Absolute continuity and numerical approximation of stochastic Cahn--Hilliard equation with unbounded noise diffusion
Jianbo Cui Jialin Hong
math.NA cs.NA 60H35, 35R60, 60H15, 65M75



	
			
		[mytitlenote]This work was supported by National Natural Science Foundation of China (NO. 91530118, NO. 91130003, NO. 11021101, NO. 91630312 and NO. 11290142).  
		
		cas]Jianbo Cuicor
		jcui82@gatech.edu
		
		cas1]Jialin Hong
		hjl@lsec.cc.ac.cn
		
		[cor]Corresponding author.
		
		[cas]  School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA
			

		[cas1]1. LSEC, ICMSEC, 
			Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing,  100190, China    2. School of Mathematical Science, University of Chinese Academy of Sciences, Beijing, 100049, China 
		









In this article, we consider  the absolute continuity and numerical approximation of the solution of the stochastic Cahn–Hilliard equation with unbounded noise diffusion. We first obtain 
the Hölder continuity and Malliavin differentiability of the solution of the stochastic Cahn–Hilliard equation by using the strong convergence of the spectral Gakerkin approximation. Then
we prove the existence and strict positivity of the density function of the law of the exact solution for the stochastic Cahn–Hilliard equation with sublinear growth diffusion coefficient, which fills a gap for the existed result when the diffusion coefficient satisfies a growth condition of order 1/3<α<1. 
To approximate the density function of the exact solution, 
we 
propose a full discretization based on the spatial spectral Galerkin approximation and the temporal drift implicit Euler scheme. Furthermore, a general framework for deriving the strong convergence rate of the full discretization is developed 
based on the variation approach and the factorization method.  Consequently, we obtain the sharp mean square convergence rates in both time and space via Sobolev interpolation inequalities and semigroup theories.  To the best of our knowledge, this is the first result on the convergence rate of full discretizations for the considered equation. 


		stochastic Cahn–Hilliard equation unbounded noise diffusion  
			 Malliavin calculus numerical approximationstrong convergence rate  
			[2010]
			60H15  60H07 60H35 35R60
	
		
	
			
		
	




§ INTRODUCTION


The stochastic  Cahn–Hilliard equation is a fundamental phase field model and can be used to describe the complicated phase separation and coarsening phenomena in a melted alloy that is quenched to a temperature at which only two different concentration phases can exist stably (see e.g. <cit.>). 
There exist a lot of works focusing on the well-posedness of the stochastic Cahn–Hilliard equation(see e.g. <cit.>).
Recently, the existence and uniqueness of the solution, as well as its regularity estimates, for the stochastic  Cahn–Hilliard equation driven by multiplicative space-time white noise with diffusion coefficient of sublinear growth are proven in <cit.>. However, 
many problems about the solution for the stochastic Cahn–Hilliard equation are still unclear. One of these problems lies on the existence, smoothness, strict positivity of the density of both the exact solution and its approximation. The authors in <cit.> obtain the existence and strict positivity of the density function for stochastic Cahn–Hilliard equation with bounded diffusion coefficient. The authors in <cit.> prove the absolute continuity of the solution for  stochastic Cahn–Hilliard equation with sublinear growth diffusion coefficient of order α<1/3. To the best of our knowledge, there still exists no result for the existence of density function for general sublinear growth diffusion coefficient of order 1/3≤α<1. Another important and unclear problem is how to construct the implementary  numerical approximation which can simulate the properties, such as the density function and longtime behaviors, of the exact solution as much as possible. And this topic is far from well-understood since there exists no result on the basic convergent problem of numerical approximations for the the stochastic Cahn–Hilliard equation with unbounded noise diffusion.


In this article, we consider the following stochastic  Cahn–Hilliard equation 
    dX(t)+A(AX(t)+F(X(t)))dt   =G(X(t))dW(t),   t∈ (0,T]
    
    X(0)   =X_0.

Here 0<T<∞, :=L^2(𝒪) with 𝒪=(0,L), L>0, 
-A:D(A)⊂→ is the  Laplacian operator under homogenous Dirichlet boundary condition, and  {W(t)}_t≥ 0 is a cylindrical Wiener process on a filtered probability space (Ω,ℱ, 
{ℱ_t}_t≥ 0,). 
The nonlinearity F is assumed to be the Nemytskii operator of f', where f is a polynomial of degree 4, i.e., c_4ξ^4+c_3ξ^3+c_2ξ^2+c_1ξ+c_0 with c_i∈, i=0,⋯,4, c_4>0.
A typical example is  the double well potential f=1/4(ξ^2-1)^2. When G=I, Eq. (<ref>) corresponds to the  Cahn–Hilliard–Cook equation.
The diffusion coefficient G is assumed to be the
Nemytskii operator of σ, where σ is a global Lipschitz function with the sublinear growth condition |σ(ξ)|≤ C(1+|ξ|^α), α<1.

To show the existence of the density of the law of the exact solution for Eq. (<ref>), we first consider the spectral Galerkin approximation proposed in <cit.> and derive its Hölder continuity estimates. With the help of truncated  and localization arguments, the Malliavin differentiability of  the spectral Galekin approximation of the truncated equation and the weak Malliavin differentiability of the exact solution are obtained. Combining  with the non-degeneracy assumption on |σ(·)|>0 and 
the properties of the Green function generated by the  bi-Laplacian operator -A^2, we further show that the law of {X(t,x_1),X(t,x_2),⋯,X(t,x_d)} admits a density, with t>0, x_i∈𝒪, i=1,⋯,d, being different points, which fills the gap on the absolute continuity of the exact solution for stochastic Cahn–Hilliard equation with general sublinear growth diffusion coefficient of order 1/3≤α<1. Furthermore,  assume in addition that σ∈𝒞_b^2(ℝ), by using the transformation on Ω proposed in <cit.>, the strictly positivity of the density function of the exact solution is shown. 

Once the existence of the density function of the solution is established, an important and natural question is how to simulate this density function numerically. To design such numerical methods to simulate the density function or the distribution of 
the exact solution, it is crucial to solve the basic strong convergent problem. Up to now, there have been a lot of works on the strong convergence problem of numerical approximations for the stochastic Cahn–Hillard equation and it is far from well understood (see e.g. <cit.>).
The authors in <cit.>
show the strong convergence of the finite element method and its implicitly full discretization for  Eq. (<ref>) driven by additive spatial regular noise. 
The authors in <cit.> derive the strong convergence rate of the spectral Galerkin method for Eq. (<ref>) in one dimension driven by additive  trace class noise.
The convergence rate of the finite element method and its implicitly full discretization is obtained in <cit.> for Eq. (<ref>) driven by additive spatial regular noise. 
For Eq. (<ref>) driven by additive space-time white noise, we are only aware that the authors in <cit.> derive the optimal strong convergence rate in space and super-convergence rate in time for a accelerated implicit Euler full  discretization. Recently, the authors in  <cit.> obtain the strong convergence and the optimal strong convergence rate of the the spatial spectral Galerkin method for Eq. (<ref>) with unbounded noise diffusion.
However, there exists no result about  the strong convergence of temporally and fully discrete numerical approximations for stochastic Cahn–Hilliard equation with unbounded noise diffusion. 

The present work
considers a full discretization based on the spatial spectral Galerkin method and the temporal drift implicit Euler method, and
makes further contribution on the strong convergence of numerical schemes for stochastic partial differential equations (SPDEs) with non-globally monotone continuous nonlinearity, especially for Eq. (<ref>).  The presence of the unbounded elliptic operator in front of the cubic nonlinearity and the roughness of the driving noise, make the convergence analysis of numerical approximation much more challenging and demanding. 
To overcome such difficulties,  several steps and techniques are introduced.
Instead of studying the strong convergence problem in  directly, we use the similar idea in <cit.> to present the convergence analysis of numerical approximation in 
a negative Sobolev space at the first step. 
To this end, the optimal spatial and temporal regularity estimates of the numerical approximation are presented by transforming Eq. (<ref>) to a random PDE and a stochastic convolution, as well as the monotonicity of -AF in ^-1 and the factorization method.
Then the optimal strong convergence error estimate in ^-1 is derived with the help of the variation approach and an appropriate error decomposition.
Based on the Sobolev interpolation inequality and the smoothing effect of the semigroup generated by the bi-Laplacian operator, we recover the optimal convergence rate in mean square sense of the numerical approximation for Eq. (<ref>) driven by multiplicative space-time white noise.
Let δ t be the time stepsize such that T=Kδ t, K∈^+, X_0∈^γ, γ∈ (0,3/2), N∈^+ and  p≥1, then
the numerical solution {X^N_k}_k≤ K, is strongly convergent to X  and  satisfies 

    X^N_k-X(t_k)_L^2(Ω;)   ≤ C(X_0,T,p,γ)(δ t^(γ/2)^-+λ_N^-(γ/2)^-)

for a positive constant C(X_0,T,p,γ).
We also remark that this approach is also available for deducing the strong convergence rates of numerical schemes for Eq. (<ref>)  driven by multiplicative regular noise and for proving the strong convergence of  numerical schemes for Eq. (<ref>) under other boundary conditions. 

The outline of this paper is as follows. In the next section,
some preliminaries and assumptions are listed.
Section <ref> is devoted to the existence and positivity of the density functions of the laws of the exact solution and its spectral Galerkin approximation, as well as their regularity estimates and Malliavin differentiability.
In Section <ref>, we derive the optimal mean square convergence rate of the implicit full discretization via a new approach based on the variation method and the factorization method.

  




§ PRELIMINARIES
 
In this section, we give some preliminaries and notations.
Given two separable Hilbert spaces (ℋ, ·_ℋ) and (H,·_H), 
(ℋ, H) and _1(ℋ, H)  are the Banach spaces of all linear bounded operators 
and  the nuclear operators from ℋ to H, respectively. 
The trace of an operator 𝒯∈_1(ℋ)
is tr[𝒯]=∑_k∈𝒯f_k,f_k_ℋ, where {f_k}_k∈ (={0,1,2,⋯}) is any orthonormal basis of ℋ.
In particular, if 𝒯≥ 0, tr[𝒯]=𝒯_ℒ_1.
Denote by _2(ℋ,H) the space 
of Hilbert–Schmidt operators from ℋ into H, equipped with the usual norm given by  ·__2(ℋ,H)=(∑_k∈· f_k^2_H)^1/2.
The following useful property and inequality hold 

    𝒮𝒯_ℒ_2(ℋ,H)≤𝒮_ℒ_2(ℋ,H)𝒯_ℒ(ℋ),  𝒯∈ℒ(ℋ),   𝒮∈ℒ_2(ℋ,H),
       tr[𝒬]=𝒬^1/2^2_L_2(ℋ)=𝒯^2_ℒ_2(H,ℋ),  𝒬=𝒯𝒯^*,  𝒯∈ℒ_2(H,ℋ),

where 𝒯^* is the adjoint operator of 𝒯.

Given a Banach space (ℰ,·_ℰ), we denote by γ( ℋ, ℰ) the space of γ-radonifying operators endowed with the norm
𝒯_γ(ℋ, ℰ)=(∑_k∈γ_k 𝒯f_k ^2_ℰ)^1/2,
where (γ_k)_k∈ is a Rademacher sequence on a
probability space (Ω,, ).
For convenience, let  L^q:=L^q(), 2≤ q<∞ and 
 equipped with the usual inner product and norm.
We also need the following BurkerHölder inequality in L^q,

    sup_t∈ [0,T]∫_0^t ϕ(r)dW(r)_L^q_L^p(Ω)   ≤ 
    C_p,qϕ_L^p(Ω;  L^2([0,T]; γ(ℋ;L^q))
       ≤  C_p,q((∫_0^T∑_k∈ (ϕ(t) f_k)^2_L^q/2dt)^p/2)^1/p,

 where {f_k}_k∈ is any orthonormal basis of ℋ.

Next, we introduce some assumptions and spaces associated with A.
 We denote by 
H^k:=H^k(𝒪) the standard Sobolev space and E:=𝒞(𝒪) the continuous function space. 
Denote A=-Δ the Dirichlet Laplacian operator with 

    D(A)={v∈ H^2(𝒪): v =0  on  ∂𝒪}.

It is known that A is a positive definite, self-adjoint and unbounded linear operator on . Thus there exists an orthonormal 
eigensystem {(λ_j,e_j)}_j∈ such that 0<λ_1≤⋯≤λ_j≤⋯ with λ_j ∼ j^ 2.
We  define 
^α, α∈ as the space of the series v:=∑_j=1^∞v_je_j, v_j∈, such that
v_^α:=(∑_j=1^∞λ_j^αv_j^2)^1/2<∞. Equipped with the norm ·_^α and corresponding inner product, the Hilbert space ^α equals D(A^α/2). Throughout this article, we denote by C a generic constant which may depend on several parameters but never on the projection parameter N and may change from occurrence to occurrence. 


To study the density function of the solution at fixed t∈[0,T], x∈𝒪, some objects of Malliavin calculus are also introduced. Denote by 𝒮 as the 
space of simple functionals, i.e., 𝔽=f(W(h_1), ⋯, W(h_m)) with f∈ℂ^∞_p(R^m) and an orthonormal sequence {h_i}_i=1^m in ℋ:=L^2(0,T;ℍ). Here 
    W(h):=∫_0^T h(s)dW(s)
 for h∈ℋ.
For 𝔽∈𝒮, the Malliavin derivative of 𝔽 is ℋ-valued random variable

    D_t,x𝔽=∑_i=1^m ∂ f/∂ x^i(W(h_1),W(h_2),⋯,W(h_m))h_i(t,x).

Similarly, the kth derivative of 𝔽 is given by 

    D_α^k𝔽=∑_i_1,⋯, i_k=1^m∂_i_1⋯∂_i_kf(W(h_1),⋯,W(h_m))h_i_1(α_1) ⋯ h_i_k(α_k),

where α:=(α_1,⋯,α_k), α_i=(t_i,x_i)∈ [0,T]×𝒪.

For p≥ 1 and k∈ℕ^+, the space 𝒟^p,k
is the completion of 𝒮 with respect to the seminorm 
    𝔽_𝒟^p,k=𝔽_L^p(Ω;ℝ)+∑_i=1^k D^i 𝔽_L^p(Ω; ℋ^⊗ k).
 We also define 𝒟^∞=∩_p≥ 1∩_k∈ℕ^+𝒟^p,k. The covariance matrix associated to a ℝ^d-valued random vector 𝔽 is denoted by C_ij=D^1𝔽^i,D^1𝔽^j, for i,j=1,⋯,d.






§ ABSOLUTE CONTINUITY OF THE EXACT SOLUTION



In this section, we first give the well-posedness result of the consider equation (<ref>), as well as the regularity estimate and strong convergence of the spectral Galerkin approximation. 
Next, we study the Malliavin differentiability of the exact solution via the spectral Galerkin approximation of the truncated equation of (<ref>).  
Then under the non-degeneracy condition of the diffusion coefficient, we prove that the absolute continuity of the exact solution with respect to the Lebsgue measure. We would like to mention that all the results on the density function of this section still hold if the boundary condition of Laplacian operator is changed to the homogenous Neumann condition. 




 §.§ Well-posedness and Galerkin approximation of the stochastic Cahn–Hilliard equation


In this subsection, we first present the well-posedness result of the considered equation  whose proof can be found in <cit.>.


Let T>0, X_0∈^γ, γ∈(0,3/2), p≥1. Then
Eq. (<ref>) possesses a unique mild solution
X in L^p(Ω; C([0,T]; )).
Moreover,   there exists C(X_0,T,p,γ)>0  such that 

    X_L^p(Ω;C([0,T];^γ))   ≤ C(X_0,T,p,γ)

and 

    X(t)-X(s)_L^p(Ω;)   ≤ C(X_0,T,p,γ)(t-s)^γ/4,

where s,t∈[0,T].



Under the condition of Theorem <ref> with γ≥3/2, by using the Sobolev embedding theorem and the factorization method, we can get that for β<1,

    X _L^p(Ω;𝒞([0,T];𝒞^β(𝒪)))<∞

and for β<3/8,

    X _L^p(Ω;𝒞^β([0,T]; E))<∞.




Denote P^N  the spectral Galerkin projection into the linear space  spanned by the first N eigenvectors  {e_1,⋯,e_N}.
Then the spectral Galerkin approximation satisfies the following SPDE

    dX^N(t)+A(AX^N(t)+P^NF(X(t)))dt   =P^NG(X^N(t))dW(t),   t∈ (0,T],
    
    X^N(0)   =P^NX_0.


Recall that the following strong convergence property in <cit.> of the spectral Galerkin approximation.


Let X_0∈^γ, γ∈ (0,3/2) and p≥1.
There exists a unique solution  X^N  of Eq. (<ref>)  satisfying 

    X^N_L^p(Ω;C([0,T];^γ))   ≤ C(X_0,T,p,γ)

and 

    X^N(t)-X^N(s)_L^p(Ω;)   ≤ C(X_0,T,p,γ)(t-s)^γ/4,

where C(X_0,T,p,γ)>0 and s,t∈ [0,T].

If in addition assume that sup_N∈^+X^N_0_E≤ C(X_0), then for β<γ, there 
exists C(X_0,T,p,β)>0 such that 

    X^N-X_L^p(Ω;C([0,T];))   ≤ C(X_0,T,p,β)λ_N^-β/2.





In the following, we first show that the exact solution of (<ref>) is absolutely continuous with respect to the Lebesgue measure if in addition assume that
σ is non-degenerate, i.e., 

    |σ(x)|≥ c_0>0,  for  x∈ℝ

and 
that σ is continuously differentiable on ℝ.
Then to study the strict positivity of the joint distribution of (X(t,x_1), X(t,x_2),⋯, X(t,x_d)), we further assume that σ∈𝒞^2 and possesses bounded and Lipschitz derivatives.  

For simplicity, throughout this rest part of section 3, we assume that X_0 is a sufficient smooth function. We also remark that if 
|σ(x)|>0, then by the similar arguments in <cit.>, all these results in this section still hold. 



 §.§ Malliavin differentiability of the solution


Our strategy to obtain the existence of the density function of X(t,x) is split into three steps:
first, we truncated the (<ref>) by adding a cut-off function θ_R on the nonlinear drift part and study the properties of truncated equation by using its semi-discretization. Then we show that the Malliavin differentiability of X_R(x,t) via the spectral Galerkin approximation, which implies the local
Malliavin differentiability of X(x,t). Finally, 
by using a localization argument and a lower bound estimate of the Malliavin matrix, we obtain  the  absolute continuity of the exact solution.

Let θ_R: ℝ→ℝ^+ be a 𝒞^∞ cut-off function satisfying θ_R≤ 1 and possessing bounded derivatives.  
For any R,
θ_R(x):=1, if |x|≤  R, and θ_R(x):=0, if |x|> R+1. Then we consider the truncated version of (<ref>),

    dX_R   =-A^2X_Rdt-A(F_R(X_R))dt+G(X_R)dW(t),
    
    X_R   =X_0,

where F_R is the Nemytskii operator of the function θ_R(·)f'(·). 

By denoting an increasing sample subset sequence {Ω_R}_R∈^+,
    Ω_R={ω∈Ω: sup_t∈ [0,T]X(t,x)_E≤ R},

due to the existence of continuous modification of  X by Remark <ref>, 
we have that lim_R→∞ P(Ω_R)= P(Ω)=1
and that 
(X_R,Ω_R) is a localization of (X,Ω).


There exists a unique solution X_R of (<ref>) 
which satisfies

    [sup_t∈ [0,T]X_R^p_E]≤ C(R,X_0,T,p),

for p≥ 2.


Proof.
We use the spectral Galerkin method to semi-discretize (<ref>) and define its solution as X_R^N.
Since F_R is Lipschitz, by the same arguments in <cit.>, we can get that for p≥2, [sup_t∈ [0,T]X_R^N^p_E] and that 

    lim_N→∞[sup_t∈ [0,T]X_R^N-X_R^p_E] =0,

which completes the proof.


Next, we show that X_R(t,x) ∈𝒟^1,2 for R∈ℝ^+, t∈ [0,T], x∈𝒪, which implies that X(t,x)∈𝒟_loc^1,2. For convenience, 
we denote G(t,x,y) as the Green function determined by the double Laplacian operator under homogeneous Dirichlet condition. In this case, G(t,x,y)=∑_i=1^∞e^-λ_i^2te_i(x)e_i(y), for x,y∈𝒪. Denote G^N(t,x,y)=∑_i=1^Ne^-λ_i^2te_i(x)e_i(y), for x,y∈𝒪 and N∈ℕ^+.


For fixed (t,x)∈ [0,T]×𝒪, X_R(t,x)∈𝒟^1,2 and satisfies 

    D_s,yX_R(t,x)   =G(t-s,x,y)σ(X_R(s,y))
         +∫_s^t∫_𝒪Δ G(t-r,x,z)(F_R'(X_R(r,z)) D_s,yX_R(r,z))dzdr
         +∫_s^t∫_𝒪G(t-r,x,z)σ'(X_R(r,z))D_s,yX_R(r,z)W(dz,dr),

where s∈ [0,t).


Proof.
Consider the spectral Galerkin approximation of X_R^N which satisfies the following stochastic differential equation

    dX^N_R=-A^2X^N_Rdt-A P^NF_R(X^N_R)dt+P^Nσ(X^N_R)dW(t).

Then the Gateaux derivative D_s^e_iX^N_R for a basis {e_i}_i∈^+ of L^2(𝒪) exists and satisfies

    dD_s^e_iX^N_R   =-A^2D_s^e_iX^N_Rdt-A P^NF_R'(X^N_R)D_s^e_iX^N_Rdt
    +P^Nσ'(X^N_R)D_s^e_iX^N_RdW(t),
    
    D_s^e_iX^N_R(s)   =P^Nσ(X^N_R)e_i.

Then we can get that D_s,yX_R^N(t,x)=∑_i∈^+D_s,·X_R^N(t,x),e_ie_i(y) formally satisfies 

    D_s,yX_R^N(t,x)   =G^N(t-s,x,y)σ(X_R^N(s,y))
         +∫_s^t∫_𝒪Δ G(t-r,x,z)P^N(F_R'(X_R(r,z)) D_s,yX_R^N(r,z))dzdr
         +∫_s^t∫_𝒪G(t-r,x,z)P^Nσ'(X_R(r,z))D_s,yX_R^N(r,z)W(dz,dr).

To show that D_s,yX_R^N(t,x) is the solution of the above integration equation, we need to show that the map defined by the right hand side of the above equality has a unique fixed point. By a rescaling argument, it sufficient to give a priori estimate on D_s,yX_R^N(t,x).


Taking L^2-norm with respect to y, we get 

    D_s,yX_R^N(t,x)_L^2_y^2
       ≤ 3 G^N(t-s,x,y)σ(X_R(s,y))_L^2_y^2
         +3∫_s^t∫_𝒪Δ G(t-r,x,z)P^N(F_R'(X_R(r,z)) D_s,y X_R(r,z) )dzdr_L^2_y^2
         +3∫_s^t∫_𝒪G(t-r,x,z)P^Nσ'(X_R(r,z))D_s,y X_R(r,z)W(dz,dr)_L^2_y^2
       :=IV_1+IV_2+IV_3.

By taking L^1([0,T]) integration and the property that D_s,y X_R^N(r,z)=0 if s>r,
we get 

    ∫_0^TD_s,yX_R^N(t,x)_ L^2_y^2ds   =∫_0^tD_s,yX_R^N(t,x)_ L^2_y^2ds
       ≤∫_0^tIV_1ds+∫_0^t IV_2ds+∫_0^tIV_3ds

Due to Lemma <ref> and the sublinear growth of σ, we have that 

    [∫_0^TIV_1ds]   ≤ C∫_0^TG^N(t-s,x,y)σ(X_R^N(s,y))_L^2_y^2ds
       ≤ C(R)<∞.

The Hölder inequality, the boundedness of the truncated  drift term F_R, the smoothy estimate of G(t,x,y) and the Fubini's theorem yield that 

    [∫_0^t IV_2ds]   ≤ 
    C[∫_0^t∫_s^t∫_𝒪Δ G(t-r,x,z)P^N(F_R'(X_R^N(r,z)) 
            D_s,yX_R^N(r,z))dzdr_L^2_y^2ds]
       ≤ C∫_0^t ∫_𝒪[(∫_s^t|∫_𝒪Δ G^N(t-r,x,z))
            F_R'(X_R^N(r,z))D_s,y X_R^N(r,z)dz|dr)^2 ]dyds
    
       ≤∫_0^t ∫_𝒪[(∫_s^t∫_𝒪 |Δ G^N (t-r,x,z))|_L_x^∞ |F_R'(X_R^N(r,z))|^2dzdr
         ×∫_s^t∫_𝒪  |Δ G^N(t-r,x,z))|_L_x^∞ 
    |D_s,y X_R^N(r,z)|^2dzdr) ]dyds
       ≤ C(R) ∫_0^t ∫_s^t  |Δ G^N (t-r,x,z))|_L_x^∞∫_𝒪[D_s,y X_R^N(r,z)_L^2_y^2] dzdrds
       ≤ C(R) ∫_0^t (t-r)^-1/2∫_0^r 
     ∫_𝒪[D_s,y X_R^N(r,z)_L^2_y^2] dzdsdr
 

The Burkhölder inequality, Fubini's theorem, the boundedness of σ' and the form of G^N imply that 

    [∫_0^tIV_3ds]
       ≤ 
    C ∫_0^t[∫_s^t∫_𝒪G^N(t-r,x,z)σ'(X_R(r,z))D_s,y X_R(r,z)W(dz,dr)_L^2_y^2]ds
       ≤ C∫_0^t ∫_𝒪[∫_s^t (t-s)^-1/2σ'(X_R(r,z))_L^∞_z^2∫_𝒪D_s,y X_R(r,z)^2 dzdr]dyds
       ≤ C∫_0^t (t-r)^-1/2∫_0^r∫_𝒪[D_s,y X_R(r,z)_L^2_y^2] dzds dr.

Thus by the Gronwall inequality, we conclude that  
sup_Nsup_t∈ [0,T]sup_x∈𝒪X_R^N(t,x)_𝒟^1,2≤ C(R) and X_R^N satisfies (<ref>).
By the strong convergence of X^N_R, we have that X_R(t,x)∈𝒟^1,2,
 and that D_s,yX_R(t,x) satisfies (<ref>).
In fact, due to the strong convergence of X^N_R, we can further show that X^N_R is strongly convergent to 
 X_R in 𝒟^1,2 for any fixed t>0,x∈𝒪. 
 Based on the above proposition, we obtain the following weak differentiability of the exact solution X∈𝒟_loc^1,2. 
 The Malliavin derivative D_s,yX is defined by the restriction on Ω_R.  
 
 For any t>0, x∈𝒪.
 The solution X(t,x) of (<ref>) belongs to 𝒟_loc^1,2.
 
 

 Similar arguments yield that X^N_R(t,x)∈𝒟^1,p, p≥ 2
 and X_R(t,x)∈𝒟^1,p, p≥ 2 and that  X(t,x)∈𝒟_loc^1,p.  With a slight modification, one can obtain that 
    D_s,yX_R(t,x)_L^p(Ω;L^2_y)≤ C(R,T,X_0)(t-s)^-1/4.



 



 §.§ Absolute continuity of the law of the exact solution


Assume that x_i∈𝒪, i=1,⋯,d are distinct points.
In this part, we show that 𝔽_R=(X_R(t,x_1), X_R(t,x_2),⋯, X_R(t,x_d)) possesses a density function on ℝ^d.
Without loss of generality, we assume that x_i is strictly increasing with respect to i.

To this end, according to Lemma 2.3.1 in <cit.>, it suffices  to show that the Malliavin 
matrix C^R of 𝔽_R, defined by C^R_ij(t)=D𝔽_R^i(t),D𝔽_R^j(t), t>0, i,j≤ d, is invertible and satisfies that for p≥ 2, there exists a small parameter ϵ_0(p) such that for all ϵ≤ϵ_0(p),

    sup_|ξ|=1ℙ(ξ^TC^R(t)ξ≤ϵ)≤ϵ^p.



For p≥ 2 and T>0, there exists ϵ_0(p) such that for all ϵ≤ϵ_0(p), t∈ [0,T],

    sup_|ξ|=1ℙ(ξ^TC^R(t)ξ≤ϵ)≤ϵ^p.



Proof.
Denote D_s,yX_R(t,x)=G(t-s,x,y)σ(X_R(s,y))+Q_s,y(t,x), where 

    Q_s,y(t,x)X_R(t,x)   =∫_s^t∫_𝒪Δ G(t-r,x,z)F'_R(X_R(r,z))
    D_s,yX_R(r,z)drdz
         +
    ∫_s^t ∫_𝒪G(t-r,x,z)σ'_R(X_R(r,z))
    D_s,yX_R(r,z)W(dz,dr).

We use the method similar to <cit.> and show the lower bound ξ^TC^R(t)ξ, with |ξ|=1. Assume that min_1≤ i≤ d-1|x_i-x_i+1|≥ L>4ϵ^α and that there exists a parameter α∈ (1/4, 1) such that (x_i-ϵ^α,x_i+ϵ^α)⊂𝒪.
Then we get 

    ξ^TC^R(t)ξ   ≥∑_i=1^d∫_t-ϵ^t∫_x_i-ϵ^α^x_i+ϵ^α
    (∑_j=1^dD_r,zX_R(t,x_j)ξ_j)^2dzdr
       ≥1/2∑_i=1^d∫_t-ϵ^t∫_x_i-ϵ^α^x_i+ϵ^α
    D_r,zX_R(t,x_i)^2ξ_i^2dzdr
         -∑_i=1^d∫_t-ϵ^t∫_x_i-ϵ^α^x_i+ϵ^α
    (∑_j≠ i^dD_r,zX_R(t,x_j)ξ_j)^2dzdr
       =:1/2V_1(ϵ)-V_2(ϵ).


We first give an upper estimate of V_2(ϵ).
Since {(x_i-ϵ^α,x_i+ϵ^α)}_i=1^d are disjointed interval, it holds that 

    sup_|ξ|=1V_2(ϵ)    ≤ C_d∑_j≠ i^d∫_t-ϵ^t∫_x_i-ϵ^α^x_i+ϵ^α
    (σ(X_R(r,z))G(t-r,x_j,z))^2dzdr
          +C_d∑_j≠ i^d∫_t-ϵ^t∫_x_i-ϵ^α^x_i+ϵ^α
    (Q_r,z(t,x_j))^2dzdr
       =: C_d∑_j≠ i^d V_21^ij(ϵ)+C_d∑_j≠ i^d V_22^ij(ϵ).




By taking p-moment on both sides of the above inequality, and using Hölder's inequality, the sublinear growth of σ,  and 
the property of the Green function that 

    |G(t,x,y)|≤ C_2t^-1/4exp(-C_1|x-y|^4/3/t^1/3),

we get for i≠ j,

    [|V_21^ij(ϵ)|^p]
       ≤ C_p,d[|∫_t-ϵ^t∫_x_i-ϵ^α^x_i+ϵ^ασ^2(X_R(r,z))(t-r)^-1/2e^-2C_1 L^4/3ϵ^-1/3dzdr|^p]
       ≤ C_p,dϵ^p(α +1/2)e^-Cϵ^-1/3.

By the definition of Q_r,z(t,x), we obtain 

    [|V_22^ij(ϵ)|^p]
       ≤ C_p,d[|∫_t-ϵ^t∫_x_i-ϵ^α^x_i+ϵ^α
    (Q_r,z(t,x_j))^2dzdr|^p]
       ≤ C_p,d[(∫_t-ϵ^t∫_x_i-ϵ^α^x_i+ϵ^α|∫_r^t∫_𝒪Δ G(t-θ,x_j,z_1)
          F'_R(X_R(θ,z_1))
    D_r,zX_R(θ,z_1)dθ dz_1|^2drdz)^p]
         +
     C_p,d[(∫_t-ϵ^t∫_x_i-ϵ^α^x_i+ϵ^α|∫_r^t∫_𝒪G(t-r,x_j,z_1)σ'_R(X_R(θ,z_1))
          
    D_r,zX_R(θ,z_1)W(dz_1,dθ)|^2drdz)^p]
       =: V_221^ij(ϵ)+ V_222^ij(ϵ).

Hölder's inequality and the estimate of the Green function yield that 

    V_221^ij(ϵ)
       ≤ Cϵ^(α+1)(p-1)∫_t-ϵ^t∫_x_i-ϵ^α^x_i+ϵ^α[(∫_r^t∫_𝒪|Δ G(t-θ,x_j,z_1)|
          |F'_R(X_R(θ,z_1))|
    |D_r,zX_R(θ,z_1)|dθ dz_1)^2p]drdz,
       ≤ 
    Cϵ^(α+1)(p-1)∫_t-ϵ^t∫_x_i-ϵ^α^x_i+ϵ^α(∫_r^t (t-θ)^-1/2
         (sup_z_1[D_r,zX_R(θ,z_1)_L^2_z^2p])^1/2p dθ)^2pdrdz
       ≤ C ϵ^(α+1)p+p/2.

Similar arguments, together with the Burkhölder inequality,
imply that 

    V_222^ij(ϵ)
       ≤ Cϵ^(α+1)(p-1)∫_t-ϵ^t∫_x_i-ϵ^α^x_i+ϵ^α[|∫_r^t∫_𝒪G(t-r,x_j,z_1)σ'_R(X_R(θ,z_1))
          
    D_r,zX_R(θ,z_1)W(dz_1,dθ)|^2p]dzdr
       ≤ C ϵ^(α+1)p+p/4.
 
Thus we conclude that [|sup_|ξ|=1V_2(ϵ)|^p]≤ C ϵ^p(α +1)+p/4.

Next, we estimate the lower bound of the term sup_|ξ|=1V_1(ϵ).
By Young inequality, we have that 

    sup_|ξ|=1V_1(ϵ)
       ≥1/2∑_i=1^d∫_t-ϵ^t∫_x_i-ϵ^α^x_i+ϵ^ασ^2(X_R(r,z))G^2(t-r,x_i,z)ξ_i^2dzdr
         -
    ∑_i=1^d∫_t-ϵ^t∫_x_i-ϵ^α^x_i+ϵ^α
    Q^2_r,z(t,x_i)ξ_i^2dzdr
       =:1/2∑_i=1^dV_11^ii(ϵ) - ∑_i=1^d V_12^ii(ϵ).

By repeating the similar procedures of estimating V_22^ij(ϵ), we get that 

    [|V_12^ii(ϵ)|^p]   ≤ϵ^(α+1)p+p/4.

For the term V_11^ii(ϵ), we have 

    V_11^ii(ϵ)
       ≥1/2∫_t-ϵ^t∫_x_i-ϵ^α^x_i+ϵ^ασ^2(X_R(r,z))G^2(t-r,x_i,x_i)ξ_i^2dzdr
         -∫_t-ϵ^t∫_x_i-ϵ^α^x_i+ϵ^α(σ(X_R(r,z))G(t-r,x_i,x_i)-σ(X_R(r,z))G(t-r,x_i,z))^2ξ_i^2dzdr
       =:1/2V_111^ii(ϵ)- V_112^ii(ϵ).


Based on the lower bound estimate of the Green function 

    G(s,x_j,x_j)≥ C_x_js^-1/4,

and the fact that x_i ∈𝒪 implies that min_i=1^dC_x_i>0,  together with the non-degeneracy of σ, we have that 

    V_111^ii(ϵ)   ≥ Cinf_x|σ(x)|∫_t-ϵ^t∫_x_i-ϵ^α^x_i+ϵ^α
    (t-r)^-1/2dzdr ξ_i^2
       ≥ Cϵ^1/2+αξ_i^2.

Thus we conclude that 
∑_i=1^dV_111^ii≥ Cϵ^1/2+α.

Then we give an upper bound estimate of V_112^ii by using the continuity of X_R. 
First, we split [|V_112^ii(ϵ)|^p] as 

    [|V_112^ii(ϵ)|^p]
       ≤ 
    2[|∫_t-ϵ^t∫_x_i-ϵ^α^x_i+ϵ^α((σ(X_R(r,x_i))(G(t-r,x_i,x_i)-G(t-r,x_i,z)))^2dzdr|^p]
         +2[|∫_t-ϵ^t∫_x_i-ϵ^α^x_i+ϵ^α((σ(X_R(r,x_i))-σ(X_R(r,z)))G(t-r,x_i,z)))^2dzdr|^p]
       =:V_1121(ϵ, p)+V_1122(ϵ, p).


The Hölder inequality and the estimate of Green function 

    |G(s,x,y)-G(s,x,y_1)|≤ C|y-y_1|s^-1/2

yield that for p≥ 2,

    [|V_112^ii(ϵ)|^p]
       ≤ C[|∫_t_0-ϵ^t_0∫_x_i-ϵ^α^x_i+ϵ^α|σ(X_R(r,z))|^2(G(t-r,z,x_i)-G(t-r,x_i,x_i))^2dzdr|^p]
       ≤ C[|∫_t_0-ϵ^t_0∫_x_i-ϵ^α^x_i+ϵ^α|σ(X_R(r,z))|^2|z-x_i|(t-r)^-3/4dzdr|^p]
       ≤ Cϵ^(1/4+2α)p.

The similar arguments, together with the continuity of X_R, the Lipschitz continuity of σ yield that
for sufficient small ϵ>0, 

    [|V_112^ii(ϵ)|^p]
       ≤ C[|∫_t_0-ϵ^t_0∫_x_i-ϵ^α^x_i+ϵ^α|σ(X_R(r,x_i))-σ(X_R(r,z))|(t-r)^-1/2dzdr|^p]
       ≤ϵ^(1/2+2α-ϵ)p.

Combining the above estimates, we get that 

    ξ^TC^R(t)ξ≥1/2V_1(ϵ)-V_2(ϵ)
       ≥
    C_p,d,Rϵ^(1/2+α)-
    (V_2(ϵ)+∑_i=1^d(V_12^ii(ϵ)+V_112^ii(ϵ))).

Thus the Chebyshev inequality yields that 
    sup_|ξ|=1ℙ(ξ^TC^R(t)ξ≤ϵ)
       ≤sup_|ξ|=1ℙ(V_2(ϵ)+∑_i=1^d(V_12^ii(ϵ)+V_112^ii(ϵ))
    ≤ϵ+C_p,d,Rϵ^1/2+α)
       ≤ (ϵ+C_p,d,Rϵ^1/2+α)^-p[|V_2(ϵ)+∑_i=1^d(V_12^ii(ϵ)+V_112^ii(ϵ))|^p]
       ≤ (ϵ+C_p,d,Rϵ^1/2+α)^-p C_p,d,Rϵ^(2α+1/4)p,

which completes the proof.



Let t>0, x_i∈𝒪, i=1,⋯,d be different points. Then the law of {X(t,x_1),X(t,x_2),⋯,X(t,x_d)} admits a density.


Proof.
Based on Proposition <ref> and the localization arguments, we have that for any p≥ 2, there exists R_0  large enough, such that  for R≥ R_0,

    sup_|ξ|=1ℙ(ξ^TC(t)ξ=0)
       =sup_|ξ|=1ℙ(ξ^TC(t)ξ=0, ω∈Ω_R^c)
         +sup_|ξ|=1ℙ(ξ^TC^R(t)ξ =0,ω∈Ω_R)
       ≤ϵ+sup_|ξ|=1ℙ(ξ^TC^R(t)ξ=0)
    
       ≤ϵ.

Let R→∞, we get 

    sup_|ξ|=1ℙ(ξ^TC(t)ξ=0)=0,

which, together with X(t,x)∈𝒟_loc^1,2, implies 
the existence of the density function of the joint distribution.


Indeed, the above argument also yields the existence and smoothness of the density function of the numerical approximation. 
By using the localization X^N_K:=X^NI_Ω_K, 

    Ω_K={ω∈Ω: sup_t∈[0,T]sup_x∈𝒪 |X^N(t,x)|≤ K},
 and using similar arguments in proving Theorem <ref>, we can get the following result.


Let t>0, x_i∈𝒪, i=1,⋯,d, be different points. Then the law of {X^N(t,x_1),X^N(t,x_2),⋯,X^N(t,x_d)} admits a density.




 §.§ Positivity of the density function


To illustrate the strategy to show the positivity of the density function, we use the idea of <cit.> and denote the probability space (Ω,ℱ, ℙ) as Ω={ϕ∈ C([0,T]×𝒪)|ϕ(0,x)=ϕ(t,0)=0} such that the canonical process is a Brownian sheet. 
Consider a sequence {h_ϵ_n∈ L^2([0,T]×𝒪)}, ϵ_n→ 0 and z∈ℝ^d. Let T_n^z be the transformation on Ω such that 

    T_n^z(ω)(t,x)
    =ω(t,x)+∑_i=1^dz_i∫_0^t∫_𝒪h_ϵ_n^i(y,s)dyds.

Define Φ_n(z):=𝔽∘ T_n^z.
The function h_ϵ_n is chosen according to the lower bound estimate in Proposition <ref>, where ϵ_n  satisfies the same condition of ϵ in Proposition <ref> for n large enough.  Let α∈ (1/4,1). 
Then we define 

    h_ϵ_n^i(s,y)=(c_n^i)^-11_[t-ϵ_n,t](s)1_(x_i-ϵ_n^α,x_i+ϵ_n^α)(y),

where 

    c_n^(i)=∫_t-ϵ_n^t∫_x_i-ϵ_n^α^x_i+ϵ_n^αG(t-r,x_i,y)dyds.

Then it suffices to prove that for some fixed R, there will exist C,C' and δ, h_ϵ_n,  r small enough,

    ℙ(Ω_R ∩{|𝔽-y|≤ r }∩{(∇Φ_n(0))≥ C}
           ∩{sup_|z|≤δ(|∇Φ_n(z)|+|∇^2 Φ_n(z)|)≤ C'})>0,

where  Φ_n(z)=𝔽∘ T_n^z.

The above result will be immediately obtained if the following two propositions hold.


Let y∈ supp(ℙ∘𝔽^-1).
There exists C,r_0>0 such that for r≤ r_0,

    lim sup_n→∞ℙ(Ω_R ∩{|𝔽-y|≤ r }∩{(∇Φ_n(0))≥ C})>0.



Proof.
For simplicity, we only give the proof for the case d=1.
Denote  the Malliavin derivative of X_R(s,x)∘ T_z^n with respect to the direction h_n as
u_z^R(s,x)=∂/∂ z X_R(s,x)∘ T_z^n.
Then we have 

    u_z^R(t,x)   =∫_0^t∫_𝒪G(t-s,x,y)σ(X_R(s,y)∘ T_z^n)h_ϵ_n(s,y)dyds
          +∫_t-ϵ^t∫_𝒪Δ G(t-s,x,y)F_R'(X_R(s,y)∘ T_z^n)u_z^R(s,y)dyds
          +∫_t-ϵ^t∫_𝒪G(t-s,x,y)σ'(X_R(s,y)∘ T_z^n)u_z^R(s,y)W(dy,ds)
          +∫_t-ϵ^t∫_𝒪G(t-s,x,y)σ'(X_R(s,y)∘ T_z^n)u_z^R(s,y)zh_ϵ_n(s,y)dyds.

For the first term in the above equality, 

    [|∫_0^t∫_𝒪G(t-s,x,y)σ(X_R(s,y)∘ T_z^n)h_ϵ_n(s,y)dyds|^p]
       ≤ C_p,Rsup_s∈ [0,t][|σ(X_R(s,y)∘ T_z^n)|^p_L^∞_y]≤ C_p,R<∞,

where we use sup_s∈[0,T][|X_R(s,y)∘ T_z^n|_L^∞_y^p]≤ C_p which can be proven by similar procedures in the estimate of X_R(s,y).
By the similar arguments in Proposition <ref>, we get that for p large enough, 
    [ |u_z^R(t,x)|^p]≤ C_p,R<∞,

and  that 

    [ |u_z^R(t,x)-∫_0^t∫_𝒪G(t-s,x,y)σ(X_R(s,y)∘ T_z^n)h_ϵ_n(s,y)dyds|^p]
       ≤ C_p,R(ϵ_n^p/4+|z|^p).

Notice that 

    ∫_0^t∫_𝒪G(t-s,x,y)σ(X_R(s,y)∘ T_0^n)h_ϵ_n(s,y)dyds
       =σ(X_R(t,x))+
    ∫_0^t∫_𝒪G(t-s,x,y)(σ(X_R(s,y)-σ(X_R(t,x)))h_ϵ_n(s,y)dyds.

Taking z=0 and using the continuity of X_R(·,·), we obtain that  

    [ |u_0^R(t,x)-σ(X_R(t,x))|^p]
       ≤ C_p,Rϵ_n^p/4.

Thus for any y ∈ supp(ℙ∘𝔽^-1), there exists r_0>0 such that for 0<r≤ r_0,
ℙ(|𝔽-y|≤ r )>0.
Furthermore, for r≤ r_0, it holds that

    lim sup_n→∞ℙ(Ω_R∩{|𝔽-y|≤ r}∩{(∇_z Φ_n(0)) ≥ C})>0.



To prove the following proposition, we in addition assume that |σ”|_L^∞(ℝ)≤ C.


There exists C'>0 and small enough δ >0 such that 

    lim_n→∞ℙ( {sup_|z|≤δ(|∇Φ_n(z)|+|∇^2 Φ_n(z)|)≥ C'}=𝒪(δ).




Proof.
For simplicity, we only show the estimate of |∇Φ_n(z)| for  d=1. For the estimate of |∇Φ_n(z)| with d>1 and the estimate of |∇^2 Φ_n(z)|, the proofs are similar.
Denote 

    v_z^R(t,x)   =∫_0^t∫_𝒪G(t-s,x,y)σ(X_R(s,y)∘ T_z^n)h_ϵ_n(s,y)dyds
           +∫_t-ϵ^t∫_𝒪G(t-s,x,y)σ'(X_R(s,y)∘ T_z^n)u_z^R(s,y)zh_ϵ_n(s,y)dyds.

Based on the estimates in the proof of Proposition <ref>, taking δ small enough, we have that 

    [|u_z^R(t,x)-u_z_1^R(t,x)|^p]+[|v_z^R(t,x)-v_z_1^R(t,x)|^p]   ≤ C_p, R|z-z_1|^p.

By using the Kolmogorov criterion of continuity theorem, we obtain
the Hölder continuity of u_z^R(t,x) and v_z^R(t,x),
and that 

    [sup_|z|≤δ|u_z^R(t,x)|^p]+[sup_|z|≤δ|v_z^R(t,x)|^p]≤ C_p,R.

Similarly, we have that 

    lim_n→∞sup_|z|≤δ [ |u_z^R(t,x)-v_z^R(t,x)|^p]=0,

which, together with the Garsia-Rodemich-Rumsey lemma, yields that  

    lim_n→∞ [sup_|z|≤δ|u_z^R(t,x)-v_z^R(t,x)|^p]=0.

Now, it suffices to show that  

    lim_n→∞ℙ( {sup_|z|≤δ|∫_0^t∫_𝒪G(t-s,x,y)
                   σ(X_R(s,y)∘ T_z^n)h_ϵ_n(s,y)dyds| ≥ C'}∩Ω_R )=0.

Since on Ω_R, 
 
    sup_s∈[0,T]sup_y∈𝒪|σ(X_R(s,y))|≤ C(1+|X_R(s,y)|^α)≤ C(1+R^α),α<1,

we have that

    |∫_0^t∫_𝒪G(t-s,x,y)σ(X_R(s,y)∘ T_z^n)h_ϵ_n(s,y)dyds|
       ≤|∫_0^t∫_𝒪G(t-s,x,y)σ(X_R(s,y))h_ϵ_n(s,y)dyds|
         +|∫_0^t∫_𝒪G(t-s,x,y)(σ(X_R(s,y)∘ T_z^n)-σ(X_R(s,y))h_ϵ_n(s,y)dyds|
       ≤ C(1+R)+|∫_0^t∫_𝒪G(t-s,x,y)(σ(X_R(s,y)∘ T_z^n)-σ(X_R(s,y))h_ϵ_n(s,y)dyds|.

By the Hölder continuity of X_R(s,y)∘ T_z^n
with respect to z and using Markov inequality, we can choose proper C'(R) such that 
(<ref>) holds.


Based on the above propositions, we obtain the following theorem, which  yields that supp(ℙ∘𝔽^-1)=ℝ^d, see e.g. <cit.>.


For t>0, distinct points x_i∈𝒪, i=1,⋯,d, there exists a continuous function 
ρ>0 such that for all ϕ∈𝒞_b^∞(ℝ^d,ℝ^+),

    [ϕ(X(t,x_1),⋯,X(t,x_d))]
    ≥∫_ℝ^dϕ(y)ρ(y)dy.
 

Proof.
Based on Propositions <ref> and 
<ref>, by the similar proof of Theorem 3.3 in <cit.>,
we have

    [1_Ω_Rϕ(X_R(t,x_1),⋯,X_R(t,x_d))]
    ≥∫_ℝ^dϕ(y)ρ_R(y)dy,

which, together with ϕ(·)∈ℝ^+, completes the proof.







§ REGULARITY ESTIMATE AND STRONG CONVERGENCE OF A FULL DISCRETIZATION




In previous section, we establish the existence and positivity of the density function of the exact solution  X. In the viewpoint of applications, it is important to design numerical approximation for computing the density function of X. However,  it  is far from well-understood since there exists no result on the basic convergent problem of numerical approximations for the stochastic Cahn–Hilliard equation with unbounded noise diffusion. 
In this section, we aim to solve the  basic strong convergence problem in terms of  the following full discretization.



Let δ t be the time stepsize such that T=Kδ t, K∈^+.  
The full discrete numerical scheme starting from X^N_0:=P^NX_0, based on the temporal drift implicit Euler scheme  and the spatial spectral Galerkin method, is define as 

    X^N_k+1   =X^N_k-A^2X^N_k+1δ t
    -AP^NF(X^N_k+1)+G(X_k^N) δ_k  W,   k≤ K-1
  where δ_k  W=W(t_k+1)-W(t_k),   k≤ K-1.
Then we also have the mild form of X^N_k,

    X^N_k+1=T_δ t X^N_k
    -δ tT_δ tA
    P^N(F(X^N_k+1))+T_δ tG(X_k^N) δ_k  W,   k≤ K-1,

where T_δ t=(I+A^2δ t)^-1.




 §.§ A priori and regularity estimates of the full discretization


In order to study the moment estimate of the numerical scheme, we decompose  X^N_k into Y^N_k and Z^N_k, which satisfy for k≤ K-1 and N∈^+,

    Y^N_k+1   =Y^N_k-A^2Y^N_k+1δ t-AF(Y^N_k+Z^N_k)δ t, 
    
    Z^N_k+1   =Z^N_k-A^2Z^N_k+1δ t+G(Y^N_k+Z^N_k) δ_k  W,

with Y^N_0=X^N_0, Z^N_0=0.
We would like to mention that the similar decomposition has been used to derive the strong convergence rates of numerical approximations for non-global Lipschitz SPDEs driven by additive noise(see e.g. <cit.>). 
The following lemmas and propositions are
 devoted to deducing the a priori estimates of Y^N_k, Z^N_k and X^N_k, k≤ K, N∈^+.


For 2≤ p≤∞ and  m > 0, it holds that 

    T_δ t^m f_L^p≤ C(mδ t)^-1/4(1/2 -1/p)f,    f∈,  t>0.



Proof.
For f∈, we have T_δ t^mf=∑_i∈^+(1/1+λ_i^2δ t)^mf,e_ie_i.
From the uniform boundedness of e_i, i∈^+ and the fact that ci^2 ≤λ _i≤ Ci^2, it follows that 

    T_δ t^mP^Nf_E   =∑_i=1^N(1/1+λ_i^2δ t)^mf,e_ie_i_ E
       ≤ (∑_i=1^N(1/1+λ_i^2δ t)^2m)^1/2f≤ C(mδ t)^-1/8f,

and 

    T_δ t^mP^Nf   ≤∑_i=1^N(1/1+λ_i^2δ t)^mf,e_ie_i≤ Cf.

The Riesz–Thorin interpolation theorem leads to the desired result. 

Let X_0∈, T>0 and q≥1.
There exists a unique solution  {X^N_k}_k≤ K  of Eq. (<ref>)  satisfying 

    sup_k≤ K[ X^N_k_^-1^q]
       ≤ C(X_0,T,q)

for a positive constant C(X_0,T,q).


Proof.
Taking inner product with Y^N_k+1 in ^-1 on both sides of Eq. (<ref>) leads to

    Y^N_k+1_^-1^2
    ≤Y^N_k_^-1^2
    -2∇  Y^N_k+1^2δ t
    -2F(Y^N_k+1+Z^N_k+1), Y^N_k+1δ t.

From the monotonicity of -F, the equivalence of norms in ^1 and H^1, and the Young inequality, it follows that for some small ϵ>0,

    Y^N_k+1_^-1^2
    +8(c_4-ϵ) Y^N_k+1_L^4^4δ t+
    (2-ϵ) Y^N_k+1_^1^2δ t
       ≤ Y^N_k_^-1^2
    +C(ϵ)(1
    +Z_k+1^N_L^4^4+A^-1/2 Z_k+1^N^2)δ t
       ≤ Y^N_k_^-1^2
    +C(ϵ)(1
    +Z_k+1^N_L^4^4)δ t.

From the mild form of Z^N_k, the Hölder inequality  and the Burkhölder inequality, it follows that for p≥ 2 and q≥ 4, 

    [Z^N_k+1_L^p^q]
       =
    [∑_i=0^k T_δ t^k+1-iG(Y^N_i+Z^N_i) δ  W_i_L^p^q]
       ≤[(∑_i=0^k∑_j∈^+ T_δ t^k+1-iG(Y^N_i+Z^N_i)e_j_L^p^2δ t)^q/2]
       ≤[(∑_i=0^k
    ((k+1-i)δ t)^-1/2+1/2p(1+Y^N_i^2α+Z^N_i^2α)δ t)^q/2]
       ≤ C(∑_i=0^k
    ((k+1-i)δ t)^-1+1/p)^q/4[(∑_i=0^k(1+Y^N_i^4α+Z^N_i^4α)δ t)^q/4].

Using the Young inequality, we obtain for 0≤ s≤ t,

    [Z^N_k+1_L^p^q]
       ≤  CT^q/4p(1+[(∑_i=0^kY^N_i^4 αδ t)^q/4]+∑_i=0^k[Z_i^N^α q]δ t)
    
       ≤ CT^q/4p(1+[(∑_i=0^kY^N_i^4 αδ t)^q/4]+∑_i=0^k[Z_i^N^q]δ t).

Gronwall's inequality yields that for  0≤ s≤ t≤ T,

    [Z^N_k+1_L^p^q]
       ≤ C(T)(1+[(∑_i=0^kY^N_i^4 αδ t)^q/4]).


Now taking m-moment, m∈^+ on (<ref>) and letting p=4,q=4m, we have 

    [(∑_i=0^kY^N_k^4_L^4δ t)^m]
       ≤ CY^N(0)_^-1^2m
    +C(ϵ,T)∑_i=0^k(1+[Z^N_i_L^4^4m])δ t
       ≤ CY^N(0)_^-1^2m+C(ϵ,T)∑_i=0^k(C(ϵ_1)+ϵ_1 [(∑_j=0^i Y^N_k_L^4^4δ t)^m]δ t),

where ϵ_1>0 is a small number such that 
C(ϵ,T)ϵ_1T<1/2. 
The above estimation leads to 

    [(∑_i=0^kY^N_k^4_L^4ds)^m]
       ≤  CY^N(0)_^-1^2m+C(m,ϵ,ϵ_1,T),

which in turn yields that for m∈^+,

    [Y^N_k+1^2m_^-1]+[(∑_i=0^k∇ Y^N_k^2δ t)^m]+[(∑_i=0^kY^N_k^4_L^4δ t)^m]
       ≤ C(X_0,T,m),

and for p≥ 2, q≥ 4,

    [Z^N_k+1^q_L^p]
       ≤ C(X_0,T,m).

Combining the above estimates, we complete the proof.

Let  X_0∈, T>0 and q≥1.
There exists a positive constant C(T,q,X_0)
such that 

    sup_k≤ K[Z^N_k^q_E]
       ≤ C(T,q,X_0).


Proof.
By the Burkholder inequality and the smoothing effect of T_δ t, we have that for q ≥ 2,

    [Z_k+1^N_ E^q]   =
    [∑_i=0^k T_δ t^k+1-iG(Y^N_k+Z^N_k) δ  W_i_E^q]
       ≤[(∑_i=0^k∑_j∈^+ T_δ t^k+1-iG(Y^N_i+Z^N_i)e_j_^1/2+ϵ^2δ t)^q/2]
       ≤[(∑_i=0^k
    ((k+1-i)δ t)^-1/2-2ϵ(1+Y^N_i^2α+Z^N_i^2α)δ t)^q/2].

The Hölder inequality yields that for any l>2, 

    [Z^N_k+1_E^q]
       ≤
    C[(∑_i=0^k((k+1-i)δ t)^-1+4ϵ/2l/(l-1)δ t)^(q(l-1)/2l)
    (∑_i=0^k(1+Y^N_i^2α l+Z^N_i^2α l)δ t)^q/2l].

Since α<1, there always exists l>2 such that 
2α l<4. Then  taking  such l and 
combining the estimates (<ref>)-(<ref>), we 
obtain 

    [Z^N_k+1_E^q]
       ≤
    C(α,T)
    [(∑_i=0^k(1+Y^N_i^4+Z^N_i^2α l)δ t)^q/2l]≤ C(X_0,T,q,α).




With the help of the factorization method, we can get a stronger priori estimates of Z^N.



Let X_0∈, T>0 and q≥1.
There exists a positive constant C(X_0,T,q)
such that 

    [sup_k≤ KZ^N_k^q_E]
       ≤  C(X_0,T,q).



Proof.
We introduce a new auxiliary process Z^N(t) defined by 

    Z^N(t)=∫_0^t S(t-s)G(Y^N_[r]+Z^N_[r])dW(r).

Then by using the factorization method and the procedures in the proof of <cit.>[Lemma 3.2] , it is not difficult to obtain that 

    [ sup_t∈[0,T]Z^N(t)_E^q]   ≤ C(X_0,T,q).

By the triangle inequality, we have that 

    [ sup_k≤ KZ^N_k_E^q]
       ≤ C_q[sup_k≤ KZ^N(kδ t)_E^q]
    +C_q[sup_k≤ KZ^N(kδ t)-Z^N_k_E^q].

Next, it suffices to show that 

    sup_k≤ K[Z^N_k-Z^N(kδ t)^q_E]≤ C(X_0,T,q)δ^η q,
 
for some η>0 and q>1 large enough. Indeed, from the proof of <cit.>, we can obtain that for η q>1,

    [sup_k≤ KZ^N(kδ t)-Z^N_k_E^q]
       ≤∑_k≤ K[Z^N(kδ t)-Z^N_k_E^q]
       ≤ KC(X_0,T,q)δ t^η q≤ C(X_0,T,q),

which, together with (<ref>), yields the desired result.
Now, we prove (<ref>).
By using the Sobolev embedding theorem, the Burkholder inequality,error estimate between T_δ t^k and e^-A^2kδ t, it holds that for some ϵ>0 and η>0,

    [Z^N(t_k)-Z_k^N_E^q]
       ≤ C_q[Z^N(t_k)-Z_k^N_^1/2+ϵ^q]
       ≤ C_q[∫_0^t_k(e^-A^2(t_k-s)-T_δ t^k-[s])G(Z_[s]^N+Y_[s]^N)dW(r)_^1/2+ϵ^q]
       ≤ C_q δ t^η q/2[(∫_0^t_k(1+(t_k-[s])^-1/2-ϵ/4-η)(1+Z_[s]^N^2α+Y_[s]^N^2α)dr)^q/2].

Applying the Hölder inequality and taking ϵ>0 small enough, we get that for some  l>2  and small η>0 which depend on α,

    [Z^N(t_k)-Z_k^N_E^q]
       ≤ C_q δ t^η q/2(∫_0^t_k(1+(t_k-[s])^-(1/2+ϵ/4+η)l/l-1)ds)^q(l-1)/2l
         ×[(∫_0^t_k(1+Z_[s]^N^2α l+Y_[s]^N^2α l)dr)^q/2l].

The above estimate, together with (<ref>) and (<ref>) yields that (<ref>). Thus we complete the proof.



Let  X_0∈, T>0, q≥1.
There exists a positive constant C(X_0,T,q)
such that 

    sup_k≤ K[X^N_k^q]
       ≤ C(X_0,T,q).



Proof.
By Lemma <ref>, it suffices to bound Y_k^N in .
Taking inner product on both sides of Eq. (<ref>) with Y^N_k+1 in  and using the Gagliardo–Nirenberg inequality,
we obtain 

    Y^N_k+1^2
       ≤ Y^N_k^2
    -(2-ϵ)δ tA Y^N_k+1^2
          +C(ϵ)
    (1+Z^N_k+1_E^4+Y^N_k+1_L^4^4)δ t.

By the Sobolev embedding theorem and Lemma <ref>, we obtain that for p≥ 1, ϵ>0 small enough,

    [∇ Z^N_k+1_L^4^4p]
       ≤ C[Z^N_k+1_^3/2-ϵ^4p]≤ C(T,p,X_0).













The above estimate, together with (<ref>)-(<ref>) and Proposition <ref>, yields that for p≥ 1,

    Y^N_k+1_L^2p(Ω; ℍ)+∑_k=0^K-1A Y^N_k+1^2δ t_L^p(Ω;ℝ)
       ≤ C(ϵ, T)∑_k=0^K-1Z^N_k+1_E^4δ t_L^p(Ω;ℝ)
         +C(ϵ, T)∑_k=0^K-1Y^N_k+1_L^4^4δ t_L^p(Ω;ℝ)+C(X_0,p,T)
    
       ≤ C(X_0,p,T),

which completes the proof.



Similarly, one can obtain the a priori estimate in E.
More precisely, it holds that for q≥1, γ≤1/2 and sufficient small ϵ>0,

    [X^N_k^q_E]
       ≤ C(X_0,T,q)(1+(kδ t)^-q/8+γ q/4-ϵ q),

and for γ > 1/2,

    [X^N_k^q_ E]
       ≤ C(X_0,T,q).



In the following, we present the discrete continuity of 
the discrete convolution {Z^N_k}_k≤ K and the numerical approximation {X^N_k}_k≤ K.




Let X_0∈, T>0, q≥1. Then for all γ∈ (0,3/2), there exists C(X_0,T,p,γ)>0 such that  

    sup_k≤ K[Z^N_k_^γ^q]
       ≤  C(X_0,T,p,γ)

and 

    [Z^N_k-Z^N_k_1^q]
       ≤  C(X_0,T,p,γ)|(k-k_1)δ t|^γ q/4,

where k_1,k≤ K.


Proof.
By the Burkhölder inequality and the smoothing effect of T_δ t, we obtain that  

    [Z^N_k^q_ℍ^γ]
       ≤ 
    C_q[(∑_i=0^k∑_j∈^+T_δ t^k+1-iG(Y^N_k+Z^N_k)e_j_ℍ^γ^2 δ t)^q/2]
       ≤ 
     C_q[(∑_i=0^k
    ((k+1-i)δ t)^-1/4-γ/2(1+Y^N_i^2α+Z^N_i^2α)δ t)^q/2].

By similar arguments in the proof of Lemma <ref>, we complete the proof for the first estimate.

For convenience, we denote [s]=[s]_δ t/δ t and [s]_δ t:=max{0,δ t,⋯,kδ t,⋯}∩ [0,s].
Define the continuous extension of Z^N_k as

    Z^N(t)=∫_0^tT_δ t^-(t/δ t-[s])P^NG(X_[s]^N)dW(s), Z^N(0)=0,

where Z^N(t_k)=Z^N_k, t_k=kδ t.
Without loss of generality, we assume that k>k_1
The mild form of Z^N and the Burkhölder inequality yield that 
for 0≤ s≤ t≤ T, q≥ 1 and sufficient small ϵ_1>0,

    [Z^N(t_k)-Z^N(t_k_1)^q]
       ≤[∫_0^t_k_1(T_δ t^-(k-[r])-T_δ t^-(k_1-[r]))P^N(G(Y^N_[r]+Z^N([r]_δ t)))dW(r)^q]
         
    +[∫_t_k_1^t_kT_δ t^-(k-[r])P^NG(Y^N_[r]+Z^N([r]_δ t))dW(r)^q]
       ≤ C[(∫_0^t_k_1∑_i∈^+T_δ t^-(k_1-[r])(T_δ t^-(k-k_1)-I)P^N(G(Y^N_[r]+Z^N([r]_δ t)))e_i^2dr)^q/2]
         
    +C[(∫_t_k_1^t_k∑_i∈^+T_δ t^-(k-[r])P^NG(Y^N_[r]+Z^N([r]_δ t))e_i^2dr)^q/2]
       ≤ 
     C[(∫_0^t_k_1 (k_1-[r]_δ t)^-γ/2-1/4-ϵ_1((k-k_1)δ t)^γ/2G(Y^N_[r]+Z^N([r]_δ t))^2dr)^q/2]
         
    +C[(∫_t_k_1^t_k(k-[r]_δ t)^-1/4-ϵG(Y^N_[r]+Z^N([r]_δ t))^2dr)^q/2].

Then combining the above estimate and Proposition <ref>, we complete the proof.




Next, we focus on the  regularity estimate of the numerical solution {X^N_k}_k≤ K.



Let X_0∈^γ, γ∈ (0,3/2), T>0 and  q≥1.
Then there exists C(X_0,T,q,γ)>0 such that

    sup_k≤ K[X^N_k_^γ^q]
       ≤ C(X_0,T,q,γ)

and 

    [X^N_k-X^N_k_1^q]
       ≤ C(X_0,T,q,γ)|(k-k_1)δ t|^γ p/4,

where k_1,k≤ K.



Proof.
Due to the above  regularity estimate in Lemma <ref>, 
it only suffices to give the regularity estimate of {Y^N_k}_k≤ K.
To this end, we first give the upper bound of Y_k^N_L^6. For convenience, we only give the proof for the case that γ∈ [1,3/2).
From the Sobolev embedding theorem, the smoothing effect of e^-A^2t, the Gagliardo–Nirenberg and Young inequalities, it follows  that 

    Y^N_k+1_L^6
       ≤T_δ t^k+1 Y_0^N_L^6
    + ∑_j=0^kT_δ t^k+1-jA F(Y_j+1^N+Z_j+1^N)_L^6δ t
       ≤ CY_0^N_^1+
    C∑_j=0^k(k+1-j)^-7/12δ t^-7/12(1+Z_j+1^N_L^6^3+ Y^N_j+1)δ t
          +C∑_j=0^k(k+1-j)^-7/9δ t^-7/9 Y^N_j^10/3δ t+∑_j=0^k(-A) Y^N_j+1^2δ t.

The a priori estimates of Y^N_k
 and Z^N_k imply that for q≥ 1,

    [sup_k≤ K Y^N_k_L^6^q]
       ≤ C(X_0,T,q).

Now, we are in the position to give the desired regularity estimate. 
From the mild form of Y^N_k and the above a priori estimate in L^6, it follows that 

    [ Y^N_k_^γ^q]
       ≤ 
    C[ Y^N_0_^γ^q]
    +Cδ t[(∑_j=0^K-1T_δ t^K-jA F(Y^N_j+1+Z^N_j+1)_^γ^q]
       ≤ C(q) X^N(0)_^γ^q+
    C(q)∑_j=0^K-1(T-t_j)^-1/2-γ/4δ tsup_j≤ K[(1+Y^N_j_L^6^3+Z^N_j_L^6^3)^q]
       ≤ C(q) X^N(0)_^γ^q+
    C(T,p)sup_j≤ K[(1+Y^N_j_L^6^3+Z^N_j_L^6^3)^q]
       ≤ C(X_0,T,q).

For convenience, we assume that  k> k_1.
The mild forms of Y^N_k and Y^N_k_1, together with  the smoothing effect of T_δ t, yield that 

    [Y^N_k-Y^N_k_1^q]
       ≤[ T_δ t^k_1(T_δ t^k-k_1-I)Y_0^N^q]
    
         +[(∑_j=0^k_1-1(T_δ t^k_1-j(T_δ t^k-k_1-I)A F(Y^N_j+1+Z^N_j+1)δ t)^q]
    
         +[(∑_j=k_1^k-1T_δ t^k-jAF(Y^N_j+1+Z^N_j+1)δ t)^q]
       ≤ C(X_0,T,q,γ)|(k-k_1)δ t|^γ q/4.

Combining the above regularity estimates together, we finish the proof for γ∈ [1,3/2).
Similar arguments, together with
 
    e^-A^2tX_0^N_L^6≤ Ct^-1/12+γ/4X_0,

lead to the 
desired result for the case of γ∈ (0,1).




Let X_0∈, T>0 and q≥1, sup_i≤ Nδ tλ_i^2≤ 1.
There exists a positive constant C(X_0,T,q)
such that 

    [sup_k≤ KZ^N_k^q_E]
       ≤  C(X_0,T,q).


Proof.
Due to the continuous extension of Z^N_k,
it suffices to estimate 
    [sup_t∈[0,T]Z^N(t)_E^q].

Notice that  for α_1>1/p+γ, p>1, γ=1/8,

    Z^N(t)
       =∫_0^t(t-s)^α_1-1T_δ t^-(t/δ t-s/δ t)Y_α_1,N(s)ds,

where Y_α_1,N(s)=∫_0^s(s-r)^-α_1T_δ t^-(s/δ t-[r])
P^NG(Y^N_[r])+Z^N([r]_δ t))dW(r).
By the Hölder inequality, we have  

    [sup_t∈[0,T]Z^N(t)_E^q]
       ≤ 
    C[Y_α_1,N_L^p([0,T];)^q].

From the Hölder and Burkhölder inequalities,  it follows that for q≥max(p,2),

    [Y_α_1,N_L^p(0,T; )^q]
       =
    [ (∫_0^T∫_0^s(s-r)^-α_1T_δ t^-(s/δ t-[r])
    P^NG(Y^N_[r]+Z^N([r]_δ t))dW(r)^pds)^q/p]
       ≤ C(T,q)∫_0^T[ ∫_0^s(s-r)^-α_1T_δ t^-(s/δ t-[r])
    P^NG(Y^N_[r]+Z^N([r]_δ t))dW(r)^q]ds
       ≤ C(T,q)∫_0^T[ (∫_0^s(s-r)^-2α_1∑_i∈^+T_δ t^-(s/δ t-[r])
    P^NG(Y^N_[r]+Z^N([r]_δ t))e_i^2dr)^q/2]ds.


To give an upper bound of the above term, the following  smoothing effect of T_δ t is required, that is, for any κ>0 and β>0,

    A^βT_δ t^κ x≤ C(κδ t)^-β/2x,   x∈.

Due to the fact |1/1+z|≤ e^-Cz for z∈ [0,1] and some C>0,
we have that T_δ t^κ e_i≤ e^-Cκ z when λ_i^2δ t≤ 1. Hence,

    sup_λ_i^2δ t≤1A^βT_δ t^κ e_i   ≤sup_λ_i^2δ t≤ 1λ_i^βe^-Cλ_i^2κδ t≤ C(κδ t)^-β/2.


Therefore, the estimate (<ref>) holds.
By Parseval's identity and the smoothing effect of T_δ t, we get for any sufficient small ϵ>0,

    [Y_α_1,N_L^p(0,T; )^q]
       ≤  C(T,q)∫_0^T[ (∫_0^s(s-r)^-2α_1∑_i∈^+∑_j∈^+
    P^NG(Y^N_[r])+Z^N([r]_δ t))e_i, 
                 T_δ t^-(s/δ t-[r])e_j^2dr)^q/2]ds
       ≤ C(T,q)∫_0^T[ (∫_0^s(s-r)^-2α_1∑_j∈^+T_δ t^-(s/δ t-[r])e_j^2G(Y^N_[r])+Z^N([r]_δ t))^2dr)^q/2]ds
       ≤ C(T,q)∫_0^T[ (∫_0^s(s-r)^-2α_1
    (s-[r]_δ t)^-1/4-ϵ(1+Y^N_[r]^2α+Z^N([r]_δ t)^2α)dr)^q/2]ds.


Since α<1, one can choose a positive number l>2 and a  large enough number p 
 such that
2α l<4 and (2α_1+1/4+ϵ)l/l-1<1.
Then by using a priori estimates (<ref>) and (<ref>), we obtain 

    [Y_α,N_L^p(0,t; )^q]
       ≤ C(T,q,α)
    ∫_0^t(∫_0^s(s-r)^-(2α_1+1/4+ϵ)l/l-1dr)^q(l-1)/2l
           ×[(∫_0^s(1+Y^N_[r]^2α l+Z^N([r]_δ t)^2α l)dr)^q/2l]ds
       ≤  C(T,q,α,Y_0^N),

which completes the proof.



Based on the above lemma, we have the following stronger a priori estimate result.

Let X_0∈^γ, γ > 1/2, T>0 and q≥1, sup_i≤ Nδ tλ_i^2≤ 1.
There exists a positive constant C(X_0,T,q)
such that 

    [sup_k≤ KX^N_k^q_E]
       ≤  C(X_0,T,q).


 It seems that it is not difficult to prove.

In the following, we present the discrete factorization method, which is useful for showing the uniform a priori estimate of the discrete stochastic process and analyzing the optimal strong convergence rate of the numerical approximation. 
Denote (E_1,·_E_1) a separable Banach space.
Denote l_δ t^p(E_1), p≥ 1, is the linear space of all E_1-vectors f={f_i}_i, i≤ M∈^+,  equipped with the norm f_l_δ t^p(E_1):=(∑_i=0^Mf_i_E_1^pδ t)^1/p, if p<∞,  f∈ l_δ t,M^p, and l_δ t^∞(E_1):=sup_i≤ Mf_i_E_1 if p=∞, f∈ l_δ t,M^∞(E_1).


Assume that p>1, γ≥ 0, α>1/p+γ, Mδ t≤ T and that E_1, E_2 are Banach spaces such that 
    T_δ t^ix_E_1≤ L (iδ t)^-γx_E_2,   0<i≤ M,  x∈ E_2,

then 

    {G_α f}_k+1   :=∑_i=0^k ((k+1-i)δ t)^α -1T_δ t^k+1-if_iδ t,    0≤ k ≤ M-1,
    {G_α f}_0   =0,

satisfies 

    G_α f_l^∞_δ t,M(E_1)   ≤ C(L,α,T)f_l_δ t,M^p(E_2).


Proof.
By the Minkowski and Hölder inequality, we have 
for 0 ≤ k≤  M-1,

    ∑_i=0^k ((k+1-i)δ t)^α -1T_δ t^k+1-if_iδ t_E_1
       ≤ L∑_i=0^k ((k+1-i)δ t)^α -1-γf_i_E_2δ t
    
       ≤ L(∑_i=0^k ((k+1-i)δ t)^(α -1-γ)qδ t)^1/qf_l^p_δ t(E_2)
       ≤ C(L,α)f_l^p_δ t,M(E_2).


As 
the factorization formula in continuous case, the following Corollary plays a important role in the discrete case.

Under the same condition of <ref>, it holds that for k≤ M-1,

    (G_α f)_k+1_E_1   ≤ C(L,α,T)f_l^p_δ t,k.




In fact, the following lemma ensure that the stochastic Fubini theorem holds for the discrete 
stochastic convolution. Thus one could use to Corollary <ref> to deduce the uniform estimate for the discrete 
stochastic convolution.


Assume that for some α∈ (0,1) and for  k≤ K-1,

    ∑_j=1^k ((k+1-j)δ t)^α-1(∑_i=0^j-1 ((j-i)δ t)^-2α[T_δ t^k+1-iΦ_i__2()^2]δ t)^1/2δ t<∞.

Then we have 

    Z_k+1   =∑_i=0^kT_δ t^k+1-iΦ_iδ W_i,  k≤ K-1,
       = ∑_j=1^k((k+1-j)δ t)^α -1 T_δ t^k+1-jY^α,k_jδ t+T_δ tΦ_kδ W_k,

 and

    Y^α,k_j   =∑_i=0^j-1((j-i)δ t)^-α
    T_δ t^j-iC(α,i,k)Φ_iδ W_i,   j≤ k,

where  C(α,i,k), i≤ k-1,k≤ K-1, is a positive constant.

Proof.
By similar arguments in the proof of <cit.>, we have

    ∑_j=1^k((k+1-j)δ t)^α -1 T_δ t^k+1-jY^α,k_jδ t
       =∑_j=1^k((k+1-j)δ t)^α -1 T_δ t^k+1-j∑_i=0^j-1((j-i)δ t)^-αC(α,i,k)
    T_δ t^j-iΦ_iδ W_iδ t
       =∑_i=0^k-1C(α,i,k)∑_j=i+1^k((k+1-j)δ t)^α-1((j-i)δ t)^-αδ t
    T_δ t^k+1-iΦ_iδ W_i
       =∑_i=0^k-1T_δ t^k+1-iΦ_iδ W_i,

where C(α,i,k)=1/∑_j=i+1^k((k+1-j)δ t)^α-1((j-i)δ t)^-αδ t≤ C, 1≤ k≤ K, 0≤ i≤ k-1.



Given a fixed i≤ K-1. The sequence C(α,i,k) is strictly decreasing with respect to k, i+1≤ k≤ K.


Proof.
Let i ≤ k-1≤ K-1. From the definition of C(α,i,k), the monotonicity
C(α,i,k)≥ C(α,i,k+1) 
is equivalent to the monotonicity of
∑_j=1^l(l+1-j)^α-1j^-α. By using the convexity of the function (1-x)^α-1x^-α, x∈ (0,1), we complete the proof.

Assume that 

    sup_j≤ K-1[(∑_i=0^j-1 ((j-i)δ t)^-2α∑_k∈^+T_δ t^(j-i)Φ_ie_k^2δ t)^p/2]<∞.

It holds that 

    [sup_k≤ KZ^N_k_E^p]
    ≤ C(T,q,Y_0^N).


Proof.
According to Lemma <ref>, for 1≤ k≤ K-1, 

    Z_k^N_ℰ   = ∑_j=1^k((k+1-j)δ t)^α -1 T_δ t^k-jY^α,k_jδ t_ℰ.

Denoting E_1= and E_2=E,  and using Lemma <ref> and Corollary <ref>, for γ=1/8,  we have for 1≤ k≤ K-1,

    Z_k^N_ℰ^p
       ≤  C(∑_j=1^k((k+1-j)δ t)^(α -1-γ)p'δ t )^p/p'(∑_j=1^k Y^α,k_j^p δ t ),
 
which implies that 

    sup_k≤ K-1Z_k^N_ℰ^p≤ C(∑_j=1^K-1sup_k≥ jY^α,k_j^p δ t ).

We claim that 

    [sup_k≥ jY^α,k_j^p]≤ C[(∑_i=0^j-1 ((j-i)δ t)^-2α∑_l∈^+T_δ t^(j-i)Φ_ie_l^2δ t)^p/2].

Indeed, 

    [sup_k≥ jY^α,k_j^p]
       =[sup_k≥ j(∑_l=1^N ∑_i=0^j-1((j-i)δ t)^-α
    T_δ t^j-iC(α,i,k)Φ_iδ W_i, e_l^2)^p/2]

By Lemma <ref>, 

Since C(α,i,k)≥  C(α,i,k+1), we have the 
following monotonicity of 

    ∑_j=1^k+1Y^α,k+1_j^p δ t
    ≤∑_j=1^kY^α,k_j^p δ t+Y^α,k+1_k+1^pδ t,   k≤ K-1.

Indeed, it only suffices to prove 

    Y^α,k_j^2
       ≥Y^α,k+1_j^2.

By the definition of Y^α,k_j, we have for 1≤ j≤ k,

    Y^α,k+1_j^2- Y^α,k_j^2
       = 
    ∑_i=0^j-1((j-i)δ t)^-α
    T_δ t^j-iC(α,i,k+1)Φ_iδ W_i^2
         -∑_i=0^j-1((j-i)δ t)^-α
    T_δ t^j-iC(α,i,k)Φ_iδ W_i^2
       =
    ∑_i_1,i_2=0^j_1-1,j_2-1C(α,i_1,k+1)((j_1-i_1)δ t)^-α
    T_δ t^j_1-i_1Φ_i_1δ W_i_2, 
          C(α,i_2,k+1)((j_2-i_2)δ t)^-α
    T_δ t^j_2-i_2Φ_i_2δ W_i_2
         -
    ∑_i_1,i_2=0^j_1-1,j_2-1C(α,i_1,k)((j_1-i_1)δ t)^-α
    T_δ t^j_1-i_1Φ_i_1δ W_i_2,  
          
    C(α,i_2,k)((j_2-i_2)δ t)^-α
    T_δ t^j_2-i_2Φ_i_2δ W_i_2
       =
    ∑_i_1,i_2=0^j_1-1,j_2-1(C(α,i_1,k+1)-C(α,i_1,k))C(α,i_2,k+1) 
         ((j_1-i_1)δ t)^-α
    T_δ t^j_1-i_1Φ_i_1δ W_i_2, 
    ((j_2-i_2)δ t)^-α
    T_δ t^j_2-i_2Φ_i_2δ W_i_2
         +
    ∑_i_1,i_2=0^j_1-1,j_2-1C(α,i_1,k)(C(α,i_2,k+1)-C(α,i_2,k))
         ((j_1-i_1)δ t)^-α
    T_δ t^j_1-i_1Φ_i_1δ W_i_2,  
    ((j_2-i_2)δ t)^-α
    T_δ t^j_2-i_2Φ_i_2δ W_i_2
       =∑_i_1,i_2=0^j_1-1,j_2-1((C(α,i_1,k+1)-C(α,i_1,k))C(α,i_2,k+1)
           +C(α,i_1,k)(C(α,i_2,k+1)-C(α,i_2,k)))
         ×((j_1-i_1)δ t)^-α
    T_δ t^j_1-i_1Φ_i_1δ W_i_2,  
    ((j_2-i_2)δ t)^-α
    T_δ t^j_2-i_2Φ_i_2δ W_i_2
       ≤ 0.


Therefore, we get 

    [sup_k≤ KZ_k_ℰ^p]
       ≤[ sup_k≤ K-1Z_k-1_ℰ^p]
    + [ Z_K_ℰ^p]
       ≤ C(T,p)[∑_j=1^K-1sup_k≥ jY^α,k_j^p δ t ]+C(T,p,Y_0^N)
       ≤ C(T,p)∑_j=1^K-1[(∑_i=0^j-1 ((j-i)δ t)^-2α∑_k∈^+T_δ t^(j-i)Φ_ie_k^2δ t)^p/2] δ t 
    +C(T,p,Y_0^N).




Let  p≥1.
There exists a positive constant C(T,q,Y^N_0)
such that 

    sup_k≤ K[Z^N_k^q_E]
       ≤ C(T,q,Y^N_0).



Proof.
By using the continuous version of factorization method, one could also recover the result in Proposition <ref>.
Indeed, we first define a process

    Z^N(t)=∫_0^t T_δ t^t/δ t-[s]_δ tΦ_[s]_δ tdW(s),  t∈ [0,T]

as a continuous version of  Z^N_k.
Then the similar arguments in the proof of  Lemma <ref> yield that  for α_1>1/p+γ, p>1, γ=1/8,

    [sup_k∈ KZ^N_k_E^q]
    ≤[sup_t∈ [0,T]Z^N(t)_E^q]
    ≤ C(q,T)[Y^α_1,N_L^p(0,T; H)^q],

where

    Z^N(t)= ∫_0^t (t-s)^α_1-1T_δ t^t/δ t-s/δ tY^α_1,N(s) ds,
 
and 

    Y^α_1,N(s)=∫_0^sC(α,r,t)(s-[r]_δ t)^-α_1 T_δ t^s/δ t-[r]_δ tΦ_[r]_δ tdW(r)

with C(α,r,t)=1/∫_r^t(t-s)^α_1-1(s-[r]_δ t)^-α_1ds. 
It is not difficult to see that C(α,r,t)



It suffices to estimate the bound of 

    sup_j≤ K[(∑_i=0^j-1 ((j-i)δ t)^-2α_1∑_l∈^+T_δ t^(j-i)G(X^N_i)e_l^2δ t)^p/2].

From the Hölder and Burkhölder inequalities,  it follows that

    [(∑_i=0^j-1 ((j-i)δ t)^-2α_1∑_l∈^+T_δ t^(j-i)G(X^N_i)e_l^2δ t)^p/2]
       ≤ C(T,p)[(∑_i=0^j-1 ((j-i)δ t)^-2α_1∑_j∈^+G(X^N_i)e_j^2T_δ t^(j-i)e_j^2δ t)^p/2]
       ≤ C(T,p)[(∑_i=0^j-1 ((j-i)δ t)^-2α_1-1/4(1+Y^N_i^2α+Z^N_i^2α)δ t].

Since α<1, one can choose a positive number l>2 and a  large enough number p 
 such that
2α l<4 and (2α_1+1/4)l/l-1<1.
Then by using a priori estimates (<ref>)  and (<ref>), we obtain 

    [(∑_i=0^j-1 ((j-i)δ t)^-2α_1∑_l∈^+T_δ t^(j-i)G(X^N_i)e_l^2δ t)^p/2]
       ≤
    C(T,p)
    (∑_i=0^j-1 ((j-i)δ t)^-(2α_1+1/4)l/l-1δ t)^l/l-1[(∑_i=0^j-1
    (1+Y^N_i^4+Z^N_i^4)δ t)^1/l]
       ≤ C(T,p,α,X_0).





: Key difficult in proving the discrete factorization lemma: sup_k≥ j Y^α,k_j 
12500235759270719W








 §.§ Mean square convergence rate of the full discretization



Based on the a priori  and  regularity estimates of the numerical approximation, we focus on the strong convergence and the strong convergence rate of the proposed scheme for Eq. (<ref>). 

For convenience, we only present the convergence analysis in mean square sense. 
Notice that X^N_k-X(t_k)≤X^N_k-X^N(t_k)+X^N(t_k)-X(t_k). 
By Lemma <ref> and <cit.>, we have that 
if X_0∈^γ,  γ∈ (0,3/2), then for α∈ (0,γ) it holds that 

    X^N-X_L^2(Ω;C([0,T];))≤ C(T,X_0,α) λ_N^-α/2.
 
for a positive constant C(T,X_0,α).
Thus we only need to estimate the error of X^N_k-X^N(t_k).
 

To deduce the mean square convergence rate in time, 
we  introduce an auxiliary process X^N_k, k≤ K with X^N_0=X^N(0), defined by 

    X^N_k+1
    =X^N_k
    -A^2δ t X^N_k+1
    -P^N A F(X^N(t_k+1))δ t
    +P^NG(X^N(t_k))δ W_k.



 Then we split the error of X^N_k-X^N(t_k) as 

    X^N_k-X^N(t_k)≤X^N(t_k)-X^N_k 
    +X^N_k-X^N_k.
 
The first error is bounded by the following lemma.
The second error will be dealt with the interpolation arguments. We would like to mention that compared with the existing literatures (see e.g. <cit.>), this is also a new approach to deducing strong convergence rates of temporal and full discretizations for SPDEs with non-monotone coefficients.  




Let X_0∈^γ, γ∈ (0,3/2) and  p≥1.
There exists a positive constant C(X_0,T,p,γ) such that

    X^N(t_k)-X^N_k_L^p(Ω;)   ≤ C(X_0,T,p,γ)δ t^γ/4.



Proof.
Denote [s]_δ t:=max{0,δ t,⋯,kδ t,⋯}∩ [0,s] and [s]=[s]_δ t/δ t.
The mild forms of X^N(t_k) and X^N_k yield that 

    X^N(t_k)-X^N_k_L^p(Ω;)
       ≤∫_0^t_k(e^-A^2(t_k-s)-T_δ t^k-[s])AP^NF(X^N(s))ds_L^p(Ω;)
         +∫_0^t_kT_δ t^k-[s]AP^N(F(X^N(s))-F(X^N([s]_δ t+δ t)))ds_L^p(Ω;)
         +∫_0^t_k(e^-A^2(t_k-s)-T_δ t^k-[s])P^NG(X^N(s))dW(s)_L^p(Ω;)
         +∫_0^t_kT_δ t^k-[s]P^N(G(X^N(s))-G(X^N([s]_δ t)))dW(s)_L^p(Ω;)
       =:I_1+I_2+I_3+I_4.

By the properties of e^-A^2t and T_δ t^k, the priori estimates of Y^N and Z^N, and e^-A^2tX_0^N_L^6≤ Ct^min(-1/12+γ/4,0)X_0, it holds that for β∈(0,min(1/2,1/4+3γ/4)),

    I_1
       ≤  C∑_j=0^k-1∫_t_j^t_j+1(t_k-[s]_δ t)^-1/2+βδ t^β(1+X^N(s)^3_L^3p(Ω;L^6))ds
       ≤ C(T,X_0,p)δ t^β.

The similar arguments in the proof of <cit.>  yield that 
for  ϵ>0 small enough,

    [X^N(t)^q_E]
       ≤ C(X_0,T,q)(1+t^min(0,-q/8+γ q/4-ϵ q)),

which implies that 

    I_2   ≤∫_0^t_kT_δ t^k-[s]AP^N(F'(X^N(s))(X^N(s)-X^N([s]_δ t+δ t)))ds_L^p(Ω;)
       ≤∫_0^t_k (t_k-[s]_δ t)^-1/2F'(X^N(s))_EX^N(s)-X^N([s]_δ t+δ t)_L^p(Ω)ds
       ≤ C(T,X_0)δ t^γ/4.

The Burkhölder inequality and the Sobolev embedding theorem lead that for small ϵ>0,

    I_3
       ≤ C(T,p) (∫_0^t_k(e^-A^2(t_k-s)-T_δ t^k-[s])P^NG(X^N(s))__2()^2ds_L^p/2(Ω))^1/2
       ≤ C(T,p) (∫_0^t_kA^1/4+ϵ+γ/2(e^-A^2(t_k-s)-T_δ t^k-[s])A^-γ/2^2
               A^-1/4-ϵP^NG(X^N(s))__2()^2ds_L^p/2(Ω))^1/2
       ≤ C(T,p) (∫_0^t_kA^1/4+ϵ+γ/2(e^-A^2(t_k-s)-T_δ t^k-[s])A^-γ/2^2 A^-1/4-ϵ G(X^N(s))__2()^2ds _L^p/2(Ω))^1/2
       ≤ C(X_0,T,p)δ t^γ/4.


Similar arguments yield that

    I_4
       ≤ C(T,p) (∫_0^t_kT_δ t^k-[s]P^N(G(X^N(s))-G(X^N([s]_δ t)))__2()^2 ds_L^p/2(Ω))^1/2
       ≤  C(T,p)  (∫_0^t_kA^1/4+ϵT_δ t^k-[s]^2 A^-1/4-ϵ(G(X^N(s))-G(X^N([s]_δ t)))__2()^2 ds_L^p/2(Ω))^1/2
       ≤ C(T,p)  (∫_0^t_k (t_k-[s]_δ t)^-1/2-2ϵX^N(s)-X^N([s]_δ t)^2 ds_L^p/2(Ω))^1/2
       ≤ C(X_0,T,p)  δ t^γ/4.


Combining (<ref>) and the above regularity estimates, we complete the proof.




To deduce the strong convergence rate in time, 
we  introduce an auxiliary process X^N_k, k≤ K with X^N_0=X^N(0), defined by 

    X^N_k+1
    =Y^N_k
    -A^2δ t Y^N_k+1
    -P^N A F(Y^N(t_k+1)+Z^N(t_k+1))δ t.

Then we split the error of Y^N_k-Y^N(t_k) as 

    Y^N_k-Y^N(t_k)≤Y^N(t_k)-Y^N_k
    +Y^N_k-Y^N_k.
 
The first error is bounded by the following lemma.
The second error will be dealt with the interpolation arguments.



Let X_0∈^γ, γ∈ (0,3/2), N∈^+ and  p≥1.
There exists a positive constant C(X_0,T,p) such that

    Y^N(t_k)-Y^N_k_L^p(Ω;)   ≤ C(X_0,T,p)δ t^γ/2.



Proof.

Denote [s]_δ t:=max{0,δ t,⋯,kδ t,⋯}∩ [0,s] and [s]=[s]_δ t/δ t.
The mild forms of Y^N(t_k) and Y^N_k yield that 

    Y^N(t_k)-Y^N_k_L^p(Ω;)
       ≤∫_0^t_k(e^-A^2(t_k-s)-T_δ t^k-[s])AP^NF(Y^N(s)+Z^N(s))ds_L^p(Ω;)
         +∫_0^t_kT_δ t^k-[s]AP^N(F(Y^N(s)+Z^N(s))-F(Y^N([s]_δ t+δ t)+Z^N([s]_δ t+δ t)))ds_L^p(Ω;)
       :=I_1+I_2.

By the properties of e^-A^2t and T_δ t^k, the priori estimates of Y^N and Z^N, and 
Corollary <ref>, the first term is estimated as for small ϵ>0, γ<3/2,

    I_1
       ≤  C∑_j=0^k-1∫_t_j^t_j+1(t_k-[s]_δ t)^-1+ϵδ t^1/2+γ/4-ϵ(1+X^N(s)_ℰ^2+∇ X^N(s)_ℰ^2+
            X^N(s)_^γ^2
    )(X^N(s)_^γ+∇ X^N(s)_ℰ)_L^p(Ω;)ds≤ C(T,X_0,p)δ t^1/2+γ/4-ϵ.

Similarly, by using the Talyor formula and the mild form of X^N, we have 

    I_2   ≤∫_0^t_kT_δ t^k-[s]AP^N(F'(X^N(s))(e^-A^2([s]_δ t+δ t-s)-I)X^N(s))ds_L^p(Ω;)
          +∫_0^t_kT_δ t^k-[s]AP^N(F'(X^N(s)∫_s^[s]_δ t+δ te^-A^2([s]_δ t+δ t-r)AP^NF(X^N(r))dr)ds_L^p(Ω;)
          +∫_0^t_kT_δ t^k-[s]AP^N(F'(X^N(s))∫_s^[s]_δ t+δ te^-A^2([s]_δ t+δ t-r)P^NdW(r))ds_L^p(Ω;)
          + ∫_0^t_k
    T_δ t^k-[s]AP^N(∫_0^1F”(λ X^N(s)+(1-λ)X^N([s]_δ t)(1-λ)dλ
           (X^N(s)-X^N([s]_δ t+δ t))^2)ds_L^p(Ω;)
    :=I_21+I_22+I_23+I_24.

Then the smoothing effect of e^-A^2t and Corollary <ref> lead to
for 1≤γ<3/2,

    I_21   ≤∫_0^t_kT_δ t^k-[s]A^1+3/4+ϵ A^-3/4-ϵ( F'(X^N(s))(e^-A^2([s]_δ t+δ t-s)-I)X^N(s))ds_L^p(Ω;)
       ≤∫_0^t_k(t_k-[s]_δ t)^-7/8-ϵ/2 F'(X^N(s))(e^-A^2([s]_δ t+δ t-s)-I)X^N(s)_L^p(Ω;^-3/2-2ϵ)ds
       ≤∫_0^t_k(t_k-[s]_δ t)^-7/8-ϵ/2(1+X^N(s)_ℰ^2+∇ X^N(s)_ℰ^2+ X^N(s)_^γ^2)
           e^-A^2([s]_δ t+δ t-s)-I)X^N(s)_^-γ_L^p(Ω;)ds
       ≤ C(T,X_0,p)δ t^γ/2.

Similarly, we have 

    I_22   ≤∫_0^t_kT_δ t^k-[s]AP^N(F'(X^N(s)∫_s^[s]_δ t+δ te^-A^2([s]_δ t+δ t-r)AP^NF(X^N(r))dr)ds_L^p(Ω;)
       ≤ C∫_0^t_k (t_k-[s]_δ t)^-1/2F'(X^N(s))_ℰ∫_s^[s]_δ t+δ te^-A^2([s]_δ t+δ t-r)A^1-γ/2
           P^NA^γ/2F(X^N(r))dr_L^p(Ω;)ds≤ C(T,X_0,p)δ t^1/2+γ/4.



From the mild form of Y^N, Z^N and Z^N_k, 
we have  the continuity in the negative Sobolev space as for s≤ t≤ T, 1≤γ<3/2, 

    Y^N(t)-Y^N(s)_^-γ
       ≤e^-A^2s(e^-A^2(t-s)-I)A^-γ/2 Y^N(0)
         +∫_0^se^-A^2(s-r)(e^-A^2(t-s)-I)A^1-γ/2 f(Y^N(r)+Z^N(r))dr
         + ∫_s^te^-A^2(t-r)A^1-γ/2 f(Y^N(r)+Z^N(r))dr
       ≤ C(t-s)^γ/2 Y^N(0)_^γ
         +C(t-s)^γ/2∫_0^s (s-r)^-1/2 f(Y^N(r)+Z^N(r))_^γdr
         +∫_s^te^-A^2(t-r)A^1-γ f(Y^N(r)+Z^N(r))_^γdr.

Taking the L^p(Ω;)-norm leads to 

    Y^N(t)-Y^N(s)_ L^p(Ω; ^-γ)   ≤ C(T,X_0,p)δ t^γ/2.

From the stochastic Fubini theorem, the Burkhölder–Davis–Gundy inequality and Hölder inequality, it follows that for p≥ 2,

    I_23   =∑_j=0^k-1∫_t_j^t_j+1T_δ t^k-[s]AF'(X^N(s))∫_s^t_j+1e^-A^2([s]_δ t+δ t-r)P^NdW(r)ds_L^p(Ω;)
       =∑_j=0^k-1∫_t_j^t_j+1∫_t_j^rT_δ t^k-[s]AF'(X^N(s))e^-A^2([s]_δ t+δ t-r)P^NdsdW(r)_L^p(Ω;)
       ≤ C_p√(∑_j=0^k-1∫_t_j^t_j+1∫_t_j^rT_δ t^k-[s]AF'(X^N(s))e^-A^2(t_j+1-r)P^Nds_L^p(Ω;_2())^2dr)
       ≤ C_pδ t^1/2√(∑_j=0^k-1∫_t_j^t_j+1∫_t_j^rT_δ t^k-[s]A^1-ϵA^ϵF'(X^N(s))e^-A^2(t_j+1-r)P^N_L^p(Ω;_2())^2dsdr)
       ≤ C(T,X_0,p)δ t^1/2+γ/4-ϵ/2.

Due to the continuity of Z^N and Y^N and Sobolev embedding theorem, we obtain for small ϵ>0,

    I_24   =∫_0^t_k
    T_δ t^k-[s]A^1+1/4+ϵA^-1/4-ϵP^N(∫_0^1F”(λ X^N(s)+(1-λ)X^N([s]_δ t)(1-λ)dλ
           (X^N(s)-X^N([s]_δ t+δ t))^2)ds_L^p(Ω;)
       ≤
    C∫_0^t_k(t_k-[s]_δ t)^-5/8+ϵ/2∫_0^1F”(λ X^N(s)+(1-λ)X^N([s]_δ t)(1-λ)dλ_L^2p(Ω;E)
           X^N(s)-X^N([s]_δ t+δ t)^2_L^4p(Ω;)ds≤ C(T,X_0,p)δ t^γ/2.

Combining (<ref>) and the above regularity estimates, we complete the proof.




Next, we show the optimal regularity of  X^N_k and give
the estimate of X^N_k-X^N_k in ^-1. By a proof similar to that of <cit.>, we have the following result on the regularity estimate of X^N_k.


Let X_0∈^γ, γ∈ (0,3/2), T>0 and p≥1.
Then X^N_k  satisfies 

    sup_k≤ K[X^N_k_^γ^p]
       ≤ C(X_0,T,p,γ).







Under the condition of Lemma <ref>, there 
exist δ t_0≤ 1 and  C(X_0,T,γ)>0 such that 
for any δ t≤δ t_0, N∈^+, we have 

    X^N_k-X^N_k_L^2(Ω;^-1)   ≤ C(X_0,T,γ)δ t^γ/4.
 

Proof.
From the definitions of  X^N_k+1 and X^N_k+1, it follows that  for small ϵ>0,

    X^N_k+1-X^N_k+1_^-1^2
       ≤X^N_k-X^N_k_^-1^2-2A^1/2(X^N_k+1-X^N_k+1)^2 δ t 
         
    -2F(X^N_k+1)
    -F(X^N(t_k+1)),X^N_k+1-X^N_k+1)δ t
    
         +X^N_k+1-X^N_k+1, (G(X_k^N)-G(X^N(t_k)))δ W_k_^-1
       ≤X^N_k-X^N_k_^-1^2
    -(2-ϵ)A^1/2(X^N_k+1-X^N_k+1)^2 δ t
         +
    2C(ϵ)X^N_k+1-X^N_k+1_^-1^2δ t
    -2C(ϵ)F(X^N_k+1)-F(X^N(t_k+1))^2_^-1δ t
         +X^N_k+1-X^N_k, (G(X_k^N)-G(X^N(t_k)))δ W_k_^-1
         +X^N_k-X^N_k, (G(X_k^N)-G(X^N(t_k)))δ W_k_^-1
         +
    X_k^N-X^N_k+1, (G(X_k^N)-G(X^N(t_k)))δ W_k_^-1
       =: X^N_k-X^N_k_^-1^2
    -(2-ϵ)A^1/2(X^N_k+1-X^N_k+1)^2 δ t
         +
    2C(ϵ)X^N_k+1-X^N_k+1_^-1^2δ t
    +II_1^k
    +II_2^k+II_3^k+II_4^k.

Taking 2C(ϵ )δ t small enough, for l≤ K-1, we get

    X^N_l+1-X^N_l+1_^-1^2+
    ∑_k=0^lA^1/2(X^N_k+1-X^N_k+1)^2 δ t
    
       ≤ 
    2C(ϵ)∑_k=0^l-1X^N_k+1-X^N_k+1_^-1^2δ t
    +
    C∑_k=0^l(II_1^k
    +II_2^k+II_3^k+II_4^k).

Taking expectation on both sides of (<ref>), we have  

    [X^N_l+1-X^N_l+1_^-1^2]+
    [∑_k=0^lA^1/2(X^N_k+1-X^N_k+1)^2 δ t ]
       ≤ 
    C(ϵ)∑_k=0^l-1[X^N_k+1-X^N_k+1_^-1^2]δ t+
    C[∑_k=0^lII_1^k]
          +C[∑_k=0^lII_2^k]
    +
    C[∑_k=0^lII_3^k]
    +
    C[∑_k=0^lII_4^k].



By using the a priori estimates of X_k+1^N, X^N, Lemma <ref> and <cit.>, we have that for small ϵ>0,

    [∑_k=0^lII_1^k]
       ≤ C[∑_k=0^l F(X^N_k+1)-F(X^N(t_k+1))_^-1^2δ t]
       ≤ C(X_0,p)[∑_j=0^l(1+X^N(t_j+1)_E^4+X^N_j+1_E^4)X^N_j+1-X^N(t_j+1)^2δ t]
       ≤ C(X_0,T,p)(∑_j=0^l((j+1)δ t)^min(-1/2+γ-ϵ,0)δ t)δ t^γ/2.

The mild form of X^N_k yields that 

    [∑_k=0^l(II_2^k+II_4^k)]
       ≤
    C[∑_k=0^l (T_δ t-I)(X^N_k-X^N_k), (G(X_k^N)-G(X^N(t_k)))δ W_k_^-1]
    
       +C
    [∑_k=0^l T_δ tA(F(X^N_k+1)-F(X^N(t_k+1)))δ t, (G(X_k^N)-G(X^N(t_k)))δ W_k_^-1]
       +C
    [∑_k=0^l T_δ t(G(X^N_k)-G(X^N(t_k)))δ W_k, (G(X_k^N)-G(X^N(t_k)))δ W_k_^-1]
       =:III_1+III_2+III_3.

The Burkhölder inequality and Sobolev interpolation inequality lead to 

    III_1   ≤[(∑_k=0^l ∑_j∈^+(T_δ t-I)(X^N_k-X^N_k), (G(X_k^N)-G(X^N(t_k))e_j_^-1^2δ t)^1/2]
       ≤[sup_k≤ l(T_δ t-I)(X^N_k-X^N_k)_^-1(∑_k=0^l X_k^N-X_k^N^2δ t)^1/2]
       <∞.

Thus III_1=0.
For the second term, 

    III_2
       ≤ 
    C[∑_k=0^l T_δ tA(F(X^N_k)-F(X^N(t_k)))δ t, (G(X_k^N)-G(X^N(t_k)))δ W_k_^-1]
    
       +
    C[∑_k=0^l T_δ tA(F(X^N_k+1)-F(X^N_k))δ t, (G(X_k^N)-G(X^N(t_k)))δ W_k_^-1]
       +
    C[∑_k=0^l T_δ tA(F(X^N(t_k+1))-F(X^N(t_k)))δ t, (G(X_k^N)-G(X^N(t_k)))δ W_k_^-1]
       =:III_21+III_22+III_23.

Similar to III_1, we have III_21=0.
The estimations of  III_22 and III_23 are similar, we only give the estimate of III_22.
The continuity of X^N_k, Lemma <ref> and  the Sobolev interpolation equality lead to

    III_22
       ≤ C[|∑_k=0^l T_δ tA(F(X^N_k+1)-F(X^N_k))δ t, (G(X_k^N)-G(X^N(t_k)))δ W_k_^-1|]
    
       ≤ C∑_k=0^l [T_δ tA(F(X^N_k+1)-F(X^N_k))δ t_^-1^2]
         
    +C∑_k=0^l
    [(G(X_k^N)-G(X^N(t_k)))δ W_k_^-1^2]
       ≤ C(T,X_0)δ t^γ/2+C∑_k=0^l
    [X_k^N-X^N_k^2]δ t
    +C∑_k=0^l
    [X_k^N-X^N(t_k)^2]δ t
       ≤ C(T,X_0)δ t^γ/2+ϵ∑_k=0^l
    [X_k^N-X^N_k_^1^2]δ t
    
         +C(ϵ) ∑_k=0^l
    [X_k^N-X^N_k_^-1^2]δ t
    +C∑_k=0^l
    [X_k^N-X^N(t_k)^2]δ t.

The Burkhölder inequality  yields that 

    III_3   ≤
    C∑_k=0^l ∑_j∈^+[(G(X^N_k)-G(X^N(t_k)))e_j_^-1^2]δ t
       ≤ C(ϵ)∑_k=0^l[X^N_k-X^N(t_k)_^-1^2] δ t
         +ϵ∑_k=0^l[(X^N_k-X^N(t_k)_^1^2] δ t.

Notice that [∑_k=0^lII_3^k]=0.
Combining all the estimates of II_1^k-II_4^k, we complete the proof.



Let X_0∈^γ,  γ∈ (0,3/2), T>0. Then there exists δ t_0≤ 1 such that for N∈^+ and δ t∈ (0,δ t_0),

    X^N_k-X^N(t_k)_L^2(Ω; )   ≤ C(T,X_0,α)δ t^α/4,

where  α∈ (0,γ) and C(T,X_0,p,α)>0. 

Proof.
The triangle inequality yields that 

    X^N_k-X^N(t_k)_L^2(Ω; )   ≤X^N_k-X^N(t_k)_L^2(Ω; )
    +X^N_k-X^N_k_L^2(Ω; )
       ≤ C(T,X_0)δ t^γ/4
    +X^N_k-X^N_k_L^2(Ω; ).

The mild forms of X^N_l and X^N_l
lead to

    X^N_l-X^N_l_L^2(Ω; )   ≤ 
    C∑_k=0^l-1T_δ t^l-kA(F(X^N_k+1)-F(X^N(t_k+1)))δ t_L^2(Ω; )
         +C∑_k=0^l-1T_δ t^l-k(G(X^N_k)-G(X^N(t_k)))δ W_k_L^2(Ω; ).

The smoothing effect of T_δ t, together with the Sobolev interpolation inequality, Remark <ref>, <cit.> and Lemma <ref>,
implies that for β∈ (0,1),

    ∑_k=0^l-1T_δ t^l-kA(F(X^N_k+1)-F(X^N(t_k+1)))δ t_L^2(Ω; )
       ≤∑_k=0^l-1(t_l-t_k)^-3/4(1+X^N_k+1_ E^2+X^N_k+1^2_^β+X^N(t_k+1)_ E^2
         +X^N(t_k+1)^2_^β) X^N_k+1-X^N(t_k+1)^β_^-1X^N_k+1)-X^N(t_k+1)^1-β_L^2(Ω)δ t
       ≤∑_k=0^l-1(t_l-t_k)^-3/4(1+X^N_k+1_E^2+X^N_k+1^2_^β+X^N(t_k+1)_E^2
           +X^N(t_k+1)^2_^β)X^N_k+1)-X^N(t_k+1)^1-β_L^2/1-β(Ω)
           ×X^N_k+1-X^N(t_k+1)^β_^-1_L^2/β(Ω)δ t
       ≤ C(T,X_0)δ t^βγ/4.

The Burkhölder inequality yields that 

    ∑_k=0^l-1T_δ t^l-k(G(X^N_k)-G(X^N(t_k)))δ W_k^2_L^2(Ω; )
       ≤∑_k=0^l-1∑_j∈^+[T_δ t^l-k(G(X^N_k)-G(X^N(t_k)))e_j^2]δ t
       ≤∑_k=0^l-1 (t_k-t_l)^-1/4[X^N_k-X^N_k^2] δ t+C(T,X_0,γ)δ t^γ/2.

The discrete Gronwall inequality leads to the desired result.


As a consequence of (<ref>) and Proposition <ref>, we present the sharp mean square convergence rate result of the proposed scheme. We also remark that the convergence analysis of the numerical approximation can be applied to more general Q-Wiener process cases. 


Let X_0∈^γ,  γ∈ (0,3/2), T
>0. Then there exists δ t_0≤ 1 such that for N∈^+ and δ t∈ (0,δ t_0),

    X^N_k-X(t_k)_L^2(Ω; )   ≤ C(T,X_0,α)(δ t^α/4+λ_N^-α/2),

where α∈ (0,γ) and C(T,X_0,α)>0.


From the above arguments in the proof of  Proposition <ref>, it is not hard to see that under the same condition of Theorem <ref>, sup_k≤ KX^N_k-X(t_k)_L^2(Ω,)≤ C(T,X_0,α)(δ t^α/4+λ_N^-α/2), α∈ (0,γ) also holds.
As another consequence of the above theorem, we have the following strong convergence of the proposed full discretization in 
E-norm.

Let X_0∈^γ,  γ≥3/2, T
>0. Then there exists δ t_0≤ 1 such that for N∈^+ and δ t∈ (0,δ t_0),

    X^N_k-X(t_k)_L^2(Ω; E)   ≤ C(T,X_0,α)(δ t^α/4+λ_N^-α/2),

where α∈ (0,1) and C(T,X_0,α)>0.


The above result implies that once the density function of X^N_k(x) exists, the density function of  X^N_k(x) weakly converges or converges in Wasserstein metric
to  the density function of the exact solution X(t_k,x). If one can further obtain the uniform Malliavin differentiability of X^N_k(x), then the weak convergence of the density functions can be improved to the strong convergence.





§ CONCLUSION

In this article, we prove the 
the existence and positivity of the density function of the law of  the exact solution for the stochastic Cahn–Hilliard equation with unbounded diffusion coefficient satisfying a growth condition of order 0<α<1. Furthermore, we propose an implicit full discretization to approximate the density function of the exact solution. With the help of  the variation approach and the factorization method, the sharp mean square convergence rate of the full discretization is established. However, to fully understand how to compute the density function numerically, there are still many   open problems.
For instance,
 the existence and 
 positivity of the density function of full discretization, and the smoothness of the density functions of both the exact solution and full discretization are  still unknown for stochastic Cahn–Hilliard equation. 



References

plain




