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Register a new userGraphon particle system: Uniform-in-time concentration bounds
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In this paper, we consider graphon particle systems with heterogeneous mean-field type interactions and the associated finite particle approximations. Under suitable growth (resp. convexity) assumptions, we obtain uniform-in-time concentration estimates, over finite (resp. infinite) time horizon, for the Wasserstein distance between the empirical measure and its limit, extending the work of Bolley--Guillin--Villani.
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Two-sided bounds are explored for concentration functions and Renyi entropies in the class of discrete log-concave probability distributions. They are used to derive certain variants of the entropy power inequalities.
This paper shows the equivalence class definition of graphons hinders a direct development of dynamics on the graphon space, and hence proposes a state-driven approach to obtain dynamic graphons. The state-driven dynamic graphon model constructs a time-index sequence of the permutation-invariant probability measures on the universal graph space by assigning i.i.d. state random processes to $mathbbm{N}$ and edge random variables to each of the unordered integer pairs. The model is justified from three perspectives: graph limit definition preservation, genericity, and analysis availability. It preserves the graph limit definition of graphon by applying a bijection between the permutation-invariant probability measures on the universal graph space and the graphon space to obtain the dynamic graphon, where the existence of the bijection is proved. Also, a generalized version of the model is proved to cover the graphon space by an application of the celebrated Aldous-Hoover representation, where generalization is achieved by adding randomness to the edge-generating functions. Finally, analysis of the behavior of the dynamic graphon is shown to be available by making assumptions on the state random processes and the edge random variables.
This paper considers the space homogenous Boltzmann equation with Maxwell molecules and arbitrary angular distribution. Following Kacs program, emphasis is laid on the the associated conservative Kacs stochastic $N$-particle system, a Markov process with binary collisions conserving energy and total momentum. An explicit Markov coupling (a probabilistic, Markovian coupling of two copies of the process) is constructed, using simultaneous collisions, and parallel coupling of each binary random collision on the sphere of collisional directions. The euclidean distance between the two coupled systems is almost surely decreasing with respect to time, and the associated quadratic coupling creation (the time variation of the averaged squared coupling distance) is computed explicitly. Then, a family (indexed by $delta > 0$) of $N$-uniform weak coupling / coupling creation inequalities are proven, that leads to a $N$-uniform power law trend to equilibrium of order ${sim}_{ t to + infty} t^{-delta} $, with constants depending on moments of the velocity distributions strictly greater than $2(1 + delta)$. The case of order $4$ moment is treated explicitly, achieving Kacs program without any chaos propagation analysis. Finally, two counter-examples are suggested indicating that the method: (i) requires the dependance on $>2$-moments, and (ii) cannot provide contractivity in quadratic Wasserstein distance in any case.
We explore asymptotically optimal bounds for deviations of Bernoulli convolutions from the Poisson limit in terms of the Shannon relative entropy and the Pearson $chi^2$-distance. The results are based on proper non-uniform estimates for densities. They deal with models of non-homogeneous, non-degenerate Bernoulli distributions.
We explore asymptotically optimal bounds for deviations of distributions of independent Bernoulli random variables from the Poisson limit in terms of the Shannon relative entropy and Renyi/Tsallis relative distances (including Pearsons $chi^2$). This part generalizes the results obtained in Part I and removes any constraints on the parameters of the Bernoulli distributions.
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