2000 Mathematics Subject Classification: 60H15, 60H40We review results obtained in [13] and [14] on a one-dimensional Burgers-type stochastic differential equation involving fractional power of the Laplacian in its linear part, perturbed by a white noise term, with Dirichlet boundary conditions. We discuss existence of weak solutions and regularity of solutions

Kolkovska, Ekaterina T.

Bulgarian Digital Mathematics Library at IMI-BAS

Pliska Stud. Math. Bulgar. 16 (2004), 89-99STUDIA MATHEMATICABULGARICAON A STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONWITH A NOISY TERMEkaterina T. KolkovskaWe review results obtained in [13] and [14] on a one-dimensional Burgers-type stochastic differential equation involving fractional power of the Lapla-cian in its linear part, perturbed by a white noise term, with Dirichlet bound-ary conditions. We discuss existence of weak solutions and regularity ofsolutions.1. IntroductionWe consider the stochastic partial differential equation∂∂tu(t, x) =∂2∂x2u(t, x) + f(t, x, u(t, x)) +∂∂xg(t, x, u(t, x))+ σ(t, x, u(t, x))∂2∂t∂xW (t, x)with Dirichlet boundary conditionsu(t, 0) = u(t, 1) = 0, t ≥ 0and initial conditionu(0, x) = u0(x), x ∈ [0, 1],2000 Mathematics Subject Classification: 60H15, 60H40Key words: Burgers equation, white noise, weak and strong solutions, Hilbert spaceregularity.90 E. T. Kolkovskawhere ∂2∂t∂xW (t, x) is a space-time white noise (see [21] for the definition andproperties of white noise), u0 ∈ L2([0, 1]) and f ≡ f(t, x, y), g ≡ g(t, x, y),σ ≡ σ(t, x, y) are Borel-measurable functions on R+ × [0, 1] × R. A solution ofthe above equation is an L2([0, 1])-valued continuous process, adapted to thefiltration generated by the white noise, which solves the equation in a weak sense(see below).When f = σ = 0 and g(t, x, y) = y2/2 the above equation is called Burgersequation. It has been proposed as a model for turbulent fluid motion (see [4, 5,11]). When g = 0 the equation is a stochastic reaction-diffusion equation whichhas been studied intensively (see e.g. [8, 21, 3, 15] and the references therein).When f = 0, g(t, x, y) = y2/2 and σ 6= 0, we obtain the Burgers equationperturbed by a space-time white noise. It has been studied by several authorsunder Lipschitz conditions on σ (see e.g. [1, 7, 6, 10] and the references therein).Burgers equation involving fractional powers ∆α := −(−∆)α/2, α ∈ (0, 2], ofthe Laplacian in its linear part has also been studied in connection with modelsof several hydrodynamical phenomena (see e.g. [20], [9], [6] and the referencestherein for applications).In [6] Biller, Funaki and Woyczynski studied existence, uniqueness, regularityand asymptotic behavior of solutions of the multidimensional fractal Burgers-typeequation∂∂tu(t, x) = ν∆αu(t, x)− a∇ur(t, x),(1)where x ∈ Rd, d ≥ 1, α ∈ (0, 2], r ≥ 1, and a ∈ Rd is a fixed vector. For α > 3/2and d = 1 they proved existence of a unique regular weak solution of (1) withinitial conditions in H1(R).In [13] it is proved existence of a weak solution of the one-dimensional stochas-tic Burgers equation perturbed by a white noise term with a non-Lipschitz coef-ficient∂∂tu(t, x) = ∆u(t, x) + λ∇u2(t, x) + γ√u(t, x)(1 − u(t, x)) ∂2∂t∂xW (t, x),u(t, 0) = u(t, 1) = 0,u(0, x) = f(x), x ∈ [0, 1],(2)where f : [0, 1] → [0, 1] is continuous and ∂2∂t∂xW (t, x) is the space-time whitenoise. Equation (2) is interpreted in the weak sense, which means that for eachϕ ∈ C2([0, 1]),∫[0,1]u(t, x)ϕ(x) dx =∫[0,1]u(0, x)ϕ(x) dx +∫[0,1]u(t, x)ϕ′′(x) dx−(3)Stochastic partial differential equation 91−λ∫ t0∫[0,1]u2(s, x)ϕ′(x) dx ds+ γ∫ t0∫[0,1]√u(s, x)(1 − u(s, x))ϕ(x)W (ds, dx).The method of proof in [13] consists in approximating (2) by finite systemsof stochastic differential equations possessing a unique strong solution. Usingbounds for the fundamental solution of the discrete Laplacian, it is shown tight-ness of the approximating systems, and that each weak limit is a weak solutionof (2).In this paper we concentrate mainly on the one-dimensional fractal Burgersequation∂∂tu(t, x) = ∆αu(t, x) + λ∇u2(t, x) + γ√u(t, x)(1 − u(t, x)) ∂2∂t∂xW (t, x),u(t, 0) = u(t, 1) = 0, x ∈ [0, 1],(4)where the random positive initial condition u(0, x) is bounded by 1.Due to the presence of non-Lipschitz coefficients, existence and uniquenessof a weak solution of (4) cannot be achieved by classical results. Following themethod of proof of [13], in this paper we consider a discrete version of (4). Usingthe method of [13], we prove existence of a strong solution of the correspondingfinite system of SDEs. We discuss also existence and regularity properties of aweak solutuons proved in [14].2. Notations and basic resultsWe recall some notations from [2]. Let S = [0, 1) and let T denote the quotientspace obtained from [0, 1] by identifying 0 and 1. We put ϕ0(x) = 1 for x ∈ [0, 1],andϕn(x) =√2 cos(pinx), ψn(x) =√2 sin(pinx), x ∈ [0, 1], n = 2, 4, . . . .This system of functions, which we also denote by em, m = 0, 1, 2, ..., is the usualorthonormal basis in L2([0, 1]). Moreover, for all n, ∆en = −pi2n2en. For anyβ ∈ R we define Hβ as the Hilbert space obtained from L2(S) by completionwith respect to the norm|f |β =(∑〈f, em〉2(1 + pi2m2)β)1/2,where 〈·, ·〉 denotes the usual inner product in L2(S).For any integer N ≥ 1, let H(N) denote the set of functions f : [0, 1] → Rwhich are constant on [ kN ,k+1N ) for k = 0, 1, 2, ..., N −1. Clearly we have H(N) ⊂L2([0, 1]).92 E. T. KolkovskaWithout loss of generality we assume that λ = γ = 1.Let us fix a positive integer N , and consider the discretized version of (4),namely the system∂∂tXN (t, r) = ∆N,αXN (t, r) +∇NXN (t, r)2 ++√NXN (t, r)(1 −XN (t, r)) dBN (t, r),XN (0, r) = X(0, r), r = 0, 1N , ...,N−1N , t ≥ 0,(5)where ∆N,α is the fractional power of the discrete Laplacian, and {BN (t, r)}ris a sequence of independent Brownian motions. Our results are the followingtheorems.Theorem 1. (a) For any positive initial random condition XN (0) boundedby 1, there exists a unique strong solution XN (t) of (5) in the spaceC([0,∞), L2([0, 1]).(b) The distributions of {XN} are relatively compact on C((0,∞) : Hβ) ifβ ≤ 0, α > β + 3/2, and on C([0,∞) : Hβ) for α > β + 3/2, β < −1/2.(c) For any α > 3/2, equation (4) has a weak solution in the space C((0,∞),L2([0, 1])).Remark 1. Theorem 1 is consistent with results obtained in [6] for the caseγ = 0.Theorem 2. The weak solution X(t) in Theorem 1 has a modification whichis Holder continuous in time: it satisfiesP(sup0<s0≤s<t≤T|X(t)−X(s)|β|t− s|δ <∞)= 1for each 0 < δ < [(2α − 2β − 3)/(2α)] ∧ 1/2, 3/2 < α ≤ 2, and β < (2α − 3)/2.Remark 2.In particular, when α = 2 and 0 ≤ β < 1/2, we can take 0 < δ <1−2β4 , and obtainP (X ∈ C((0,∞) : Hβ)) = 1.Thus X(t) is smoother than an L2([0, 1]) function for t > 0. This is due to theregularization property of the Laplacian.Remark 3.We are going to prove only part (a) of Theorem 1. The remainingresults, including Theorem 2, are proved in [14].Stochastic partial differential equation 933. Proof of Theorem 1(a)Let us write xNr (t) = XN (t, r). The system (5) then can be written in the morecompact formdxNi (t) = N∑j=1aNijxNj (t) + bNijxNj (t)2 dt+√NxNi (t) (1− xNi (t)) dBi(t)(6)wherebNij =N if j = i+ 1,−N if j = i,0 otherwise,and aNij are the coefficients of ∆N,α.For example, in the case α = 2, (5) takes the form:∂∂tXN(t,kN)= ∆NXN(t,kN)+∇N(XN(t,kN)2)(7)+√NXN(t,kN)(1−XN(t,kN))dBk(t),1 ≤ k ≤ N, t ≥ 0.Here {Bk(t)}1≤k≤N is an infinite system of independent one-dimensional Brow-nian motions and ∇N and ∆N are, respectively, the discrete approximations ofthe first and second derivative with respect to the variable x:∆NXN(t,kN)=XN(t, k+1N)− 2XN (t, kN )+XN (t, k−1N )1N2,∇N h(s,kN)=h(s, k+1N)− h (s, kN )1N, 1 ≤ k ≤ N.Substituting the above expressions in equation (7), we obtain the finite-dimensio-nal system of stochastic differential equationsdxNi (t) = N2[xNi+1(t)− 2xNi (t) + xNi−1(t)]+NxNi+1(t)2 −NxNi (t)2+√NxNi (t)(1− xNi (t))dBi(t), i = 1, ..., N,94 E. T. Kolkovskawhich can be written as (6) with withaNij =N2 if j = i+ 1, i − 1,−2N2 if j = i,0 otherwiseandbNij =N if j = i+ 1,−N if j = i,0 otherwise.In the general case 0 < α ≤ 2 we also will have ∑i aNij = 0 because the operator∆N,α is symmetric.Note that for system (6) we cannot apply standard results on existence anduniqueness of solution because Lipschitz assumptions on the drift and diffusioncoefficients fail. We prove the following result.Lemma 1. For any initial random condition XN (0) = (xN1 , ..., xNN ) ∈ [0, 1]N ,the systemdxNi (t) =∑jaNijxNj (t) +∑jbNijxNj (t)2 dt+(8)+√NxNi (t)(1− xNi (t)) dBi(t)xNi (0) = xi, i = 1, ..., N,admits a unique strong solution XN (t) = (xN1 (t), . . . , xNN (t)) ∈ C([0,∞), [0, 1]N ).P r o o f. Let us consider the re-scaled systemdxNi (t) =∑jaNijxNj (t) +∑jbNijxNj (t)2 dt+√g(xNi (t)) dBi(t)(9)xNi (0) = xi, i = 1, ..., N,where g : R → R is defined by g(x) = Nx(1 − x) for 0 ≤ x ≤ 1, and g(x) = 0otherwise. Since the coefficients of (9) are continuous, by Skorohod’s existencetheorem ([19, 12]) we conclude that on some probability space there exists aweak solution XN (t) of (9). We will prove that each weak solution XN (t) =(xN1 (t), . . . , xNN (t)) of this system is bounded: xNi (t) ∈ [0, 1] for all i = 1, . . . , Nand t ≥ 0, thus showing that XN (t) also solves (8).Stochastic partial differential equation 95First we show that xNi (t) ≥ 0 for each i = 1 . . . , N . Since the coefficients ofthe system are non-Lipschitz, the solution may explode in finite time. Let τ1 ≤ ∞denote the explosion time of the solution. If some of the solution coordinates arenegative, then there exists a random time 0 < τ2 ≤ ∞ such that for 0 < t ≤ τ2all such coordinates are between −1 and 0. This is so because there is only finitenumber of coordinates, and they are continuous.In order to obtain pathwise uniqueness of weak solutions we shall use thelocal time techniques of Le Gall combined by the classical method of Ikeda andWatanabe (see e.g. [18], Chapter V, §43) We state the following result of Le Gall([16]).Lemma 2. Let Z ≡ {Z(t), t ≥ 0} be a real-valued semimartingale. Supposethat there exists a function ρ : [0,∞) → [0,∞) such that ∫ ε0 duρ(u) = +∞ for allε > 0, and∫ t01{Zs>0}ρ(Zs)d〈Z〉s <∞ for all t > 0 a.s. Then the local time at zero ofZ, L0t (Z), is identically zero for all t a.s.Applying Lemma 2 to xNi (t) with ρ(u) = u, and using the Tanaka’s formula(see [17]), after summation we obtain for xNi (t)− := max[0,−xNi (t)],N∑i=1xNi (t)− = −∫ t0N∑i=11xNi(s)<0N∑j=1(aNijxj(s) + bNijxj(s)2) ds≤∫ t0N∑i,j=11xNi(s)<0aNijxj(s)− ds+N∫ t0N∑i=11xNi(s)<0xi(s)2 ds≤∫ t0N∑i,j=1aNijxj(s)− ds+N∫ t0N∑i=1xNi (s)− ds= N∫ t0N∑i=1xNi (s)− ds,where we used that∑i aNij = 0 to obtain the last equality. Then by Gronwall’slemma we obtain that∑Ni=1 xNi (t)− = 0, and hence that the solution is non-negative for each t ≥ 0. By a similar argument applied to (1−xNi (t))−, it followsthat xNi (t) ≤ 1 for each 1 ≤ i ≤ N.Let X1,N = (x1,N1 , . . . , x1,NN ) and X2,N = (x2,N1 , . . . , x2,NN ) be two weak so-lutions of (8) with the same initial conditions and the same Brownian motions.96 E. T. KolkovskaThen,x1,Ni (t)− x2,Ni (t) ==∫ t0∑jaNij(x1,Nj (s)− x2,Nj (s))+ bNij(x1,Nj (s)2 − x2,Nj (s)2) ds++∫ t0[√Nx1,Ni (s)(1− x1,Ni (s))−√Nx2,Ni (s)(1− x2,Ni )(s)]dBi(s),i = 1, . . . , N.Since〈X〉t =∫ t0[√Nx1,Ni (s)(1− x1,Ni (s))−√Nx2,Ni (s)(1− x2,Ni )(s)]2dsand∫ t0[√Nx1,Ni (s)(1− x1,Ni (s))−√Nx2,Ni (s)(1− x2,Ni )(s)]2x1,Ni (s)− x2,Ni (s)××1x1,Ni(s)−x2,Ni(s)>0ds ≤∫ t02N1x1,Ni(s)−x2,Ni(s)>0ds < 2Nt(where we used that (√x(1− x) −√y(1− y) )/(x − y) < 2 for x, y ∈ [0, 1],x > y, which follows from L’Hospital rule), we can apply Lemma 3.2 to Z(t) =x1,Ni (t) − x2,Ni (t) with ρ(x) = x. Therefore, L0t(x1,Ni (s)− x2,Ni (s))= 0 for alli ∈ {1, . . . , N}.Applying Tanaka’s formula again,∣∣∣x1,Ni (t)− x2,Ni (t)∣∣∣ =∫ t0sgn(x1,Ni (s)− x2,Ni (s))××∑jaNij(x1,Nj (s)− x2,Nj (s))+ bNij(x1,Nj (s)2 − x2,Nj (s)2) ds++∫ t0sgn(x1,Ni (s)− x2,Ni (s))[√Nx1,Ni (s)(1− x1,Ni (s))−−√Nx2,Ni (s)(1− x2,Ni (s))]· dBi(s), i = 1, . . . , N.Stochastic partial differential equation 97Since aNij and bNij are bounded, it follows thatEN∑i=1∣∣∣x1,Ni (t)− x2,Ni (t)∣∣∣≤∫ t0EN∑i=1∣∣∣∣∣∣∑jaNij(x1,Nj (s)− x2,Nj (s))+ bNij(x1,Nj (s)2 − x2,Nj (s)2)∣∣∣∣∣∣ ds≤∫ t0K(N) EN∑i=1∣∣∣x1,Ni (s)− x2,Ni (s)∣∣∣ ds,where K(N) is a constant depending on N. From Gronwall’s inequality we con-clude thatEd∑i=1∣∣∣x1,Ni (t)− x2,Ni (t)∣∣∣ = 0for all t ≥ 0, thus proving pathwise uniqueness.By a classical theorem of Yamada and Watanabe [22], this is sufficient forexistence of a unique strong solution of (8).Acknowledgement The author wishes to thank the organizers of the So-zopol Probability School 2003 for their hospitality and for giving her the oppor-tunity to participate.REFERE NCES[1] L. Bertini, N. Cancrini and G. Jona-Lasinio. The stochastic Burgersequation. Comm. Math. Phys. 164 (1994), 211–232.[2] D. Blount. Fourier analysis applied to SPDEs, Stocastic Processes Appl.62 (1996), 223–242.[3] D. Blount. Density dependant limits for a non-linear reaction-diffusionmodel. Ann. Prob. 22 (1994) 2040–2070.[4] J. M. Burgers. A mathematical model illustrating the theory of turbu-lence. Adv. Appl. Mech. 1 (1948), 171-199.[5] J. Burgers. The nonlinear diffusion equation. Asymptotic solutions andstatistical problems. D. Reidel Publishing, Dordrecht, 1974.98 E. T. Kolkovska[6] P. Biler, T. Funaki and W. Woyczinski. Fractal Burgers equations.Journal of Differential Equations 148 (1998), 9–46.[7] G. Da Prato and D. Gatarek. Stochastic Burgers equation with cor-related noise. Stoc. Stochastics reports 52 (1995), 29–41.[8] T. Funaki. Random motion of strings and related stochastic evolutionequations. Nagoya Math. J. 89 (1983), 129–193.[9] T. Funaki, D. Surgailis and W. Woyczynski. Gibbs-Cox randomfields and Burgers turbulence. Ann. Appl. Prob. 5 (1995), 701–735.[10] I. Gyongy. Existence and uniqueness results for semilinear stochasticpartial differential equations. Stoc. Proc. Appl. 73 (1998), 271–299.[11] E. Hopf. The partial differential equation ut + uux = µuxx. Commun.Pure Appl. Math. 3 (1950), 201–230.[12] N. Ikeda and S. Watanabe. Stochastic differential Equations and Dif-fusion Processes, North-Holland, 1989.[13] E. Kolkovska. On a stochastic Burgers equation with Dirichlet bound-ary conditions. To appear in Intern. Journal of Math. Math. Sciences 43(2003).[14] E. Kolkovska. On a fractal Burgers equation with a stochastic noisyterm. Submitted. (2003).[15] P. Kotelenez. High density limit theorems for nonlinear quemical reac-tions with diffusion. Prob.Th. Rel. Fields 93 (1998), 11–37.[16] J. F. Le Gall. Applications du temps local aux equations differen-tielles stochastiques unidimentionnelles. Sem. Prob. XVII, Lecture Notesin Math. 986 (1983), 15–31.[17] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion,2nd Ed., Springer, 1994.[18] L. C. G. Rogers and D. Williams. Diffusions, Markov Processes andMartingales, 2nd Ed., Cambridge University Press, 2000.[19] A. V. Skorohod. Studies in the theory of random processes, AddisonWesley, 1965.Stochastic partial differential equation 99[20] M. F. Shlesinger, G.M. Zaslavsky and U. Frisch (Eds). Le´vy Flightsand related topics in physics. Lecture Notes in Physiscs, Vol. 450, Springer-Verlag, Berlin 1995.[21] J. B. Walsh. An Introduction to stochastic partial differential equations.Lecture Notes in Mathematics, Vol. 1180, Springer-Verlag, New York, 1986.[22] T. Yamada and S. Watanabe. On the uniqueness of solutions of stochas-tic differential equations. J. Math. Kyoto Univ. 11 (1971), 155–167.Ekaterina KolkovskaCentro de Investigacio´n en Matema´ticasApartado postal 402, 36000 Guanajuato, Mexicoe-mail: todorova@cimat.mx

On a Stochastic Partial Differential Equation with a Noisy Term

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On a Stochastic Partial Differential Equation with a Noisy Term

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