Invariant and Ergodic Nonlinear Expectations for G-diffusion Processes *

In this paper we study the problems of invariant and ergodic expectations under G-expectation framework. In particular, the stochastic differential equations driven by G-Brownian motion (G-SDEs) have the unique invariant and ergodic expectations. Moreover, the invariant and ergodic expectations of G-SDEs are also sublinear expectations. However, the invariant expectations may not coincide with the ergodic expectations, which is different from the classical case.


Introduction
Recently, Peng systemically established a time-consistent fully nonlinear expectation theory (see [11,14,15] and the references therein), which is an effective tool to study the problems of model uncertainty, nonlinear stochastic dynamical systems and fully nonlinear partial differential equations (PDEs).As a typical and important case, Peng introduced the G-expectation theory.In the G-expectation framework, the notion of G-Brownian motion and the corresponding stochastic calculus of Itô's type were also established.Moreover, Peng [14] and Gao [4] obtained the existence and uniqueness theorem of G-SDEs.
It is well known that invariant measure plays an important role in the theory of stochastic dynamical systems and ergodic theory.In particular, the invariant measure can be thought of as describing the long-term behaviour of a dynamical system, which has many important applications in, for example, PDEs and financial mathematics.So far, there are many papers in the literature which were devoted to studying the invariant measures of Markov processes, in both finite and infinite dimensional spaces (see [1] and the references therein).
The paper is organized as follows.In section 2, we present some notations and results which will be used in this paper.The existence and uniqueness theorem of invariant expectations of G-diffusion processes is established in section 3.In section 4, we shall study the relationships between invariant expectations and ergodic expectations under the G-expectation framework.

Preliminaries
The main purpose of this section is to recall some basic notions and results of Gexpectation, which are needed in the sequel.The readers may refer to [5], [6], [12], [13], [14] for more details.Definition 2.1.Let Ω be a given set and let H be a vector lattice of real valued functions defined on Ω, namely c ∈ H for each constant c and |X| ∈ H if X ∈ H. H is considered as the space of random variables.A sublinear expectation Ê on H is a functional Ê : H → R satisfying the following properties: for all X, Y ∈ H, we have Definition 2.2.Let X 1 and X 2 be two n-dimensional random vectors defined respectively in sublinear expectation spaces where X is an independent copy of X, i.e., X d = X and X⊥X.Here the letter G denotes the function where S d denotes the collection of d × d symmetric matrices.
Let Ω = C 0 ([0, ∞); R d ), the space of R d -valued continuous functions on [0, ∞) with ω 0 = 0, be endowed with the distance and B = (B i ) d i=1 be the canonical process.For each T > 0, denote For any given monotonic and sublinear function G :  ([3, 7]).There exists a weakly compact set P ⊂ M 1 (Ω), the set of all probability measures on (Ω, B(Ω)), such that P is called a set that represents Ê.
Let P be a weakly compact set that represents Ê.For this P, we define the capacity c(A) := sup P ∈P

P (A), A ∈ B(Ω).
A set A ⊂ B(Ω) is polar if c(A) = 0.A property holds "quasi-surely (q.s.) if it holds outside a polar set.In the following, we do not distinguish two random variables X and Y if X = Y q.s.. Definition 2.6.Let M 0 G (0, T ) be the collection of processes in the following form: for a given partition {t 0 , s ) 0≤t≤T are well defined, see Li-Peng [10] and Peng [14].

Invariant nonlinear expectation
In this section, we shall study the invariant expectations of G-diffusion processes.Let G : S d → R be a given monotonic and sublinear function and B t = (B where b, h ij : R n → R n , σ : R n → R n×d are deterministic continuous functions.In particular, denote X x = X 0,x .Consider also the following assumptions: (H1) There exists a constant L > 0 such that for some constants η > 0, where σ i is the i-th row of σ.
We have the following estimates of G-SDEs which can be found in Chapter V in Peng [14].
where the constant C depends on L, G, m, n and T .
The following result is important for our future discussion (see also [8]).Especially, the constant C is independent of T .Lemma 3.2.Under assumptions (H1) and (H2), if ξ, ξ ∈ L 2p G (Ω t ), then there exists a constant C depending on G, L, p, n and η, such that: To simplify presentation, we shall prove only the case when n = d = 1, as the higher dimensional case can be treated in the same way without difficulty.Set ECP 20 (2015), paper 30. where Note that On the other hand, by Lemma 3.1,

Ê[(
Then the right side of inequality (3.2) is a G-martingale.Thus we conclude that By a similar analysis as Lemma 4.1 of [8], we can also obtain the second inequality, which completes the proof.
In particular, for each t, there exists a constant C 1 depending on G, η, L, K f , n and p such that Proof.For a fixed x and each f ∈ C 2p,Lip (R n ), from Lemma 3.2, we can find some constant C depending on C and K f such that Then there exists a sequence T n → ∞ such that Ê[f (X x Tn )] → λf for some constant λf .From the uniqueness of solutions to G-SDEs, we obtain )] for each t and t with t ≤ t, then we have Applying Hölder's inequality and Lemma 3.2, we obtain that ECP 20 (2015), paper 30.
where the constant C 1 depending on p G, η, n, L, and K f may vary from line to line.Consequently, for each t, we get For each x, x ∈ R n , applying Lemma 3.2 (i) yields that which completes the proof.
The following result is a direct consequence of Theorem 3.3.
From the nonlinear Feynman-Kac formula in [14], we obtain u f (t, which is the unique viscosity solution to the following fully nonlinear PDE. where Then by Lemma 3.3, we get the following large time behaviour of solution to the fully nonlinear parabolic PDE (3.3).
Proof.The proof is immediate from Theorem 3.3 and the definition of G-expectation.
Proof.For each fixed N > 0, where Applying Lemma 3.2, there exits a constant C 1 depending on G, f 1 , p, n and η such that, It follows from f i ↓ 0 and Dini's theorem that k N i ↓ 0. Thus we have . Since N can be arbitrarily large, we get Λ[f i ] ↓ 0.
Remark 3.8.From the above proof, in general we cannot get the result for {f i } ∞ i=1 ⊂ C 2p,Lip (R n ).Theorem 3.9.Suppose (H1) and (H2) hold.Then there exists a family of weakly compact probability measures {m θ } θ∈ Θ defined on (R n , B(R n )) such that λf = sup Proof.By the representation theorem (Theorem 2.1 of Chapter 1 in [14]), for the sub-

dx).
Let P = {m θ : θ ∈ Θ} be the family of all probability measures on (R n , Then from the above result, we obtain that ECP 20 (2015), paper 30.Now we prove that P is weakly compact.Set f i (x) = (|x| − i) + ∧ 1, it is easy to check that f i ⊂ C 2p−1,Lip (R n ) and f i ↓ 0. Then by Lemma 3.7, we obtain Thus P is tight.Let m θi , i ≥ 1, converge weakly to m.Then by the definition of weak convergence, we can get for any f Thus by the monotone convergence theorem under m, we obtain Thus m ∈ P, which completes the proof.
In the classical case, i.e., Λ[•] is a linear expectation, it is easy to check that Θ consists of a single element θ 0 .In particular, the probability measure m θ0 is the unique invariant measure for the diffusion process X.Under the G-expectation framework, we can also give the following definition.Definition 3.10.A sublinear expectation Ẽ on (R n , C 2p,Lip (R n )) is said to be an invariant expectation for the G-diffusion process X if The family of probability measures that represents Ẽ on (R n , C 2p−1,Lip (R n )) is called invariant for the G-diffusion process X.
Remark 3.11.For the invariant expectation Ẽ[•], it corresponds to the family of probability measures, which can be explained as the uncertainty of the initial distribution.Given this uncertainty of the initial distribution, the left-hand side of the equality in the above definiton can be explained as the uncertainty of the distribution of X t .Thus under the invariant expectation Ẽ[•], the distribution uncertainty of the G-diffusion process X is invariant in time.
Theorem 3.12.Assume (H1) and (H2) hold.Then there exists a unique invariant expectation Ẽ for the G-diffusion process X.
. By Lemma 3.2 and Theorem 3.3, we can find some constant C 1 such that ECP 20 (2015), paper 30.
which concludes that Λ is an invariant expectation for the G-diffusion process X.
Uniqueness: Assume Λ is also an invariant expectation for the G-diffusion process X.Then for each f ∈ C 2p,Lip (R n ) and t ≥ 0, we obtain .
Consequently, we derive that and this completes the proof.
If there exists a point t 0 > 0 such that, then Ẽ is the unique invariant expectation for X.
Proof.Denote f (x) := Ê[f (X x t0 )].Then using the same method as in the proof of Theorem 3.12, we have In a similar way, we obtain for each integer n ≥ 1, Then by Theorem 3.3, we get which is the desired result.
Consider the following Ornstein-Uhlenbeck process driven by G-Brownian motion: where α > 0 is a given constant.It is obvious that assumption (H2) holds for each p ≥ 1 in this case.
Lemma 3.15.The invariant expectation for G-Ornstein-Uhlenbeck process Y is the G-normal distribution of Proof.From the G-Itô formula, we get exp(αs)dB s , for all t ≥ 0.
For each integer N , denote Then it is obvious that Thus, for each p ≥ 1 and f ∈ C p,Lip (R d ) , we have Applying Lemma 3.2 yields that Thus by Theorem 3.12, we obtain which is the desired result.
Next we shall consider the following G-diffusion process: for each x ∈ R, where α > 0 is a given constant.Applying the G-Itô formula, we get

Ergodic nonlinear expectation
In this section, we shall only consider non-degenerate G-Brownian motion, i.e., there exists some constant σ 2 > 0 such that, for any We begin with the following lemma, which is essentially from [8].
Proof.The proof is immediate from Lemma 3.2, Theorems 5.4 and 5.5 of [8].
By a similar analysis as in Lemma 3.3, it is easy to check that Λ is a sublinear expectation on (R n , C 2p,Lip (R n )).
Lemma 4.2.Assume (H1) and (H2) hold.Then we obtain In addition, we have the following result.
Proof.The proof is similar to that of Theorem 3.9.The family of probability measures that represents Ẽ is called ergodic for the G-diffusion process X.Note that Ê[ T 0 f (X x s )ds] ≤ T 0 Ê[f (X x s )]ds.Then it follows from Corollary 3.4 that λ f ≤ λf and Θ ⊂ Θ.In the classical case, it is obvious that Λ = Λ.In particular, if Θ only has a single element, it is easy to check that λ f = λf .However, in general we cannot get Λ = Λ under G-framework.In the linear case, i.e., G(a) = 1 2 a, it is easy to check that Then by the ergodic theory, we obtain However, under the nonlinear expectation framework, there is no such relationship for fully nonlinear PDE (4.3).
Remark 4.10.In the linear expectation case, ergodic theory and related problems are connected with the invariant expectation.However, from the above results, this relationship may not hold true under the nonlinear expectation framework.Thus we should study nonlinear ergodic problems via ergodic expectation Λ instead of invariant expectation Λ.In particular, [8] obtained the links between ergodic expectation and large time behaviour of solutions to fully nonlinear PDEs.

Definition 4 . 4 .
A sublinear expectation Ẽ on (R n , C 2p,Lip (R n )) is said to be an ergodic expectation for the G-diffusion process X if x s )ds], ∀f ∈ C 2p,Lip (R n ).
[3]for p ≥ 1. Denis et al.[3]proved that the completions of C b (Ω) (the set of bounded continuous function on Ω) and L ip (Ω) under • L p G are the same.Similarly, we can define Brownian motion.For a given integer p ≥ 1, a realvalued function f defined on R n is said to be inC p,Lip (R n ) if there exists a constant K f on f such that |f (x) − f (x )| ≤ K f (1 + |x| p−1 + |x | p−1 )|x − x |.Consider the following type of G-SDEs (in this paper we always use Einstein convention): for each t ≥ 0 and ξ depending [8]ariant and ergodic nonlinear expectations for G-diffusion processes Thus f (x) ∈ C 2p,Lip (R n ).From Theorem 3.3 and Lemma A.3 of[8], we get Page 8/15 ecp.ejpecp.org Example 3.16.Suppose B is a 1-dimensional G-Brownian motion.For each x ∈ R, let − exp(−2t))B 1 + (1 − exp(−t)) B 1 are identically distributed.Then for each p ≥ 1 and f ∈ C p,Lip (R) , we have dB s , for all t ≥ 0.
From Theorems 3.3, 3.13 and Lemma 3.15, we have the following.Corollary 3.17.Given a sublinear space (R, C p,Lip (R), Ẽ) and denote ζ(x) = x for x ∈ R, then Ẽ is the invariant expectation for G-process Y x if and only if for some point t > 0 and x ∈ R, exp(−α B t )ζ + exp(−α B t ) t 0 exp(α B s )dB s and ζ are identically distributed, where (B t ) t≥0 is independent from ζ.