# American Institute of Mathematical Sciences

February  2019, 12(1): 119-128. doi: 10.3934/dcdss.2019008

## Asymptotic behaviour for operators of Grushin type: Invariant measure and singular perturbations

 1 Dipartimento di Matematica "Tullio Levi-Civita", Università degli Studi di Padova, Via Trieste 63, 35121, Padova, Italy 2 Dipartimento di Ingegneria dell'Informazione, Università degli Studi di Padova, Via Gradenigo 6/b, 35131, Padova, Italy 3 Université Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France

Received  February 2017 Revised  October 2017 Published  July 2018

This paper concerns singular perturbation problems where the dynamics of the fast variable evolve in the whole space according to an operator whose infinitesimal generator is formed by a Grushin type second order part and a Ornstein-Uhlenbeck first order part.

We prove that the dynamics of the fast variables admits an invariant measure and that the associated ergodic problem has a viscosity solution which is also regular and with logarithmic growth at infinity. These properties play a crucial role in the main theorem which establishes that the value functions of the starting perturbation problems converge to the solution of an effective problem whose operator and initial datum are given in terms of the associated invariant measure.

Citation: Paola Mannucci, Claudio Marchi, Nicoletta Tchou. Asymptotic behaviour for operators of Grushin type: Invariant measure and singular perturbations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (1) : 119-128. doi: 10.3934/dcdss.2019008
##### References:
 [1] O. Alvarez and M. Bardi, Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equations, Mem. Amer. Math. Soc., 204 (2010), vi+77 pp. doi: 10.1090/S0065-9266-09-00588-2. Google Scholar [2] M. Bardi, A. Cesaroni and L. Manca, Convergence by viscosity methods in multiscale financial models with stochastic volatility, SIAM J. Financial Math., 1 (2010), 230-265. doi: 10.1137/090748147. Google Scholar [3] A. Bensoussan, Perturbation Methods in Optimal Control, Wiley/Gauthier-Villars Series in Modern Applied Mathematics. John Wiley & Sons, Chichester; Gauthier-Villars, Montrouge, 1988. Google Scholar [4] I. Capuzzo Dolcetta and A. Cutrì On the Liouville property for the sub-Laplacians, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 25 (1997), 239–256. Google Scholar [5] M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second-order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5. Google Scholar [6] L. C. Evans, The perturbed test-function method for viscosity solutions of nonlinear PDE, Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), 359-375. doi: 10.1017/S0308210500018631. Google Scholar [7] W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, Berlin, 1993. Google Scholar [8] Y. Fujita, H. Ishii and P. Loreti, Asymptotic solutions of viscous Hamilton-Jacobi equations with Ornstein-Uhlenbeck operator, Comm. Partial Differential Equations, 31 (2006), 827-848. doi: 10.1080/03605300500358087. Google Scholar [9] D. Ghilli, Viscosity methods for large deviations estimates of multi scale stochastic processes, ESAIM Control Optim. Calc. Var., to appear. doi: 10.1051/cocv/2017051. Google Scholar [10] H. Ishii and P.-L. Lions, Viscosity solution of fully nonlinear second-order elliptic partial differential equations, J. Differential Equations, 83 (1990), 26-78. doi: 10.1016/0022-0396(90)90068-Z. Google Scholar [11] P. L. Lions and M. Musiela, Ergodicity of Diffusion Processes, unpublished.Google Scholar [12] P. Mannucci, C. Marchi and N. Tchou, The ergodic problem for some subelliptic operators with unbounded coefficients, NoDEA Nonlinear Differential Equations Appl., 23 (2016), Art. 47, 26 pp. doi: 10.1007/s00030-016-0401-2. Google Scholar [13] P. Mannucci, C. Marchi and N. A. Tchou, Singular perturbations for a subelliptic operator, ESAIM Control Optim. Calc. Var., to appear. doi: 10.1051/cocv/2017063. Google Scholar

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##### References:
 [1] O. Alvarez and M. Bardi, Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equations, Mem. Amer. Math. Soc., 204 (2010), vi+77 pp. doi: 10.1090/S0065-9266-09-00588-2. Google Scholar [2] M. Bardi, A. Cesaroni and L. Manca, Convergence by viscosity methods in multiscale financial models with stochastic volatility, SIAM J. Financial Math., 1 (2010), 230-265. doi: 10.1137/090748147. Google Scholar [3] A. Bensoussan, Perturbation Methods in Optimal Control, Wiley/Gauthier-Villars Series in Modern Applied Mathematics. John Wiley & Sons, Chichester; Gauthier-Villars, Montrouge, 1988. Google Scholar [4] I. Capuzzo Dolcetta and A. Cutrì On the Liouville property for the sub-Laplacians, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 25 (1997), 239–256. Google Scholar [5] M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second-order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5. Google Scholar [6] L. C. Evans, The perturbed test-function method for viscosity solutions of nonlinear PDE, Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), 359-375. doi: 10.1017/S0308210500018631. Google Scholar [7] W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, Berlin, 1993. Google Scholar [8] Y. Fujita, H. Ishii and P. Loreti, Asymptotic solutions of viscous Hamilton-Jacobi equations with Ornstein-Uhlenbeck operator, Comm. Partial Differential Equations, 31 (2006), 827-848. doi: 10.1080/03605300500358087. Google Scholar [9] D. Ghilli, Viscosity methods for large deviations estimates of multi scale stochastic processes, ESAIM Control Optim. Calc. Var., to appear. doi: 10.1051/cocv/2017051. Google Scholar [10] H. Ishii and P.-L. Lions, Viscosity solution of fully nonlinear second-order elliptic partial differential equations, J. Differential Equations, 83 (1990), 26-78. doi: 10.1016/0022-0396(90)90068-Z. Google Scholar [11] P. L. Lions and M. Musiela, Ergodicity of Diffusion Processes, unpublished.Google Scholar [12] P. Mannucci, C. Marchi and N. Tchou, The ergodic problem for some subelliptic operators with unbounded coefficients, NoDEA Nonlinear Differential Equations Appl., 23 (2016), Art. 47, 26 pp. doi: 10.1007/s00030-016-0401-2. Google Scholar [13] P. Mannucci, C. Marchi and N. A. Tchou, Singular perturbations for a subelliptic operator, ESAIM Control Optim. Calc. Var., to appear. doi: 10.1051/cocv/2017063. Google Scholar
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