    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, , and finishing by Wu's papers -, 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 & Continuous Dynamical Systems - B, 2006, 6 (4) : 803-828. doi: 10.3934/dcdsb.2006.6.803
  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  Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations & Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013  Federico Bassetti, Lucia Ladelli. Large deviations for the solution of a Kac-type kinetic equation. Kinetic & Related Models, 2013, 6 (2) : 245-268. doi: 10.3934/krm.2013.6.245  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 & Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017  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  Thierry Horsin, Peter I. Kogut. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I. Existence result. Mathematical Control & Related Fields, 2015, 5 (1) : 73-96. doi: 10.3934/mcrf.2015.5.73  Jan Haskovec, Nader Masmoudi, Christian Schmeiser, Mohamed Lazhar Tayeb. The Spherical Harmonics Expansion model coupled to the Poisson equation. Kinetic & Related Models, 2011, 4 (4) : 1063-1079. doi: 10.3934/krm.2011.4.1063  Marco A. Fontelos, Lucía B. Gamboa. On the structure of double layers in Poisson-Boltzmann equation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1939-1967. doi: 10.3934/dcdsb.2012.17.1939  Luca Lussardi. On a Poisson's equation arising from magnetism. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 769-772. doi: 10.3934/dcdss.2015.8.769  Hyung Ju Hwang, Juhi Jang. On the Vlasov-Poisson-Fokker-Planck equation near Maxwellian. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 681-691. doi: 10.3934/dcdsb.2013.18.681  Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami. Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1285-1301. doi: 10.3934/cpaa.2012.11.1285  Laurent Bernis, Laurent Desvillettes. Propagation of singularities for classical solutions of the Vlasov-Poisson-Boltzmann equation. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 13-33. doi: 10.3934/dcds.2009.24.13  Jean Dolbeault. An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 361-380. doi: 10.3934/dcds.2002.8.361  Qian Shen, Na Wei. Stability of ground state for the Schrödinger-Poisson equation. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020095  Anaïs Crestetto, Nicolas Crouseilles, Mohammed Lemou. Kinetic/fluid micro-macro numerical schemes for Vlasov-Poisson-BGK equation using particles. Kinetic & Related Models, 2012, 5 (4) : 787-816. doi: 10.3934/krm.2012.5.787  Angela Pistoia, Tonia Ricciardi. Sign-changing tower of bubbles for a sinh-Poisson equation with asymmetric exponents. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5651-5692. doi: 10.3934/dcds.2017245  Renjun Duan, Tong Yang, Changjiang Zhu. Boltzmann equation with external force and Vlasov-Poisson-Boltzmann system in infinite vacuum. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 253-277. doi: 10.3934/dcds.2006.16.253  Håkon Hoel, Anders Szepessy. Classical Langevin dynamics derived from quantum mechanics. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020135  Miguel Abadi, Sandro Vaienti. Large deviations for short recurrence. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 729-747. doi: 10.3934/dcds.2008.21.729  D. Blömker, S. Maier-Paape, G. Schneider. The stochastic Landau equation as an amplitude equation. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 527-541. doi: 10.3934/dcdsb.2001.1.527

2019 Impact Factor: 1.27