Singular perturbations for a subelliptic operator

Abstract

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 ǫ → 0, the value functions of the singular perturbation problems converge locally uniformly to the solution of an effective problem whose operator and data are explicitly given in terms of the invariant measure for the ergodic operator.

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

“…By our choice of the matrix σ, we obtain the desired contradiction. ✷ Remark 3.1 As in [13], for b(x) = (−α 1 x 1 , −α 2 x 2 ), we obtain the same result when α 1 > 1, α 2 > 0 andL > L/l where l = min{α 1 − 1, α 2 }. and…”

“…Proof. The proof follows the same arguments of the proof of [13,Theorem 3.2]. For completeness, we briefly sketch the main steps.…”

“…Problems of this type arise from models where the variables Y evolve much faster than the variables X. We refer to [2] and [13] for the financial models which inspired this research.…”

“…Let us recall from [13, Lemma 3.3] the following result on the growth of u δ ; for the proof, we refer the reader to [13].…”

An application of the theorem on Sums to viscosity solutions of degenerate fully nonlinear equations

“…Key words and phrases. Degenerate elliptic operators, Nonlinear elliptic operators, Carnot groups, viscosity solutions, Theorem on sums, Hölder regularity.The author is supported by MURST, Italy, and INDAM-GNAMPA project 2017: Regolarità delle soluzioni viscose per equazioni a derivate parziali non lineari degeneri.The author wishes to thank P. Mannucci for pointing out her joint papers [15] and [16].…”