Long-time behavior of stochastic model with multi-particle synchronization

We consider a basic stochastic particle system consisting of $N$ identical particles with isotropic $k$-particle synchronization, $k\geq 2$. In the limit when both number of particles $N$ and time $t=t(N)$ grow to infinity we study an asymptotic behavior of a coordinate spread of the particle system. We describe three time stages of $t(N)$ for which a qualitative behavior of the system is completely different. Moreover, we discuss the case when a spread of the initial configuration depends on $N$ and increases to infinity as $N\to \infty$.Comment: 9 pages, 1 figure; one reference and one figure added, typos corrected, minor improvements of the proof

Positivity of transition probabilities of infinite-dimensional diffusion processes on ellipsoids

We consider diffusion processes in Hilbert spaces with constant non-degenerate diffusion operators and show that, under broad assumptions on the drift, the transition probabilities of the process are positive on ellipsoids associated with the diffusion operator. This is an infinite-dimensional analogue of positivity of densities of transition probabilities. Our results apply to diffusions corresponding to stochastic partial differential equations.Comment: 12 pages; to appear in Theory of Stochastic Processe

Controlling of clock synchronization in WSNs: structure of optimal solutions

Energy-saving optimization is very important for various engineering problems related to modern distributed systems. We consider here a control problem for a wireless sensor network with a single time server node and a large number of client nodes. The problem is to minimize a functional which accumulates clock synchronization errors in the clients nodes and the energy consumption of the server over some time interval $[0,T]$. The control function $u=u(t)$, $0\leq u(t)\leq u_{1}$, corresponds to the power of the server node transmitting synchronization signals to the clients. For all possible parameter values we find the structure of optimal trajectories. We show that for sufficiently large $u_{1}$ the solutions contain singular arcs.Comment: 14 pages, 6 figure

Time Scales in Probabilistic Models of Wireless Sensor Networks

We consider a stochastic model of clock synchronization in a wireless network consisting of N sensors interacting with one dedicated accurate time server. For large N we find an estimate of the final time sychronization error for global and relative synchronization. Main results concern a behavior of the network on different time scales $t=t_N \to \infty$, $N \to \infty$. We discuss existence of phase transitions and find exact time scales on which an effective clock synchronization of the system takes place.Comment: 31 page

Intrinsic scales for high-dimensional LEVY-driven models with non-Markovian synchronizing updates

We propose stochastic $N$-component synchronization models $(x_{1}(t),...,x_{N}(t))$, $x_{j}\in\mathbb{R}^{d}$, $t\in\mathbb{R}_{+}$, whose dynamics is described by Levy processes and synchronizing jumps. We prove that symmetric models reach synchronization in a stochastic sense: differences between components $d_{kj}^{(N)}(t)=x_{k}(t)-x_{j}(t)$ have limits in distribution as $t\rightarrow\infty$. We give conditions of existence of natural (intrinsic) space scales for large synchronized systems, i.e., we are looking for such sequences $\{b_{N}\}$ that distribution of $d_{kj}^{(N)}(\infty)/b_{N}$ converges to some limit as $N\rightarrow\infty$. It appears that such sequence exists if the Levy process enters a domain of attraction of some stable law. For Markovian synchronization models based on $\alpha$-stable Levy processes this results holds for any finite $N$ in the precise form with $b_{N}=(N-1)^{1/\alpha}$. For non-Markovian models similar results hold only in the asymptotic sense. The class of limiting laws includes the Linnik distributions. We also discuss generalizations of these theorems to the case of non-uniform matrix-based intrinsic scales. The central point of our proofs is a representation of characteristic functions of $d_{kj}^{(N)}(t)$ via probability distribution of a superposition of $N$ independent renewal processes.Comment: 50 page

Optimal chattering solutions for longitudinal vibrations of a nonhomogeneous bar

We consider a control problem for longitudinal vibrations of a nonhomogeneous bar with clamped ends. The vibrations of the bar are controlled by an external force which is distributed along the bar. For the minimization problem of mean square deviation of the bar we prove that the optimal control has an infinite number of switchings in a finite time interval, i.e., the optimal control is the chattering control.Comment: 12 page

Estimates for Kantorovich functionals between solutions to Fokker -- Planck -- Kolmogorov equations with dissipative drifts

We obtain estimates for the Kantorovich functionals between solutions to different Fokker -- Planck -- Kolmogorov equations for measures with same diffusion part but different drifts and different initial conditions. We show possible applications of such estimates to the study of the well-posedness for nonlinear equations.Comment: 13 pages; revised, extended to a more general case, minor mistaked correcte

Asymptotic analysis of a particle system with mean-field interaction

We study a system of $N$ interacting particles on $\bf{Z}$. The stochastic dynamics consists of two components: a free motion of each particle (independent random walks) and a pair-wise interaction between particles. The interaction belongs to the class of mean-field interactions and models a rollback synchronization in asynchronous networks of processors for a distributed simulation. First of all we study an empirical measure generated by the particle configuration on $\bf{R}$. We prove that if space, time and a parameter of the interaction are appropriately scaled (hydrodynamical scale), then the empirical measure converges weakly to a deterministic limit as $N$ goes to infinity. The limit process is defined as a weak solution of some partial differential equation. We also study the long time evolution of the particle system with fixed number of particles. The Markov chain formed by individual positions of the particles is not ergodic. Nevertheless it is possible to introduce relative coordinates and to prove that the new Markov chain is ergodic while the system as a whole moves with an asymptotically constant mean speed which differs from the mean drift of the free particle motion.Comment: 31 pages; added references, section 5 and appendix; corrected typo

Phase transitions in the time synchronization model

We continue the study of the time synchronization model from arXiv:1201.2141 . There are two types $i=1,2$ of particles on the line $R$, with $N_{i}$ particles of type $i$. Each particle of type $i$ moves with constant velocity $v_{i}$. Moreover, any particle of type $i=1,2$ jumps to any particle of type $j=1,2$ with rates $N_{j}^{-1}\alpha_{ij}$. We find phase transitions in the clusterization (synchronization) behaviour of this system of particles on different time scales $t=t(N)$ relative to $N=N_{1}+N_{2}$.Comment: 10 pages; first published in Theory Probab. Appl. 50, pp. 134-141 (2006); small corrections in text of the paper; bibliography update

Dynamics of Phase Boundary with Particle Annihilation

Infinitely many particles of two types ("plus" and "minus") jump randomly along the one-dimensional lattice $\mathbf{Z}_{\varepsilon}=\varepsilon\mathbf{Z}$. Annihillations occur when two particles of different time occupy the same site. Assuming that at time $t=0$ all "minus" particles are placed on the left of the origin and all "plus" particles are on the right of it, we study evolution of $\beta_\varepsilon(t)$, the boundary between two types. We prove that in large density limit $\epsilon\to 0$ the boundary $\beta_\varepsilon(t)$ converges to a deterministic limit. This particle system can be interpreted as a microscopic model of price formation on economic markets with large number of players