We study some classes of singular perturbation problems where the dynamics of the fast variables evolve in the whole space obeying to an infinitesimal operator which is subelliptic and ergodic. We prove that the corresponding ergodic problem admits a solution which is globally Lipschitz continuous and it has at most a logarithmic growth at infinity. The main result of this paper establishes that, as ϵ → 0, the value functions of the singular perturbation problems converge locally uniformly to the solution of an effective problem whose operator and terminal data are explicitly given in terms of the invariant measure for the ergodic operator.
Accepté le :
DOI : 10.1051/cocv/2017063
Mots-clés : Subelliptic equations, Heisenberg group, invariant measure, singular perturbations, viscosity solutions, degenerate elliptic equations
@article{COCV_2018__24_4_1429_0,
author = {Mannucci, Paola and Marchi, Claudio and Tchou, Nicoletta},
title = {Singular perturbations for a subelliptic operator},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1429--1451},
publisher = {EDP-Sciences},
volume = {24},
number = {4},
year = {2018},
doi = {10.1051/cocv/2017063},
zbl = {1414.35019},
mrnumber = {3922437},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv/2017063/}
}
TY - JOUR AU - Mannucci, Paola AU - Marchi, Claudio AU - Tchou, Nicoletta TI - Singular perturbations for a subelliptic operator JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2018 SP - 1429 EP - 1451 VL - 24 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2017063/ DO - 10.1051/cocv/2017063 LA - en ID - COCV_2018__24_4_1429_0 ER -
%0 Journal Article %A Mannucci, Paola %A Marchi, Claudio %A Tchou, Nicoletta %T Singular perturbations for a subelliptic operator %J ESAIM: Control, Optimisation and Calculus of Variations %D 2018 %P 1429-1451 %V 24 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2017063/ %R 10.1051/cocv/2017063 %G en %F COCV_2018__24_4_1429_0
Mannucci, Paola; Marchi, Claudio; Tchou, Nicoletta. Singular perturbations for a subelliptic operator. ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 4, pp. 1429-1451. doi : 10.1051/cocv/2017063. http://archive.numdam.org/articles/10.1051/cocv/2017063/
[1] and , Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equations. Mem. Amer. Math. Soc. 204 (2010) 960 | MR | Zbl
[2] and , Singular perturbations of nonlinear degenerate parabolic PDEs: a general convergence result. Arch. Ration. Mech. Anal. 170 (2003) 17–61 | DOI | MR | Zbl
[3] , and , Multiscale problems and homogenization for second-order Hamilton Jacobi equations. J. Differ. Equ. 243 (2007) 349–387 | DOI | MR | Zbl
[4] and , Optimal control and viscosity solutions of Hamilton-Jacobi Bellman equations. Systems and Control: Foundations and Applications. Birkhauser, Boston (1997) | MR | Zbl
[5] and , Optimal control with random parameters: a multiscale approach. Eur. J. Control 17 (2011) 30–45 | DOI | MR | Zbl
[6] , and , Convergence by viscosity methods in multiscale financial models with stochastic volatility. SIAM J. Financial Math. 1 (2010) 230–265 | DOI | MR | Zbl
[7] , and , Some results on partial differential equations and Asian options. Math. Models Methods Appl. Sci. 11 (2001) 475–497 | DOI | MR | Zbl
[8] , and , User’s guide to viscosity solutions of second-order partial differential equations. Bull. Amer. Math. Soc. 27 (1992) 1–67 | DOI | MR | Zbl
[9] and , Uniqueness results for second-order Bellman-Isaacs equations under quadratic growth assumptions and applications. SIAM J. Control Optimiz. 45 (2006) 74–106 | DOI | MR | Zbl
[10] and , Controlled Markov Processes and Viscosity Solutions. Springer–Verlag, Berlin (1993) | MR | Zbl
[11] , and , Derivatives in financial markets with stochastic volatility. Cambridge University Press, Cambridge (2000) | MR | Zbl
[12] , and , Asymptotic solutions of viscous Hamilton-Jacobi equations with Ornstein-Uhlenbeck operator. Comm. Partial Diff. Equ. 31 (2006) 827–848 | DOI | MR | Zbl
[13] , Viscosity methods for large deviations estimates of multiscale stochastic processes. ESAIM: COCV 24 (2018) 605–637 | Numdam | MR | Zbl
[14] , Weak convergence methods and singularly perturbed stochastic control and filtering problems. Birkhauser, Boston (1990) | DOI | MR | Zbl
[15] and , Two-scale stochastic systems. Asymptotic analysis and control. In Vol. 49 of Stochastic Modelling and Applied Probability. Springer–Verlag, Berlin (2003) | MR | Zbl
[16] , On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions. Funkcial. Ekvac. 38 (1995) 101–120 | MR | Zbl
[17] and , Gradient bounds for nonlinear degenerate parabolic equations and applications to large time behaviour of systems. Nonlin. Anal. 130 (2016) 76–101 | DOI | MR | Zbl
[18] P.L. Lions, Lectures at Collège de France 2014–2015. Available at: http://www.college-de-france.fr.
[19] P.L. Lions and M. Musiela, Ergodicity of diffusion processes. Cahiers du CEREMADE (2002) Available at: https://www.ceremade.dauphine.fr/
[20] , and , The ergodic problem for some subelliptic operators with unbounded coefficients. Nonlin. Differ. Equ. Appl. 23 (2016) 47 | DOI | MR | Zbl
[21] P. Mannucci, C. Marchi and N.A. Tchou, Asymptotic behaviour for operators of Grushin type: invariant measure and singular perturbations. Discrete Contin. Dyn. Syst.- S 12 (2019) 119–128 | MR
[22] and , Periodic homogenization under a hypoellipticity condition. Nonlin. Differ. Equ. Appl. 22 (2015) 579–600 | DOI | MR | Zbl
[23] , and , The mathematics of financial derivatives. Cambridge University Press, Cambridge (1995) | DOI | MR | Zbl
Cité par Sources :
