Pathwise solution to rough stochastic lattice dynamical system driven by fractional noise

The fBm-driving rough stochastic lattice dynamical system with a general diffusion term is investigated. First, an area element in space of tensor is desired to define the rough path integral using the Chen-equality and fractional calculus. Under certain conditions, the considered equation is proved to possess a unique local mild path-area solution.


1.
Introduction. Due to the spatially discrete structure, the deterministic lattice dynamical systems occur in a wide variety of applications and various properties of solutions to lattice dynamical systems have been studied, see e.g. [11,28,2,33,34,30,35] and references therein. As we know, dynamical systems are often under random perturbation, and the noise can drastically modify the deterministic dynamics, and even induce new types of dynamical behavior. Therefore, stochastic lattice dynamical systems (SLDS) arise naturally when taking into account these random influences or uncertainties. Since Bates et al. firstly studied the existence of a compact global random attractor within the set of tempered random bounded sets of the one-dimensional SLDS [1], many works in the literature have been extensively done regarding the existence of global random attractors for first (or second)-order SLDS with additive white noises [27,26,25,31,7], or a multiplicative white noise [6,22,23,8,3,5].
In physics, the uncertainties are used to describe the interaction between the (small) system and its (large) environment. The driving noises could be white or colored, Markovian or non-Markovian, Gaussian or non-Gaussian, and semimartingale or non-semimartingale. There is no a-priori reason to assume that the stochastic forces are independent of disjoint time intervals. This fact motivates us to choose the so-called fractional Brownian motion (fBm) as driving process for the SLDS. In this respect, the random attractor of the SLDS driven by an additive fBm with Hurst parameter bigger than 1/2 was shown to be a singleton sets under usual dissipativeness conditions [19,20]. The synchronous phenomena of the solutions to system driven by fractional environmental noises on finite lattice was studied in [21] when H > 1/2. Later on, the existence of a global forward attracting set of a stochastic lattice system with a Caputo fractional time derivative was established in the weak mean-square topology [32]. Recently, stochastic lattice dynamical systems driven by fractional Brownian motion with H ∈ (1/2, 1) was proved to have a unique pathwise mild solution, which is exponentially stable under suitable conditions [4]. There has, however, been little mention about the SLDS in the rough paths case, say, H ∈ (0, 1/2]. To define the stochastic integral of rough functions, Hu and Nualart formulated a second equation for the so-called area in the space of tensors and proved an existence and uniqueness result for finite-dimensional stochastic differential equations [24], in which the fBm is such that H ∈ (1/3, 1/2]. Later on, Garrido-Atienza et al. have extended this idea to infinite-dimensional setting and proved the existence of a unique local mild solution consisting in a path and an area component [13]. Under some more restrictive conditions, they further proved the existence and uniqueness of a global mild pathwise solution [14]. To the best of our knowledge, the existence and uniqueness of a mild pathwise solution to the SLDS driven by an fBm with H ∈ (0, 1/2] is a challenging and rather open problem up to now.
In this paper, we consider the stochastic lattice equation driven by fractional Brownian motion with H ∈ (1/3, 1/2]: is a one-dimensional two-sided fBm with Hurst parameter H ∈ (1/3, 1/2], and f i , g i are smooth functions satisfying proper conditions, which will be made precise below. This paper is organized as follows. In Section 2 we present a brief introduction to some spaces, definitions, and preliminary results. In Section 3 we consider the local existence and uniqueness of mild solution to (1). Specifically, in Subsection 3.1 we give assumptions of f i and g i , and then rewrite (1) into a stochastic integral equation at first. In Subsection 3.2 we define the mild path-area solution in a limit points space. At last, in Subsection 3.3 we present the local existence and uniqueness theorem and prove it with some key technique estimates.
2. Preliminaries. We now present a brief introduction to some spaces, definitions, and preliminary facts, which are used throughout this paper.
Let V = (V, (·), | · |) be a separable Hilbert space, and V × V , V ⊗ V be the cartesian product and the tensor product of V respectively. For convenience, we denote the norm of V ⊗V by |·| V ⊗V and the rank-one tensor by x⊗ V y for x, y ∈ V . Then (e i ⊗ V e j ) i,j∈N is a complete orthogonal system of V ⊗ V where (e i ) i∈N can be any complete orthogonal system for V . For 0 ≤ T 1 < T 2 < ∞ and 0 < β < 1, we use C β ([T 1 , T 2 ]; V ) to denote the Banach space of V -valued Hölder continuous function with the norm As shown in [13,Lemma 4], C β+β (∆ T1,T2 ; V ⊗ V ) is Banach space. In order to treat the stochastic integral of rough functions, we introduce several main features of fractional calculus [29]. To do this, assume that for some 0 and v ∈ C β+β (∆ T1,T2 ; V ⊗ V ) for 1/3 < β < β < 1/2, and that these three elements satisfy the Chen-equality [9], given by for T 1 < s ≤ r ≤ t ≤ T 2 . Let V 1 , V 2 be two separable Hilbert spaces and L 2 (V 1 ; V 2 ) be the space of Hilbert-Schmidt operators from V 1 to V 2 . Throughout this paper, we assume that 0 < α < 1.
Definition 2.2. The generalized fractional derivative of the tensor valued element v is defined for T 1 ≤ r < t ≤ T 2 by , the compensated fractional derivative of order α of g(u(·)) is given bŷ It is easy to check that D α s + g(u(·))[r] is not well-defined provided that β < α, g(u(·)) ∈ C β ([T 1 , T 2 ]; V ) and Dg is bounded. Hence the classical Young integral 814 CAIBIN ZENG, XIAOFANG LIN, JIANHUA HUANG AND QIGUI YANG does not make sense in this situation. Following the idea in [13], however, one can introduce another stochastic integral for the rough case. In fact, provided that u, v, ω satisfy the Chen-equality (2) and β < α < 2β, α + β > 1, β + 1 > 2α. It is worth mentioning that the expression of v is needed. Hence we need a second equation for the so call area component v. Next, we turn to introduce the driving process. Denote by the separable Hilbert space of square summable sequences, equipped with the norm and the inner product Additionally, we denote by 2 κ the subspace of 2 as 2 κ := u ∈ 2 : Let us consider the infinite sequence (e i ) i∈Z in 2 , which has 1 in position i and 0 in other positions. Then (e i ) i∈Z is a complete orthonormal basis of 2 , and ( ei (1+|i|) κ ) i∈Z is the complete orthonormal basis of 2 κ . One can check that 2 κ is a separable Hilbert space with inner product and norm Throughout this paper, we will consider a fractional Brownian motion (fBm) with values in 2 and Husrt parameter H ∈ (1/3, 1/2]. On a probability space (Ω, F, P), let us denote (B H i ) i∈Z by an independent and identically distributed sequence of fBm with same Hurst parameter H. This means that each B H i is a centered Gaussian process on R with the covariance Let Q be a linear operator on 2 such that Qe i = σ 2 i e i , σ = (σ i ) i∈Z . Thus Q is a non-negative, symmetric, and trace class operator. Then a continuous 2 -valued fBm B H with covariance operator Q and Hurst parameter H is defined by In the sequel, we will work with the canonical version of the fBm. Let Ω = C 0 (R; 2 ) be the space of continuous paths in 2 with values zero at zero equipped with the compact open topology. F is defined as the Borel-σ-algebra and P H is the distribution of B H (t). Let us consider the Wiener shift given by θ t ω(·) = ω(· + t) − ω(t) for t ∈ R and ω ∈ C 0 (R; 2 ). It follows from [18, Theorem 1] that the quadruple (Ω, F, P H , θ) is an ergodic metric dynamical system. Also, we limit this metric dynamical system to the set Ω of β -Hölder continuous paths on [−m, m] for any m ∈ N and β ∈ (1/3, H) and denote the restricted metric dynamical system again by (Ω, F, P H , θ) with a slight abuse of notation. In particular, let us also identify B H (·, ω) and ω(·) in the following discussion.
3. Existence and uniqueness of local mild solution. (1). Let A be a linear bounded operator

Reformulation of the equation
It is easy to check that and thus Au, u ≥ 0 for any u ∈ 2 . Let us further consider the linear bounded operator where λ is defined in (1). Then A λ u, u ≥ 0, which implies that −A λ is a negative defined and bounded operator, and thus it generates a uniformly continuous semigroup S := e −A λ t on 2 . Then we have the following estimates for the semigroup S: for all 0 ≤ s ≤ t. The first one can be obtained by the energy inequality and the last two follow by the mean value theorem, see also [4]. Furthermore, using these inequalities we can deduce the following properties of S(t). In fact, for any 0 ≤ q ≤ r ≤ s ≤ t these inequalities hold We are now in position to formulate the needed assumptions for the nonlinear functions f i and g i : The process ω is a canonical 2 -valued continuous fBm with covariance Q, defined by (3). In particular, the parameters are chosen to satisfy that (A3): g i ∈ C 3 (R; R), and there exist constants D g , M g , F g ≥ 0 such that for any ζ ∈ R, i ∈ Z, the following inequalities hold: Let u = (u i ) i∈Z be an element of 2 . Then by (A2) and (A3) one can define the Then, similar to [4, Lemma 3.2], f and g are proved to be well-posed in the next result, as well as their main regularity properties are given. (ii) The operator g : 2 × 2 → 2 κ is well-defined and continuously differentiable. Moreover, its first derivative Dg, second derivative D 2 g and the third derivative D 3 g are bounded with the bounds D g , M g and F g , respectively. Furthermore, for u, v, w, z ∈ 2 we have the following regularity properties: Proof. In fact, it follows that for h, y ∈ 2 which implies the required estimate (6f). Following the same procedure, one can obtain (6g). In particular, other desired estimates were proved in [4,Lemma 3.2]. Therefore, the proof of this Lemma is completed.
With the above discussion at hand, we are going to reformulate the eqaution (1) as the following evolution equation with values in 2 : For our purpose, we look for a mild solution to equation (7), namely, for the existence of u(t) = (u i (t)) i∈Z ∈ 2 satisfying the integral operator equation 3.2. Path-area solution to (8). In order to understand the notion of path-area solution to (8), we first consider the case where the driving noise is regular, to later on consider the Hölder case which we are interested in. In particular, the fractional integration techniques allow us to shed light on this issue. Inspired by [13], we give the second equation for the so-called area as follows. To this, assume where the element w = (u ⊗ (ω ⊗ S ω)) is defined by (ω S (r, t) − ω S (s, t))edω(r), for s ≤ τ ≤ t. Herein, the driving noise ω was assumed to be a smooth enough function, which does not include the fBm with H ∈ (1/3, 1/2]. To avoid this embarrassing situation, it is desirable to import a linear approximation of a given fixing ω. Precisely, we introduce one more assumption condition: (A4): Let (ω n ) n∈N be a sequence of piecewise linear function with values in 2 such that each element of ((ω n ⊗ S ω n )) n∈N is defined before. Also, the sequence (ω n , (ω n ⊗ S ω n )) n∈N converges to (ω, ω ⊗ S ω) in C β ([0, T ]; 2 ) × C 2β (∆ 0,T ; L 2 (L 2 ( 2 ; 2 κ ); 2 ⊗ 2 )) for any β < H on a set of full measure. Remark 1. In Appendix A, we have proven that when n → ∞, and ω n → ω in C β ([0, T ]; 2 ), the sequence (ω n ⊗ S ω n ) n∈N is convergent in C 2β (∆ 0,T ; L 2 (L 2 ( 2 ; 2 κ ); 2 ⊗ 2 ) on a set of full measure, that says the assumption (A4) is available.
In fact,Ŵ 0,T is a complete metric space depending on ω with the norm In addition, each element from U = (u, v) ∈Ŵ 0,T satisfies the Chen-quality (2) with respect to ω. Letting U = (u, v) ∈Ŵ 0,T , we consider the operator T (U, ω, (ω ⊗ S ω), u 0 ) = (T 1 (U, ω, u 0 ), T 2 (U, ω, (ω ⊗ S ω), u 0 )) defined onŴ 0,T by the expressions and Having the above discussion in mind, we are ready to define a path-area solution to (8) as follows. Before proceeding with the existence of the solution to the considered eqaution, we pause to provide some key estimates, which are very useful for proving our main results. For simplicity and without causing ambiguity, we denote · L2( 2 ; 2 κ ) , · L( 2 × 2 ; 2 κ ) and all norms of operator by · . Throughout the whole paper we will write very often a positive constant c, which can change from line to line but is always chosen independent of time parameter T .
By performing a suitable change of variable to Beta function, the following integral formula is straightforward. Precisely, for any 0 < s < t ≤ T , µ, ν > −1, there exists a constant c = c(µ, ν) > 0 such that We now in position to state and prove several key estimations.
Moreover, similar to Lemma 3.4, we have Next we state a useful estimate about T 1 based on Lemmas 3.4 and 3.5.
We have finished the calculation of T 1 up to now. In the following, we are going to estimate the another component T 2 of T . In order to achieve this goal, we must give the following Lemma treating the operator w.
Proof. According to the expression of ω S given before, it is straightforward that the first two inequalities hold sure. For the last one, we first separate w into three parts I 1 , I 2 and I 3 , Herein we adopt the similar idea in proving [13,Lemma 20] but just point out the main differences. As for I 2 , there are two substantial distinctions But for I 3 , there are two different estimations Substituting their counterpart with these different estimations above into the proof of [13,Lemma 20] step by step, we obtain , and Thus the required claim is sure.
With above estimates at hand, it is enough to prove Theorem 3.3 for us now.
Proof of Theorem 3.3. Inspired by Lemmas 3.6 and 3.8, one can find a small enough T 1 such that the operator T onŴ 0,T1 ⊆Ŵ 0,T is a self mapping and contractive.