July  2006, 6(4): 803-828. doi: 10.3934/dcdsb.2006.6.803

Examples of moderate deviation principle for diffusion processes

1. 

CEREMADE, Université Paris Dauphine and TSI, Ecole nationale des Telecommunications, France

2. 

Electrical Engineering Systems, Tel Aviv University, 69978 - Ramat Aviv, Tel Aviv, Israel

Received  January 2005 Revised  November 2005 Published  April 2006

Taking into account some likeness of moderate deviations (MD) and central limit theorems (CLT), we develop an approach, which made a good showing in CLT, for MD analysis of a family

$S^\kappa_t=\frac{1}{t^\kappa}\int_0^tH(X_s)ds, t\to\infty$

for an ergodic diffusion process $X_t$, provided that $0.5<\kappa<1$, and appropriate $H$. We use a well known decomposition with "corrector'':

$\frac{1}{t^\kappa}\int_0^tH(X_s)ds=$corrector$+\frac{1}{t^\kappa}$ ${M_t}$martingale.

and show that, as in the CLT analysis, the corrector is negligible, and the main contribution in the MD brings the family "$ \frac{1}{t^\kappa}M_t, \ t\to\infty. $'' Starting from Freidlin, [7], and finishing by Wu's papers [33]-[37], in the MD study Laplace's transform dominates. In the paper, we replace this technique by "Stochastic exponential'' one, enabling to formulate the MDP conditions in terms of "drift-diffusion'' parameters and $H$. However, a verification of these conditions heavily depends on a specificity of a diffusion model. That is why the paper is named "Examples ...''.

Citation: A. Guillin, R. Liptser. Examples of moderate deviation principle for diffusion processes. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 803-828. doi: 10.3934/dcdsb.2006.6.803
[1]

Manuel de León, Juan Carlos Marrero, David Martín de Diego. Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics. Journal of Geometric Mechanics, 2010, 2 (2) : 159-198. doi: 10.3934/jgm.2010.2.159

[2]

Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations and Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013

[3]

Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control and Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017

[4]

Sebastián Ferrer, Martin Lara. Families of canonical transformations by Hamilton-Jacobi-Poincaré equation. Application to rotational and orbital motion. Journal of Geometric Mechanics, 2010, 2 (3) : 223-241. doi: 10.3934/jgm.2010.2.223

[5]

Thierry Horsin, Peter I. Kogut. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I. Existence result. Mathematical Control and Related Fields, 2015, 5 (1) : 73-96. doi: 10.3934/mcrf.2015.5.73

[6]

Federico Bassetti, Lucia Ladelli. Large deviations for the solution of a Kac-type kinetic equation. Kinetic and Related Models, 2013, 6 (2) : 245-268. doi: 10.3934/krm.2013.6.245

[7]

Xingxing Liu. Stability in the energy space of the sum of $ N $ solitary waves for an equation modelling shallow water waves of moderate amplitude. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022105

[8]

Jan Haskovec, Nader Masmoudi, Christian Schmeiser, Mohamed Lazhar Tayeb. The Spherical Harmonics Expansion model coupled to the Poisson equation. Kinetic and Related Models, 2011, 4 (4) : 1063-1079. doi: 10.3934/krm.2011.4.1063

[9]

Marco A. Fontelos, Lucía B. Gamboa. On the structure of double layers in Poisson-Boltzmann equation. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1939-1967. doi: 10.3934/dcdsb.2012.17.1939

[10]

Luca Lussardi. On a Poisson's equation arising from magnetism. Discrete and Continuous Dynamical Systems - S, 2015, 8 (4) : 769-772. doi: 10.3934/dcdss.2015.8.769

[11]

Dongfen Bian, Huimin Liu, Xueke Pu. Modulation approximation for the quantum Euler-Poisson equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4375-4405. doi: 10.3934/dcdsb.2020292

[12]

Hyung Ju Hwang, Juhi Jang. On the Vlasov-Poisson-Fokker-Planck equation near Maxwellian. Discrete and Continuous Dynamical Systems - B, 2013, 18 (3) : 681-691. doi: 10.3934/dcdsb.2013.18.681

[13]

Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami. Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1285-1301. doi: 10.3934/cpaa.2012.11.1285

[14]

Laurent Bernis, Laurent Desvillettes. Propagation of singularities for classical solutions of the Vlasov-Poisson-Boltzmann equation. Discrete and Continuous Dynamical Systems, 2009, 24 (1) : 13-33. doi: 10.3934/dcds.2009.24.13

[15]

Jean Dolbeault. An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 361-380. doi: 10.3934/dcds.2002.8.361

[16]

Qian Shen, Na Wei. Stability of ground state for the Schrödinger-Poisson equation. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2805-2816. doi: 10.3934/jimo.2020095

[17]

Caixia Chen, Aixia Qian. Multiple positive solutions for the Schrödinger-Poisson equation with critical growth. Mathematical Foundations of Computing, 2022, 5 (2) : 113-128. doi: 10.3934/mfc.2021036

[18]

Zhiyan Ding, Qin Li. Constrained Ensemble Langevin Monte Carlo. Foundations of Data Science, 2022, 4 (1) : 37-70. doi: 10.3934/fods.2021034

[19]

Anaïs Crestetto, Nicolas Crouseilles, Mohammed Lemou. Kinetic/fluid micro-macro numerical schemes for Vlasov-Poisson-BGK equation using particles. Kinetic and Related Models, 2012, 5 (4) : 787-816. doi: 10.3934/krm.2012.5.787

[20]

Angela Pistoia, Tonia Ricciardi. Sign-changing tower of bubbles for a sinh-Poisson equation with asymmetric exponents. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5651-5692. doi: 10.3934/dcds.2017245

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (140)
  • HTML views (0)
  • Cited by (21)

Other articles
by authors

[Back to Top]