdam, Am Neuen Palais   •   April 1 - 3, 2019

Abstracts

Here you will be able to find the abstracts of the talks.

Multidimensional pseudo-differential equation with spatial p-adic variables and corresponding stochastic processes (slides)

The p-adic Mathematics is widely used in the Theoretical Physics and Biology. It attracts a great deal of interest in quantum mechanics, string theory, quantum gravity, spin-glass theory and system biology.

The concept of a hierarchical energy landscape is very important from the point of view of the description of relaxation phenomena in complex systems, in particular, glasses, clusters and proteins. This concept can be outlined as follows. A complex system is assumed to have a large number of metastable configurations which realize local minima on the potential energy surface. The local minima are clustered in hierarchically nested basins of minima, namely, each large basin consists of smaller basins, each of these consisting of even smaller ones, and so on. Thus we may say that the hierarchy of basins possesses ultrametric geometry and transitions between the basins determine the rearrangements of the system configuration for different time scales. Thus the key points of the concept of a hierarchical structure which is typical for p-adic world is the main advantage which can be used for the description of the complex phenomenon.

Using a p-adic interpretation of a porous medium and the hydrodynamic description of fluids we give an example of a non-Archimedean mathematical model of fluid propagating through a porous medium. This is done in terms of a non-homogeneous Markov process. Some other examples of the analysis and SDE on the p-adic space will be given.

On exponential convergence of a distance between solutions of an SDE
with discontinuous drift (slides)

We consider an SDE with a Gaussian noise and a non-regular drift term. We prove that the distance between two solutions starting from different initial points tends to zero exponentially as time tends to infinity.

Wave equation driven by general stochastic measure (slides)

The Cauchy problem for the wave equation driven by a general stochastic measure is considered in d-dimensional case (d=1,2,3). The existence and uniqueness of the mild solution are proved. Holder regularity of its paths in time and spatial variables is obtained. The asymptotic behavior of the mild solution is investigated.

The modified Euler scheme for a weak approximation of solutions of stochastic differential equations driven by a Wiener process (slides)

Our goal is to provide direct and clear way for obtaining the weak approximation of diffusion processes of the form \(X_t=x+\int\limits_0^ta(X_s)ds+\int\limits_0^t\sigma(X_s)dW_s, \quad 0\leq t\leq T,\)  where  \(W=(W^1, \ldots, W^m)\)is a Wiener process, \(x\in\mathbb{R}^d\),\(a:\mathbb{R}^d\to\mathbb{R}^d,\)  \(\sigma:\mathbb{R}^d\to\mathbb{R}^{d\times m}.\)In the classical approach the Ito-Taylor expansion is used for such a reason (see, for example, book [1]). But in this case we need to simulate multiple Ito integrals which is quite complicated problem. Instead of such multiple integrals
the random variables which have to satisfy some moment conditions can be used (for details, see [1], Corollary 5.12.1).
And it is not so convenient, because in multi-dimensional case for weak approximation of higher order the choosing of such variables is not
very clear. We propose another way for obtaining the weak approximation of diffusion processes which avoids all mentioned above difficulties.

This report is based on the joint research with prof. A. Kulik.

[1] Kloeden P. E., Platen E. Numerical solution of stochastic differential equations.  Springer, 1995.

Semilinear Dirichlet problem for the fractional Laplacian (slides)

We give a general framework and results on existence, representation, and uniqueness of solutions to the semilinear problem for the fractional Laplacian with Dirichlet conditions on the complement and at the boundary of general open sets. (In cooperation with Sven Jahrohs and Edita Kania)

Uniqueness and decay of correlation for Gibbs point processes, using disagreement percolation (slides)

In this talk I will present a result proving uniqueness of Gibbs measure and exponential decay of the pair correlation function at low activity.

The result is applicable to general interactions satisfying natural conditions (for instance finite range interactions).

To prove this result we are using a disagreement percolation technique which is controlling the influence of the boundary, "placing it" into a percolation cluster of a dominating Poisson Boolean model.

 

Statistical research of models generated by discretely observed solution to SDE's (slides)

The talk is divided into two parts. The first one is devoted to research asymptotic properties of estimators for discretely observed solution to  Lévy driven SDE's
\(d X_t=a_\theta(X_t)dt +d Z_t,\) where \(Z\) is a  Lévy process without diffusion component. For this model the following tasks are solved:

  • Estimation of the Fischer information.   
  • Effectiveness of the estimation method.   
  • Software implementation of the algorithm.   


We consider one parameter case and multi parameters one.

In the second part we consider the statistical model generated by SDE \(dX_t=a(\alpha; X_t)dt+\sigma(\beta; X_t)dW_t.\)
Here, \(W\) is the one dimensional Wiener process and the coefficients \(a\) and \(\sigma\) are differentiable functions with bounded derivatives. The following problems are investigated:
 

  • Estimation of the    transition probability density.   
  • Construct QMLE.   
  • Software implementation.   
  1. Ivanenko D.O. (2014) Second derivative of the log-likelihood in the model given by a Lévy driven SDE's. Visn. Kyiv Nat. Univ., arxiv org, 1410.2880, vol. 2, p.~18 --22.
  2. Ivanenko D.O.,  Kulik A.M. (2015) Malliavin calculus approach to statistical inference for Levy driven SDE's. Meth. and Comp. in Appl. Prob. , vol. 17, p.~107--123.
  3. Ivanenko D.O.,  Bodnarchuk S.V. (2015) Efficiency of estimating method, in statistical models controlled by Levy noise. Th. of Prob. and Math. Stat., vol. 92, p.~9 -- 22.

   
 

Langevin dynamics with boundary conditions (slides)

This talk will be dedicated to the modeling of boundary conditions, and the theoritical problems related to such modeling, in Langevin dynamics. Generically, these Langevin dynamics describe, according to a second order SDE, the time evolution of the position and velocity of a generic fluid particle, endowed with some internal force
which ensure that a restriction of the position paths within a fixed confinement domain and which model the possible interactions of the particle in front of a solid wall located at the boundary of this domain. In their simpliest form, these interactions are characterized by the Maxwellian boundary conditions, introduced in the kinetic theory of gas, describing different situations: total or elastic reflection; absorbing or diffusive boundary; mixed boundary conditions ... The resulting dynamics drastically differ from classical reflected diffusion processes and lead to some original wellposedness and numerical approximation problems. After exposing the main properties of these Langevin dynamics and their links with constrained (deterministic) mechanical systems and trace problems for partially degenerated parabolic equation, the talk will focus on the presentation of some
wellposedness results and a particular application for the simulation of turbulent flows. These results are part of a serie of joint works with Mireille Bossy (INRIA Sophia-Antipolis Méditerranée) and Christophe Profeta (LaMME, Univ. d'Evry Val-d'Essonne). 

Noise sensitivity of Lévy driven stochastic differential equations: estimates and applications (slides)

The topic of this talk is induced by the following question: can the deviation between the solutions of two different Lévy driven SDE’s be controlled in terms of the characteristics of the underlying Lévy processes? In the case of SDE’s with additive noise we give the estimate for the deviation between the solutions in terms of the coupling distance for Lévy measures, which is based on the notion of the Wasserstein distance. Such estimate can be applied, for example, to the analysis of the low-dimensional conceptual climate models with paleoclimate data.

Regression approximation and sensitivity analysis for diffusions and Lévy-type processes, with applications to statistics and simulation (slides)

Lévy-type processes are heuristically understood as a (sort of) Lévy processes with the characteristic triplets dependent on the current state of a  process; e.g. [1]. This definition has the same spirit with  the classical Kolmogorov's definition of  a diffusion process as a location-dependent Brownian motion with a drift, and  widely  used nowadays in huge variety of models in  physics, biology,  finance etc., where the random noise - by different reasons - can not be assumed Gaussian, and thus the entire model does not fit to the diffusion framework, likewise to the celebrated Ditlevsen's model of the millennial climate changes [2]. The talk contains a survey of the set of methods for a quantitative description of transition probabilities and semigroups, available for diffusions and Lévy-type processes, and a discussion of natural  applications to statistics and simulation.

[1] B. Böttcher, R. Schilling, and J. Wang (2013).  Lévy-type processes: construction, approximation and sample path properties, Lévy matters. III,  Springer, Cham.

[2]  P. D. Ditlevsen (1999). Observation of α-stable noise induced millenial climate changes from an ice record. Geophysical Research Letters, 26, no. 10, 1441--1444.

 

On the "heterogeneous diffusion process" (slides)

We determine solutions of the irregular/singular Stratonovich SDE \(dX = |X|^\alpha \circ dB\), \(\alpha\in(-1,1)\), which are strong Markov processes spending zero time in 0. The process \(X\) was introduced by Cherstvy et al. in "New Journal of Physics" 15, 2013, under the name of "heterogeneous diffusion process" and can be seen as the Stratonovich version of the famous Girsanov (Itô) SDE \(dX = |X|^\alpha dB\).

This is a joint work with G. Shevchenko (Kiev University), http://arxiv.org/abs/1812.0532<wbr />4

 

Limit behavior of random walks with elastic screens (slides)

Consider a random walk \(\tilde S(n)\), that behaves as a simple random walk except 0. When it reaches 0 the \(i\)- time, the walk stops for a random amount of time \(\eta_i \geq 0\) (i.i.d.). Then the random walk continues its way as before. We study the limit behaviour of \(\frac{\tilde S(n t)}{\sqrt n}\). The examples of limit processes may be Wiener process, Wiener process stopped at 0 or sticky Wiener process.

 

Functional limit theorems for perturbed random walks

Let \(\{X_k, k\geq0\}\) be an integer-valued random walk perturbed at a finite set A, i.e., \(\{X_k\}\) is a Markov chain such that \(P(X_{n+1}=i+j | X_n=i) = P(\xi=i), i\notin A,\) where \(\xi\) is a fixed random variable. Distribution of jumps from \(A\) may be arbitrary.

We discuss functional limit theorems for a proper scaling of perturbed random walk \(\{X_k\}\).

For example, if \(P(\xi=\pm1)=1/2, \ A=\{0\},\) and \(P(X_{n+1}=1 | X_n=0) = 1-P(X_{n+1}=-1 | X_n=0)= p\in[0,1],\) then the classical result of Harrison and Shepp \cite{HS} states
that a sequence
\(\{ \frac{ X_{[nt]}}{\sqrt{n}}, t\in[0,T]\}\) converges in distribution   to a  skew Brownian motion as \(n\to\infty.\)


Different approaches of investigations of perturbed random walks will be discussed. In particular, the result of Harrison and Shepp is generalized to the case when random walks  are perturbed  at a finite number of points.
 

 

Correlation of clusters: partially truncated correlation functions and their decay (slides)

Partially truncated correlation functions (PTCF) of infnite continuous systems of classical point particles with pair interaction are investigated. We derive Kirkwood-Salsburg (KS)-type equations for the PTCF and write the solutions of these equations as a sum of contributions labelled by certain special graphs (forests), the connected components of which are tree graphs. We generalize the method introduced by Minlos and Pogosyan [1] in the case of truncated correlations. These solutions make it possible to derive strong cluster properties for PTCF which were obtained earlier for lattice spin systems. The report is based on the article [2].

[1] R. A. Minlos, S. K. Pogosyan. Estimates of Ursell functions, group functions, and their derivatives, Theor. Math. Phys., 31 (1977), # 2, 408-418.

[2] T. C. Dorlas, A. L. Rebenko, B. Savoie. Correlation of Clusters: Partially Truncated Correlation Functions and Their Decay. Preprint, arXiv:1811.12342, 2018.

Gibbsian theory as framework for the study of interacting diffusions (slides)

In this talk, we will propose an overview of several links between the Gibbs theory and random dynamics.

 

Entropy method for Gibbs point processes

In 1988, Hans-Otto Georgii introduced a new method of proving the existence of an infinite-volume Gibbs measures, that uses the specific entropy as a Lyapunov function.

In this talk, we will use this approach in the framework of marked Gibbs point processes with unbounded interaction. Some examples from stochastic geometry will also be discussed.