We introduce a stochastic partial differential equation (SPDE) with elliptic operator in divergence form, with measurable and bounded coefficients and driven by space-time white noise. Such SPDEs could be used in mathematical modelling of diffusion phenomena in medium consisting of different kinds of materials and undergoing stochastic perturbations. We characterize the solution and, using the Stein–Malliavin calculus, we prove that the sequence of its recentered and renormalized spatial quadratic variations satisfies an almost sure central limit theorem. Particular focus is given to the interesting case where the coefficients of the operator are piecewise constant.
Many diffusion phenomena in various fields of real life are modelled by the following type of partial differential equations (PDEs)
∂u(t,x)∂t=Lu(t,x),
where L is the elliptic divergence form operator defined by
L=1r(x)ddx(R(x)ddx),R and r are two measurable and bounded functions defined on R and satisfying
μ1≤R(x)andμ2≤r(x)for allx∈R
where μ1 and μ2 are two strictly positive real constants, and ddx denotes the derivative in the distributional sense. More information on PDEs of the type (1) and their applications can be found, e.g., in [5, 15, 19] and references therein. One interesting example of such PDEs is the one defined by
Lp=12ρ(x)ddx(ρ(x)A(x)ddx),A(x)=a11{x≤0}+a21{0<x}andρ(x)=ρ11{x≤0}+ρ21{0<x},ai,ρi (i=1,2) are strictly positive constants. Operators of the kind (3) are the infinitesimal generators of diffusion processes that have been widely studied in literature (see [7, 12] and references therein). The discontinuity of the coefficients A and ρ reflects the heterogeneity of the media in which the modelled process under study propagates.
In the present paper, we introduce a stochastic partial differential equation (SPDE), that can be considered as a stochastic counterpart of PDE (1). More specifically, we consider
∂u(t,x)∂t=Lu(t,x)+W˙(t,x);t>0,x∈R,u(0,.):=0,0,x\in \mathbb{R},\\ {} u(0,.)& :=& 0,\hspace{0.2778em}\end{array}\right.\]]]>
where L is defined by (2) and W˙ denotes the formal derivative of a space-time white noise. That is, W is a centered Gaussian field W={W(t,C);t∈[0,T],C∈Bb(R)} with covariance
E(W(t,C)W(s,D))=(t∧s)λ(C∩D),
where λ denotes the Lebesgue measure and Bb(R) is the set of λ-bounded Borel sub-sets of R. So W behaves as a Wiener process both in time and in space. The solution to Equation (5) is a random field {u(t,x),t≥0,x∈R}, where t represents the time variable and x is the space variable. In the particular case where the functions r and R are constants r:=2 and R:=1, the operator L is reduced to 12∂2∂x2. So Equation (5) also represents a natural extension of the stochastic heat equation driven by the space-time white noise, which has been widely studied in the literature (see [22] and the references therein). This can be considered as an important motivation for the investigation of such equation’s solution.
This paper has a twofold objective: the first is to lay the first milestone towards the investigation of the stochastic process solution to (5). We prove its existence, and we investigate its spatial quadratic variation. In fact, the study of quadratic variation is motivated by its numerous applications in many fields. For example, in the estimation theory, the analysis of the asymptotic behaviour of the quadratic variations of self-similar processes play an important role in the construction of consistent estimators for the self-similarity parameter (see, e.g., [24] and references therein). In stochastic analysis, quadratic variations are as well one of the main tools used to characterize the semi-martingale property for some mixed Gaussian processes (see, e.g., [10, 14, 29]). Examples of applications of quadratic variation investigation include also the theory of the Itô stochastic calculus with respect to martingales [21] and mathematical finance [3]. We refer to the monograph [22] for a more complete exposition on variations of stochastic processes in general, and of solutions to certain SPDEs in particular. In this paper, under some conditions on the fundamental solution to PDE (1), we fix t and study the limit behaviour in distribution of the sequence (∑j=0N−1(u(t,j+1N)−u(t,jN))2)N≥1. More precisely, using some elements of the Stein–Malliavin calculus, we show that, after recentralization and renormalization, the above sequence satisfies an Almost Sure Central Limit Theorem (in short: ASCLT). Similar study has been done in the case of stochastic heat equation (see [20, 23]) and also in the case of stochastic wave equation (see [11]). But no similar study has been carried out on SPDEs (5). For more information on ASCLT, see [2] and references therein. The second objective of this paper is to make a further study of the SPDE defined by
∂u(t,x)∂t=Lpu(t,x)+W˙(t,x);t>0,x∈R,u(0,.):=0,0,x\in \mathbb{R},\\ {} u(0,.)& :=& 0,\hspace{0.2778em}\end{array}\right.\]]]>
where Lp is the operator defined by (3). We note that Equation (7) is a particular case of (5), and it could be a good model for diffusion phenomena in a medium consisting of two kind of materials, undergoing stochastic perturbations. Equation (7) has been introduced in [30] where the authors proved the existence of the solution and they presented explicit expressions of its covariance and variance functions. Some regularity properties of the solution sample paths have also been analyzed. In [31], Zili and Zougar expanded the quartic variations in time and the quadratic variations in space of the solution to Equation (7). Both expansions allowed them to deduce an estimation method of the parameters a1 and a2 appearing in (4). We make here another step in the study of SPDE (7) by showing that its solution satisfies all conditions under which we can use the ASCLT. In addition to the Stein–Malliavin calculus, our proofs require many integration techniques, calculation, and analysis tools.
The paper is organized as follows. In the next section, we prove the existence of the mild solution to Equation (5) and we give some characterizations of its spatial increments. In Section 3, using some elements of the Stein–Malliavin theory, we establish an almost sure central limit theorem which applies to the solution to SPDE (5) under some conditions on the fundamental solution associated to the operator L. The last section focuses on a further investigation of the solution to SPDE (7). In which case, we show that the ASCLT is statisfied.
Existence and some characteristics of the solution
The notion of solution to (5) is defined in the mild sense. We call a mild solution to (5) the stochastic process
u(t,x)=∫0t∫RG(t−s,x,y)W(ds,dy),t∈[0,T],x∈R,
where W is the Gaussian noise with covariance given by (53), G is the fundamental solution of the operator L and the integral in (8) is the Wiener integral with respect to the Gaussian noise W. The existence and many properties of the fundamental solution G of the operator L have been obtained in many papers (see, e.g., [13] and [19]).
It is well known (see, e.g., [25]) that the mild solution to (5) exists when the Wiener integral (8) is well-defined and this happens when the function (s,y)⟼G(t−s,x,y) belongs to H0=L2([0,T]×R), the canonical Hilbert space associated with the Gaussian process W. In fact, H0 is none other than the closure of the linear span generated by the indicator functions 1[0,t]×C, t∈[0,T], C∈Bb(R), with respect to the inner product
<1[0,t]×C,1[0,s]×D>H0=(t∧s)λ(C∩D)._{{\mathcal{H}_{0}}}}=(t\wedge s)\lambda (C\cap D).\]]]>
Moreover, the process (u(t,x),t∈[0,T],x∈R), when it exists, is a centered Gaussian process.
The following proposition deals with the existence of the mild solution to Equation (5).
The centered Gaussian process(u(t,x),t∈[0,T],x∈R)defined by (8), as a solution to Equation (5), exists and satisfiessupt∈[0,T],x∈RE(u(t,x)2)<+∞.
The existence and some bounds of the fundamental solution to PDE (1) have been established in [1] and [9]. In particular, it has been proved that there exist positive constants C1 and C2 such that
G(t,x,y)≤C12πtexp(−C2(x−y)2t),
for any t∈[0,T] and (x,y)∈R2. Thus,
∫0t∫RG2(t−s,x,y)dyds≤∫0t∫RC122π(t−s)exp(−2C2(x−y)2t−s)dyds≤C3∫0t1t−sds≤C4T,
where C3 and C4 denote two strictly positive constants. This with Remark 1 and Wiener’s isometry allow us to get the existence of the mild solution to (5) and to show that
E(u(t,x)2)=∫0t∫RG2(t−s,x,y)dyds≤C4T,
for every t∈[0,T] and x∈R. □
Now we consider an interval I in R and denote
ΔhG(u,x,z)=G(u,x+h,z)−G(u,x,z)
and
‖ΔhG(t−.,x,.)‖L2([0,t]×R)2=∫0t∫R(ΔhG(t−σ,x,y))2dσdy
for every u,t∈(0,T], h>00$]]> and x,z∈I. We also consider the conditions:
The following lemma will play an important role in this paper.
Let u be the mild solution to Equation (5).
If ConditionH1(I)is satisfied then, for everyt>00$]]>, there exists a positive constantC8such that∀x,y∈I,C8|y−x|≤E(u(t,y)−u(t,x))2.
If ConditionH2(I)is satisfied then, for everyt>00$]]>, there exists a positive constantC9such that∀x,y∈I,E(u(t,y)−u(t,x))2⩽C9|y−x|.
If ConditionH3(I)is satisfied then, for everyt>00$]]>,∀h>00$]]>,∀x,y∈I,E((u(t,x+h)−u(t,x))(u(t,y+h)−u(t,y)))⩽C7h2.
We first note that if x=y, then Inequalities (10) and (11) are trivial. We also note that the proofs in the cases x>yy$]]> and y>xx$]]> are similar. So we consider only the case y>xx$]]>. Using Wiener’s isometry we get
E(u(t,y)−u(t,x))2=E(∫(0,t)×RG(t−u,y,z)W(du,dz)−∫(0,t)×RG(t−u,x,z)W(du,dz))2=E(∫(0,t)×R(G(t−u,y,z)−G(t−u,x,z))W(du,dz))2=∫(0,t)×R(G(t−u,y,z)−G(t−u,x,z))2dudz=‖Δy−xG(t−.,x,.)‖L2([0,t]×R)2.
Equality (13) and Condition H1(I) [respectively H2(I)] allow us to get the two first assertions in Lemma 1.
As for the third one, using again Wiener’s isometry we get
E((u(t,x+h)−u(t,x))(u(t,y+h)−u(t,y)))=E(∫(0,t)×RΔhG(t−u,x,z)W(du,dz)×∫(0,t)×RΔhG(t−u,y,z)W(du,dz))=∫(0,t)×RΔhG(t−u,x,z)ΔhG(t−u,y,z)dudz.
Using Condition H3(I) the proof of the third assertion in Lemma 1 is achieved. □
From Assertion 2 in Lemma 1 and by Kolmogorov’s criterion of continuity, we easily get the following corollary.
Let u be the mild solution to (5). If ConditionH2(I)is satisfied, then, for everyt∈[0,T], the process(u(t,x))x∈Iis Hölder continuous of order γ with0<γ<12.
From Corollary 1, under Condition H2(I), the process u being a solution to (5) keeps the same Hölder regularity in space as the solution to the standard stochastic heat equation driven by a time-space white noise (see [20] and references therein).
Almost sure central limit theorem
Let us start this section with the following definition.
Let (GN)N≥1 be a sequence of real-valued random variables defined on a common probability space (Ω,F,P). We say that the sequence (GN)N≥1 satisfies an almost sure central limit theorem (ASCLT), if, almost surely, for every bounded and continuous function φ:R→R, we have:
1logN∑i=1Nφ(Gi)i⟶E(φ(Z))asN⟶∞,
where Z is an N(0,1) random variable.
For fixed t∈(0,T], we consider the Gaussian process (u(t,x))x∈[0,1] being the mild solution to Equation (5). We also consider the partition 0=x0<x1<⋯<xN=1 of the interval [0,1] defined by xi=iN for every i=0,1,…,N. We define the centered re-normalized quadratic variation statistic in the following way:
VN=∑i=0N−1[(u(t,xi+1)−u(t,xi))2E(u(t,xi+1)−u(t,xi))2−1]andV˜N=12NVN.
The aim of this section is to show that the sequence (V˜N)N≥1 satisfies the ASCLT. Let us first recall briefly some basic elements of the Stein–Malliavin theory (see [16]) that will be useful in our proof.
Elements of the Stein–Malliavin theory
Consider a real separable Hilbert space (H,<.,.>H)_{\mathcal{H}}})$]]> and an isonormal Gaussian process (B(φ),φ∈H) on a probability space (Ω,F,P), which is a centered Gaussian family of random variables such that
E(B(φ),B(ψ))=<φ,ψ>H,_{\mathcal{H}}},\]]]>
for every φ,ψ∈H. For q≥1, let H⊗q be the qth tensor product of H and denote H⊙q the associated qth symmetric tensor product.
Denote by Iq the qth multiple stochastic integral with respect to B. This Iq is actually an isometry between the Hilbert space H⊙q equipped with the scaled norm 1q!‖.‖H⊗q and the Wiener chaos of order q, which is defined as the closed linear span of the random variables Hq(B(φ)), where φ∈H,‖φ‖H=1 and Hq is the Hermite polynomial of degree q⩾1 defined by
Hq(x)=(−1)qq!exp(x22)dqdxq(exp(−x22)),x∈R.
The isometry of multiple integrals can be written as follows: for p,q⩾1,f∈H⊗p and g∈H⊗q,
E(Ip(f)Iq(g))=q!<fˆ,gˆ>H⊗qifp=q0otherwise._{{\mathcal{H}^{\otimes q}}}}\hspace{1em}& \text{if}\hspace{0.1667em}p=q\\ {} 0\hspace{1em}& \text{otherwise}.\end{array}\right.\]]]>
It holds that
Iq(f)=Iq(fˆ),
where fˆ denotes the canonical symmetrization of f defined by
fˆ(x1,…,xq)=1q!∑σ∈Sqf(xσ(1),…,xσ(q)),
where the sum runs over all permutations σ of {1,…,q}.
We recall that any square-integrable random variable F, which is measurable with respect to the σ-algebra generated by B, can be expanded into an orthogonal sum of multiple stochastic integrals:
F=E(F)+∑q=1∞Iq(fq),
where the series converges in the L2(Ω)-sense and the kernels fq, belonging to H⊙q, are uniquely determined by F.
Consider now the class of smooth random variables F that can be written in the form
F=g(B(φ1),…,B(φn)),
where n⩾1, g:Rn⟼R is a C∞-function with compact support and φ1,…,φn∈H. The Malliavin derivative of a smooth random variable F of the form (19) is the H-valued random variable given by
DF=∑i=1n∂g∂xi(B(φ1),…,B(φn))φi.
The following formula for multiplication of Wiener chaos integrals of any orders p,q will play a basic role in the next section. For any symmetric integrands f∈H⊙p and g∈H⊙p, we have
Ip(f)Iq(g)=∑r=0p∧qr!(rp)(rq)Ip+q−2r(f⊗rg),
where, in the particular case when H=L2([0,T]), for r=1,…,p∧q, the rth contraction f⊗rg is the element of H⊗(p+q−2r) defined by
(f⊗rg)(s1,…,sp−r,t1,…,tq−r)=∫[0,T]rdu1...durf(s1,…,sp−r,u1,…,ur)g(t1,…,tq−r,u1,…,ur).
The following theorem gives a description of the normal approximation of multiple stochastic integrals. We refer to [16–18] and references therein for the proof.
Fixq⩾1. Assume that(GN)N⩾1:=(Iq(gN))N⩾1withgN∈H⊙qis a sequence of random variables belonging to the qth Wiener chaos such thatlimN→∞E(GN2)=σ2.Hence,GNconverges in law toZ∼N(0,1)if and only iflimN→∞‖DGN‖H2=qσ2.Furthermore, if we denote by d one of the metrics on the space of probability measures onR, including the Kolmogorov, Wasserstein and Total Variation measures, then for N large enough:d(GN,N(0,1))⩽C(Var(‖DGN‖H2)+E(‖DGN‖H2)−qσ2).
The following theorem has been introduced in [4]. It gives a sufficient condition for extending Theorem 1 to an ASCLT for multiple stochastic integrals.
Fixq⩾2, and let(GN)N⩾1be a sequence of random variables defined byGN:=(Iq(gN))N⩾1;gN∈H⊙q.Suppose that:
For everyN⩾1,E(GN2)=1.
For everyr=1,…,q−1,limN→∞‖gN⊗rgN‖H⊗2(q−r)2=0.
For everyr=1,…,q−1,∑N⩾21Nlog2N∑l=1N1l‖gl⊗rgl‖H⊗2(q−r)2<∞.
∑N⩾21Nlog3N∑i,j=1N|E(GiGj)|ij<∞.
Then, the sequence(GN)N≥1satisfies an ASCLT.
We finish this section with the following useful reduction lemma. For its proof see Lemma 2.2 in [2].
Consider a real-valued sequence(an)n≥1converging toa∞≠0. Consider also a sequence of real valued random variables(Gn)n≥1. Then the sequence(Gn)n≥1satisfies an ASCLT if, and only if,(anGn)n≥1does.
Limiting behavior of the re-normalized quadratic variation of the spatial solution process
For fixed t∈(0,T], we denote by H the canonical Hilbert space associated to the Gaussian process (u(t,x))x∈[0,1] being a mild solution to Equation (5). This Hilbert space is defined as the closure of the linear span generated by the indicator functions 1[0,x],x>00$]]>, with respect to the inner product
E(u(t,x)u(t,y))=<1[0,x],1[0,y]>H._{\mathcal{H}}}.\]]]>
We also denote by Iq, q≥1, the multiple stochastic integral with respect to the Gaussian process (u(t,x))x∈[0,1]. So for every x<y we have
u(t,y)−u(t,x)=I1(1[x,y]).
We start our study of the limit behavior in distribution of the sequence (V˜N)N≥1 by the following main theorem.
Let u be the mild solution to Equation (5), G be the fundamental solution associated to the operatorLandV˜Nbe given by (14). If G satisfies ConditionsH1([0,1])andH3([0,1]), thenlimN→∞E(V˜N2)=1.
By using Formula (21), we can write
VN=∑j=0N−1[(u(t,xj+1)−u(t,xj))2E(u(t,xj+1)−u(t,xj))2−1]=∑j=0N−1[I12(1[xj,xj+1])E(u(t,xj+1)−u(t,xj))2−1]=∑j=0N−1I2(1[xj,xj+1]⊗2)E(u(t,xj+1)−u(t,xj))2.
By virtue of the isometry formula (16), we get
E(VN2)=E(∑j=0N−1I2(1[xj,xj+1]⊗2)E(u(t,xj+1)−u(t,xj))2)2=∑j,k=0N−1E(I2(1[xj,xj+1]⊗2)I2(1[xk,xk+1]⊗2))E(u(t,xj+1)−u(t,xj))2E(u(t,xk+1)−u(t,xk))2=2∑j,k=0N−1<1[xj,xj+1],1[xk,xk+1]>H2E(u(t,xj+1)−u(t,xj))2E(u(t,xk+1)−u(t,xk))2._{\mathcal{H}}^{2}}}{\mathbb{E}{(u(t,{x_{j+1}})-u(t,{x_{j}}))^{2}}\hspace{0.2778em}\mathbb{E}{(u(t,{x_{k+1}})-u(t,{x_{k}}))^{2}}}.\end{array}\]]]>
Thus,
E(VN2)=T1,N+T2,N,
where
T1,N=2∑j=0N−1<1[xj,xj+1],1[xj,xj+1]>H2[E(u(t,xj+1)−u(t,xj))2]2_{\mathcal{H}}^{2}}}{{[\mathbb{E}{(u(t,{x_{j+1}})-u(t,{x_{j}}))^{2}}]^{2}}}\]]]>
and
T2,N=2∑j,k=0;j≠kN−1<1[xj,xj+1],1[xk,xk+1]>H2E(u(t,xj+1)−u(t,xj))2E(u(t,xk+1)−u(t,xk))2._{\mathcal{H}}^{2}}}{\mathbb{E}{(u(t,{x_{j+1}})-u(t,{x_{j}}))^{2}}\hspace{0.2778em}\mathbb{E}{(u(t,{x_{k+1}})-u(t,{x_{k}}))^{2}}}.\]]]>
On the one hand we clearly have T1,N=2N. On the other hand, since Conditions H1([0,1]) and H3([0,1]) are satisfied, by virtue of Lemma 1 we get
T2,N⩽2N2∑j,k=0;j≠kN−1<1[xj,xj+1],1[xk,xk+1]>H2⩽CN2∑j,k=0;j≠kN−1(1N2)2≤C,_{\mathcal{H}}^{2}}\leqslant C\hspace{0.1667em}{N^{2}}\hspace{0.1667em}{\sum \limits_{j,k=0;j\ne k}^{N-1}}{\bigg(\frac{1}{{N^{2}}}\bigg)^{2}}\le C,\]]]>
where C denotes a universal positive constant. Thus, we deduce that the dominant term for E(V˜N2) is obviously T1,N. Consequently, we obtain, for a fixed t∈(0,T],
E(V˜N2)=12NE(VN2)⟶1asN⟶∞.
□
In the following theorem we establish the convergence in law of the sequence (V˜N)N.
Consider the sequence of random variablesV˜Ndefined in (14). If G satisfies ConditionsH1([0,1])andH3([0,1]), thenV˜N⟶LawN(0,1).Moreover, if we denote by d one of the metrics on the space of probability measures onR, including the Kolmogorov, Wasserstein and Total Variation measures, then for N large enoughd(V˜N,N(0,1))⩽CN.
By virtue of Formula (20) we get
DV˜N=12N∑j=0N−1I1(1[xj,xj+1])1[xj,xj+1]E(u(t,xj+1)−u(t,xj))2.
Hence, for every fixed t∈[0,T], using Formula (21), we get
‖DV˜N‖H2=2N∑j,k=0N−1I2(1[xj,xj+1]⊗1[xk,xk+1])<1[xj,xj+1],1[xk,xk+1]>HE(u(t,xj+1)−u(t,xj))2E(u(t,xk+1)−u(t,xk))2+E(‖DV˜N‖H2),_{\mathcal{H}}}}{\mathbb{E}{(u(t,{x_{j+1}})-u(t,{x_{j}}))^{2}}\hspace{0.2778em}\mathbb{E}{(u(t,{x_{k+1}})-u(t,{x_{k}}))^{2}}}\\ {} & & \displaystyle +\mathbb{E}\big(\| D{\tilde{V}_{N}}{\| _{\mathcal{H}}^{2}}\big),\end{array}\]]]>
and consequently,
Var(‖DV˜N‖H2)=E[‖DV˜N‖H2−E(‖DV˜N‖H2)]2=E[2N∑j,k=0N−1I2(1[xj,xj+1]⊗1[xk,xk+1])<1[xj,xj+1],1[xk,xk+1]>HE(u(t,xj+1)−u(t,xj))2E(u(t,xk+1)−u(t,xk))2]2=8N2∑j,k,m,l=0N−1E(I2(1[xj,xj+1]⊗1[xk,xk+1])I2(1[xm,xm+1]⊗1[xl,xl+1]))E(u(t,xj+1)−u(t,xj))2E(u(t,xk+1)−u(t,xk))2×<1[xj,xj+1],1[xk,xk+1]>H<1[xm,xm+1],1[xl,xl+1]>HE(u(t,xm+1)−u(t,xm))2E(u(t,xl+1)−u(t,xl))2=8N2∑j,k,m,l=0N−1<1[xj,xj+1]⊗˜1[xk,xk+1],1[xm,xm+1]⊗˜1[xl,xl+1]>H⊗2‖1[xj,xj+1]‖H2‖1[xk,xk+1]‖H2×<1[xj,xj+1],1[xk,xk+1]>H<1[xm,xm+1],1[xl,xl+1]>H‖1[xm,xm+1]‖H2‖1[xl,xl+1]‖H2,_{\mathcal{H}}}}{\mathbb{E}{(u(t,{x_{j+1}})-u(t,{x_{j}}))^{2}}\hspace{0.2778em}\mathbb{E}{(u(t,{x_{k+1}})-u(t,{x_{k}}))^{2}}}\Bigg]^{2}}\\ {} & \displaystyle =& \displaystyle \frac{8}{{N^{2}}}{\sum \limits_{j,k,m,l=0}^{N-1}}\frac{\mathbb{E}({\mathrm{I}_{2}}({\mathbf{1}_{[{x_{j}},{x_{j+1}}]}}\otimes {\mathbf{1}_{[{x_{k}},{x_{k+1}}]}})\hspace{0.1667em}{\mathrm{I}_{2}}({\mathbf{1}_{[{x_{m}},{x_{m+1}}]}}\otimes {\mathbf{1}_{[{x_{l}},{x_{l+1}}]}}))}{\mathbb{E}{(u(t,{x_{j+1}})-u(t,{x_{j}}))^{2}}\hspace{0.2778em}\mathbb{E}{(u(t,{x_{k+1}})-u(t,{x_{k}}))^{2}}}\\ {} & & \displaystyle \times \frac{<{\mathbf{1}_{[{x_{j}},{x_{j+1}}]}},{\mathbf{1}_{[{x_{k}},{x_{k+1}}]}}{>_{\mathcal{H}}}\hspace{0.1667em}<{\mathbf{1}_{[{x_{m}},{x_{m+1}}]}},{\mathbf{1}_{[{x_{l}},{x_{l+1}}]}}{>_{\mathcal{H}}}}{\mathbb{E}{(u(t,{x_{m+1}})-u(t,{x_{m}}))^{2}}\hspace{0.2778em}\mathbb{E}{(u(t,{x_{l+1}})-u(t,{x_{l}}))^{2}}}\\ {} & \displaystyle =& \displaystyle \frac{8}{{N^{2}}}{\sum \limits_{j,k,m,l=0}^{N-1}}\frac{<{\mathbf{1}_{[{x_{j}},{x_{j+1}}]}}\tilde{\otimes }{\mathbf{1}_{[{x_{k}},{x_{k+1}}]}},{\mathbf{1}_{[{x_{m}},{x_{m+1}}]}}\tilde{\otimes }{\mathbf{1}_{[{x_{l}},{x_{l+1}}]}}{>_{{\mathcal{H}^{{\otimes ^{2}}}}}}}{\| {\mathbf{1}_{[{x_{j}},{x_{j+1}}]}}{\| _{\mathcal{H}}^{2}}\hspace{0.2778em}\| {\mathbf{1}_{[{x_{k}},{x_{k+1}}]}}{\| _{\mathcal{H}}^{2}}}\\ {} & & \displaystyle \times \frac{<{\mathbf{1}_{[{x_{j}},{x_{j+1}}]}},{\mathbf{1}_{[{x_{k}},{x_{k+1}}]}}{>_{\mathcal{H}}}\hspace{0.1667em}<{\mathbf{1}_{[{x_{m}},{x_{m+1}}]}},{\mathbf{1}_{[{x_{l}},{x_{l+1}}]}}{>_{\mathcal{H}}}}{\| {\mathbf{1}_{[{x_{m}},{x_{m+1}}]}}{\| _{\mathcal{H}}^{2}}\hspace{0.2778em}\| {\mathbf{1}_{[{x_{l}},{x_{l+1}}]}}{\| _{\mathcal{H}}^{2}}},\end{array}\]]]>
where f⊗˜g denotes the symmetrization of the tensor product f⊗g that satisfies
f⊗˜g=12(f⊗g+g⊗f)
and
<f⊗˜g,f′⊗˜g′>H=12(<f,f′>H<g,g′>H+<f,g′>H<g,f′>H)._{\mathcal{H}}}=\frac{1}{2}\big(_{\mathcal{H}}}_{\mathcal{H}}}+_{\mathcal{H}}}_{\mathcal{H}}}\big).\]]]>
Therefore,
Var(‖DV˜N‖H2)=8N2∑j,k,m,l=0N−1<1[xj,xj+1],1[xk,xk+1]>H<1[xm,xm+1],1[xl,xl+1]>H‖1[xj,xj+1]‖H2‖1[xk,xk+1]‖H2×<1[xj,xj+1],1[xk,xk+1]>H<1[xm,xm+1],1[xl,xl+1]>H‖1[xm,xm+1]‖H2‖1[xl,xl+1]‖H2=D4,N+D3,N+D2,N+D1,N_{\mathcal{H}}}\hspace{0.1667em}<{\mathbf{1}_{[{x_{m}},{x_{m+1}}]}},{\mathbf{1}_{[{x_{l}},{x_{l+1}}]}}{>_{\mathcal{H}}}}{\| {\mathbf{1}_{[{x_{j}},{x_{j+1}}]}}{\| _{\mathcal{H}}^{2}}\hspace{0.2778em}\| {\mathbf{1}_{[{x_{k}},{x_{k+1}}]}}{\| _{\mathcal{H}}^{2}}}\\ {} & & \displaystyle \times \frac{<{\mathbf{1}_{[{x_{j}},{x_{j+1}}]}},{\mathbf{1}_{[{x_{k}},{x_{k+1}}]}}{>_{\mathcal{H}}}\hspace{0.1667em}<{\mathbf{1}_{[{x_{m}},{x_{m+1}}]}},{\mathbf{1}_{[{x_{l}},{x_{l+1}}]}}{>_{\mathcal{H}}}}{\| {\mathbf{1}_{[{x_{m}},{x_{m+1}}]}}{\| _{\mathcal{H}}^{2}}\hspace{0.2778em}\| {\mathbf{1}_{[{x_{l}},{x_{l+1}}]}}{\| _{\mathcal{H}}^{2}}}\\ {} & \displaystyle =& \displaystyle {D_{4,N}}+{D_{3,N}}+{D_{2,N}}+{D_{1,N}}\end{array}\]]]>
where Di,N, for every i∈{1,2,3,4}, contains all the terms with i equal indices. So, D4,N contains all the summands above with j=k=m=l; that is
D4,N=8N2∑j=0N−11=8N.
As for D3,N, it contains all the terms corresponding to j=k=l≠m; so, since G satisfies Conditions H1([0,1]) and H3([0,1]), using Lemma 1 we get
D3,N≤8N2∑l,m=0N−1‖1[xl,xl+1]‖H4<1[xl,xl+1],1[xm,xm+1]>H2‖1[xl,xl+1]‖H6‖1[xm,xm+1]‖H2≤CN2∑l,m=0N−1(1N2)2(1N)2=CN2._{\mathcal{H}}^{2}}}{\| {\mathbf{1}_{[{x_{l}},{x_{l+1}}]}}{\| _{\mathcal{H}}^{6}}\hspace{0.2778em}\| {\mathbf{1}_{[{x_{m}},{x_{m+1}}]}}{\| _{\mathcal{H}}^{2}}}\\ {} & \displaystyle \le & \displaystyle \frac{C}{{N^{2}}}{\sum \limits_{l,m=0}^{N-1}}\frac{{(\frac{1}{{N^{2}}})^{2}}}{{(\frac{1}{N})^{2}}}=\frac{C}{{N^{2}}}.\end{array}\]]]>
By the same way, and using again Lemma 1, we show that
D2,N⩽CN2andD1,N⩽CN2.
All this allow us to get
Var(‖DV˜N‖H2)⩽CN.
Moreover, we have
E(‖DV˜N‖H2)=2E(V˜N)2=E(VN2)N=1N(T1,N+T2,N)=2+T2,NN
where T1,N and T2,N are defined by (24) and (25). This and Inequality (26) allow us to deduce that
E(‖DV˜N‖H2)−2⩽CN.
By virtue of Theorem 1, the proof of Theorem 4 is completed. □
Almost sure central limit theorem
The following theorem is a kind of extension of Theorem 4.
If G satisfies ConditionsH1([0,1])andH3([0,1]), then the sequence(V˜N)N≥1satisfies an ASCLT.
Denoting σN=E(V˜N2), for every N≥1, according to Theorem 3, we have limN→∞σN=1. So without loss of generality, we assume that infN≥1σN=σ0>00$]]> and we consider GN=V˜NσN, for every N≥1.
According to Lemma 2, to obtain Theorem 5 it suffices to show that the sequence (GN)N≥1 satisfies an ASCLT. To this end, since for every N≥1 we have GN=I2(gN) with
gN:=1σN2N∑j=0N−11[xj,xj+1]⊗2E(u(t,xj+1)−u(t,xj))2,
and since we obviously have E(GN2)=1, for every N≥1, it suffices to check the three last assumptions in Theorem 2.
By the 1st contraction defined by (22), we obtain
gl⊗1gl=12σl2l∑j,k=0l−11[xj,xj+1]⊗2⊗11[xk,xk+1]⊗2E(u(t,xj+1)−u(t,xj))2E(u(t,xk+1)−u(t,xk))2=12σl2l∑j,k=0l−1<1[xj,xj+1],1[xk,xk+1]>H1[xj,xj+1]⊗1[xk,xk+1]E(u(t,xj+1)−u(t,xj))2E(u(t,xk+1)−u(t,xk))2._{\mathcal{H}}}{\mathbf{1}_{[{x_{j}},{x_{j+1}}]}}\otimes {\mathbf{1}_{[{x_{k}},{x_{k+1}}]}}}{\mathbb{E}{(u(t,{x_{j+1}})-u(t,{x_{j}}))^{2}}\hspace{0.1667em}\mathbb{E}{(u(t,{x_{k+1}})-u(t,{x_{k}}))^{2}}}.\end{array}\]]]>
Therefore,
‖gl⊗1gl‖H⊗22=14σl4l2∑j,k,m,p=0l−1<1[xj,xj+1],1[xk,xk+1]>H<1[xm,xm+1],1[xp,xp+1]>HE(u(t,xj+1)−u(t,xj))2E(u(t,xk+1)−u(t,xk))2×<1[xj,xj+1]⊗˜1[xk,xk+1],1[xm,xm+1]⊗˜1[xp,xp+1]>HE(u(t,xm+1)−u(t,xm))2E(u(t,xp+1)−u(t,xp))2=14σl2l2∑j,k,m,p=0l−1<1[xj,xj+1],1[xk,xk+1]>H<1[xm,xm+1],1[xp,xp+1]>HE(u(t,xj+1)−u(t,xj))2E(u(t,xk+1)−u(t,xk))2×<1[xj,xj+1],1[xm,xm+1]>H<1[xk,xk+1],1[xp,xp+1]>HE(u(t,xm+1)−u(t,xm))2E(u(t,xp+1)−u(t,xp))2._{\mathcal{H}}}\hspace{0.1667em}<{\mathbf{1}_{[{x_{m}},{x_{m+1}}]}},{\mathbf{1}_{[{x_{p}},{x_{p+1}}]}}{>_{\mathcal{H}}}}{\mathbb{E}{(u(t,{x_{j+1}})-u(t,{x_{j}}))^{2}}\hspace{0.1667em}\mathbb{E}{(u(t,{x_{k+1}})-u(t,{x_{k}}))^{2}}}\\ {} & & \displaystyle \times \frac{<{\mathbf{1}_{[{x_{j}},{x_{j+1}}]}}\tilde{\otimes }{\mathbf{1}_{[{x_{k}},{x_{k+1}}]}},{\mathbf{1}_{[{x_{m}},{x_{m+1}}]}}\tilde{\otimes }{\mathbf{1}_{[{x_{p}},{x_{p+1}}]}}{>_{\mathcal{H}}}}{\mathbb{E}{(u(t,{x_{m+1}})-u(t,{x_{m}}))^{2}}\hspace{0.1667em}\mathbb{E}{(u(t,{x_{p+1}})-u(t,{x_{p}}))^{2}}}\\ {} & \displaystyle =& \displaystyle \frac{1}{4{\sigma _{l}^{2}}{l^{2}}}{\sum \limits_{j,k,m,p=0}^{l-1}}\frac{<{\mathbf{1}_{[{x_{j}},{x_{j+1}}]}},{\mathbf{1}_{[{x_{k}},{x_{k+1}}]}}{>_{\mathcal{H}}}\hspace{0.1667em}<{\mathbf{1}_{[{x_{m}},{x_{m+1}}]}},{\mathbf{1}_{[{x_{p}},{x_{p+1}}]}}{>_{\mathcal{H}}}}{\mathbb{E}{(u(t,{x_{j+1}})-u(t,{x_{j}}))^{2}}\hspace{0.1667em}\mathbb{E}{(u(t,{x_{k+1}})-u(t,{x_{k}}))^{2}}}\\ {} & & \displaystyle \times \frac{<{\mathbf{1}_{[{x_{j}},{x_{j+1}}]}},{\mathbf{1}_{[{x_{m}},{x_{m+1}}]}}{>_{\mathcal{H}}}\hspace{0.1667em}<{\mathbf{1}_{[{x_{k}},{x_{k+1}}]}},{\mathbf{1}_{[{x_{p}},{x_{p+1}}]}}{>_{\mathcal{H}}}}{\mathbb{E}{(u(t,{x_{m+1}})-u(t,{x_{m}}))^{2}}\hspace{0.1667em}\mathbb{E}{(u(t,{x_{p+1}})-u(t,{x_{p}}))^{2}}}.\end{array}\]]]>
Since 1σl4≤1σ04 for every l≥1, and since G satisfies Conditions H1([0,1]) and H3([0,1]), proceeding in the same way as in the proof of Theorem 4, we get
‖gl⊗1gl‖H⊗22⩽Cl,
and consequently the second assumption in Theorem 2 is satisfied.
From Inequality (27) we also deduce that
∑N⩾21Nlog2N∑l=1N1l‖gl⊗1gl‖H⊗22⩽C∑N⩾21Nlog2N∑l=1∞1l2⩽C∑N⩾21Nlog2N<∞,
that means that the third assumption in Theorem 2 is also satisfied.
Let us now check the last assumption in Theorem 2. Since we have
E(GiGj)=2<gi,gj>H⊗2_{{\mathcal{H}^{{\otimes ^{2}}}}}}\]]]>
and since Conditions H1([0,1]) and H3([0,1]) are satisfied, using Lemma 1 we get:
Ifi=j,<gi,gi>H⊗2⩽Ciandifi>j,<gi,gj>H⊗2⩽Cji._{{\mathcal{H}^{{\otimes ^{2}}}}}}\leqslant \frac{C}{i}\hspace{1em}\text{and}\hspace{1em}\text{if}\hspace{2.5pt}i>j,\hspace{1em}<{g_{i}},{g_{j}}{>_{{\mathcal{H}^{{\otimes ^{2}}}}}}\leqslant C\hspace{0.1667em}\sqrt{\frac{j}{i}}.\]]]>
Therefore,
∑N⩾21Nlog3N∑i,j=1N|E(GiGj)|ij=∑N⩾21Nlog3N[∑i≠j=1N|E(GiGj)|ij+∑i=1N|E(Gi2)|i2]⩽2∑N⩾21Nlog3N[2∑i>j=1N|<gi,gj>H⊗2|ij+∑i=1N|<gi,gi>H⊗2|i2]⩽C∑N⩾22Nlog3N[∑i>j=1N2iij+∑i=1N1i3]<∞.j=1}^{N}}\frac{|<{g_{i}},{g_{j}}{>_{{\mathcal{H}^{{\otimes ^{2}}}}}}|}{ij}+{\sum \limits_{i=1}^{N}}\frac{|<{g_{i}},{g_{i}}{>_{{\mathcal{H}^{{\otimes ^{2}}}}}}|}{{i^{2}}}\Bigg]\\ {} & \displaystyle \leqslant & \displaystyle C\hspace{0.1667em}\sum \limits_{N\geqslant 2}\frac{2}{Nlo{g^{3}}N}\hspace{0.1667em}\Bigg[{\sum \limits_{i>j=1}^{N}}\frac{2}{i\sqrt{ij}}+{\sum \limits_{i=1}^{N}}\frac{1}{{i^{3}}}\Bigg]\\ {} & \displaystyle <& \displaystyle \infty .\hspace{280.0pt}\end{array}\]]]>
□
Stochastic heat equation with piecewise constant coefficients
The study done in the previous section allows us to make a new step in the investigation of the solution to the SPDE (7). Equation (7) is obviously a particular case of (5). Indeed, the operator Lp defined by (3) can be written in the form (2) with
r(x)=2ρ(x)andR(x):=ρ(x)A(x).
In the following proposition we present the expression of the fundamental solution associated to the operator Lp. For a proof see, e.g., [8, 26, 27] and [28].
There exists a unique fundamental solutionG(t−s,x,y)associated to the operatorLp. It can be explicitly expressed asG(u,x,z)=m(u)[1a1A−(u,x,z)1{z⩽0}+1a2A+(u,x,z)1{z>0}]0\}}}\bigg]\]]]>withm(u)=12πu1{u>0},0\}}},\]]]>A−(u,x,z)=E−(u,x,z)−βE+(u,x,z),A+(u,x,z)=E−(u,x,z)+βE+(u,x,z),E−(u,x,z)=exp−(f(z)−f(x))22u,E+(u,x,z)=exp−(|f(z)|+|f(x)|)22u,f(z)=za11{z⩽0}+za21{z>0}andβ=ρ2a2−ρ1a1ρ2a2+ρ1a1.0\}}}\hspace{1em}\mathit{and}\hspace{1em}\beta =\frac{{\rho _{2}}\sqrt{{a_{2}}}-{\rho _{1}}\sqrt{{a_{1}}}}{{\rho _{2}}\sqrt{{a_{2}}}+{\rho _{1}}\sqrt{{a_{1}}}}.\]]]>
In this section, by making an in-depth study of the terms f,A− and A+ defined in Expressions (30) and (32), we will prove the following theorem.
Let u be the mild solution to Equation (7) andV˜Nbe the sequence given by (14). Suppose that the coefficients A and ρ defined in (4) satisfymax(1,a1a2)≤ρ2ρ1.Then the following is valid:
V˜N⟶LawN(0,1).Moreover, if we denote by d one of the metrics on the space of probability measures onR, including the Kolmogorov, Wasserstein and Total Variation measures, then for N large enoughd(V˜N,N(0,1))⩽CN.
The sequence(V˜N)N≥1satisfies an ASCLT.
If a1=a2=1 and ρ1=ρ2=2, then Condition (33) is well satisfied. Thus, the result of Theorem 6 applies to the standard stochastic heat equation with the time-space white noise and it corresponds exactly to that obtained in [23].
To prove Theorem 6, we shall first establish the following lemmas.
Preliminary lemmas
Consider f, the function defined in (32). For everyx,y∈R,min(1a2,1a1)|y−x|≤|f(y)−f(x)|≤max(1a2,1a1)|y−x|.
Expression (32) allows to get
f(y)−f(x)=y−xa2ify>0x>0,y−xa1ify≤0x≤0,ya1−xa2ify≤0x>0,ya2−xa1ify>0x≤0.0\hspace{0.2778em}x>0,\\ {} \displaystyle \frac{y-x}{\sqrt{{a_{1}}}}\hspace{1em}& \text{if}\hspace{2.5pt}& y\le 0\hspace{0.2778em}x\le 0,\\ {} \displaystyle \frac{y}{\sqrt{{a_{1}}}}-\displaystyle \frac{x}{\sqrt{{a_{2}}}}\hspace{1em}& \text{if}\hspace{2.5pt}& y\le 0\hspace{0.2778em}x>0,\\ {} \displaystyle \frac{y}{\sqrt{{a_{2}}}}-\displaystyle \frac{x}{\sqrt{{a_{1}}}}\hspace{1em}& \text{if}\hspace{2.5pt}& y>0\hspace{0.2778em}x\le 0.\end{array}\right.\]]]>
If xy≥0, both inequalities in Lemma 3 are directly obtained from (34). If y>00$]]> and x<0,
max(1a2,1a1)|y−x|−|f(y)−f(x)|=max(1a2,1a1)(y−x)−ya2+xa1=y[max(1a2,1a1)−1a2]−x[max(1a2,1a1)−1a1]>0& \displaystyle 0\end{array}\]]]>
and
min(1a2,1a1)|y−x|−|f(y)−f(x)|=min(1a2,1a1)(y−x)−ya2+xa1=y[min(1a2,1a1)−1a2]−x[min(1a2,1a1)−1a1]<0.
The proof of both inequalities in the case where y<0 and x>00$]]> is similar. □
There exists a universal positive constant C, such that∫0t12πu∫R|E+(u,y,z)−E+(u,x,z)|2dzdu⩽C|y−x|and∫0t12πu∫R|E−(u,y,z)−E−(u,x,z)|2dzdu⩽C|y−x|for everyt>00$]]>andx,y∈R.
We present only the proof of the first inequality; the proof of the second one is similar. By using Expression (31), we get
∫0t12πu∫R|E+(u,y,z)−E+(u,x,z)|2dzdu=∫0t∫R[12πuexp(−(|f(z)|+|f(y)|)22u)−12πuexp(−(|f(z)|+|f(x)|)22u)]2dzdu=∫0t∫0∞[12πuexp((z/a2+|f(y)|)22u)−12πuexp(−(z/a2+|f(x)|)22u)]2dzdu+∫0t∫−∞0[12πuexp(−(−z/a1+|f(y)|)22u)−12πuexp(−(−z/a1+|f(x)|)22u)]2dzdu.
The changes of variables Z=z/a2+|f(x)| in the first integral and Z=−z/a1+|f(x)|, in the second one give
∫0t12πu∫R|E+(u,y,z)−E+(u,x,z)|2dzdu=a2∫0t∫|f(x)|+∞[12πuexp(−(Z+(|f(y)|−|f(x)|))22u)−12πuexp(−Z22u)]2dZdu+a1∫0t∫|f(x)|+∞[12πuexp(−(Z+(|f(y)|−|f(x)|))22u)−12πuexp(−Z22u)]2dZdu.
Therefore,
∫0t12πu∫R|E+(u,y,z)−E+(u,x,z)|2dzdu⩽2max(a1,a2)∫0t∫R[12πuexp(−(Z+H˜)22u)−12πuexp(−Z22u)]2dZdu=2max(a1,a2)∫0t∫R[pu(Z+H˜)−pu(Z)]2dZdu,
where H˜=|f(y)|−|f(x)| and pu denotes the heat kernel defined by
pt(x)=12πtexp(−x22t),for everyt>0andx∈R.0\hspace{2.5pt}\text{and}\hspace{2.5pt}x\in \mathbb{R}.\]]]>
It is known that the Fourier transform of pu is
F(pt)(ξ)=e−tξ2/2∀ξ∈R,t>0.0.\]]]>
By virtue of the Plancherel theorem we can write
∫0tds∫R[ps(v+h)−ps(v)]2dv=12π∫0tds∫−∞∞|e−sξ2/2+iξh−e−sξ2/2|2dξ=1π∫0tds∫−∞∞e−sξ2(1−cos(hξ))dξ
for every h∈R. Applying Fubini’s Theorem and using the fact that the functions cosine and ξ⟼1−cos(hξ)ξ2 are even we get:
∫0tds∫R[ps(v+h)−ps(v)]2dv=1π∫−∞∞[∫0te−sξ2ds](1−cos(hξ))dξ=1π∫−∞∞(1−e−tξ2)1−cos(hξ)ξ2dξ=2π∫0∞(1−e−tξ2)1−cos(|h|ξ)ξ2dξ.
Suppose that h≠0. By a simple change of variables in (38), using the fact that
∀θ≥01−exp(−θ)⩽1,
we obtain
∫0tds∫R[ps(v+h)−ps(v)]2dv=2|h|π∫0∞(1−e−tξ2h2)1−cos(ξ)ξ2dξ⩽2|h|π∫0∞1−cos(ξ)ξ2dξ.
The function g:ξ⟼1−cos(ξ)ξ2 is continuous on the interval (0,+∞) and consequently it is locally integrable. In addition, on the one hand, limξ→0g(ξ)=12, which implies that g is integrable in a neighbourhood of 0. On the other hand, |g(ξ)|≤2ξ2 for every ξ>11$]]> and ∫1+∞1ξ2dξ<∞, which entails the integrability of g on a neighbourhood of +∞. From all this, one can deduce that the integral ∫0∞1−cos(ξ)ξ2dξ is convergent and, consequently, by using (39),
∫0tds∫R[ps(v+h)−ps(v)]2dv⩽C|h|,
for every real h≠0. Morover, Inequality (40) is obviously true for h=0. Thus, (40) is statisfied for every h∈R.
This and Inequality (35) imply that
∫0t12πu∫R|E+(u,y,z)−E+(u,x,z)|2dzdu≤C||f(y)|−|f(x)||≤C|f(y)−f(x)|≤C|y−x|,
where in the last inequality we used Lemma 3. □
For everyA>00$]]>andt∈[0,T], there exists a positive constant c such that∫0t12πu∫R|E−(t−s,y,z)−E−(t−s,x,z)|2dzdu⩾c|y−x|and∫0t12πu∫R|E+(t−s,y,z)−E+(t−s,x,z)|2dzdu⩾c|y−x|for everyx,y∈[0,A].
We present the proof just for the first inequality. The second is obtained in the same way. We have
∫0t12πu∫R|E−(t−s,y,z)−E−(t−s,x,z)|2dzdu=∫0t∫R[12πuexp(−(f(z)−f(y))22u)−12πuexp(−(f(z)−f(x))22u)]2dzdu=∫0t∫0∞[12πuexp(−(z/a2−f(y))22u)−12πuexp(−(z/a2−f(x))22u)]2dzdu+∫0t∫−∞0[12πuexp(−(z/a1−f(y))22u)−12πuexp(−(z/a1−f(x))22u)]2dzdu.
Applying the changes of variables Z=z/a2−f(x) in the first integral and Z=z/a1−f(x) in the second one we obtain
∫0t12πu∫R|E−(t−s,y,z)−E−(t−s,x,z)|2dzdu=a2∫0t∫−f(x)∞[12uexp(−(Z−(f(y)−f(x)))22u)−12πuexp(−Z22u)]2dZdu+a1∫0t∫∞−f(x)[12uexp(−(Z−(f(y)−f(x)))22u)−12πuexp(−Z22u)]2,dZdu⩾min(a1,a2)∫0t∫R[12uexp(−(Z−(f(y)−f(x)))22u)−12πuexp(−Z22u)]2dZdu=min(a1,a2)∫0t∫R[pu(Z−K˜)−pu(Z)]2dZdu,
where K˜=f(y)−f(x) and pu is the heat kernel defined by (37). Without loss of generality we can suppose that x<y. So,
0<K˜=y−xa2<Aa2.
Applying the same technique as that used in (38), we get
∫0t∫R[pu(Z−h)−pu(Z)]2dZdu=12π∫0tds∫−∞∞|e−sξ2/2−iξh−e−sξ2/2|2dξ=1π∫0tds∫−∞∞e−sξ2(1−cos(hξ))dξ=2π∫0∞(1−e−tξ2)1−cos(|h|ξ)ξ2dξ
for every h∈R. Thus,
∫0tds∫R[ps(Z−K˜)−ps(Z)]2dy=2π∫0∞(1−e−tz2)1−cos(K˜z)z2dz=2π∫01K˜(1−e−tz2)1−cos(K˜z)z2dz+2π∫1K˜∞(1−e−tz2)1−cos(K˜z)z2dz≥2π∫1K˜∞(1−e−tz2)1−cos(K˜z)z2dz,
where in the last inequality we used the fact that
2π∫01K˜(1−e−tz2)1−cos(K˜z)z2dz≥0.
Since 1−e−tz2≥1−e−tK˜−2 for every z≥1K˜, from (41) we get
∫0tds∫R[ps(Z−K˜)−ps(Z)]2dy≥2π(1−e−tK˜−2)∫1K˜∞1−cos(K˜z)z2dz≥K˜2π(1−e−tK˜−2)∫1∞1−cos(ξ)ξ2dξ,
where the last inequality is obtained after applying the change of variables ξ=K˜z.
Now, since
0<K˜=y−xa2<Aa2,
we have
1−e−tK˜−2≥1−e−ta2A−2
and consequently
∫0t∫R[pu(Z−K˜)−pu(Z)]2dZdu⩾c|y−x|
with
c=2a2π(1−e−ta2A−2)∫1∞1−cos(z)z2dz.
□
Proof of Theorem <xref rid="j_vmsta139_stat_019">6</xref>
Since Equation (7) is a particular case of (5), by Theorems 4 and 5, to get Theorem 6, it suffices to show that the fundamental solution associated to the operator Lp satisfies Conditions Hi([0,1]), for i=1,2,3. Consider x,y∈[0,1];x<y and t∈[0,T].
According to [6, page 54], we know that
‖f−g‖L2(R)2⩾14(‖f‖L2(R)+‖g‖L2(R))2‖f‖f‖L2(R)−g‖g‖L2(R)‖L2(R)2
for every f,g∈L2(R);f≠0 and g≠0 a.e. Thus,
‖Δy−xG(t−s,x,.)‖L2([0,t]×R)2⩾14min(1a1,1a2)∫0tI(s)2π(t−s)×(‖E−(t−s,y,.)−E−(t−s,x,.)‖+|β|‖E+(t−s,y,.)−E+(t−s,x,.)‖)2ds,
where
I(s)=‖|E−(t−s,y,.)−E−(t−s,x,.)|‖E−(t−s,y,.)−E−(t−s,x,.)‖−|β||E+(t−s,y,.)−E+(t−s,x,.)|‖E+(t−s,y,.)−E+(t−s,x,.)‖‖2,
and ‖.‖ denotes ‖.‖L2(R). On the one hand we have
I(s)=1+β2−2|β|‖E−(t−s,y,z)−E−(t−s,x,z)‖‖E+(t−s,y,z)−E+(t−s,x,z)‖×∫R|E−(t−s,y,z)−E−(t−s,x,z)||E+(t−s,y,z)−E+(t−s,x,z)|dz.
On the other hand, applying Hölder’s Inequality, we get
∫R|E−(t−s,y,z)−E−(t−s,x,z)||E+(t−s,y,z)−E+(t−s,x,z)|dz≤‖E−(t−s,y,.)−E−(t−s,x,.)‖‖E+(t−s,y,.)−E+(t−s,x,.)‖.
Hence,
I(s)⩾1+β2−2|β|=(1−|β|)2
and therefore,
‖Δy−xG(t−s,x,.)‖L2([0,t]×R)2⩾(1−|β|)24min(1a1,1a2)∫0t12π(t−s)×(‖E−(t−s,y,.)−E−(t−s,x,.)‖+|β|‖E+(t−s,y,.)−E+(t−s,x,.)‖)2ds⩾(1−|β|)24min(1a1,1a2)∫0t12πu×(‖E−(u,y,.)−E−(u,x,.)‖2+β2‖E+(u,y,.)−E+(u,x,.)‖2)du,
where in the last inequality we used the fact that x2+y2≤(x+y)2 for every non-negative real numbers x and y.
This and Lemma 5 show that Hypothesis H1([0,1]) is satisfied.
Using the expressions of A− and A+ given in (30) we get
‖Δy−xG(t−.,x,.)‖L2([0,t]×R)2=∫0t∫R|G(t−s,y,z)−G(t−s,x,z)|2dsdz=∫0t[12a1π(t−s)∫−∞0|A−(t−s,y,z)−A−(t−s,x,z)|2dz]ds+∫0t[12a2π(t−s)∫0∞|A+(t−s,y,z)−A+(t−s,x,z)|2dz]ds≤∫0t12π(t−s)∫−∞0Δmax(s,z)dzds+∫0t12π(t−s)∫0∞Δmax(s,z)dzds,
where
Δmax(s,z)=max(1a1|A−(t−s,y,z)−A−(t−s,x,z)|2,1a2|A+(t−s,y,z)−A+(t−s,x,z)|2)=max(1a1((E−(t−s,y,z)−E−(t−s,x,z))−β(E+(t−s,y,z)−E+(t−s,x,z)))2,1a2((E−(t−s,y,z)−E−(t−s,x,z))+β(E+(t−s,y,z)−E+(t−s,x,z)))2)
for every t∈(0,T] and (x,y)∈[0,1]2;y>xx$]]>.
Since
max(γ1(a−b)2,γ2(a+b)2)⩽2max(γ1,γ2)(a2+b2)
for any (a,b)∈R2 and any γ1,γ2>00$]]>, we have
Δmax(s,z)≤2max(1a1,1a2)(|E−(t−s,y,z)−E−(t−s,x,z)|2+β2|E+(t−s,y,z)−E+(t−s,x,z)|2).
Consider x,x′∈[0,1] and h>00$]]>. We have
∫0t∫RΔhG(t−s,x,z)ΔhG(t−s,x′,z)dzds=∫0t∫R(G(t−s,x+h,z)−G(t−s,x,z))(G(t−s,x′+h,z)−G(t−s,x′,z))dzds=∫0t[m2(t−s)a1∫−∞0(A−(t−s,x+h,z)−A−(t−s,x,z))(A−(t−s,x′+h,z)−A−(t−s,x′,z))dz]ds+∫0t[m2(t−s)a2∫0∞(A+(t−s,x+h,z)−A+(t−s,x,z))(A+(t−s,x′+h,z)−A+(t−s,x′,z))dz]ds=L+K.
Using the expression of A− (30), denoting x˜=xa2, x˜′=x′a2 and h˜=ha2, then making the change of variables z′=za1, we get
L=∫0t[(1−β)22πa1u∫−∞0(E−(u,x+h,z)−E−(u,x,z))(E−(u,x′+h,z)−E−(u,x′,z))dz]du=1a2∫0ta2a1(1−β)22πu[∫−∞0(exp(−(z′−x˜−h˜)22u)−exp(−(z′−x˜)22u))×(exp(−(z′−x˜′−h˜)22u)−exp(−(z′−x˜′)22u))dz′]du.
Now, using the expression of A+ (30) and making the change of variable z′=za2, the integral K can be written in the form
K=∫0t12a2πu[K1+βK2+βK3+β2K4]du,
where
K1=∫0+∞(exp(−(z′−x˜−h˜)22u)−exp(−(z′−x˜)22u))×(exp(−(z′−x˜′−h˜)22u)−exp(−(z′−x˜′)22u))dz′,K2=∫0+∞(exp(−(z′−x˜−h˜)22u)−exp(−(z′−x˜)22u))×(exp(−(z′+x˜′+h˜)22u)−exp(−(z′+x˜′)22u))dz′,K3=∫0+∞(exp(−(z′+x˜+h˜)22u)−exp(−(z′+x˜)22u))×(exp(−(z′−x˜′−h˜)22u)−exp(−(z′−x˜′)22u))dz′,K4=∫0+∞(exp(−(z′+x˜+h˜)22u)−exp(−(z′+x˜)22u))×(exp(−(z′+x˜′+h˜)22u)−exp(−(z′+x˜′)22u))dz′.
By using the change of variable z=−z′, we get
K2+K3=∫R(exp(−(z′+x˜+h˜)22u)−exp(−(z′+x˜)22u))×(exp(−(z′−x˜′−h˜)22u)−exp(−(z′−x˜′)22u))dz′
and
K1+β2K4=∫R(exp(−(z′−x˜−h˜)22u)−exp(−(z′−x˜)22u))×(exp(−(z′−x˜′−h˜)22u)−exp(−(z′−x˜′)22u))dz′+(β2−1)∫−∞0(exp(−(z′−x˜−h˜)22u)−exp(−(z′−x˜)22u))×(exp(−(z′−x˜′−h˜)22u)−exp(−(z′−x˜′)22u))dz′.
Therefore,
L+K=L1+βL2+(a2a1(1−β)2+β2−1)L3,
where
L1=∫0tdu2a2πu∫R(exp(−(z′−x˜−h˜)22u)−exp(−(z′−x˜)22u))×(exp(−(z′−x˜′−h˜)22u)−exp(−(z′−x˜′)22u))dz′,L2=∫0tdu2a2πu∫R(exp(−(z′+x˜+h˜)22u)−exp(−(z′+x˜)22u))×(exp(−(z′−x˜′−h˜)22u)−exp(−(z′−x˜′)22u))dz′
and
L3=∫0tdu2a2πu∫−∞0(exp(−(z′−x˜−h˜)22u)−exp(−(z′−x˜)22u))×(exp(−(z′−x˜′−h˜)22u)−exp(−(z′−x˜′)22u))dz′.
We first investigate the sign of the third integral. On the one hand, z′≤0, x˜≥0 and h˜≥0; thus, by virtue of the fact that the function x⟼exp(−x2) is increasing on the interval (−∞,0], we see that L3≥0. On the other hand, using the expression of β, given in (32), by the fact that ρ2≥ρ1 (see Condition (33)), we get
a2a1(1−β)2+β2−1=−4ρ1a1a2(ρ2a2+ρ1a1)2(ρ2−ρ1)≤0.
Therefore,
(a2a1(1−β)2+β2−1)L3≤0.
Now we will calculate explicitely the integral L1. With the notation
T(x,y,u):=∫Rexp(−(v−y)22u)exp(−(v−x)22u)dv,
for every x,y∈R and u>00$]]>, L1 can be written in the form
L1=∫0t12a2πu{T(x˜+h˜,x˜′+h˜,u)−T(x˜,x˜′+h˜,u)−T(x˜+h˜,x˜′,u)+T(x˜,x˜′,u)}du.
By the changes of variables V=v−x and W=y−x−2v2u, we get
T(x,y,u)=∫Rexp(−v22u)exp(−((y−x)−v)22u)dv=exp(−(y−x)24u)∫Rexp(−((y−x)−2v)24u)dv=πuexp(−(y−x)24u).
Thus, applying an integration by parts then the change of variables w=y−x2u, we get
∫0t12πuT(x,y,u)du:=∫0t12πuexp(−(y−x)24u)du=tπexp(−(y−x)24t)−(y−x)24π∫0tu−3/2exp(−(y−x)24u)du=tπexp(−(y−x)24t)−12(y−x)erfc(y−x2t).
Hence,
L1=ta2π{2exp(−(x˜′−x˜)24t)−exp(−(x˜′−x˜+h˜)24t)−exp(−(x˜′−x˜−h˜)24t)}−12a2{2(x˜′−x˜)erfc(x˜′−x˜2t)−(x˜′−x˜+h˜)erfc(x˜′−x˜+h˜2t)−(x˜′−x˜−h˜)erfc(x˜′−x˜−h˜2t)}.
The function h˜⟼L1(h˜)=L1 is clearly twice differentiable and via a simple calculation we get
L1′(h˜)=−12a2{erfc(x˜′−x˜−h˜2t)−erfc(x˜′−x˜+h˜2t)}
and
L1″(h˜)=−12a2πt[exp(−(x˜′−x˜−h)24t)+exp(−(x˜′−x˜+h)24t)].
It’s easy to check that L1(0)=L1′(0)=0 and that L1″ is bounded. From all this and by Taylor’s formula, we obtain
L1(h˜)⩽Ch˜2≤Ch2
for every h>00$]]>, where C denotes a positive universal constant.
Applying the same techniques used in the above argunents, and since β≥0 (see the expression of β given in (32) and Assumption (33)), we get
βL2(h˜)⩽Ch˜2≤Ch2
for every h>00$]]>. Combining (44), (31), (46), (48) and (49), the proof of H3([0,1]) is finished, and consequently, the proof of Theorem 6 is also finshed.
Considering an integer d≥1, one can extend Equation (5) to the d-dimensional case by introducing the following SPDE:
∂u(t,x)∂t=Ldu(t,x)+W˙d(t,x);t>0,x=(x1,…,xd)∈Rd,u(0,.):=0,0,\hspace{0.2778em}x=({x_{1}},\dots ,{x_{d}})\in {\mathbb{R}^{d}},\\ {} u(0,.)& :=& 0,\hspace{0.2778em}\end{array}\right.\]]]>
with
Ld=∑i,j=1d1rij(x)∂∂xi(Rij(x)∂∂xj),
where x⟼Rij(x) and x⟼rij(x) are two measurable and bounded real-valued functions satisfying
rij(x)=rji(x),Rij(x)=Rji(x),
and there exists a constant ν>00$]]> such that
rij(x)ξiξj≥ν‖ξ‖d2andRij(x)ξiξj≥ν‖ξ‖d2
for every x∈Rd, ξ=(ξ1,…,ξd)∈Rd and i,j∈{1,…,d}. In (52), ‖.‖d denotes the Euclidean norm in Rd, and in (51) ∂∂xi denotes the partial derivative in the distributional sense. The noise Wd is a centered Gaussian field Wd={Wd(t,C);t∈[0,T],C∈Bb(Rd)} with covariance
E(Wd(t,C)Wd(s,D))=(t∧s)λd(C∩D),
where λd denotes the Lebesgue measure on Rd and Bb(Rd) is the set of λd-bounded Borel sub-sets of Rd. In the particular case where d=1, SPDE (50) is clearly reduced to Equation (5).
According to [1], if we denote by Gd the fundamental solution associated to the operator Ld, then there exist two constants D1>00$]]> and D2>00$]]> such that
Gd(t,x,y)≥D1td/2exp(−D2‖x−y‖d2t)
for every t∈[0,T] and (x,y)∈Rd. It follows then that
∫0t∫RdGd2(t−s,x,y)dyds≥∫0t∫RdD12(t−s)dexp(−2D2‖x−y‖d2t−s)dyds.
Denoting by I the right-hand side of Inequality (54), we have
I=∫0tD12(t−s)d∏i=1d(∫Rexp(−2D2(xi−yi)2t−s)dyi)ds=∫0tD12(t−s)d(π(t−s)2D2)dds=D12(π2D2)d/2∫0tds(t−s)d/2,
where the second equality in (55) is obtained by the change of variables yi′=(xi−yi)2D2t−s. Since the term ∫0tds(t−s)d/2 is finite if, and only if d<2, from (54) and (55) we deduce that, for every d≥2 we have
∫0t∫RdGd2(t−s,x,y)dyds=+∞.
Therefore, for d≥2, the Wiener integral ∫0t∫RdGd(t−s,x,y)Wd(ds,dy) is not well-defined and consequently, the mild solution to Equation (50) exists if, and only if d=1.
Acknowledgement
We thank very much the unknown referee for his valuable and profound comments.
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