scholarly journals Large deviations for the empirical measure of the zig-zag process

2021 ◽  
Vol 31 (6) ◽  
Author(s):  
Joris Bierkens ◽  
Pierre Nyquist ◽  
Mikola C. Schlottke
Keyword(s):  
Large Deviations ◽  
Empirical Measure

scholarly journals Large deviations for randomly connected neural networks: II. State-dependent interactions

2018 ◽  
Vol 50 (3) ◽  
pp. 983-1004 ◽  
Author(s):  
Tanguy Cabana ◽  
Jonathan D. Touboul
Keyword(s):  
Neural Networks ◽  
Large Deviations ◽  
Rate Function ◽  
Network Size ◽  
Kuramoto Model ◽  
Empirical Measure ◽  
Large Deviations Principle ◽  
Stochastic Version ◽  
State Dependent ◽  
Spatially Extended

Abstract We continue the analysis of large deviations for randomly connected neural networks used as models of the brain. The originality of the model relies on the fact that the directed impact of one particle onto another depends on the state of both particles, and they have random Gaussian amplitude with mean and variance scaling as the inverse of the network size. Similarly to the spatially extended case (see Cabana and Touboul (2018)), we show that under sufficient regularity assumptions, the empirical measure satisfies a large deviations principle with a good rate function achieving its minimum at a unique probability measure, implying, in particular, its convergence in both averaged and quenched cases, as well as a propagation of a chaos property (in the averaged case only). The class of model we consider notably includes a stochastic version of the Kuramoto model with random connections.

scholarly journals Large deviations for randomly connected neural networks: I. Spatially extended systems

2018 ◽  
Vol 50 (3) ◽  
pp. 944-982 ◽  
Author(s):  
Tanguy Cabana ◽  
Jonathan D. Touboul
Keyword(s):  
Neural Networks ◽  
Large Deviations ◽  
Rate Function ◽  
Network Size ◽  
Empirical Measure ◽  
Interaction Coefficients ◽  
Large Deviations Principle ◽  
Implicit Equation ◽  
Random Interaction ◽  
Spatially Extended

Abstract In a series of two papers, we investigate the large deviations and asymptotic behavior of stochastic models of brain neural networks with random interaction coefficients. In this first paper, we take into account the spatial structure of the brain and consider first the presence of interaction delays that depend on the distance between cells and then the Gaussian random interaction amplitude with a mean and variance that depend on the position of the neurons and scale as the inverse of the network size. We show that the empirical measure satisfies a large deviations principle with a good rate function reaching its minimum at a unique spatially extended probability measure. This result implies an averaged convergence of the empirical measure and a propagation of chaos. The limit is characterized through a complex non-Markovian implicit equation in which the network interaction term is replaced by a nonlocal Gaussian process with a mean and covariance that depend on the statistics of the solution over the whole neural field.

scholarly journals Large deviations for the empirical measure of heavy-tailed Markov renewal processes

2016 ◽  
Vol 48 (3) ◽  
pp. 648-671 ◽  
Author(s):  
Mauro Mariani ◽  
Lorenzo Zambotti
Keyword(s):  
Large Deviations ◽  
Finite Graph ◽  
Rate Function ◽  
Empirical Measure ◽  
Markov Renewal Process ◽  
Large Deviations Principle ◽  
Arrival Times ◽  
Heavy Tailed Distributions ◽  
Markov Renewal Processes ◽  
Heavy Tailed

Abstract A large deviations principle is established for the joint law of the empirical measure and the flow measure of a Markov renewal process on a finite graph. We do not assume any bound on the arrival times, allowing heavy-tailed distributions. In particular, the rate function is in general degenerate (it has a nontrivial set of zeros) and not strictly convex. These features show a behaviour highly different from what one may guess with a heuristic Donsker‒Varadhan analysis of the problem.

scholarly journals Gradient and GENERIC Systems in the Space of Fluxes, Applied to Reacting Particle Systems

Entropy ◽  
2018 ◽  
Vol 20 (8) ◽  
pp. 596
Author(s):  
D. Renger
Keyword(s):  
Large Deviations ◽  
Particle Flux ◽  
General Setting ◽  
Particle Systems ◽  
Empirical Measure ◽  
Spatial Diffusion ◽  
Free Energies ◽  
Gradient Systems ◽  
Generic System ◽  
Macroscopic Quantity

In a previous work we devised a framework to derive generalised gradient systems for an evolution equation from the large deviations of an underlying microscopic system, in the spirit of the Onsager–Machlup relations. Of particular interest is the case where the microscopic system consists of random particles, and the macroscopic quantity is the empirical measure or concentration. In this work we take the particle flux as the macroscopic quantity, which is related to the concentration via a continuity equation. By a similar argument the large deviations can induce a generalised gradient or GENERIC system in the space of fluxes. In a general setting we study how flux gradient or GENERIC systems are related to gradient systems of concentrations. This shows that many gradient or GENERIC systems arise from an underlying gradient or GENERIC system where fluxes rather than densities are being driven by (free) energies. The arguments are explained by the example of reacting particle systems, which is later expanded to include spatial diffusion as well.

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