MSTA Modern Stochastics: Theory and Applications 2351-6054 2351-6046 2351-6046 VTeXMokslininkų g. 2A, 08412 Vilnius, Lithuania MSTA141 10.15559/19-MSTA141 Research Article Estimation of the drift parameter for the fractional stochastic heat equation via power variation Mahdi KhalilZeinazeina_kh@outlook.fr TudorCiprianciprian.tudor@univ-lille.fr∗ Laboratoire Paul Painlevé, Université de Lille, CNRS, UMR 8524, F-59655 Villeneuve d’Ascq, France Corresponding author. 2019 310201964397417 842019 2272019 1192019 © 2019 The Author(s). Published by VTeX2019 Open access article under the CC BY license.

We define power variation estimators for the drift parameter of the stochastic heat equation with the fractional Laplacian and an additive Gaussian noise which is white in time and white or correlated in space. We prove that these estimators are consistent and asymptotically normal and we derive their rate of convergence under the Wasserstein metric.

 Stochastic heat equation fractional Brownian motion fractional Laplacian q variation drift parameter estimation 60G15 60H05 60G18 C. Tudor is partially supported by Labex Cempi (ANR-11-LABX-0007-01) and MATHAMSUD Project SARC (19-MATH-06). Introduction

The purpose of this work is to estimate the drift parameter θ>0 0$]]> of the fractional stochastic heat equation ∂uθ∂t(t,x)=−θ(−Δ)α2uθ(t,x)+W˙(t,x),t≥0,x∈R, with vanishing initial conditions, where (−Δ)α2 denotes the fractional Laplacian of order α∈(1,2] , θ>0 0$]]> and W is a Gaussian noise which is white in time and white or correlated in space.

The parameter estimation for stochastic partial differential equations (SPDEs in the sequel) constitutes a research direction of wide interest in probability theory and mathematical statistics. We refer, among many others, to the recents surveys [16] and [4]. On the other side, there are relatively few works that consider the solution to a SPDE observed at discrete points in time and/or in space. Among the first works in this direction, we refer to [18] and [17] for the maximum likelihood and least square estimators for parabolic, respectively elliptic type SPDEs driven by a space-time white noise. The study in [17] has been then extended in [2], by adding a time-varying volatility in the noise term and by using power variation techniques to estimate the parameter of the model. Other recent works on parameter estimates for discretely sampled SPDEs via power variations are [5, 3, 1, 21] and [24].

In this paper, we extend the above results into two directions. Firstly, we replace the standard Laplacian operator used in all the above references by a fractional Laplacian. On the other hand, we consider a simpler form, comparing to [2, 17], of the differential operator. Secondly, we also consider a noise term which is correlated in space. Our purpose is to propose power variation type estimators for the drift parameter in the stochastic model (1), based on discrete observations of the solution in time or in space, and to analyze the consistency and the limit distribution of the estimators by taking advantage of the link between the solution and the fractional Brownian motion. Our approach to construct and analyze the estimators for the drift parameter is based on the asymptotic behavior of the q-variations of the mild solution to (1). It is well known (see, e.g., [7, 13, 23]) that there exists a strong link between the law of this mild solution with θ=1 and the fractional Brownian motion and related processes. We will use this connection in order to deduce the behavior of the q-variations (of suitable order q) of the solutions to (1) and to prove the consistency, asymptotic normality and Berry–Esséen bounds under the Wasserstein distance for the associated estimators. For the situation when W is a space-time white noise, we will obtain two estimators for the drift parameter: one based on the temporal variations and one based of the spatial variations of the mild solution uθ . Similarly, two estimators are defined when the Gaussian noise W is white in time and colored in space (with the spatial covariance given by the Riesz kernel). Even if the order of the variations which appear in the definition of the estimator is different in the four cases (this order may depend on the parameter α of the fractional Laplacian and/or on the spatial correlation), all the estimators are asymptotically normal, they have the same rate of convergence of order n−12 and they have the same distance to the Gaussian distribution. The case of the standard Laplacian (i.e., α=2 ) has been studied in [21].

We organize the paper as follows. In Section 2 we present general facts on the stochastic heat equation with the fractional Laplacian and the behavior of the variations of the perturbed fractional Brownian motion. In Section 3 we discuss the drift parameter estimation for the fractional heat equation with a space-time white noise while in Section 4 we treat the case when the noise is correlated in space.

We will denote by c, C a generic positive constant that may change from line to line (or even inside of the the same line). By →(d) we denote the convergence in distribution while ≡(d) stands for the equivalence of finite dimensional distributions.

 The fractional heat equation driven by a space-time white noise

We start by treating the fractional stochastic heat equation with a space-time white noise. We recall the basic properties of the solution, its relation with the fractional Brownian motion and then we discuss the estimation of the drift parameter θ via the q-variations.

General properties of the solution

On the standard probability space Ω,F,P , we consider a centered Gaussian field W(t,A),t≥0,A∈Bb(R) with covariance EW(t,A)W(s,B)=(s∧t)λ(A∩B)for everys,t≥0,A,B∈Bb(R), where λ denotes the Lebesgue measure on R and Bb(R) is the class of bounded Borel subsets of R . The Gaussian field W is usually called the space-time white noise.

We will consider the stochastic heat equation ∂uθ∂t(t,x)=−θ(−Δ)α2uθ(t,x)+W˙(t,x),t≥0,x∈R, with vanishing initial condition u(0,x)=0 for every x∈R . In the above equation, (−Δ)α2 represents the fractional Laplacian of order α. We will assume in the sequel that α∈(1,2] . We refer to [6, 11, 10, 12] for the precise definition and other properties of the fractional Laplacian operator. We will denote its Green kernel (or the fundamental solution) by Gα , which represents the deterministic kernel that solves the heat equation without noise ∂∂tu(t,x)=−(−Δ)α2u(t,x) . It is know from the above references that for t>0 0$]]>, x∈R Gα(t,x)=∫Reitξ−t|ξ|αdξ.

It is an immediate conclusion that the fundamental solution associated to the operator −θ(−Δ)α2uθ(t,x) is Gα(θt,x) .

The solution to (3) is understood in the mild sense, i.e., uθ(t,x)=∫0t∫RGα(θ(t−s),x−y)W(ds,dy), where the stochastic integral W(ds,dy) is the usual Wiener integral with respect to the space-time white noise, which satisfies the isometry E∫0T∫RH(s,y)W(ds,dy)2=∫0T∫RH(s,y)2dyds for every T>0 0$]]> and for every measurable square integrable function H.

For θ=1 , the solution to the heat equation (3) has been studied in [13]. This solution exists only if the spatial dimension is d=1 , and it is connected to the bifractional Brownian motion. Recall that (see [9, 22]), given constants H∈(0,1) and K∈(0,1] , the bifractional Brownian motion (bi-fBm for short) (BtH,K)t≥0 is a centered Gaussian process with covariance RH,K(t,s):=R(t,s)=12Kt2H+s2HK−|t−s|2HK,s,t≥0. In particular, for K=1 , BH,:=BH,1 is the fractional Brownian motion (fBm in the sequel) with the Hurst parameter H∈(0,1) .

Let us recall some of the results in [13] which will be needed in the sequel.

The mild solution (5) is well-defined. For every x∈R , the process ( u1(t,x) , t≥0 ) coincides in distribution, modulo a constant, with the bifractional Brownian motion, i.e., u1(t,x),t≥0≡(d)c2,αBt12,1−1α,t≥0, where B12,1−1α is a bifractional Brownian motion with the Hurst parameters H=12 and K=1−1α and c2,α2=c1,α21−1αwithc1,α=12π(α−1)Γ1α.

For every t≥0 , we have (see Proposition 3.1 in [7]) u1(t,x),x∈R≡(d)mαBα−12(x)+St(x),x∈R, where Bα−12 is a fractional Brownian motion with the Hurst parameter α−12∈[0,12] , (St(x))x∈R is a centered Gaussian process with C∞ sample paths and mα is an explicit numerical constant.

The above facts, combined with the decomposition (18) of the bifractional Brownian motion, show that the solution to the heat equation can be expressed as the sum of a fBm and a smooth process (we will call this sum as a perturbed fractional Brownian motion).

 Variations of the perturbed fractional Brownian motion

Since the process (5) is connected to the perturbed fBm (i.e., the sum of a fBm and a smooth Gaussian process), let us recall some facts concerning the asymptotic behavior of the variation of the perturbed fBm. Some of the below results are directly taken from [13] while those concerning the rate of convergence under the Wasserstein distance are deduced from [19].

We first define the notion of (exact) q-variation for stochastic processes.

Let A1<A2 , and for n≥1 , let ti=A1+in(A2−A1) for i=0,…,n . A continuous stochastic process (Xt)t≥0 admits a q-variation (or a variation of order q) over the interval [A1,A2] if the sequence S[A1,A2]n,q(X):= ∑i=0n−1Xti+1−Xtiq converges in probability as n→∞ . The limit, when it exists, is called the exact q-variation of X over the interval [A1,A2] .

If [A1,A2]=[0,t] , we will simply denote Stn,q(X):=S[0,t]n,q(X) . Moreover, if t=1 , we denote Sq,n(X):=Stn,q(X) . In the case q=2 the limit of S2,n is called the quadratic variation, while for q=3 we have the cubic variation.

Let us recall the following result (see [13]) concerning the exact variation of the perturbed fractional Brownian motion, i.e., the sum of a fBm and a smooth Gaussian process. In the rest of this section, we will fix an interval [A1,A2] with A1<A2 and a partition tj=A1+jn(A2−A1) , n≥1 , j=0,…,n , of this interval. Also, we denote by Z a standard normal random variable, and μq=EZq for q≥1 . Define σH,q2=q!∑v∈ZρH(v)q , with ρH(v)=12|v+1|2H+|v−1|2H−2|v|2H for v∈Z .

Let (BtH)t≥0 be a fBm with H∈(0,12] and consider a centered Gaussian process (Xt)t≥0 such that EXt−Xs2≤C|t−s|2for everys,t≥0. Define YtH=aBtH+Xtfor everyt≥0 with a≠0 .

The process Y has 1H -variation over the interval [A1,A2] which is equal to a−1HE|Z|1/H(A2−A1).

Let Vq,n(YH):= ∑i=0n−1nHq(A2−A1)qHaq(Yti+1H−YtiH)q−μq. Then, if H∈(0,12) and q≥2 is an integer, 1nVq,n(YH)=1n ∑i=0n−1nHq(A2−A1)qHaq(Yti+1H−YtiH)q−μq→(d)N(0,σH,q2). If H=12 , q=2 and the process (Xt)t≥0 is adapted to the filtration generated by B, then 1nV2,n(YH)=1n ∑i=0n−1n(A2−A1)a2(Yti+112−Yti12)2−1→(d)N(0,σ12,22).

Using the recent Stein–Malliavin theory, it is also possible to deduce the rate of convergence in the above Central Limit Theorem (CLT in the sequel) under the Wasserstein distance. Before stating and proving the result, let us briefly recall the definition of the Wasserstein distance. The Wasserstein distance between the laws of two Rd -valued random variables F and G is defined as dW(F,G)=suph∈AEh(F)−Eh(G) where A is the class of Lipschitz continuous function h:Rd→R such that ‖h‖Lip≤1 , where ‖h‖Lip=supx,y∈Rd,x≠y|h(x)−h(y)|‖x−y‖Rd.

Assume H≤12 . Let YH be as in Lemma 1 and let Vq,n(YH) be given by (10). Then for n large and with σH,q from (11), dW1nVq,n(YH),N(0,σH,q2)≤C1n.

From the proof of Lemma 2.1 in [13], we can express the variation of YH and the variation of the fBm BH plus a rest term, i.e., 1nVq,n(YH)=1nVq,n(BH)+Rn, where Rn satisfies, for every n≥1 , E|Rn|≤cnH−1. By the definition of the Wasserstein distance, we can write dW1nVq,n(YH),N(0,σH,q2)≤dW1nVq,n(BH),N(0,σH,q2)+dW1nVq,n(YH),1nVq,n(BH)≤dW1nVq,n(BH),N(0,σH,q2)+E|Rn|. In order to estimate dW(1nVq,n(BH),N(0,σH,q2)) , we will use the chaos expansion of the random variable Vq,n(BH) and several results in [19]. Notice that (see, e.g., the proof of Corollary 3 in [20]), Vq,n(BH)= ∑k=1qk!Cqkμq−k ∑i=0n−1HknHK(A2−A1)HKBti+1H−BtiH, where Hk is the k-th probabilists’ Hermite polynomial Hk(x)=(−1)ke−x22dndxne−x22 for k≥1 with H0(x)=1 . We know from [19] that the vector (F1,n,F2,n,…,Fq,n):=1n ∑i=0n−1HknHK(A2−A1)HKBti+1H−BtiHk=1,…,q converges in distribution to a centered Gaussian vector with diagonal covariance matrix C (the explicit expression of C can be found in [19], it is not needed in our work). Moreover, Proposition 6.2.2 and Corollary 7.4.3 in [19] imply that dW(Fk,n)k=1,…,q,N(0,C)≤c1n. This will easily lead to dW1nVq,n(BH),N(0,σH,q2)≤c1n. Since H≤12 , we obtain the conclusion via (14) and (15).  □

 Estimators for the drift parameter

Our purpose is to estimate the parameter θ>0 0$]]> based on the observations of the process  uθ . We will define two estimators: the first is based on the temporal variations of the process uθ while the second is constructed via its variation in space. Their behavior is strongly related to the law of the process uθ , therefore we start by analyzing the distribution of this Gaussian process.

The law of the solution

Let Gα(t,x) be the Green kernel associated to the operator −(−Δ)α2 . Then the Green kernel associated to the operator operator −θ(−Δ)α2 is Gα(θt,x).

Suppose that the process uθ(t,x),t≥0,x∈R satisfies (3). Define vθ(t,x):=uθtθ,x,t≥0,x∈R. Then the process vθ(t,x),t≥0,x∈R satisfies ∂vθ∂t(t,x)=−(−Δ)α2vθ(t,x)+(θ)−12W˜˙(t,x),t≥0,x∈R, with vθ(0,x)=0 for every x∈R , where W˜˙ is a space-time white noise, i.e., a centered Gaussian random field with covariance (2).

From (5), we have for every t≥0 , x∈R , vθ(t,x)=uθtθ,x=∫0tθ∫RGα(t−θs,x−y)W(ds,dy)=∫0t∫RGα(t−s,x−y)W(dsθ,dy)=θ−12∫0t∫RGα(t−s,x−y)W˜(ds,dy), where, for t≥0 , A∈B(R) , we denoted W˜(t,A):=θ12Wtθ,A . Notice that W˜ has the same finite dimensional distributions as W, due to the scaling property of the white noise.  □

We can deduce the law of the process uθ in time and space.

For every x∈R and θ>0 0$]]>, we have uθ(t,x),t≥0≡(d)θ−12αc2,αBt12,1−1α,t≥0, where B12,1−1α is a bifractional Brownian motion with parameters H=12 and K=1−1α and c2,α is given by (7).

Fix x∈R and θ>0 0$]]>. Then for every s,t≥0 , we have Euθ(t,x)uθ(s,x)=Evθ(θt,x)vθ(θs,x)=θ−1Eu1(θt,x)u1(θs,x)=θ−1c1,α(θt+θs)1−1α−|θt−θs|1−1α=θ−1αc2,α2EBt12,1−1αBs12,1−1α.  □

For every t≥0 , θ>0 0$]]>, we have the following equality in distribution uθ(t,x),x∈R≡(d)θ−12mαBα−12(x)+Sθt(x),x∈R, where Bα−12 is a fractional Brownian motion with the Hurst parameter α−12∈(0,12] , (Sθt(x))x∈R is a centered Gaussian process with C∞ sample paths and mα from (8).

The result is immediate since for every t>0 0$]]>, θ>0 0$]]> uθ(t,x),x∈R=vθ(θt,x),x∈R≡(d)θ−12u1(θt,x),x∈R ≡(d)θ−12mαBα−12(x)+Sθt(x),x∈R, where we used (8).  □

Notice that the Hurst parameter of the fBm in Proposition 3 may be 12 if α=2 .

 Estimators based on the temporal variation

Proposition 2 indicates that the process uθ behaves as a bi-fBm in time. Recall the following connection between the fBm and the bi-fBm (see [14]): Let H∈(0,1) , K∈(0,1] . If (BtHK)t≥0 is a fBm with the Hurst parameter HK and (BtH,K)t≥0 is a bi-fBm, then C1XtH,K+BtH,K,t≥0≡(d)C2BtHK,t≥0, with C1>0 0$]]> and C2=21−K2 . In (18), XH,K is a Gaussian process, independent of BH,K with C∞ sample paths. In particular, it satisfies (9). Therefore, the bi-fBm is a perturbed fBm and the same holds true for the solution (uθ(t,x),t≥0) , by Proposition 2. Therefore, we obtain, by using the notation tj=A1+jn(A2−A1) , n≥1 , j=0,…,n , the following lemma.

Let uθ be the solution to (3). Then for every x∈R , S[A1,A2]n,2αα−1:= ∑i=0n−1uθ(tj+1,x)−uθ(tj,x)2αα−1→n→∞c2,α2αα−121α−1μ2αα−1(A2−A1)|(θ)|−1α−1 in probability.

Relation (19) motivates the definition of the following estimator for the parameter θ>0 0$]]> of the model (3): θˆn,1=c2,α2αα−121α−1μ2αα−1(A2−A1)−1 ∑i=0n−1uθ(tj+1,x)−uθ(tj,x)2αα−11−α=c2,α2αα−121α−1μ2αα−1(A2−A1)α−1Sn,2αα−1(uθ(·,x))1−α, and so θˆn,111−α=1c2,α2αα−121α−1μ2αα−1(A2−A1) ∑i=0n−1uθ(tj+1,x)−uθ(tj,x)2αα−1.

We will prove the consistency and the asymptotic normality of the above estimator.

Assume q:=2αα−1 is an even integer and consider the estimator θˆn,1 defined by (20). Then θˆn,1→n→∞θ in probability and nθˆn,111−α−θ11−α→(d)N(0,s1,θ,α2)withs1,θ,α2=σ1q,q2θ21−αμ2αα−1−2. Moreover, for n large enough dWnθˆn,111−α−θ11−α,N(0,sθ,α2)≤c1n.

From Proposition 2 and the relation between the fBm and the bi-fBm (18), we obtain that uθ(t,x)+c2,αθ−12αXt≡(d)c2,αθ−12α212αBα−12α, where Bα−12α is a fBm with the Hurst parameter α−12α∈(0,12) . Therefore, uθ is a perturbed fBm and we obtain, by taking H=α−12α and q=1H=2αα−1, 1n ∑i=0n−1nθ1α−1c2,α2αα−121α−1(A2−A1)uθ(tj+1,x)−uθ(tj,x)2αα−1−θ11−α→(d)N(0,σ1q,q2). This means nμ2αα−1θ1α−1θˆn,111−α−θ11−α→(d)N(0,σ1q,q2) which is equivalent to (22).  □

Using the so-called delta-method, we can get the asymptotic behavior of the estimator θˆn . Recall that if (Xn)n≥1 is a sequence of random variables such that n(Xn−γ0)→(d)N(0,σ2) and g is a function such that g′(γ0) exists and does not vanish, then n(g(Xn)−g(γ0))→(d)N(0,σ2g′(γ0)2).

Consider the estimator (20) and let s1,θ,α be given by (22). Then as n→∞ , n(θˆn,1−θ)→N(0,s1,θ,α2(1−α)2θ2αα−1), and for n large enough, dWn(θˆn,1−θ),N(0,s1,θ,α2(1−α)2θ2αα−1)≤c1n.

By applying the delta-method for the function g(x)=x1−α , Xn=θˆn,111−α and γ0=θ11−α , we immediately obtain the convergence (24). Concerning the rate of convergence, we can write, with γ0˜ a random point located between Xn and γ0 , n(g(Xn)−g(γ0))=ng′(γ0˜)(Xn−γ0)=g′(γ0)n(Xn−γ0)+n(Xn−γ0)(g′(γ0˜)−g′(γ0))=:g′(γ0)n(Xn−γ0)+Tn. We have, for n large, E|Tn|=En(Xn−γ0)(g′(γ0˜)−g′(γ0))≤En(Xn−γ0)212E(g′(γ0˜)−g′(γ0))212≤cE(g′(γ0˜)−g′(γ0))212≤cEθˆn,1αα−1−θαα−1212≤cEθˆn,11α−1−θ1α−1212≤c1n where we used the assumption α>1 1$]]> for the first inequality of the line above and relation (22) (which gives in particular the L2(Ω) -convergence of θˆn,11α−1 to θαα−1 as n→∞ ) for the second inequality on the same line. Therefore, by the triangle inequality and Proposition 4, for n large enough, dWn(θˆn,1−θ),N(0,s1,θ,α2(1−α)2θαα−1)≤cdWn(Xn−γ0),N(0,s1,θ,α2)+E|Tn|≤c1n.  □

 Estimators based on the spatial variation

It is possible to define an estimator for the parameter θ based on the spatial variations of the solution (5). The result in Proposition 3 says that the process uθ(t,x),x∈R is a perturbed fBm, so we know its exact variation in space. Below xj=A1+jn(A2−A1) , j=0,…,n , will denote a partition of the interval [A1,A2] .

Let uθ be given by (3). Then ∑i=0n−1uθ(t,xj+1)−uθ(t,xj)2α−1→n→∞mα2α−1μ2α−1(A2−A1)|θ|−1α−1 and if q:=2α−1 is an integer, 1n ∑i=0n−1nmα2α−1(A2−A1)θ1α−1uθ(t,xi+1)−uθ(t,xi)2α−1−μ2α−1→(d)N(0,σα−12,2α−12).

Proposition 6 leads to the definition of the estimator θˆn,2=(mα2α−1μ2α−1(A2−A1))−1 ∑i=0n−1uθ(t,xj+1)−uθ(t,xj)2α−11−α, and we can immediately deduce from Proposition 3 its asymptotic proprieties.

The estimator (25) converges in probability as n→∞ to the parameter θ. Moreover, if q:=2α−1 is an even integer, nθˆn,211−α−θ11−α→(d)N(0,s2,θ,α2)withs2,θ,α2=σα−12,2α−12μ2α−1−2θ21−α. Moreover, for n large, dWnθˆn,211−α−θ11−α,N(0,s2,θ,α2)≤c1n.

Using the law of the process uθ(t,x),x∈R obtained in Proposition 3, we deduce that the Gaussian process θ12mα−1uθ(t,x),x∈R is a perturbed fractional Brownian motion. Therefore, by relation (11) in Lemma 1, 1n ∑i=0n−1nθ1α−1(A2−A1)mα2α−1uθ(t,xj+1)−uθ(t,xj)2α−1−μ2α−1=nμ2α−1θ1α−1θˆn,211−α−θ11−α→n→∞(d)N0,σα−12,2α−12. Moreover, Proposition 1 implies that dWnμ2α−1θ1α−1θˆn,211−α−θ11−α,N(0,σα−12,2α−12)≤c1n and this obviously leads to the desired conclusion.  □

By using the delta-method, we can obtain the asymptotic distribution of θˆn,2 .

Let θˆn,2 be given by (25). Then, with s2,θ,α from (26), as n→∞ , n(θˆn,2−θ)→(d)N0,s2,θ,α2(1−α)2θ2αα−1, and for n large enough, dWn(θˆn,2−θ),N(0,s2,θ,α2(1−α)2θ2αα−1)≤c1n.

It suffices to apply (23) to the function g(x)=x1−α and γ0=θ11−α and to follow the proof of Proposition 5.  □

The estimators (20) and (25) coincide with the estimators in [21] in the case of the standard Laplacian α=2 .

The distance of the estimators (20) and (25) to their limit distribution is of the same order, although they involve q-variations with different q.

Heat equation with the fractional Laplacian and a white-colored noise

In this section, we will consider the stochastic heat equation with an additive Gaussian noise which behaves as a Wiener process in time and as a fractional Brownian motion in space, i.e. its spatial covariance is given by the so-called Riesz kernel. We will again study the distribution of the solution, its connection with the fractional and bifractional Brownian motion and we apply the q-variation method to obtain an asymptotically normal estimator for the drift parameter.

General properties of the solution

We will consider the stochastic heat equation ∂∂tuθ(t,x)=−θ(−Δ)α2uθ(t,x)+W˙γ(t,x),t≥0,x∈Rd, with uθ(0,x)=0 for every x∈Rd . In (27), −(−Δ)α2 denotes the fractional Laplacian with exponent α2 , α∈(1,2] , and Wγ is the so-called white-colored noise, i.e. Wγ(t,A),t≥0 , A∈B(Rd) , is a centered Gaussian field with covariance EWγ(t,A)Wγ(s,B)=(t∧s)∫A∫Bf(x−y)dxdy, where f is the so-called Riesz kernel of order γ given by f(x)=Rγ(x):=gγ,d‖x‖−d+γ,0<γ<d, where gγ,d=2d−γπd/2Γ((d−γ)/2)/Γ(γ/2) . As usual, the mild solution to (27) is given by uθ(t,x)=∫0t∫RdGα(θ(t−s),x−z)Wγ(ds,dz), where the above integral Wγ(ds,dz) is a Wiener integral with respect to the Gaussian noise Wγ .

We know the following facts concerning the mild solution (30) when θ=1 .

The mild solution (27) is well-defined as a square integrable process satisfying supt∈[0,T],x∈RdE|u1(t,x)|2<∞ if and only if d<γ+α. In particular, condition (31) shows that the solution exists in any spatial dimension d, via suitable choice of the parameter γ.

Assume (31) is satisfied. Then for every x∈Rd , we have the following equivalence in distribution u1(t,x),t≥0≡(d)c2,α,γBt12,1−d−γα,t≥0, where B12,1−d−γα is a bifractional Brownian motion with the Hurst parameters H=12 and K=1−d−γα and c2,α,γ2=c1,α,γ21−d−γα with c1,α,γ=(2π)−d∫Rddξ‖ξ‖−γe−‖ξ‖α12(1−d−γα).

For every t≥0 , we have (see Proposition 4.6 in [13]) u(t,x),x∈Rd≡(d)mα,γBα+γ−d2(x)+St(x),x∈Rd, where Bα+γ−d2 is an isotropic d-dimensional fractional Brownian motion (see the next section) with the Hurst parameter α+γ−d2 , (St(x))x∈Rd is a centered Gaussian process with C∞ sample paths and mα,γ2 is an explicit numerical constant.

Perturbed isotropic fractional Brownian motion

Since the law of the solution (30) is related to the isotropic fBm, let us recall the definition of this process. The isotropic d-parameter fBm (also known as the Lévy fBm) (BdH(x),x∈Rd) with the Hurst parameter H∈(0,1) is defined as a centered Gaussian process, starting from zero, with covariance function E(BdH(x)BdH(y))=12‖x‖2H+‖y‖2H−‖x−y‖2Hfor everyx,y∈Rd, where ‖·‖ denotes the Euclidean norm in Rd . It can be also represented as a Wiener integral with respect to the Wiener sheet, see [8, 15].

As in the one-parameter case, we define the q-variation of the isotropic fBm as the limit in probability as n→∞ of the sequence S[A1,A2]n,q(BH)= ∑i=0n−1BdH(xi+1)−BdH(xi)q, where xi=(xi(1),…,xi(d)) with xi(j)=A1+in(A2−A1) for i=0,…,n and j=1,…,d . And from [13] we know that the isotropic fBm (BH(x))x∈Rd has 1H -variation over [A1,A2] which is equal to (A2−A1)E|BdH(1)|1/H=(A2−A1)dE|Z|1/H.

The q-variation of the isotropic fBm perturbed by a regular multiparameter process has been obtained in [13], Lemma 4.1.

Let (BH(x))x∈Rd be a d-parameter isotropic fBm and consider a d-parameter stochastic process (X(x))x∈Rd , independent of BH , that satisfies E|X(x)−X(y)|2≤C‖x−y‖2,for everyx,y∈Rd. Define Y(x)=BdH(x)+X(x)for everyx∈Rd.

Then:

The process (Y(x))x∈Rd has 1H -variation which is equal to (A2−A1)dE|Z|1/H.

If H∈(0,12) and q≥2 , 1nVq,n(YH):=1n ∑i=0n−1nHqd−Hq/2(A2−A1)qH(YH(xi+1)−YH(xi))q−μq→(d)N(0,σH,q2).

It is immediate to deduce the rate of convergence in the above central limit theorem. Recall that we denoted by dW the Wasserstein distance.

Let YH be as in the statement of Lemma 4. Then for n large, dW1nVq,n(YH),N(0,σH,q2)≤C1n.

We notice that the Gaussian vector BdH(xi+1)−BdH(xi)0,1,…,n−1 has the same law as dH/2(BH(xj+1)−BH(xj))0,1,…,n−1 where B is a one-parameter fBm with the Hurst parameter H and we then apply Lemma 1. Therefore, the distribution of the sequence 1nVq,n(BdH) is independent of d≥1 and we can use the same argument as in the proof of Proposition 1 above.  □

 Estimators for the drift paramater

Throughout this section we will assume (31). As in the previous section, we will construct and analyze estimators for the drift parameter θ by using the limit behavior of the variations (in time and in space) of the process (30).

The law of the solution

Let us start by analyzing the distribution of the solution to (27) and its link with the (bi)fractional Brownian motion.

For every x∈Rd and θ>0 0$]]>, we have uθ(t,x),t≥0≡(d)θ−d−γ2αc2,α,γBt12,1−d−γα,t≥0, where B12,1−d−γα is a bifractional Brownian motion with parameters H=12 and K=1−d−γα and the constant c2,α,γ is defined by (33).

Denote vθ(t,x)=uθtθ,xfor everyt≥0,x∈Rd. Then, as in Lemma 2, vθ solves the equation ∂vθ∂t(t,x)=−(−Δ)α2vθ(t,x)+(θ)−12W˜γ˙(t,x),t≥0,x∈Rd, with vθ(0,x)=0 for every x∈Rd , where W˜γ˙ is a white colored Gaussian noise (i.e. a Gaussian process with zero mean and covariance (28)).

Fix x∈Rd and θ>0 0$]]>. For every s,t≥0 , we have Euθ(t,x)uθ(s,x)=Evθ(θt,x)vθ(θs,x)=θ−1Eu1(θt,x)u1(θs,x)=θ−1c1,α,γ(θt+θs)1−d−γα−|θt−θs|1−d−γα=θ−d−γαc2,α,γ2EBt12,1−d−γαBs12,1−d−γα.  □

For the behavior with respect to the space variable, we obtain the following result.

For every t≥0 , θ>0 0$]]>, we have the following equality in distribution uθ(t,x),x∈Rd≡(d)θ−12mα,γBα+γ−d2(x)+Sθt(x),x∈Rd, where Bα+γ−−d2 is a fractional Brownian motion with the Hurst parameter α+γ−d2∈(0,12] , (Sθt(x))x∈Rd is a centered Gaussian process with C∞ sample paths and mα,γ from (34).

The result is immediate since for a fixed time t>0 0$]]> uθ(t,x),x∈Rd=vθ(θt,x),x∈Rd≡(d)θ−12u1(θt,x),x∈Rd ≡(d)θ−12mα,γBα+γ−d2(x)+Sθt(x),x∈Rd.  □

 Estimators based on the temporal variation

Again tj=A1+jn(A2−A1) , j=0,…,n , will denote a partition of the interval [A1,A2] .

Assume (31). Let uθ be the solution to (27). Then for every x∈Rd , the process uθ(t,x),t≥0 admits 2αα+γ−d -variation over the interval [A1,A2] , i.e. S[A1,A2]n,2αα+γ−d:= ∑i=0n−1uθ(tj+1,x)−uθ(tj,x)2αα+γ−d→n→∞c2,α,γ2αα+γ−d2d−γα+γ−dμ2αα+γ−d(A2−A1)|θ|γ−dα+γ−d in probability.

Clearly, for fixed x∈Rd , ∑i=0n−1uθ(tj+1,x)−uθ(tj,x)2αα+γ−d= ∑i=0n−1v(θtj+1,x)−v(θtj,x)2αα+γ−d, where (vθ(t,x),t≥0)≡(d)(θ−12u1(t,x),t≥0) . And from Proposition 4.3 in [13] we know that u1 admits a variation of order 2αα+γ−d which is equal to c2,α,γ2αα+γ−dC12,1−d−γα(A2−A1) with C12,1−d−γα=2d−γα+γ−dμ2αα+γ−d and it means that ∑i=0n−1uθ(tj+1,x)−uθ(tj,x)2αα+γ−d→n→∞c2,α,γ2αα+γ−d2d−γα+γ−dμ2αα+γ−d(θA2−θA1)|θ−12|2αα+γ−dc2,α,γ2αα+γ−d2d−γα+γ−dμ2αα+γ−d(A2−A1)|θ|γ−dα+γ−d.  □

From relation (39) we can naturally define the following estimator for the parameter θ>0 0$]]> of the stochastic partial differential equation (27) θˆn,3=c2,α,γ2αα+γ−d2d−γα+γ−dμ2αα+γ−d(A2−A1)−1× ∑i=0n−1uθ(tj+1,x)−uθ(tj,x)2αα+γ−dα+γ−dγ−d=c2,α,γ2αα+γ−d2d−γα+γ−dμ2αα+γ−d(A2−A1)d−γα+γ−d×Sn,2αα+γ−d(uθ(·,x))α+γ−dγ−d, and so θˆn,3γ−dα+γ−d=1c2,α,γ2αα+γ−d2d−γα+γ−dμ2αα+γ−d(A2−A1) ∑i=0n−1uθ(tj+1,x)−uθ(tj,x)2αα+γ−d.

We have the following asymptotic behavior.

Assume 2αα+γ−d:=q is an even integer and consider the estimator θˆn,3 in (40). Then θˆn,3→n∞θ in probability and nθˆn,3γ−dα+γ−d−θγ−dα+γ−d→(d)N(0,s3,θ,α,γ2)withs3,θ,α,γ2=σ1q,q2θ2(γ−d)α+γ−dμ2αα+γ−d−2, and for n large enough, dWnθˆn,3γ−dα+γ−d−θγ−dα+γ−d,N(0,sθ,α2)≤c1n.

From Proposition 10 and the relation between the fractional and bifractional Brownian motion (see (18)), we can see that, as n→∞ , c2,α,γ−12d−γ2αθd−γ2αuθ(t,x),t≥0 converges to a perturbed fBm with Hurst parameter H=α−d+γ2α . By taking H=α+γ−d2α and q=1H=2αα+γ−d in Lemma 1, we get 1n ∑i=0n−1nθd−γα+γ−dc2,α,γ2αα+γ−d2d−γα+γ−d(A2−A1)uθ(tj+1,x)−uθ(tj,x)2αα+γ−d−μ2αα+γ−d→N(0,σ1q,q2) or, equivalently nμ2αα+γ−dθd−γα+γ−dθˆn,3γ−dα+γ−d−θγ−dα+γ−d→N(0,σ1q,q2), which is equivalent to (22). The bound (43) follows easily from Proposition 1.  □

We finally obtain the asymptotic normality and the rate of convergence for the estimator θˆn,3 .

Let θˆn,3 be given by (40) and s3,θ,α,γ be given by (42). Then as n→∞ , nθˆn,3−θ→(d)N0,s3,θ,α,γα+γ−dγ−d2θ2αα+γ−d and dWnθˆn,3−θ,N0,s3,θ,α,γα+γ−dγ−d2θ2αα+γ−d≤c1n.

It suffices to apply (23) with g(x)=xα+γ−dγ−d and γ0=θγ−dγ+α−d and to follow the proof of Proposition 5.  □

 Estimators based on the spatial variation

We will repeat the method employed in the previous parts of our work in order to define an estimator expressed in terms of the variations in space of the process (30) for the parameter θ in (27).

Recall that we proved in Proposition 11 that for every fixed time t>0 0$]]>, θ12mα,γ−1uθ(t,x),x∈Rd is a perturbed multiparameter isotropic fractional Brownian motion as defined in Lemma 4. Then we can deduce the variation in space of uθ recalling that xi=(xi(1),…,xi(d)) with xi(j)=A1+in(A2−A1) for i=0,…,n and j=1,…,d .

Let uθ be given by (30). Then ∑i=0n−1uθ(t,xj+1)−u(θt,xj)2α+γ−d→n→∞mα,γ2α+γ−d(A2−A1)dμ2α+γ−dθ−1α+γ−d

We use Lemma 4, point 1.  □

For every n≥1 , define θˆn,4=[(mα,γ2α+γ−dμ2α+γ−dd(A2−A1))−1× ∑i=0n−1uθ(t,xj+1)−uθ(t,xj)2α+γ−d]−(α+γ−d), and so θˆn,4−1α+γ−d=1mα,γ2α+γ−dμ2α+γ−dd(A2−A1) ∑i=0n−1uθ(t,xj+1)−uθ(t,xj)2α+γ−d.

We can deduce the asymptotic properties of the estimator by using Lemma 4 and Proposition 9.

The estimator (44) converges in probability as n→∞ to the parameter θ. Moreover, if 2α+γ−d is an even integer, then nθˆn,4−1α+γ−d−θ−1α+γ−d→N(0,s4,θ,α,γ2)withs4,θ,α,γ2=σα+γ−12,2α+γ−d2μ2α+γ−d−2θ−2α+γ−d. We also have, for n large enough, dWnθˆn,4−1α+γ−d−θ−1α+γ−d,N(0,s4,θ,α,γ2)≤c1n.

Finally, we get the following proposition.

With θˆn,4 from (44), as n→∞ , nθˆn,4−θ→(d)N0,s4,θ,α,γα+γ−dγ−d2θ2αα+γ−d and dWn,4θˆn−θ,N0,s4,θ,α,γα+γ−dγ−d2θ2αα+γ−d≤c1n.

Apply again (23) with g(x)=xα+γ−dγ−d and γ0=θγ−dγ+α−d .  □

Notice that in the case γ=1 (i.e., there is no spatial correlation and in this case d has to be 1), we retrieve the results of Section 2. Observe, as in Section 2, that the distance of the estimators (40) and (44) to their limit distribution is of the same order, although they involve q-variations of different orders.

 Conclusion

To conclude, in this paper we provide estimators based on power variation for the drift parameter θ of the solution to the fractional stochastic heat equation (3). The novelty of our approach is that it allows, comparing with the literature on statistical inference for SPDEs (see [4, 17, 2], etc.), to consider the case of a Gaussian noise with non-trivial spatial correlation and to treat the situation when the differential operator in the heat equation (3) is the fractional Laplacian instead of the standard Laplacian. The proofs of the asymptotic behavior of the estimators are relatively simple and they are based on the link between the law of the solution and the fractional Brownian motion, using known results on the behavior of the power variations of the fBm. Our approach also gives the rate of convergence of the estimators under the Wasserstein distance via some recent results in Stein–Malliavin calculus (see [19]). We assumed for simplicity a vanishing initial condition in (3) but the case of a notrivial initial value, whose power variations are dominated by those of the fBm, can be also treated by our approach. Another open problem of interest that could by treated via our techniques is adding an unknown volatility parameter in the disturbance term and jointly estimating the drift and the volatility parameters. The case of the fractional heat equation on bounded domains is also interesting but in this case the fundamental solution and implicitly the law of the mild solution changes. Consequently, the relation between the law of the solution and the fBm is not obvious and therefore new techniques are needed.

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