Quenched large deviations for multidimensional random walk in random environment with holding times

We consider a random walk in random environment with random holding times, that is, the random walk jumping to one of its nearest neighbors with some transition probability after a random holding time. Both the transition probabilities and the laws of the holding times are randomly distributed over the integer lattice. Our main result is a quenched large deviation principle for the position of the random walk. The rate function is given by the Legendre transform of the so-called Lyapunov exponents for the Laplace transform of the first passage time. By using this representation, we derive some asymptotics of the rate function in some special cases.Comment: This is the corrected version of the paper. 24 page

Involvement of the population of the region in online participation: experience of non-profit organisations

The study examines the practices and problems of online involvement of the population in the activities of non-profit sector organisations depending on socio-demographic characteristics based in statistical observation, and data from a sociological survey on the territory of the Vologda region. It has been substantiated that when implementing socially significant projects and events, it is advisable for non-profit organisations to assess the territorial features of digitalisation, the possibilities of potential and target audiences. A toolkit for assessing the involvement of the population in the activities of non-profit organisations in the digital environment has been proposed based on the analysis of three parameters: awareness, goal-setting, and the effectiveness of online platforms and its approbation has been carried out. A portrait of a user of the network resources of non-profit organisations has been presented. It has been concluded that at present society is more focused on the implementation of the practices of civic participation in real, and not in the virtual space, which is shown by the experience of non-profit organisations

A Sublinear Variance Bound for Solutions of a Random Hamilton Jacobi Equation

We estimate the variance of the value function for a random optimal control problem. The value function is the solution $w^\epsilon$ of a Hamilton-Jacobi equation with random Hamiltonian $H(p,x,\omega) = K(p) - V(x/\epsilon,\omega)$ in dimension $d \geq 2$. It is known that homogenization occurs as $\epsilon \to 0$, but little is known about the statistical fluctuations of $w^\epsilon$. Our main result shows that the variance of the solution $w^\epsilon$ is bounded by $O(\epsilon/|\log \epsilon|)$. The proof relies on a modified Poincar\'e inequality of Talagrand

Student youth participation in the development of a comfortable urban environment

Currently, the problem of youth participation in the development of urban space is extremely relevant both from the perspective of transforming the living environment with the resources of local communities, and the socialization of urban youth itself, expanding opportunities for personal growth through involvement in socially useful activities. The purpose of the article is to identify the conditions and readiness to participate in the development of a comfortable urban environment for students. The study was conducted in Vologda, Russian Federation, through a quantitative methodological strategy (questionnaire survey among university students, n = 207). The paper presents theoretical and methodological approaches to understanding the urban environment and a comfortable urban environment. It is indicated that a comfortable urban environment is a socio-economic category, reflecting the interactions within the urban space, which is most comfortable for the life of citizens. It is concluded that the student youth expresses their willingness to actively participate in decision-making on the development and improvement of the urban environment. In conclusion, the directions of enhancing the civic participation of young people in the formation of a comfortable urban space on the basis of dialogue platforms are considered

On the nonequilibrium entropy of large and small systems

Thermodynamics makes definite predictions about the thermal behavior of macroscopic systems in and out of equilibrium. Statistical mechanics aims to derive this behavior from the dynamics and statistics of the atoms and molecules making up these systems. A key element in this derivation is the large number of microscopic degrees of freedom of macroscopic systems. Therefore, the extension of thermodynamic concepts, such as entropy, to small (nano) systems raises many questions. Here we shall reexamine various definitions of entropy for nonequilibrium systems, large and small. These include thermodynamic (hydrodynamic), Boltzmann, and Gibbs-Shannon entropies. We shall argue that, despite its common use, the last is not an appropriate physical entropy for such systems, either isolated or in contact with thermal reservoirs: physical entropies should depend on the microstate of the system, not on a subjective probability distribution. To square this point of view with experimental results of Bechhoefer we shall argue that the Gibbs-Shannon entropy of a nano particle in a thermal fluid should be interpreted as the Boltzmann entropy of a dilute gas of Brownian particles in the fluid

Homogenization of weakly coupled systems of Hamilton--Jacobi equations with fast switching rates

We consider homogenization for weakly coupled systems of Hamilton--Jacobi equations with fast switching rates. The fast switching rate terms force the solutions converge to the same limit, which is a solution of the effective equation. We discover the appearance of the initial layers, which appear naturally when we consider the systems with different initial data and analyze them rigorously. In particular, we obtain matched asymptotic solutions of the systems and rate of convergence. We also investigate properties of the effective Hamiltonian of weakly coupled systems and show some examples which do not appear in the context of single equations.Comment: final version, to appear in Arch. Ration. Mech. Ana

Variational formulas and cocycle solutions for directed polymer and percolation models

We discuss variational formulas for the law of large numbers limits of certain models of motion in a random medium: namely, the limiting time constant for last-passage percolation and the limiting free energy for directed polymers. The results are valid for models in arbitrary dimension, steps of the admissible paths can be general, the environment process is ergodic under spatial translations, and the potential accumulated along a path can depend on the environment and the next step of the path. The variational formulas come in two types: one minimizes over gradient-like cocycles, and another one maximizes over invariant measures on the space of environments and paths. Minimizing cocycles can be obtained from Busemann functions when these can be proved to exist. The results are illustrated through 1+1 dimensional exactly solvable examples, periodic examples, and polymers in weak disorder

Stochastic homogenization of a class of nonconvex viscous HJ equations in one space dimension

We prove homogenization for a class of nonconvex (possibly degenerate) viscous Hamilton-Jacobi equations in stationary ergodic random environments in one space dimension. The results concern Hamiltonians of the form G(p) + V (x, omega), where the nonlinearity G is the minimum of two or more convex functions with the same absolute minimum, and the potential V is a bounded stationary process satisfying an additional scaled hill and valley condition. This condition is trivially satisfied in the inviscid case, while it is equivalent to the original hill and valley condition of A. Yilmaz and O. Zeitouni [32] in the uniformly elliptic case. Our approach is based on PDE methods and does not rely on representation formulas for solutions. Using only comparison with suitably constructed super- and sub- solutions, we obtain tight upper and lower bounds for solutions with linear initial data x ? theta x. Another important ingredient is a general result of P. Cardaliaguet and P. E. Souganidis [13] which guarantees the existence of sublinear correctors for all theta outside "flat parts " of effective Hamiltonians associated with the convex functions from which G is built. We derive crucial derivative estimates for these correctors which allow us to use them as correctors for G.(c) 2022 Elsevier Inc. All rights reserved