We study the law of the solution to the stochastic heat equation with
additive Gaussian noise which behaves as the fractional Brownian motion in time
and is white in space. We prove a decomposition of the solution in terms of the
bifractional Brownian motion

Bourguin, Solesne

Tudor, Ciprian A.

English

arXiv

We study the law of the solution to the stochastic heat equation with additive Gaussian noise which behaves as the fractional Brownian motion in time and is white in space. We prove a decomposition of the solution in terms of the bifractional Brownian motion

HAL-Paris1

On the law of the solution to a stochastic heat equationwith fractional noise in timeSolesne Bourguin, Ciprian A. TudorTo cite this version:Solesne Bourguin, Ciprian A. Tudor. On the law of the solution to a stochastic heat equationwith fractional noise in time. 2011. <hal-00631250>HAL Id: hal-00631250https://hal.archives-ouvertes.fr/hal-00631250Submitted on 12 Oct 2011HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.On the law of the solution to a stochastic heat equation withfractional noise in time ∗Solesne Bourguin† and Ciprian A. Tudor‡§Universite´ Paris 1 and Universite´ Lille 1Abstract: We study the law of the solution to the stochastic heat equation with additive Gaussiannoise which behaves as the fractional Brownian motion in time and is white in space. We provea decomposition of the solution in terms of the bifractional Brownian motion. Our result is anextension of a result in the paper [7].Keywords: Stochastic heat equation, Gaussian noise, bifractional Brownian motion.2010 AMS Classification Numbers: 60H15, 60H05.1 IntroductionThe purpose of this note is to make some remarks on the law of the solution to a heat equa-tion with a fractional noise in time. Let us briefly describe the context and the motivation.Consider a centered Gaussian field(W (t, x), t ∈ [0, T ], x ∈ Rd) with covariance given byE (W (t, x)W (s, y)) = (s ∧ t)(x ∧ y) (1)where for x = (x1, .., xd), y = (y1, ..., yd) ∈ Rd and where we denoted x∧ y =∏di=1 (xi ∧ yi).Also consider the stochastic partial differential equation with additive noise∂u∂t=12∆u+ W˙ , t ∈ [0, T ], x ∈ Rd (2)u0,x = 0, x ∈ Rd,where the noise W is defined by (1). The noise W is usually refered to as a space-timewhite noise. It is well-known (see for example the seminal paper by Dalang [3]) that theheat equation (2) admits a unique mild solution if and only if d = 1. This mild solution isdefined asu(t, x) =∫ T0∫RdG(t− s, x− y)W (ds, dy), t ∈ [0, T ], x ∈ Rd (3)∗Dedicated to the memory of Constantin Tudor†SAMM, Universite´ de Paris 1 Panthe´on-Sorbonne, 90, rue de Tolbiac, 75634, Paris, France. Email:solesne.bourguin@univ-paris1.fr‡Laboratoire Paul Painleve´, Universite´ de Lille 1, F-59655 Villeneuve d’Ascq, France. Email:tudor@math.univ-lille1.fr§Associate member of the team Samm, Universite´ de Paris 1 Panthe´on-Sorbonne. The second author ispartially supported by the ANR grant ”Masterie” BLAN 012103.1where the above integral is a Wiener integral with respect to the Gaussian process W (seee.g. [1] for details) and G is the Green kernel of the heat equation given byG(t, x) ={(4pit)−d/2 exp(− |x|24t)if t > 0, x ∈ Rd0 if t ≤ 0, x ∈ Rd.(4)Consequently, the process (u(t, x), t ∈ [0, T ], x ∈ R) is a centered Gaussian process. Thequestion we address in this paper comes from the following observation: it has been provenin [7] for d = 1 that the covariance of the solution (3) satisfies the following assertion : forevery x ∈ R we haveE (u(t, x)u(s, x)) =1√2pi(√t+ s−√|t− s|), for every s, t ∈ [0, T ]. (5)This fact establishes an interesting connection between the law of the solution (3) andthe so-called bifractional Brownian motion. Recall that the bifractional Brownian motion(BH,Kt )t∈[0,T ] is a centered Gaussian process, starting from zero, with covarianceRH,K(t, s) := R(t, s) =12K((t2H + s2H)K − |t− s|2HK) , s, t ∈ [0, T ] (6)with H ∈ (0, 1) and K ∈ (0, 1]. We refer to [4] and [6] for the definition and the basicproperties of this process. Note that, if K = 1 then BH,1 is a fractional Brownian motionwith Hurst parameter H ∈ (0, 1). When K = 1 and H = 12 then it reduces to the stan-dard Brownian motion. Relation (5) implies that the solution (u(t, x), t ∈ [0, T ], x ∈ R) toequation (2) is a bifractional Brownian motion with parameters H = K = 12 multiplied bythe constant 2−K 1√2pi. This is not the only connection between the solution of the heatequation and fractional processes. Another interesting link has been proven in [5].Our purpose is to study the law of the linear heat equation driven by a fractional noise intime. That is, we will consider a Gaussian field(WH(t, x), t ∈ [0, T ], x ∈ Rd) with covarianceE(WH(t, x)WH(s, y))=12(t2H + s2H − |t− s|2H)(x ∧ y) (7)where we will assume throughout the paper that the Hurst parameter H is contained in theinterval (12 , 1). Let us consider the linear stochastic heat equationut = ∆u+ W˙H , t ∈ [0, T ] , x ∈ Rd (8)with u(., 0) = 0, where (W (t, x))t∈[0,T ],x∈Rd is a centered Gaussian noise with covariance(7). The solution of (8) can be written in mild form asU(t, x) =∫ t0∫RdG(t− s, x− y)WH(ds, dy) (9)where the above integral is a Wiener integral with respect to the noise WH (see e.g [1] forthe definition) and G is given (4).The following result has been proven in [1] :2Theorem 1 The process (U(t, x))t∈[0,T ],x∈Rd exists and satisfiessupt∈[0,T ],x∈RdE(U(t, x)2)< +∞if and only if d < 4H.This implies that, in contrast to the white-noise case, we are allowed to consider the spatialdimension d to be 1, 2 or 3.We are interested in characterizing the law of the Gaussian process U(t, .) defined by (9)and to see its relation with the bifractional Brownian motion. In other words, we try tounderstand how relation (5) changes when the noise becomes fractional in time. We provethat the solution to the heat equation with additive noise fractional in time and white inspace can be decomposed in terms of the bifractional Brownian motion with Hurst param-eters H = K = 12 but in this case some additional Gaussian processes appear. Concretely,we show that for every x ∈ Rd,(U(t, x) + Y H , t ∈ [0, T ]) Law= (C0B 12 , 12t +XHt , t ∈ [0, T ]),where C0 > 0 is an explicit constant and XH , Y H are independent Gaussian processes, alsoindependent from U , with covariance functions (12) and (13) respectively. When H = 12 , aninteresting phenomenon happens: the process Y H vanishes while the constant C0 convergesto 12√2piwhen H is close to one half and the process XH converges to 12√2piB12, 12 for Hclose to one half (see Remark 2). In other words, in the fractional case H 6= 12 the solution“keeps” half of the bifractional Brownian motion B12, 12 while the other half “spreads” intotwo parts XH and Y H .2 Analysis of the covariance of the process U(t, .)Suppose that s ≤ t and denote byR(t, s) = E (U(t, x)U(s, x))where x ∈ Rd is fixed. We can write, from the proof of Theorem 2.7 in [1],R(t, s) = αH12√2pi∫ t0∫ s0|u− v|2H−2 ((t+ s)− (u+ v))− d2 dvdu (10)with αH = H(2H − 1).The purpose of this paragraph is to analyze the covariance (10) of the solution U(t, x) andto understand its relation with the bifractional Brownian motion. The following propositiongives a decomposition of the covariance function of U(t, .) in the case d 6= 2. The lines ofthe proof below will explain why the case d = 2 has to be excluded.3Proposition 1 Suppose d 6= 2. The covariance function R(t, s) can be decomposed asfollowsR(t, s) =12√2piαHCdβ(2H − 1,−d2+ 2)[(t+ s)2H−d2 − (t− s)2H− d2]+R(d)1 (t, s)where Cd =22−d , β(x, y) is the Beta function defined for x, y > 0 by β(x, y) =∫ 10 tx−1(1 −t)y−1dt andR(d)1 (t, s) =12√2piαHCd[∫ s0daa2H−2[((t+ s)− a)− d2+1 − ((t− s) + a)− d2+1]−∫ s0da(s − a)− d2+1 [(t− a)2H−2 + (t+ a)2H−2]] .Proof: Fix t > s. By performing the change of variables u − v = a and u + v = b witha+ b = 2u ∈ (0, 2t) and b− a = 2v ∈ (0, 2s) in (10), we getR(t, s) =12√2piαH∫ t−s|a|2H−2∫ (2t−a)∧(2s+a)a∨(−a)((t+ s)− b)− d2 dbda=12√2piαH[∫ 0−s(−a)2H−2∫ 2s+a−a((t+ s)− b)− d2 dbda+∫ t−s0a2H−2∫ 2s+aa((t+ s)− b)− d2 dbda+∫ tt−sa2H−2∫ 2t−aa((t+ s)− b)− d2 dbda].By performing the change of variables a 7→ (−a) in the first summand, we getR(t, s) =12√2piαH[∫ s0a2H−2∫ 2s−aa((t+ s)− b)− d2 dbda+∫ t−s0a2H−2∫ 2s+aa((t+ s)− b)− d2 dbda+∫ tt−sa2H−2∫ 2t−aa((t+ s)− b)− d2 dbda].Remark 1 We can notice why the case d = 2 must be treated separately in the latterequation. The integral with respect to db involves logarithms and it cannot lead to thecovariance of the bifractional Brownian motion.4By explicitly computing the inner integrals, we obtainR(t, s) =12√2piαHCd[∫ s0a2H−2[−((t+ s)− b)− d2+1]b=2s−ab=ada+∫ t−s0a2H−2[−((t+ s)− b)− d2+1]b=2s+ab=ada+∫ tt−sa2H−2[−((t+ s)− b)− d2+1]b=2t−ab=ada]=12√2piαHCd[∫ s0a2H−2((t+ s)− a)− d2+1da−∫ s0a2H−2((t− s) + a)− d2+1da]+αHCd[∫ t−s0a2H−2((t+ s)− a)− d2+1da−∫ t−s0a2H−2((t− s)− a)− d2+1da]+αHCd[∫ tt−sa2H−2((t+ s)− a)− d2+1da−∫ tt−sa2H−2(a− (t− s))− d2+1da]=12√2piαHCd[∫ t+s0a2H−2((t+ s)− a)− d2+1da−∫ t−s0a2H−2((t− s)− a)− d2+1da]+R(d)1 (t, s)whereR(d)1 (t, s)=12√2piαHCd[∫ s0a2H−2((t+ s)− a)− d2+1da−∫ s0a2H−2((t− s) + a)− d2+1da−∫ tt−sa2H−2(a− (t− s))− d2+1da−∫ t+sta2H−2((t+ s)− a)− d2+1da]. (11)At this point, we perform the change of variable a 7→ at+s and we obtain∫ t+s0a2H−2((t+ s)− a)− d2+1da = (t+ s)2H− d2∫ 10a2H−2(1− a)− d2+1da= β(2H − 1,−d2+ 2)(t+ s)2H−d2and in the same way, with the change of variable a 7→ at−s , we obtain∫ t−s0a2H−2((t− s)− a)− d2+1da = (t− s)2H− d2∫ 10a2H−2(1− a)− d2+1da= β(2H − 1,−d2+ 2)(t− s)2H− d2 .5As a consequence, we obtainR(t, s) = αHCdβ(2H − 1,−d2+ 2)[(t+ s)2H−d2 − (t− s)2H− d2]+R(d)1 (t, s)with R(d)1 given by (11). Let us further analyze the function denoted by R(d)1 (t, s). Notethat for every s, t ∈ [0, T ](12√2pi)−1R(d)1 (t, s) = A(t, s) +B(t, s)whereA(t, s) = αHCd[∫ s0a2H−2((t+ s)− a)− d2+1da−∫ s0a2H−2((t− s) + a)− d2+1da]andB(t, s) = αHCd[−∫ tt−sa2H−2(a− (t− s))− d2+1da−∫ t+sta2H−2((t+ s)− a)− d2+1da].By the change of variables a− t = a˜, we can express B asB(t, s) = αHCd[−∫ 0−s(a+ t)2H−2(a+ s)−d2+1da−∫ s0(a+ t)2H−2(s − a)− d2+1da]= −αHCd∫ s0da(s − a)− d2+1 [(t− a)2H−2 + (t+ a)2H−2]and the desired conclusion is obtained.Let us point out that the constant Cd is positive for d = 1 and negative for d = 3. Thispartially explains why we obtain different decompositions in these two cases. Thanks to thedecomposition in Proposition 1, we can prove the following proposition.Theorem 2 Assume d = 1 and let U be the solution of the heat equation with fractional-white noise (8). Let B12, 12 be a bifractional Brownian motion with parameters H = K = 12 .Consider (XHt )t∈[0,T ] to be a centered Gaussian process with covariance, for s, t ∈ [0, T ]RXH(t, s) = 212√2piαH∫ s0(s− a)2H−2[(t+ a)12 − (t− a) 12]da= H12√2pi∫ s0(s− a)2H−1[(t+ a)−12 + (t− a)− 12]da, (12)and let (Y Ht )t∈[0,T ] be a centered Gaussian process with covarianceRYH(t, s) = 212√2piαH∫ s0(s− a) 12 [(t+ a)2H−2 + (t− a)2H−2] da= H12√2pi∫ s0(s− a)− 12 [(t+ a)2H−1 − (t− a)2H−1] da. (13)6Suppose that U,XH , and Y H are independent. Then for every x ∈ Rd,(U(t, x) + Y H , t ∈ [0, T ]) Law= (C0B 12 , 12t +XHt , t ∈ [0, T ]),where C20 =12√2piαHCdβ(2H − 1,−d2 + 2).Proof: Let us first justify that RXHis a covariance function. Clearly, it is symmetric andit can be written, for every s, t ∈ [0, T ], as(12√2pi)−1RXH(t, s) = H∫ s∧t0(t ∧ s− a)2H−1[((t ∨ s) + a)− 12 + ((t ∨ s)− a)− 12]= H∫ ∞01[0,t](a)1[0,s](a)(t ∧ s− a)2H−1((t+ a)−12 ∧ (s+ a)− 12)da+H∫ ∞01[0,t](a)1[0,s](a)(t ∧ s− a)2H−1((t− a)− 12 ∧ (s− a)− 12)daand both summands above are positive definite (the same argument has been used in [2],proof of Theorem 2.1.). Similarly, the function RYHis a covariance. If d = 1, we haveCd = 2 andR(t, s) = 2αHβ(2H − 1, 32)[(t+ s)12 − (t− s) 12]+R(1)1 (t, s)with (12√2pi)−1R(1)1 (t, s) = 2αH∫ s0(s− a)2H−2[(t+ a)12 − (t− a) 12]da−2αH∫ s0(s− a) 12 [(t+ a)2H−2 + (t− a)2H−2] da= H∫ s0(s− a)2H−1[(t+ a)−12 + (t− a)− 12]da−2αH∫ s0(s− a) 12 [(t+ a)2H−2 + (t− a)2H−2] dawhere we used integration by parts in the first integral.In the case d = 3 we have the following.Theorem 3 Assume d = 3. Let B12, 12 be a bifractional Brownian motion with H = K = 12and let (ZHt )t∈[0,T ] be a centered Gaussian process with covariance R(3)1 (t, s). Then(U(t, x) + C0B12, 12 , t ∈ [0, T ])Law= (ZHt , t ∈ [0, T ]),with C0 defined in Theorem 2.7Proof: We have C3 = −2. In this case we can writeR(t, s) + 2αHβ(2H − 1, 12)[(t+ s)2H−32 − (t− s)2H− 32]= R(3)1 (t, s)with (12√2pi)−1R(3)1 (t, s) = −2αH∫ s0(s− a)2H−2[(t+ a)−12 − (t− a)− 12]da+2αH∫ s0(s− a)− 12 [(t+ a)2H−2 + (t− a)2H−2] .Note that R(3)1 is a covariance function because it is the sum of two covariance functions.Remark 2 Let us understand what happens with the decompositions in Theorems 2 and 3when H is close to 12 . We focus on the case d = 1. The phenomenon is interesting. Wefirst notice that the process Y H vanishes in this case. The covariance of the process XHbecomesRX12 (t, s) =14√2pi∫ s0((t+ a)−12 − (t− a)− 12)da =12√2pi((t+ s)12 − (t− s) 12).The constant C0 =1√2piαHβ(2H − 1, 32) is not defined for H = 12 because of the pres-ence of 2H − 1 in the argument of the beta function. But the following happens: since2αH1(0,1)(u)(1 − u)2H−2 is an approximation of the unity, it follows that αHβ(2H − 1, 32)converges to 12 when H goes to12 . Therefore C0 becomes12√2pi. Therefore we retrieve theresult in [7] and recalled in relation (5). In other words, in the fractional case H 6= 12 the so-lution “keeps” half of the bifractional Brownian motion B12, 12 while the other half “spreads”into two parts.3 Analysis of the processes XH and Y HIn this paragraph, we will prove some properties of the processes XH and Y H appearingin Theorem 2. It would be nice to find that these processes are more regular than thebifractional Brownian motion B12, 12 and therefore reduce the study of the solution U to thestudy of B12, 12 whose properties are relatively well-known.We start with the following remark.Proposition 2 The processes (XHt )t≥0, (Y Ht )t≥0 and (ZHt )t≥0 from Theorems 2 and 3 are2H − 12 self-similar.8Proof: Indeed, for every t ≥ s and for every c > 0(12√2pi)−1RXH(ct, cs) = H∫ cs0(cs− a)2H−1[(ct+ a)−12 + (ct− a)− 12]da= Hc2H−12(12√2pi)−1RXH(t, s)where we used the change of variables a′ = ac . The same calculation can be done for YHand ZH .Let us now estimate the L2 norm of the increments of the process XH in order to studythe regularity of its paths.Proposition 3 For t > s we haveE(|XHt −XHs |2) ≤ C|t− s| 12 and E (|Y Ht − Y Hs |2) ≤ C|t− s| 12with C a strictly positive constant. Consequently, XH and Y H each admit a version that isHo¨lder continuous of order δ ∈ (0, 14).Proof: Using the expression of the covariance of the process XH ,E|XHt −XHs |2 = RXH(t, t)− 2RXH (t, s) +RXH (s, s)= H∫ t0(t− a)2H−1[(t+ a)−12 + (t− a)− 12]da+H∫ a0(s− a)2H−1[(s+ a)−12 + (s− a)− 12]da−2H∫ s0(s− a)2H−1[(t+ a)−12 + (t− a)− 12]da:= T1(t, s) + T2(t, s) + T3(t, s)whereT1(t, s) = H∫ ts(t− a)2H−1[(t+ a)−12 + (t− a)− 12]da,T2(t, s) = H∫ s0((t− a)2H−1 − (s− a)2H−1) [(t+ a)− 12 + (t− a)− 12 ] daandT3(t, s) = H∫ s0(s− a)2H−1[(s + a)−12 + (s − a)− 12 − (t+ a)− 12 − (t− a)− 12]da.We claim that, for every t, s ∈ [0, T ], t > s, we haveC1|t− s|2H−12 ≤ T1(t, s) ≤ C2|t− s|2H−12 ,9T2(t, s) ≤ C3|t− s|2H−12 and T3(t, s) ≤ C4|t− s|12with C1, C2, C3, C4 strictly positive constants. Indeed, since (t+a)− 12 ≤ (t−a)− 12 we obtainT1(t, s) ≤ 2H∫ ts(t− a)2H−1(t− a)− 12 da = 2H2H − 12(t− s)2H− 12 .For the lower bound, since (t− a)− 12 ≥ (t− s)− 12 ,T1(t, s) ≥ H(t− s)−12∫ ts(t− a)2H−1da = H2H − 1(t− s)2H− 12 .Concerning the summands T2 and T3 we can writeT2(t, s) ≤ (t− s)2H−1H∫ s0(t− a)− 12 da = H2(t− s)2H− 12 .andT3(t, s) ≤ HT 2H−1∫ s0[(s + a)−12 + (s − a)− 12 − (t+ a)− 12 − (t− a)− 12]da=H2((2s)12 − (t+ s) 12 + (t− s) 12)≤ H(t− s) 12 .The calculations are similar for Y H . The Ho¨lder continuity of the paths follows by usingthe Kolmogorov criterium.Remark 3 A detailed study of the term denoted by T3 above shows that its Ho¨lder ordercannot be improved. Therefore the regularity of the paths of XH and Y H is the same asthose of the bifractional Brownian motion B12, 12 . Recall that B12, 12 is Ho¨lder continuous oforder strictly less than 14 .References[1] Balan, R.M. and Tudor, C.A. 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