TO THE HEAT EQUATION
WITH ADDITIVE TIME-SPACE WHITE NOISE
CIPRIAN A. TUDOR and MARIA TUDOR
Communicated by Lucian Beznea
We study the asymptotic behavior of the spatial quadratic variations for the solution to the stochastic heat equation with additive Gaussian white noise. We show that these variations satisfy a Central Limit Theorem and we compute the speed of convergence to the limit.
AMS 2010 Subject Classication: 60F05, 60H05, 91G70.
Key words: stochastic heat equation, Gaussian noise, quadratic variations, bi(fractional) Brownian motion, multiple Wiener-Ito integrals.
1. INTRODUCTION
The study of stochastic partial dierential equations consitutes an impor- tant research direction in probability theory. One of the most known examples is the heat equation which arises in various areas of applications (we mention, among others, models for roughening of surfaces [2], continuum limits of par- ticle processes [11] or the Anderson model and intermitency [3]). The study of the properties of the solution to such equations is of particular interest.
In this work, we study the quadratic variations of the solution to the lin- ear heat equation driven by a Gaussian noise which behaves as a Brownian motion both in time and in space. This noise is usually called time-space white noise. The solution to the white noise driven heat equation has been widely studied. It is well-known that it is Holder continuous of order 0< δ < 14 with respect to its time variable and of order 0 < δ < 12 with respect to its space variable. Actually, its law is strongly connected with some fractional processes, such as the fractional Brownian motion and the bifractional Brownian motion.
The quadratic (or higher order) variations of the solution has been the object of study of the works [8] or [10]. Our purpose is to make a deeper analysis of these variations. We actually prove a Central Limit Theorem for the spatial variations and we derive the speed of convergence to the limit. We organized our paper as follows. Section 2 contains some basic elements related to the
REV. ROUMAINE MATH. PURES APPL. 58 (2013), 4, 453462
heat equation with white noise. In Section 3, we recall the behavior of the temporal variations of the solution and in Section 4, we compute the asymptotic behavior of the spatial variations. In Appendix, we describe the tools from the stochastic calculus that we need in our paper.
2. STOCHASTIC HEAT EQUATION WITH LINEAR WHITE NOISE
Let us consider a centered Gaussian process (W(t, A), t≥0, A∈ Bb(R)) with covariance
(1) EW(t, A)W(s, B) = (t∧s)λ(A∩B)
for every s, t≥0 and for everyA, B ∈ Bb(R). The processW is usually called time-space white noise because it behaves as a Brownian motion both with respect to the time variable and to the space variable. Consider the linear heat equation
(2) ut= 1
2∆u+ ˙W
with vanishing initial conditions u(0, x) = 0 for everyx∈R, where ∆denotes the Laplacian on R and W is the Gaussian process with covariance (1). It is well-known that the heat equation (2) admits a mild solution inL2(Ω)that can be written, for t≥0 and x∈R, as
(3) u(t, x) =
Z t
0
G(t−u, x−y)W(ds,dy)
if and only if the spatial dimension isd= 1. The kernelGis (3) is fundamental solution of the heat equation given by
(4) G(t, x) = (
(2πt)−d/2exp
−|x|2t2
if t >0, x∈Rd
0 if t≤0, x∈Rd.
The integral in (3) is a Wiener integral with respect to the Gaussian process W. Moreover, the covariance of the solution (3) can be expressed as (5) Eu(t, x)u(s, y) = 1
√2π Z t∧s
0
du(t+s−2u)−12e− |x−y|
2 2(t+s−2u)
for every s, t≥0and for every x, y∈R.
The path properties of the process u (3) have been widely studied. In particular, the process u admits a bicontinuous version in (t, x) it is Holder continuous of order until 14 in time and until 12 in space. Moreover, its law is related with some fractional Gaussian processes, see the next section.
3. TEMPORAL VARIATIONS
The variations with respect to the time variable of the process ugiven by (3) can be easily described. Actually, for x=y in (5),
(6) Eu(t, x)u(s, x) = 1
√2π Z t∧s
0
du(t+s−2u)−12, s, t≥0, x∈R.
This shows that for xed x∈ R, modulo a constant, the process u(t, x), t≥0has, modulo a constant, the same law as a bifractional Brownian motion with parameters H = K = 12. Recall that the bifractional Brownian motion (BtH,K)t≥0 is a centered Gaussian process, starting from zero, with covariance (7) RH,K(t, s) :=R(t, s) = 1
2K
t2H +s2HK
− |t−s|2HK ,
with H ∈(0,1)and K ∈(0,1]. Note that, ifK = 1 then BH,1 is a fractional Brownian motion with Hurst parameter H ∈ (0,1). We refer to [4] and [9]
for the basic properties of this process. In particular, it is self-similar of order HK and hence, the solution to (2) is self-similar of order 14. Therefore, we can direcly apply the results already known for the bifractional Brownian motion.
Concerning its temporal variations, we get the following (see [1]).
Theorem 1. Let 0 = t0 < t1 < ... < tN = 1 be a partition of [0,1] with ti = Ni for i = 0, ..., N. Fix x ∈ R. Dene the centered quadratic variation statistics
(8) TN =
N−1
X
i=0
"
(u(ti+1, x)−u(ti, x))2 E(u(ti+1, x)−u(ti, x))2 −1
# .
Then, with c0 > 0 a normalizing constant, the sequence T˜N := 1
c0
√ NTN converges in distribution, as N → ∞, to a standard normal lawN(0,1)and
d( ˜TN, N(0,1))≤C 1
√N, where C is a strictly positive constant.
Similar results, including weak convergence in the Skorohod topology, has been proved in [10].
The purpose of this work is to analyze the spatial variations of the solution to (2) and to compare their behavior to the result in Theorem 1. This will be done in the next section.
4. SPATIAL VARIATIONS
Take t=s in (5). Then, for everyt≥0and x, y∈Rwe have Eu(t, x)u(t, y) = 1
√2π Z t
0
du(2t−2u)−12e−|x−y|
2 4(t−u)
= 1
2√ π
Z t
0
duu−12e−|x−y|4u2 and by the change of variablesu˜= |x−y|4u 2 we obtain
Eu(t, x)u(t, y) = 1 4√
π|x−y|
Z ∞
|x−y|2 4t
u−32e−udu.
This expression is not very convenient for computation because the inte- gral R∞
0 u−32e−udu is divergent. Let us integrate now by parts. We get Eu(t, x)u(t, y) =
√t
√πe−|x−y|4t 2 − 1 2√
π|x−y|
Z ∞
|x−y|2 4t
u−12e−udu.
Since we analize the behavior of the process uwith respect to x, we may assume that t = 1. Consequently, from now on, we will consider a centered Gaussian process (U(x), x∈R)with covariance
(9) EU(x)U(y) = 1
√πe−|x−y|4 2 − 1 2√
π|x−y|
Z ∞
|x−y|2 4
u−12e−udu.
Let us consider 0 = x0 < x1 < .... < xN = 1 a partition of the unit interval [0,1] withti = Ni for everyi= 0, .., N. Dene the centered quadratic variation statistic
(10) VN =
N−1
X
i=0
"
U(i+1N )−U(Ni )2
E U(i+1N )−U(Ni )2 −1
# .
Our aim is to nd the asymptotic behavior of the renormalized sequence VN as N goes to innity. Let us rst evaluate the L2 of the increments of the process U.
Lemma 1. For every N ≥1 and for everyi= 0, ..., N−1, we have limN N E
U(i+ 1
N )−U( i N)
2
= 1.
Proof. By formula (9) E
U(i+ 1
N )−U( i N)
2
= 2
√π
1−e−4N12 + 1
√π 1 N
Z ∞
1 4N2
duu−12e−u
and clearly, since 1−e−4N12 behaves as (4N2)−1 limN N E
U(i+ 1
N )−U( i N)
2
= lim
N
√1 π
Z ∞
1 4N2
duu−12e−u = 1
√πΓ 1
2
= 1.
We denoted byΓ the Gamma function and we usedΓ 12
=√
π. We will need the following lemma in order to bound the joint increments of the processU.
Lemma 2. For every N ≥ 1 and for every 1 ≤ i, j ≤ N with i 6= j we have
E
U(i+ 1
N )−U( i
N) U(j+ 1
N )−U(j N)
≤C 1 N2. Proof. It follows from [8], Section 2.1.
Let us now estimateEVN2 withVN given by (10). Denote byU the canon- ical Hilbert space associated to the Gaussian processU. This Hilbert space is dened as the closure of the linear space generated by the indicator functions 1[0,t], t≥0with respect to the inner product
h1[0,t],1[0,s]iU =EU(s)U(t).
In the sequel, we will denote h·,·iU byh·,·i. Note also that the increment U(x)−U(y)can be expressed asI1(1[x,y]) for everyx < y withI1 the multiple integral of order 1 with respect to U. See the Appendix.
We prove the following result.
Proposition 1. Let VN be given by (10). Then E
1
√ NVN
2
→N→∞ 2.
Proof. Using the product formula for multiple integrals (14), we can write VN =
N−1
X
i=0
E
U(i+ 1
N )−U( i N)
−2
I2
1⊗2
[Ni,i+1N ]
and therefore, by the isometry of multiple Wiener-Ito integrals (12), E 1
NVN2 = 2
N−1
X
i,j=0
E
U(i+ 1
N )−U( i N)
−2
h1[i
N,i+1N ],1[j
N,j+1N ]i2
∼ 2N
N−1
X
i,j=0
h1[i
N,i+1N ],1[j
N,j+1N ]i2,
where we used Lemma 1 and the convention thataN ∼bN means that the two sides have the same limit when N → ∞. By separating the diagonal and the non-diagonal parts, we can write
E 1
NVN2 ∼ 2N
N−1
X
i=0
h1[i
N,i+1N ],1[i
N,i+1N ]i2 +2N
N−1
X
i,j=0;i6=j
h1[i
N,i+1N ],1[j
N,j+1N ]i2.
Let us rst note that the non-diagonal part converges to zero. Indeed, since h1[i
N,i+1N ],1[j
N,j+1N ]i = E U(i+1N )−U(Ni )
U(j+1N )−U(Nj)
, by Lemma 2,
N
N−1
X
i,j=0;i6=j
h1[i
N,i+1N ],1[j
N,j+1N ]i2 ≤CN
N−1
X
i,j=0;i6=j
1 N2
2
≤C 1
N →N 0.
On the other hand, By Lemma 1 2N
N−1
X
i=0
h1[i
N,i+1N ],1[i
N,i+1N ]i2∼2N
N−1
X
i=0
1 N
2
→N→∞2 and this gives the conclusion.
Denote by V˜N := √1
2NVN. We already showed in Proposition 1 that EV˜N2 →N 1. Our purpose is to show that this sequence converges in law to standard normal law and to derive the speed of this convergence. Our approach is based on Stein's method combined with the Malliavin calculus, see [5]. Below we use the letter d to denote one of several metrics on the space of probability measures on R, including the Kolmogorov, Wasserstein, and Total Variation metrics dKol, dW and dT V, respectively. We also abuse notation by using random variables, rather than their laws, as arguments for this metric. For instance, dKol(X, Y) := supz∈R|P[X≤z]−P[Y ≤z]|, and it is known that dKol metrizes certain convergences in distribution: ifF has a cummulative distribution function that is continuous, then FN converges toF
in distribution if and only iflimN→∞dKol(FN, F) = 0. See ([5], Appendix C) for other denitions and properties. The following theorem is a consequence of [6]; see ([5], Theorem 5.2.6).
Theorem 2. Let Iq(f) be a multiple integral of order q ≥ 1. Assume Eh
Iq(f)2i
=σ2. Then d Iq(f), N(0, σ2)
≤ c q2
Var
1
qkDIq(f)k2L2([0,1])
12 ,
wherec= 1/σ2 whend=dKol, andc= 1/σwhend=dW, and nallyc= 2/σ2 for d=dT V.
We can apply this theorem to V˜N given by (11) since it is a multiple integral of order 2, obtaining the following normal convergence result.
We state now the main result of this section.
Theorem 3. Denote by
(11) V˜N := 1
√
2NVN,
for every N ≥ 1. Then, as N → ∞, the sequence (VN)N≥1 converges in distribution to a standard normal random variable. Moreover,
d
V˜N, N(0,1)
≤C 1
√N. Proof. We have
DV˜n=
√
√2 N
N−1
X
i=0
E
U(i+ 1
N )−U( i N)
−2
I1
1[i
N,i+1N ]
1[i
N,i+1N ]
and therefore, kDV˜nk2U = 2
N
N−1
X
i,j=0
E
U(i+ 1
N )−U( i N)
−2 E
U(j+ 1
N )−U( j N)
−2
×I1 1[i
N,i+1N ]
I1
1[j
N,j+1N ]
h1[i
N,i+1N ],1[j
N,j+1N ]i
= 2 N
N−1
X
i,j=0
E
U(i+ 1
N )−U( i N)
−2 E
U(j+ 1
N )−U( j N)
−2
×I1
1⊗2
[Ni,i+1N ]
h1[i
N,i+1N ],1[j
N,j+1N ]i +EkDV˜nk2U
:= RN+EkDV˜nk2U.
where we used again the product formula for multiple integrals.
Let us rst show that the term denoted byRN converges to zero inL2(Ω) asN → ∞and to estimate its rate of convergence. We have
ER2N = 4 N2
N−1
X
i,j,k.l=0
E
U(i+ 1
N )−U( i N)
−2 E
U(j+ 1
N )−U( j N)
−2
×
E
U(k+ 1
N )−U(k N)
−2 E
U(l+ 1
N )−U( l N)
−2
× h1[i
N,i+1N ],1[j
N,j+1N ]ih1[i
N,i+1N ],1[k
N,k+1N ]ih1[k
N,k+1N ],1[l
N,l+1N ]ih1[j
N,j+1N ],1[l N,l+1N ]i
∼16N2
N−1
X
i,j,k.l=0
h1[i
N,i+1N ],1[j
N,j+1N ]ih1[i
N,i+1N ],1[k N,k+1N ]i h1[k
N,k+1N ],1[l
N,l+1N ]ih1[j
N,j+1N ],1[l
N,l+1N ]i ≤C 1 N by using Lemmas 1 and 2 (the fastest term is the one withi=j=k=l). Now let us analyze EkDV˜nk2U−2. Actually, sinceV˜N is a multiple integral of order 2, we know that EkDV˜nk2U = 2EV˜N2 → 2 but we need to understand how fast this sequence goes to zero. From the proof of Proposition 1 and of Lemma 1, we have that
EV˜N2 −1 ≤ 1
√π Z ∞
1 4N2
u−12e−udu−1
!2
+C 1 N
= 1
√π Z ∞
1 4N2
u−12e−udu− 1
√πΓ 1
2 !2
+C 1 N
= 1
√π Z 1
4N2
0
u−12e−udu
!2
+C 1 N
= 1
√π
γ1
2
1 4N2
2
+C 1 N
∼ C 1
N 2
+C 1
N ≤C 1 N, where we denoted byγ1
2 the incomplete gamma functionγa(x) =Rx
0 ua−1e−udu for a > 0 and we used the fact that γa(x) ∼ a−1xa for x close to zero. This gives the desired conclusion.
Remark 1. From Theorems 1 and 3 we notice that the speed of convergence is the same for the temporal and spatial variations.
5. APPENDIX: MULTIPLE WIENER INTEGRALS AND MALLIAVIN DERIVATIVES
Here, we describe the elements from stochastic analysis that we will need in the paper. Consider Ha real separable Hilbert space and(B(ϕ), ϕ∈ H) an isonormal Gaussian process on a probability space (Ω,A, P), which is a cen- tered Gaussian family of random variables such thatE(B(ϕ)B(ψ)) =hϕ, ψiH. Denote In the multiple stochastic integral with respect to B (see [7]). This In
is actually an isometry between the Hilbert spaceHn(symmetric tensor prod- uct) equipped with the scaled norm √1n!k · kH⊗n and the Wiener chaos of order nwhich is dened as the closed linear span of the random variables Hn(B(ϕ)) whereϕ∈ H,kϕkH= 1 and Hn is the Hermite polynomial of degree n≥1
Hn(x) = (−1)n n! exp
x2 2
dn dxn
exp
−x2 2
, x∈R.
The isometry of multiple integrals can be written as: for m, n positive integers,
E(In(f)Im(g)) = n!hf ,˜ ˜giH⊗n if m=n, E(In(f)Im(g)) = 0 if m6=n.
(12)
It also holds that
In(f) =In f˜ ,
wheref˜denotes the symmetrization offdened byf(˜x1, ..., xn) =n!1P
σ∈Snf(xσ(1), . . . , xσ(n)).
We recall that any square integrable random variable which is measur- able with respect to the σ-algebra generated by B can be expanded into an orthogonal sum of multiple stochastic integrals
(13) F =X
n≥0
In(fn),
wherefn∈Hnare (uniquely determined) symmetric functions andI0(f0)=E[F]. Let Lbe the Ornstein-Uhlenbeck operator
LF =−X
n≥0
nIn(fn) if F is given by (13) and it is such thatP∞
n=1n2n!kfnk2H⊗n <∞.
For p >1 and α ∈R we introduce the Sobolev-Watanabe space Dα,p as the closure of the set of polynomial random variables with respect to the norm
kFkα,p =k((I−L)F)α2kLp(Ω),
where I represents the identity. We denote by D the Malliavin derivative operator that acts on smooth functions of the form F =g(B(ϕ1), . . . , B(ϕn)) (g is a smooth function with compact support andϕi ∈ H)
DF =
n
X
i=1
∂g
∂xi(B(ϕ1), . . . , B(ϕn))ϕi. The operatorD is continuous fromDα,p intoDα−1,p(H).
We will need the general formula for calculating products of Wiener chaos integrals of any ordersp, qfor any symmetric integrandsf ∈ Hp andg∈ Hq; it is
(14) Ip(f)Iq(g) =
p∧q
X
r=0
r!
p r
q r
Ip+q−2r(f⊗rg)
as given for instance in D. Nualart's book ([7], Proposition 1.1.3); the contrac- tion f⊗rg is the element ofH⊗(p+q−2r) dened by
(f ⊗`g)(s1, . . . , sn−`, t1, . . . , tm−`) (15)
=R
[0,T]m+n−2`f(s1, . . . , sn−`, u1, . . . , u`)g(t1, . . . , tm−`, u1, . . . , u`)du1. . .du`.
Acknowledgments. Supported by the CNCS grant PN-II-ID-PCCE-2011-2-0015.
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Received 26 February 2013 Universite de Lille 1, Laboratoire Paul Painleve, F-59655 Villeneuve d'Ascq, France
Academy of Economical Studies, Bucharest, Romania tudor@math.univ-lille1.fr
