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

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.


Introduction
This paper is devoted to study with PDE's methods some asymptotic features of processes described by the dynamics where W t is a 2-dimensional Brownian motion while the matrix σ is degenerate and of Grushin type and the drift b is of Ornstein-Uhlenbeck type, namely The columns of σ in (1.2) satisfy Hörmander condition: X 1 = (1, 0) and [X 1 , X 2 ] = (0, 1) span all R 2 . Hence, we have that [X 1 , X 2 ] = ∂ x 2 .
In particular, we shall investigate: 1) existence and uniqueness of the invariant measure of this process; 2) existence, uniqueness and regularity of the solution for the ergodic problem of the infinitesimal generator L (see (2.1)); 3) the asymptotic behaviour as ǫ → 0 of the value function of optimal control problems driven by The paper is organized as follows: in Section 2 we prove existence and uniqueness of the invariant measure. In Section 3 we establish our main result on perturbation problem, to this end, we introduce the approximated ergodic problems and investigate the regularity of their solutions.

Existence of the invariant measure
We consider the stochastic dynamics (1.1) with coefficients as in (1.2). The main aim of this section is to prove existence and uniqueness of the invariant measure m associated to the process (1.1). To this goal, we use a Liouville property for the infinitesimal generator of (1.1) (see [4] for other Liouville properties for Grushin operator in a semilinear framework with a superlinear growth for the zeroth order term).
Let us recall from [3] that a probability measure m on R 2 is an invariant measure for process (1.1) is the solution to the parabolic Cauchy problem is the infinitesimal generator of process (1.1). For the sake of completeness, let us recall the result in [12, Example 5.1].
Theorem 2.1 The diffusion process (1.1) admits exactly one invariant probability measure m.
Proof. Under our assumptions it is easy to check that the matrix Then L ρ W = −tr(σ ρ σ T ρ D 2 u) + b · Du ≥ 1 is equivalent to the following condition for ρ sufficiently small. Then following the procedure used in [12, Proposition 2.1], using the function W , there exists an unique invariant measure m ρ for the process with diffusion σ ρ , and arguing as in [12, Theorem 2.1 (proof)] and using again the function W we obtain the existence of the invariant measure associated to the process (1.1). For the proof of uniqueness, we refer the reader to [ for β ≥ 0, R > 0 and suitably chosen C i > 0 (i = 1, 2).

Remark 2.3
As applications of the existence of an invariant measure we obtain, arguing as in [12] the following results: where m is the invariant measure of and u δ , u and v are the solutions respectively of and L is defined in (2.1).

Asymptotic behaviour for a singular perturbation problem
In this section, we investigate the limit of the value function where E denotes the expectation, U is the set of progressively measurable processes with values in a compact metric set U and a is a fixed positive parameter and (X t , Y t ) are driven by (1.3) (note that V ǫ depends on ǫ through the coefficients of the dynamics). Throughout this section, we shall assume ii) the function g is continuous in (x, y) and there exits C g such that iii)φ(x, y, u) andσ(x, y, u) are Lipschitz continuous and bounded in (x, y) uniformly on u: 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.
By standard theory (see [7]), the value function V ǫ is the unique (viscosity) solution to the following Cauchy problem where L is the operator defined in (2.1) and H(x, y, p, X, Z) := min Our aim is to establish that, as ǫ → 0 + , the function V ǫ converges locally uniformly to a function V = V (t, x) (which will be independent of y) which can be characterized as the unique (viscosity) solution to the effective Cauchy problem on R n .
The effective Hamiltonian and the effective terminal datum are given by and m is the invariant measure established in Theorem 2.1. As a matter of facts, H(x, p, X) is the ergodic constant λ of the cell problem (3.5) −tr(σ(y)σ T (y)D 2 w(y))−b(y)Dw(y)+H(x, y, p, X, 0) = λ y ∈ R 2 , (the solution w to this equation is called corrector) while g(x) is the constant obtained in the long time behaviour of the parabolic Cauchy problem The main issues of this setting are: 1) the fast variables evolve in the whole space, 2) the infinitesimal generator of their operator is degenerate with unbounded coefficients, 3) the variables y lacks a group structure. In order to overcome these issues, we shall use the following tools: 1) there exists a superlinear Lyapunov function, 2) a Liouville type result applies to operator L, 3) there exists an invariant measure, 4) the cell problem admits a regular solution (we shall first prove that it is globally Lipschitz continuous and then we make a bootstrap argument) with an at most logarithmic growth.
In order to prove the existence and the properties of (λ, w) satisfying (3.5), we introduce the approximated problems where δ > 0 and F (y) := −H(x, y, p, X, 0) with (x, p, X) fixed. In the next subsection we investigate the properties of the approximated correctors u δ ; in the last subsection these properties will be inherited by the corrector w.

Regularity of the approximated correctors
In this section we shall establish two results on the regularity of u δ in two different setting for F : a global Lipschitz continuity and a local Hölder continuity. In our opinion, both these results have their own interest because we apply two different techniques: the former follows the ones of [5,10] while the latter one follows the ones of [8]. However, in the rest of the paper we shall only need the former one. Throughout this section we assume 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].

Local Hölder continuity of the approximated corrector
Proposition 3.2 Assume b as in (1.2) with α > 1, (3.7) and Let u δ be the unique continuous solution of (3.6) which satisfies (3.8). Then there is a constant C > 0, independent on δ such that Proof. We follow the procedure of [8,Theorem 4.3]. We define the functions w δ (x, y) = u δ (x) − u δ (y) andg(x, y) = C F |x − y| γ (Φ(x) + Φ(y) + A) where A will be chosen suitably large. If we prove that w δ ≤g in R 2 then we obtain (3.10) with a suitable C, since Φ > 1.
At this point to find a contradiction we compute δg(x,ŷ) − Ξg(x,ŷ) + αxD xg (x,ŷ) + αyD yg (x,ŷ) directly by the definition ofg. Denoting by t = |x − y| 2 , let us introduce g(x, y) as We compute now Ξg(x, y) = tr(Σ(x, y)D 2 g(x, y)). We have Denoting by A ij the 2 × 2 minor of Σ we have that Using the explicit derivatives written here above and the definition of Φ we obtain where we denoted by ∆ G u(z) := tr(σ(z)σ T (z)u(z)), i.e. the horizontal Grushin Laplacian operator. We note that, by elementary calculations, it is possible to find a constant L α such that
Proof. In this proof, β denotes a constant which may change from line to line. The corrector w solves (3.15) − tr(σ(y)σ T (y)D 2 w(y)) + αyDw(y) = G(y) with G(y) := λ − H(x, y,p,X, 0). First let us get the global Lipschitz continuity of ∂w ∂y 2 and ∂ 2 w ∂y 2 2 . Deriving equation (3.15) with respect to y 2 (remark that this is possible because G is regular enough thanks to (3.13)) we obtain that the function u := ∂w ∂y 2 is bounded by Proposition 3.1 and it solves in the sense of distributions (3.16) − tr(σσ T D 2 u) + αyDu + αu = ∂G ∂y 2 .
From Proposition 3.1 and [13, Lemma 3.5], we get that ∂w ∂y 2 is globally Lipschitz continuous in R 2 . Deriving again equation (3.16) with respect to y 2 we obtain that the function ∂ 2 w ∂y 2 2 is globally Lipschitz continuous in R 2 .

Conclusion.
We adapt the classical perturbed test function method (see [1,6,9]) to prove the convergence. To this end, we argue as in [13, Theorem 2.1] using the Liouville property for L, the regularity of the corrector and the existence of a Lyapunov function (W (y) = y 2 1 + y 2 2 ). ✷