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Markov processes, ergodic properties of stochastic processes
Stochastic averaging and stratified spaces
Stability issues of degenerate Hamiltonian-type systems
Ballistic points of finite-mode Kolmogorov flows (with Mike Cranston).
Brief description of the physical background of this problem: Flows in nature have tendency of becoming turbulent and hectic. Reliable descriptions of the wide variety of dynamical systems, like ocean currents or atmospheric patterns, depend in part on how well we can describe turbulence. Despite the intense work in order to understand the physics of turbulence, and despite the long-term (since 1823) knowledge of Navier-Stokes equations describing turbulent flow of an incompressible fluid, the turbulence is still a mystery at large.
For many dynamical systems, the most probable distribution is their equilibrium distribution. Unfortunately, an isolated turbulent fluid cannot maintain an equilibrium distribution, since turbulence is dissipative in nature. To remain turbulent, a system needs constant supply of energy. To study turbulence, we need to keep the turbulence go, and so we need to `stir' it! If we supply energy at the rate at which it dissipates, and so that the resulting turbulence is isotropic and homogeneous, the resulting system may keep its turbulent state. In 1941, Kolmogorov proposed a theory to calculate the energy spectrum of such turbulent system, and began a new era in the theory of turbulence.
It does not appear to be possible to study turbulence as a deterministic system. Treating it as a random system instead, we investigate the dispersion properties of a flow whose turbulent velocity field experiences fluctuations around its deterministic mean. Without loss of generality, we assume that the mean of the velocity distribution is zero. We will also accept some common assumptions for random models of fluid mechanics. Namely, we assume that the distribution of the velocity field is Gaussian, is invariant under the shifts in the space variable (i.e., the field is homogeneous in space), and under the shifts in the time variable (i.e., the field is stationary), and also assume that it is Markovian in time. We restrict our attention to turbulence of incompressible fluids, requiring that the velocity field is divergence-free, and has prescribed spectrum of the Kolmogorov's type.
In the last decade, this flow has received a lot of attention. Our interest is in studying spatial spreading of the finite number of tracers for this flow. In particular, we are interested in showing that a point moving with linear speed is guaranteed to be found.
Stability of a stochastic process on a cone-shaped Hamiltonian.
Abstract: We investigate the time evolution of the laws of a stochastic process whose trajectories are governed by a two-dimensional weakly dissipative dynamical system with a cone-shaped Hamiltonian, with the critical point at the origin. Studying the Wasserstein $L_p$-distance as the shortest distance between these laws on the corresponding canonical probability spaces, we show that it is Markovian and non-expansive (up to a constant) in time.
Weak convergence of stratified processes: Wasserstein distance and Khasminskii's coordinates.
Abstract: We consider laws of the graph-valued processes whose trajectories are governed by two-dimensional weakly dissipative dynamical systems with a degenerate Hamiltonian. Using a particular Wasserstein distance as the shortest distance between the measures on the corresponding canonical probability spaces, we estimate the rate of the weak convergence of the laws of the original process to the unique law of a Markov process evolving on a stratified graph. In one of our previous papers, we have computed the distance in the related classical case, and showed that the corresponding distance in the stratified case is at least of the order of the distance in the related classical case, and at most of the order delivered by the total variation of these measures over the paths crossing the boundary of the critical set of the degenerate Hamiltonian.
In this work we investigate whether this total variation is connected with the gluing conditions. The analysis of the convergence of these measures relies on a finer set of so-called Khasminskii's coordinates used about the boundary of the critical set. Is it possible to obtain an explicit bound on the total variation of these laws using Khasminskii's coordinates? This question is likely to be connected to the previous one, and if answered positively, will reveal the role that gluing plays in this matter.
Rate of convergence of random integro-differential systems with integration over a long time interval.
Abstract: We consider a type of integro-differential equations which arise naturally in those cases where the current behavior of a system depends not only on its present state, but also on the whole history of its development since some fixed time. The integro-differential operators of our concern involve integration in time, and are different from another popular type of integro-differential operators, where integration is done in a space variable, like those for example in Hamilton-Jacobi-Bellman equations. For such random integro-differential equations, we consider their corresponding averaged deterministic equations and obtain the explicit large-deviation-type estimates. This work also treats the multidimensional case.
Papers Submitted and Published
Abstract: We consider laws of a stochastic process whose trajectories are governed by a two-dimensional weakly dissipative dynamical system with a degenerate Hamiltonian. Using Wasserstein distance as the shortest distance between the measures on the corresponding canonical probability spaces, we estimate the rate of the weak convergence of the laws of the original process to the unique law of a Markov process evolving on a stratified space. Additionally, for a related (non-stratified) example of a process on a cylinder, the corresponding estimates are obtained using two different methods. One of the methods revealed the explicit expression for the Wasserstein distance.
By presenting this work, we make the first explicit attempt to use minimal metrics to estimate the rate of convergence of stochastic stratified processes.
Double-Level Averaging on a Stratified Space. Stochastic processes and functional analysis, 277--294, Lecture Notes in Pure and Appl. Math., 238, Dekker, New York, 2004, MR2059912.
A Noisy System with a Flattened Hamiltonian and Multiple Time Scales, Stochastics and Dynamics, Vol. 3, No. 1 (2003) 1-54, © World Scientific Publishing Company, doi:10.1142/S0219493703000668
Abstract: We consider a two-dimensional weakly dissipative dynamical system with time-periodic coefficients. Their time average is governed by a degenerate Hamiltonian whose set of critical points has an interior. The dynamics of the system is studied in the presence of three time scales. Using the martingale problem approach and separating the involved time scales, we average the system to show convergence to a Markov process on a stratified space. The corresponding strata of the reduced space are a two-sphere, a point, and a line segment. Special attention is given to the domain of the limiting generator, including the analysis of the gluing conditions at the point where the strata meet. The gluing conditions resulting from the hierarchy of the time scales are similar to the conditions on the domain of skew Brownian motion, and are related to the description of spider martingales. The dynamics of the reduced process can also be understood through the Fokker-Planck equation, and the gluing coefficients arise in the corresponding conservation of flux and in the continuity equation.
A problem from Hamiltonian mechanics with time-periodic coefficients, small noise, and degeneracy. PhD Thesis, University of Illinois at Urbana-Champaign, 2002
Miser Classes and Asymptotic Behavior of Potentials of Ergodic Markov Processes. Theory Probab. Math. Statist, No. 52 (1996), 125 – 127, MR 97m:60105.
Abstract: For certain classes of functions, a theorem about the existence of potentials of ergodic Markov processes is proved.
(with V.S. Korolyuk, M.Yo. Yadrenko, M.V. Kartashov) On the Creative Mathematical Activity of Valentin Mikhailovich Shurenkov. Theory Probab. Math. Statist, No. 52 (1996), 1 - 8, MR 98f:01073a.
Abstract: This paper contains a brief description of the mathematical interests and activities of V. M. Shurenkov, his bibliography, and a list of his students.
(with V.M. Shurenkov) One More Remark on the Central Limit Theorem for Ergodic Chains. (Russian) Ukraïn. Mat. Zh. 47 (1995), no. 1, 118 – 120; translation in Ukrainian Math. J. 47 (1995), no. 1, 138 – 141, MR 96g:60036.
Abstract: We consider the sum $n^{-1/2} \sum_{k=0}^{n-1} f(X_k)$, where $X_n, n \geq 0,$ is an ergodic Markov chain, and obtain conditions on the function $f$ to guarantee that the sum is asymptotically normal.
(with V.M. Shurenkov) The Central Limit Theorem for Stochastically Additive Functionals of Ergodic Chains. (Russian) Ukraïn. Mat. Zh. 46 (1994), no. 10, 1421 – 1423; translation in Ukrainian Math. J. 46 (1994), no. 10, 1570 – 1572.
Abstract: We prove the Central Limit Theorem for stochastically additive functionals of ergodic Markov chains.
(with V.M. Shurenkov) Central Limit Theorem for Special Classes of Functions of Ergodic Chains. (Russian) Ukraïn. Mat. Zh. 46 (1994), no. 8, 1202 – 1205; translation in Ukrainian Math. J. 46 (1994), no. 8, 1092 – 1094, MR 97m:60028.
Abstract: We prove the Central Limit Theorem for a certain class of bounded functions of ergodic Markov chains. The paper also contains two useful corollaries of this result.
(with V.M. Shurenkov) On the Potential of Ergodic Markov Chains. (Ukrainian) Ukraïn. Mat. Zh.46 (1994), no. 4, 446 - 449; translation in Ukrainian Math. J. 46 (1994), no. 4, 475 - 479, MR 95g:60086.
Abstract: We study the behavior of potentials of ergodic Markov chains, and prove two theorems describing certain classes of functions for which these potentials exist.
(with V.M. Shurenkov) The Central Limit Theorem for Centered Frequencies of a Countable Ergodic Markov Chain. (Russian) Ukraïn. Mat. Zh. 45 (1993), no. 12, 1713 – 1715; translation in Ukrainian Math. J. 45 (1993), no. 12, 1928 – 1931, MR 96i:60019.
Abstract: We prove the Central Limit Theorem for certain countable ergodic Markov chains; the proof is based on our earlier results about the behavior of the corresponding potentials. As a corollary, the Central Limit Theorem for centered frequencies is obtained without the hypothesis that the time of the first return to a state is finite.
(with V.M. Shurenkov) On the Asymptotics of the Potential of a Countable Ergodic Markov Chain. (Russian) Ukraïn. Mat. Zh. 45 (1993), no. 2, 265 - 269; translation in Ukrainian Math. J. 45 (1993), no. 2, 284 - 289, MR 94e:60058.
Abstract: We prove the existence of potentials of certain uniformly ergodic Markov chains in a countable state space. The paper also contains several corollaries useful from the point of view of the possible application to the Central Limit Theorem for Markov chains. In particular, we find a condition equivalent to the hypothesis that the time of the first return to a state is finite.
(with B.V. Bondarev) Averaging in the Stochastic Goursat Problem. (Russian) Dokl. Akad. Nauk Ukrain. SSR Ser. A 1990, no. 4, 3 – 5, MR 91h:35340
Abstract: We average a random Goursat problem. The corresponding averaged equation turns out to be deterministic. Working in the uniform metric, we establish Bernstein-type exponential estimates for the sums of certain independent random variables. We consider cases of fields that satisfy uniformly strong mixing and strong mixing conditions. Based on the established large-deviation-type inequalities, we can construct the interval of reliability for a solution of the original problem.
Averaging in Stochastic Integro-Differential Equations. Theory of Random Processes and Its Applications (Russian), 106 – 115, “Naukova Dumka”, Kiev, 1990
Abstract: We construct an inequality of large deviations of the solution of the given random integro-differential equation from a solution of the corresponding deterministic averaged equation.
Brownian motion: what is it all about? (50 minute talk for general academic audience), Lyman Briggs School of Science, MSU, East Lansing, February 2005
Weak convergence of stochastic Hamiltonian laws (50-minute talk), Probability and Statistics Department, Michigan State University, East Lansing, February 2005
Weak convergence of stochastic stratified laws (50-minute talk), probability seminar, University of Californian at San Diego, October 2004
Rate of the weak convergence of a stochastic stratified process
(20-minute talk), AIMS' Fifth International Conference on
Dynamical Systems and Differential Equations,
California State Polytechnic University, Pomona, June 16 - 19, 2004
Stability of Stratified Hamiltonian Laws (15-minute talk), Seminar on Stochastic Processes - 2003, University of British Columbia, Canada, May 2004
Rate of Convergence of Stochastic Hamiltonian Laws (50-minute talk), probability seminar, University of California - Irvine, May 2004
Wiener Sausage of Brownian Sheet (50-minute colloquium talk), California State University at San Marcos, math department colloquium, September 2003
Asymptotic Problems in Stochastic Processes and PDE's (A conference celebrating the 65th birthday of Mark I. Friedlin), University of Maryland at College Park, May 2003
Wiener Sausages and Wiener Patties (50-minute talk), probability seminar, University of California - Irvine, May 2003
Abstract: A Wiener sausage is the path traced by a ball whose center moves along a Brownian trajectory. It was first considered in the mid-60's by Kesten, Spitzer, and Whitman. Most of the asymptotic results about Brownian sausage were motivated by physical problems. For example, in 1974, Kac and Luttinger used some of the conjectured results about the volume of the Brownian sausage to discuss Bose-Einstein condensation in the presence of impurities. The proof of those results was soon given by Donsker and Varadhan.
In this expository talk we discuss some known properties of this object, as well as some open problems. The talk will be presented on a level also suitable for graduate students.
Seminar on Stochastic Processes - 2003, University of Washington, Seattle, March 2003
Double-Level Averaging on a Stratified Space (50-minute talk), probability seminar, University of Southern California, LA, March 2003; probability seminar, University of California - Irvine, January 2003
A Noisy Problem with a Degenerate Hamiltonian and Multiple Time Scales (50-minute talk), probability seminar, University of California - San Diego, October 2002
A Problem from Hamiltonian Mechanics with Time-Periodic Coefficients, Small Noise, and Degeneracy (50-minute talk), thesis defense, University of Illinois at Urbana-Champaign, June 2002
A Problem from Hamiltonian Mechanics with Small Noise and Degeneracy, Contributed Paper (20-minute talk), Joint Mathematics Meetings, San Diego, January 2002
23rd Midwest Probability Colloquium, University of Chicago, Illinois, October 2001
Martingale Problem and Averaging: Issues of Uniqueness (70-minute talk), Graduate Student Summer Seminar in Stochastic Processes, University of Illinois at Urbana-Champaign, June 2001
Large Deviations: Multivariate Random Walk and Brownian Sheet (50-minute talk), Department of Mathematics, University of Illinois at Urbana-Champaign, May 2001