Some properties for Itô processes driven by G -Brownian motion *

In this paper, we use partial differential equation (PDE) techniques and probabilistic approaches to study the lower capacity of the ball for the Itô process driven by G Brownian motion ( G -Itô process). In particular, the lower bound on the lower capacity of certain balls is obtained. As an application, we prove a strict comparison theorem in G -expectation framework.


Introduction
The G-Brownian motion is a continuous process with independent and stationary increments in a nonlinear expectation space called G-expectation space. These notions and related Itô's calculus have been systemically established by Peng [15,16,17]. In particular, Gao [5] and Peng [17] have obtained the existence and uniqueness theorem for stochastic differential equations driven by G-Brownian motion. For more research on this field, we refer the reader to [1,6,7,13] and the references therein.
The G-expectation can be also viewed as a upper expectation. Indeed, Denis et al. [2] have proved that G-expectation is the supremum of linear expectations over a weakly compact subset of the probability measures of the stochastic integrals with respect to a standard Brownian motion (see Remark 2.2). This representation theorem allows us to introduce upper and lower capacities related to G-expectation. Moreover, it provides an alternative approach for the study of the probability measures of the classical Itô processes. Note that these measures may be mutually singular and do not have probability density functions (see Fabes and Kenig [4]).
In the classical probability space, Martini [14] has applied a tricky probabilistic approach to get that the marginal law of the stochastic integral with uniformly elliptic and bounded integrand does not weight points. In his paper, the main idea is to study the aforementioned upper capacity instead of the probability measure of the stochastic integral (see the operatorĈ in [14]). Indeed, he has obtained that the upper capacity of G-Brownian motion does not weight points. In sprit of G-stochastic analysis methods and PDE techniques, Hu et al. [9] have studied the upper capacity of general G-Itô process.
In their paper, the authors have got some upper bound on the upper capacity of the ball for the G-Itô process, which implies that the upper capacity of the G-Itô process also does not weight points (see Remark 3.9). Moreover, it provides a tool for the research of the path properties of the classical Itô integral (see Examples 3.9 and 3.13 in [9]).
The present paper is devoted to investigating the lower capacity for the G-Itô processes. The key ingredient of our approach is based on certain estimates of viscosity solutions of PDE introduced in [9], from which we show that the ball for G-Itô process has a positive lower capacity. Moreover, we obtain a lower bound on the lower capacity of some balls for G-Brownian motion. In particular, it also provides a lower bound on the marginal law of the balls for the Itô integral (see Remark 3.8). In contrast to the one in [9], the G-Itô process considered in this paper are more general. Indeed, we do not assume the boundedness of the integrands for dt and d B parts. Moreover, the diffusion term can be unbounded from above.
A direct application of this paper is to discuss the strict comparison theorem in Gexpectation framework, which is non-trivial due to the nonlinearity. A strict comparison theorem for G-Brownian motion has been proved by Li [12] based on the Krylov and Safonov estimates [11]. In this paper, we establish a strict comparison theorem for the G-Itô process through the above estimate of lower capacity, which extends the one in [12] to general case. In particular, the necessary and sufficient conditions for the strict comparison theorem are also stated.
The rest of this paper is organized as follows. In Section 2, we present some preliminaries for G-expectation, which are needed in the sequel. In Section 3, we state our main results: The ball for G-Itô process has a positive lower capacity. Some estimates on the lower capacity of the ball for G-Brownian motion are also obtained.
As an application, Section 4 is devoted to tackling the strict comparison theorem in G-expectation framework.

Preliminaries
In this section, we shall recall some basic notions and results about G-expectation and related capacities. The readers may refer to [17] for more details.
For convenience, every element x ∈ R n is identified to a column vector with i-th component x i and the corresponding Euclidian norm is denoted by |x|. Let Ω be the space of all R d -valued continuous paths (ω t ) t≥0 starting from origin, equipped with the Denote by B(Ω) the Borel σ-algebra of Ω. Let B t (ω) := ω t be the canonical process and is the collection of all bounded and Lipschitz functions defined on R k×d . Let Ω t = {ω ·∧t : ω ∈ Ω}, we can define L ip (Ω t ) analogously.
For each given monotonic and sublinear function G : S d → R, we could construct a sublinear expectation space (Ω, L ip (Ω),Ê, In this space the corresponding canonical process B t is called G-Brownian motion.
. In this paper, we always assume that G is non-degenerate, i.e., there exist two constants 0 < σ ≤σ < ∞ such that Then there exists a bounded and closed subset Γ ⊂ S + d such that The following representation theorem has been proved firstly in Denis et al. [2] (see also [8]).

Main results
Now we consider the following R n -valued G-Itô diffusion process which starts from ξ ∈ L 2 G (Ω t ) at time t ≥ 0 (for convenience, we always use Einstein's summation convention): are continuous deterministic functions. These coefficients are supposed to satisfy the following assumptions: (H1) There exist two positive constants L andL such that for (H2) There exists some constant λ > 0 such that for each (t, x) ∈ [0, ∞) × R n , λI n×n ≤ σ(t, x)(σ(t, x)) , if n ≤ d and λI d×d ≤ (σ(t, x)) σ(t, x), if n > d.
Then the G-SDE (3.1) admits a unique solution (X t,ξ s ) t≤s≤T ∈ M 2 G (t, T ) for each T > t (see [17]). Next, we shall use PDE approach to estimate the lower capacity of certain balls for the G-Itô process X t,ξ s . For a fixed real number T > 0, we consider the following PDE: where the function Φ ∈ C l,Lip (R n ). Here, C l,Lip (R n ) is the space of all real valued functions ϕ on R n such that |ϕ(x) − ϕ(y)| ≤ C(1 + |x| K + |y| K )|x − y| for some positive constants C and K depending on ϕ. Then it holds that T )] for each x ∈ R n . Suppose moreover that Φ ∈ C b,Lip (R n ). Then for each ξ ∈ L 2 G (Ω t ), we have The following lemma is the key point to prove our main results, whose proof is based on [9] with some modifications.
Proof. Without loss of generality, assume that T ≤ 1. In the sequel we shall use the same notations as in Lemma 3.
Thus we need to estimate the −Ê[− exp{− mθ|X t,ξ T −a| 2 2 }] term. The remainder of proof will be divided into two steps.
1. T − t ≤ ε. Recalling Theorem 3.1 and Lemma 3.2, we deduce that for each m ≥ 8κ, In sprit of Jensen's inequality and note thatÊ[−·] ≥ −Ê[·], we obtain that Combining the above inequalities indicates that for each m ≥ 8κ, The assumption (H2) is necessary for our method. For example, if b = x and h jk = σ = 0, then it is easy to check that v(|X 0,y t − (y + 1)e t | ≤ 1 2 ) = 0 for each (t, y) ∈ (0, ∞) × R n . Remark 3.6. Note that when the diffusion term is uniformly elliptic and bounded, Theorem 3.4 can be also derived from Krylov and Safanov estimates ( [11]) or Harnack's inequality (see Imbert and Silvestre [10]). Indeed, [12] has used this approach to get that the ball for G-Brownian motion has a positive lower capacity. By the method of this paper, we can also study the case that the diffusion term may be unbounded from above and degenerate elliptic (n > d). Moreover, we could estimate the lower bound on the lower capacity of the balls for G-Brownian motion.
Remark 3.8. By Remark 2.2, for each a ∈ R and δ > 0 we have, where C a,δ,t is given in Theorem 3.7.
Remark 3.9. The upper bound on the upper capacity of the ball for G-Brownian motion has been given in [9], which is where ρ = σ 2 (2σ 2 ) −1 and θ = (dσ 2 ) −1 . Unfortunately, we cannot obtain such concise estimate for the lower capacity.

Applications
As an application, we shall study strict comparison theorem in G-expectation framework. Suppose that ϕ, ψ ∈ C l,Lip (R n ). If ϕ ≤ ψ, then it is easy to check thatÊ[ϕ(X t,x T )] ≤ E[ψ(X t,x T )] for each (t, x) ∈ [0, T ] × R n . Now we are interested in finding some conditions on ϕ and ψ under which the strict comparison theorem holds. We first consider the following simple case. Lemma 4.1. Let ϕ ∈ C l,Lip (R n ) such that ϕ ≤ 0 and given (t,x) ∈ [0, T ) × R n . Then E[ϕ(Xt ,x T )] < 0 if and only if there exists some point x 0 ∈ R n such that ϕ(x 0 ) < 0.
Due to the subadditivity of G-expectation, we immediately have the following strict comparison theorem. The following result is a direct consequence of Theorems 3.1 and 4.2, which can be seen as a strict comparison theorem of PDE (3.2). Corollary 4.3. Let ϕ, ψ ∈ C l,Lip (R n ) such that ϕ ≤ ψ. If ϕ(x 0 ) < ψ(x 0 ) for some x 0 ∈ R n , then u ϕ (t, x) < u ψ (t, x), ∀(t, x) ∈ [0, T ) × R n , where u f is the viscosity solution to PDE (3.2) with terminal condition u f (T, x) = f (x), f = ϕ, ψ.
In particular, taking b = h jk = 0 and σ = I d×d , we have the following strict comparison theorem for G-Brownian motion.