scholarly journals Large deviations of the exit measure through a characteristic boundary for a Poisson driven SDE

2020 ◽  
Vol 24 ◽  
pp. 148-185
Author(s):  
Etienne Pardoux ◽  
Brice Samegni-Kepgnou
Keyword(s):  
Large Deviations ◽  
Basin Of Attraction ◽  
Exit Point ◽  
Large Deviations Principle ◽  
Characteristic Boundary ◽  
Large Numbers ◽  
Lecture Notes ◽  
Exit Measure ◽  
The Law ◽  
The Basin Of Attraction

Let O be the basin of attraction of a given equilibrium of a dynamical system, whose solution is the law of large numbers limit of the solution of a Poissonian SDE as the size of the population tends to +∞. We consider the law of the exit point from O of that Poissonian SDE. We adapt the approach of Day [J. Math. Anal. Appl. 147 (1990) 134–153] who studied the same problem for an ODE with a small Brownian perturbation. For that purpose, we will use the large deviations principle for the Poissonian SDE reflected at the boundary of O, studied in our recent work Pardoux and Samegni [Stoch. Anal. Appl. 37 (2019) 836–864]. The main motivation of this work is the extension of the results concerning the time of exit from the set O established in Kratz and Pardoux [Vol. 2215 of Lecture Notes in Math.. Springer (2018) 221–327] and Pardoux and Samegni [J. Appl. Probab. 54 (2017) 905–920] to unbounded open sets O. This is done in sections 4.2.5 and 4.2.7 of Britton and Pardoux [Vol. 2255 of Lecture Notes in Math. Springer (2019) 1–120], see also The SIR model with demography subsection below.

scholarly journals Large deviations for a stochastic Cahn–Hilliard equation in Hölder norm

2020 ◽  
Vol 23 (02) ◽  
pp. 2050010
Author(s):  
Lahcen Boulanba ◽  
Mohamed Mellouk
Keyword(s):  
Weak Convergence ◽  
Large Deviations ◽  
White Noise ◽  
Space Time ◽  
Large Deviations Principle ◽  
Hilliard Equation ◽  
Hölder Norm ◽  
The Law ◽  
Cahn Hilliard Equation ◽  
Weak Convergence Approach

We consider a stochastic Cahn–Hilliard equation driven by a space–time white noise. We prove that the law of the solution satisfies a large deviations principle in the Hölder norm. Our proof is based on the weak convergence approach for large deviations.

scholarly journals Run-and-Tumble Motion: The Role of Reversibility

2021 ◽  
Vol 183 (3) ◽  
Author(s):  
Bart van Ginkel ◽  
Bart van Gisbergen ◽  
Frank Redig
Keyword(s):  
Free Energy ◽  
Diffusion Coefficient ◽  
Large Deviations ◽  
Energy Function ◽  
Internal State ◽  
Rate Function ◽  
Free Energy Function ◽  
Large Deviations Principle ◽  
Preferred Direction ◽  
State Process

AbstractWe study a model of active particles that perform a simple random walk and on top of that have a preferred direction determined by an internal state which is modelled by a stationary Markov process. First we calculate the limiting diffusion coefficient. Then we show that the ‘active part’ of the diffusion coefficient is in some sense maximal for reversible state processes. Further, we obtain a large deviations principle for the active particle in terms of the large deviations rate function of the empirical process corresponding to the state process. Again we show that the rate function and free energy function are (pointwise) optimal for reversible state processes. Finally, we show that in the case with two states, the Fourier–Laplace transform of the distribution, the moment generating function and the free energy function can be computed explicitly. Along the way we provide several examples.

scholarly journals A large‐deviations principle for all the cluster sizes of a sparse Erdős–Rényi graph

2021 ◽  
Author(s):  
Luisa Andreis ◽  
Wolfgang König ◽  
Robert I. A. Patterson
Keyword(s):  
Large Deviations ◽  
Large Deviations Principle

scholarly journals A new proof for the generalized law of large numbers under Choquet expectation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Jing Chen ◽  
Zengjing Chen
Keyword(s):  
Probability Theory ◽  
Law Of Large Numbers ◽  
Moment Condition ◽  
Large Numbers ◽  
Choquet Expectation ◽  
The Law ◽  
Linear Probability ◽  
First Moment ◽  
Independent And Identically Distributed

Abstract In this article, we employ the elementary inequalities arising from the sub-linearity of Choquet expectation to give a new proof for the generalized law of large numbers under Choquet expectations induced by 2-alternating capacities with mild assumptions. This generalizes the Linderberg–Feller methodology for linear probability theory to Choquet expectation framework and extends the law of large numbers under Choquet expectation from the strong independent and identically distributed (iid) assumptions to the convolutional independence combined with the strengthened first moment condition.

Killing the Law of Large Numbers: Mortality Risk Premiums and the Sharpe Ratio

2006 ◽  
Vol 73 (4) ◽  
pp. 673-686 ◽  
Author(s):  
M. A. Milevsky ◽  
S. D. Promislow ◽  
V. R. Young
Keyword(s):  
Mortality Risk ◽  
Law Of Large Numbers ◽  
Sharpe Ratio ◽  
Risk Premiums ◽  
Large Numbers ◽  
The Law

scholarly journals Complete Convergence and the Law of Large Numbers

1947 ◽  
Vol 33 (2) ◽  
pp. 25-31 ◽  
Author(s):  
P. L. Hsu ◽  
H. Robbins
Keyword(s):  
Complete Convergence ◽  
Law Of Large Numbers ◽  
Large Numbers ◽  
The Law

scholarly journals The law of large numbers for the maximum of almost Gaussian log-correlated fields coming from random matrices

2018 ◽  
Vol 173 (1-2) ◽  
pp. 157-209 ◽  
Author(s):  
Gaultier Lambert ◽  
Elliot Paquette
Keyword(s):  
Random Matrices ◽  
Law Of Large Numbers ◽  
Large Numbers ◽  
The Law

SYNCHRONIZATION IN RANDOMLY DRIVEN SYSTEMS

1995 ◽  
Vol 09 (16) ◽  
pp. 985-988 ◽  
Author(s):  
A.M. JAYANNAVAR
Keyword(s):  
Simple Model ◽  
Law Of Large Numbers ◽  
Driven Systems ◽  
Large Numbers ◽  
Long Time ◽  
The Law

We have solved analytically a simple model of evolution of particles driven by identical noise. We show that the trajectories of all particles collapse into a single trajectory at long time. This synchronization also leads to violation of the law of large numbers.

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