O THE BURGERS EQUATIO WITH A STOCHASTIC STEPPIG STOE OISY TERM Eaterna T. Kolovsa Comuncacón Técnca o I-2-14/11-7-22 PE/CIMAT
On the Burgers Equaton wth a stochastc steppng-stone nosy term Eaterna T. Kolovsa CIMAT, Guanajuato, Mexco Abstract We consder the one-dmensonal Burgers equaton perturbed by a whte nose term wth Drchlet boundary condtons and a non-lpschtz coeffcent. We obtan exstence of a wea soluton provng tghtness for a sequence of polygonal approxmatons for the equaton and solvng a martngale problem for the wea lmt. Key words: Burgers equaton, whte nose, strong solutons, polygonal approxmatons, martngale problem. 1 Introducton The one-dmensonal Burgers equaton where = 2 x 2 [3, 4, 8]. t ut, x = ν ut, x λ 2 u2 t, x, and = x, has been proposed as a model for turbulent flud moton see Burgers equatons perturbed by space-tme whte noses wth Lpschtz coeffcents have been studed recently by several authors see e.g. [1, 6, 2] and the references theren. Our am n ths paper s to study a one-dmensonal Burgers equaton perturbed by a stochastc nose term wth a non-lpschtz coeffcent, namely, t ut, x = ut, x λ u2 t, x γ 2 ut, x1 ut, x W t, x, t x ut, = ut, 1 =, 1.1 u, x = fx, x [, 1], where fx : [, 1] [, 1] s a contnuous functon and 2 t xw t, x s the space-tme whte nose see [16] for the defnton and propertes of the whte nose. The stochastc term n Research partally supported by COACyT Grant o. 3241E 1
ths equaton corresponds to contnuous-tme steppng-stone models n populaton genetcs [5, 12], where ut, x models the gene frequency n colones. Equaton 1.1 s nterpreted n the wea sense, whch means that for each C 2 [, 1], ut, xx dx = u, xx dx ut, x x dx 1.2 [,1] λ γ [,1] [,1] [,1] [,1] u 2 s, x x dx ds us, x1 us, xx W ds, dx. Snce the coeffcents of 1.1 are non-lpschtz, the standard results on exstence and unqueness of solutons cannot be appled. In ths paper we prove exstence of a non-negatve wea soluton of 1.1 Theorem 4.2. Our method of proof s brefly descrbed as follows. Followng Funa [7], n Secton 1 we defne a dscrete verson of 1.1, whch s a fnte-dmensonal system 2.2 of stochastc dfferental equatons. In Secton 2 we prove exstence of a wea soluton for ths system, and use the method of Le Gall [11] to obtan pathwse unqueness of wea solutons. Ths yelds exstence of a unque strong soluton x t of 2.2. ext, n Secton 3, we defne a system of polygonal approxmatons u t, x of x t, and use the multdmensonal Kolmogorov- Toto crteron to obtan tghtness of {u t, x, = 1, 2,...}. In the last secton we use a martngale problem to conclude the proof of exstence of a wea soluton of equaton 1.1. 2 Exstence of a soluton of the dscretzed verson Let us fx an nteger 1 and consder the dscretzed verson of 1.1 on the set {, 1 }: t X t, X, = X t, X 2 t, X t, 1 X t, db t, = f, 1, t. 2.1 Here s the set of the non-negatve nteger numbers, {B t} 1 s an nfnte system of ndependent one-dmensonal Brownan motons and and are, respectvely, the dscrete approxmatons of the frst and second dervatve wth respect to the varable x: X t, = X t, 1 2X t, 2 X t, 1 1, 2
h s, = h s, 1 h s, 1, 1. Let us wrte x t = X t,. Substtutng the above expressons n equaton 2.1 we obtan the fnte-dmensonal system of stochastc dfferental equatons dx t = 2 [ x 1t 2x t x 1t ] x 1t x t x t 1 x t db t, = 1,...,, whch can be wrtten n the more compact form dx t = a j x j t b j x j t 2 dt where and x = f/, 1, j, 2 f j = 1, 1, a j = 2 2 f j =, otherwse f j = 1, b j = f j =, otherwse. x t 1 x t db t 2.2 ote that for ths system we cannot apply the standard results on exstence and unqueness of soluton because Lpschtz assumptons on the drft and dffuson coeffcents fal. We prove the followng result. Theorem 2.1. For any ntal random condton X = x 1,..., x [, 1], the system dx t = a j x j t b j x j t 2 dt x t1 x t db t 2.3 j j x = x, = 1,...,, admts a unque strong soluton X t = x 1 t,..., x t C[,, [, 1]. Proof. Let us consder the re-scaled system dx t = j a j x j t j b j x j t 2 dt gx t db t 2.4 x = x, = 1,...,, 3
where g : R R s defned by gx = x1 x for x 1, and gx = otherwse. Snce the coeffcents of 2.4 are contnuous, by the Sorohod s exstence theorem [14, 1] we conclude that there exsts on some probablty space a wea soluton X t of 2.4. We wll prove that for each wea soluton X t = x 1 t,..., x t of ths system, x t [, 1] for all = 1,..., and t, thus showng that X t s a soluton of 2.3. Frst we show that x t for each = 1...,. Snce the coeffcents of the system are non-lpschtz, the soluton may explode n fnte tme. Let τ 1 denote the exploson tme of the soluton. If some of the soluton coordnates are negatve, then there exsts a random tme < τ 2 such that for < t τ 2 all such coordnates are between 1 and. Ths s so because 2.4 s fnte-dmensonal, and ts soluton s contnuous. We shall use the followng lemma [11]. Lemma 2.2. Let Z {Zt, t } be a real-valued semmartngale. Suppose that there exsts a functon ρ : [, [, such that ε du ρu = for all ε >, and 1 {Zs>} ρz s d Z s < for all t > a.s. Then the local tme at zero of Z, L t Z, s dentcally zero for all t a.s. Applyng Lemma 2.2 to x t wth ρu = u and usng the Tanaa formula [13], we obtan for x t := max[, x t], =1 x t = = where we used that a j 1 x s< =1 a j x j s b j x j s 2 ds 1 x s< a j x j s ds, a j x j s ds, x s ds, =1 1 x s< x s 2 ds =1 x s ds =1 = to obtan the last equalty. Then by Gronwall s lemma we obtan that =1 x t =, and hence that the soluton s non-negatve for each t. By a smlar argument appled to 1 x t, t follows that x t 1 for each 1. Usng Lemma 2.2 we shall prove pathwse unqueness of wea solutons of 2.3. X 1, = x 1, 1,..., x 1, and X2, = x 2, 1,..., x 2, be two solutons of 2.3 wth the same ntal condtons and the same Brownan motons. We defne d X l, t Let = a j x l, j t b j x l, j t 2, t, l = 1, 2. 4
Then x 1, t x 2, t = [ d X 1, s d X 2, s ] ds [ x 1, = 1,...,, s1 x 1, s x 2, s1 x 2, ] s db s, Snce and X t = [ x 1, 21 x 1, [ x 1, s1 x 1, s x 2, s1 x 1, x 1, s x 2, s x 2, s ds < 2t s> 2 s x 2, s1 x 2, s] ds s1 x 2, ] 2 s 1 x 1, s x 2, s> ds where we used that x1 x y1 y /x y < 2 for x, y [, 1], x > y, whch follows from L Hosptal rule, we can apply Lemma 2.2 to Xt = x 1, t x 2, t wth ρx = x. Therefore, L t x 1, s x 2, s = for all {1,..., }. By Tanaa s formula, x 1, t x 2, t = sgn x 1, s x 2, d s X 1, s d X 2, s ds sgn x 1, s x 2, db s, = 1,...,. [ s x 1, s 1 x 1, s x 2, s 1 x 2, ] s Snce a j and b j are bounded, t follows that E x 1, t x 2, t =1 E d X 1, s d X 2, s ds =1 K E x 1, s x 2, s ds, =1 5
where K s a constant dependng on. From Gronwall s nequalty we conclude that d E x 1, t x 2, t = =1 for all t, thus provng pathwse unqueness. By a classcal theorem of Yamada and Watanabe [17], ths s suffcent for exstence of a unque strong soluton of 2.3. 3 Tghtness of the approxmatng processes As a consequence of Theorem 2.1, there exsts a strong soluton of the system approxmatng 1.1 dx t = a j x j t b j x j t 2 x t 1 x t db t 3.1 x = f 1 j 1 j, = 1, 2,...,, where s fxed. We denote by u t, x the polygonal approxmaton of x t, u t, x = X t, [x] 1 x [x] X t, [x] [x] 1 x, 3.2 t, x [, 1], where by defnton [y] = for y < 1. Therefore we have X t, = x t = u t,, 1, and u t, x 1 for all t, x [, 1]. Let p t,, j, t,, j 1, be the fundamental soluton of,.e., t p t,, j p,, j = p t, = δ j,, j, t >, 1, j wth the boundary condtons p t,, j = p t, 1, j =, t >, 1 j. Then 3.1 s equvalent to the system see [9] x 1 t = p t,, j x j 6 1 p t s,, j b, jx j s 2 ds
[p t s,, j x ] j s 1 x j s db j s, 1, where n the last ntegral we used that { 1 B js, 1 j } s an ndependent system of Brownan motons whch we also denote by {B j s}. Let us defne the re-scaled polygonal nterpolaton G of p n [, 1] by G t, x, j = p t, [x] 1, j x [x] p t, [x], j [x] 1 x. Usng 3.2 and 3.3 we obtan the followng representaton for the approxmate soluton. For x [, 1, u t, x = 1 G t, x, j u, j 1 p := u 1 [ 1 p t s, 1, j b 1, jx j s 2 x [x] t s,, j ] b, jx j s 2 [x] 1 x ds G t s, x, j u s, j 1 u s, j db j s ds t, x u2 t, x u3 t, x. Then u t, x satsfes the boundary condtons n 1.1. Theorem 3.1. For each T >, the sequence {u t, x, 1} s tght n the space C[, T ], A, where A = C[, 1], [, 1]. Proof. Usng the fact that u t, x [, 1], we obtan, as n the proof of Lemma 2.2 and Proposton 2.1 n [7], that for each T < and p there exsts C = CT, p > such that E u 3 t 1, x u 3 t 2, y 2p C t 1 t 2 p/2 x y p/2 3.3 for every t 1, t 2 [, T ], x, y [, 1] and, and that lm sup t,y [,T ] [,1] where ut, y s the fundamental soluton of. Snce u 2 t, = u 1 t, y ut, y =, 3.4 [p t s,, 1 x 1 s2 p t s,, ] x s2 ds 7
and p s a fundamental soluton of, t follows that t u2 t, = u 2 t, u t, 1 2 u t, 2. Also, u 2 t, = u2 t, 1 = by the boundary condton 3.3. Integraton by parts and an applcaton of Gronwall s nequalty gve, as n [9] Thm. 4.2, max 1 u2 t, t e 2t max 1 u1 s, u 3 s, ds. Hence, from the polygonal form of u 2 and 3.3, 3.4 we obtan that for every T < and p there exsts C 1 = C 1 T, p > such that E u 2 t 1, x u 2 t 2, y 2p C 1 t 1 t 2 p/2 x y p/2 3.5 for every t 1, t 2 [, T ], x, y [, 1] and. It follows from the multdmensonal Kolmogorov s-toto crteron [15] that u t, x C[, T ], A and that the famly {u t, x, } s tght n C[, T ], A for each postve T. 4 The martngale problem for the SPDE Snce the sequence {u t, x, 1} s tght by Theorem 3.1, there exsts a subsequence, whch we denote agan by {u t, x}, that converges wealy n C[, T ], A to a lmt vt, x. By the Sorohod s representaton theorem we can construct processes {v t, x}, ut, x on some probablty space Ω, F, {F t }, P, such that {u } D = {v }, u D = v, and {v t, x} converges to ut, x unformly on compact subsets of [, T ] R for any T > as. Obvously ut, x satsfes the boundary condtons n 1.1. We shall show that ut, x s a wea soluton of 1.1 by solvng the correspondng martngale problem. Theorem 4.1. For any C 2 c, M t := [,1] ut, xx dx u, xx dx ut, x x dx [,1] [,1] [,1] u 2 s, x x dx s an {F t }-martngale wth M t = [,1] us, x1 us, x2 x dx ds. Proof. Usng that n Z a n b n1 b n n Z 8 b n1 a n1 a n =
and multplyng both sdes of 2.1 by 1, we obtan for fxed 1, M t := =1 = = u t, 1 =1 =1 u s, 1 1 v t, v s, 1 v s, u, 1 v, 1 {u 2 =1 1 v 2 1 v s, s, } 1 s, 1 db s. 4.1 Hence M t s a martngale because by 4.1 M t s the sum of a fnte number of martngales. Moreover, { M t } s unformly ntegrable because sup EM t 2 < unformly n t [, T ]. Indeed, snce 2 s ntegrable, EM t 2 = 1 2 T 1 2 < C, T, [v s, 1 v s, ] ds where C, T s a fnte constant dependng only on and T, but not on. Therefore M t M t as, where M t = [,1] 1 vt, xx dx vt, x dx [,1] 1 vs, x x dx ds v 2 s, x x dx ds 4.2 s a martngale. From 4.1 we obtan the quadratc varaton of M t, whch s gven by M t = = 1 v s, v s, 1 v s, db s 1 v s, 9 1 2 2 ds. t
Hence lm M t = [,1] vs, x 1 vs, x 2 x dx ds = M t, and the theorem s proved. ow we proceed to the proof of the man result. Theorem 4.2. ut, x s a wea soluton of the stochastc partal dfferental equaton 1.1. Proof. To the quadratc varaton M t there corresponds a martngale measure Mdt, dx wth quadratc measure νdx dt = ut, x1 ut, x dx dt see [16]. Let W be a whte nose ndependent of M defned possbly on a extended probablty space. Let us defne W t = [,1] [,1] 1 us, x 1 us, x 1 {us,x {,1}}xMds, dx 1 {us,x {,1}} x W ds, dx. Then W t corresponds to a space -tme whte nose W ds, dx such that M t = [,1] us, x1 us, xx W ds, dx. From 4.2 we conclude that ut, x satsfes 1.2, and hence that ut, x s a wea soluton of 1.1. The theorem s proved. References [1] L. Bertn,. Cancrn and G. Jona-Lasno 1994. The stochastc Burgers equaton. Comm. Math. Phys. 164, 211-232. [2] P. Bler, T. Funa, and W. Woyczyns 1998. Fractal Burgers equatons. J.Dff. Equatons, 148, 9-46. [3] J.M. Burgers 1948. A mathematcal model llustratng the theory of turbulence. Adv. Appl. Mech., 1, 171-199. [4] J.M. Burgers 1974. The nonlnear dffuson equaton, D.Redel Publshng Company, Dordbrecht. [5] J.F. Crow and M. Kmura An Introducton to Populaton Genetcs Theory, Harper Row, 197. [6] G. Da Prato and D. Gatare 1995. Stochastc Burgers equaton wth correlated nose. Stoc. Stochastcs reports, 52, 29-41. 1
[7] T. Funa 1983. Random moton of strngs and related stochastc evoluton equatons. agoya Math. J. 89, 129-193. [8] E. Hopf 195. The partal dfferental equaton u t uu x = µu xx. Commun. Pure Appl. Math. 3, 21-23. [9] K. Iwata 1987. An nfnte dmensonal stochastc dfferental equaton wth state space CR. Prob.Th. Rel. Felds 74, 141-159. [1].Ieda and S.Watanabe Stochastc fferental Equatons and Dffuson Processes, orth-holland, 1989. [11] J.F. Le Gall 1983. Applcatons du temps local aux equatons dfferentelles stochastques undmentonnelles. Sem. Prob. XVII, Lecture otes n Math. 986, 15-31. [12] T. Maruyama, Stochastc problems n populaton genetcs Lecture otes n Math. 17, Sprnger-Verlag. [13] D. Revuz and M. Yor, Contnuous Martngales and Brownan Moton, 2nd Ed., Sprnger, 1994. [14] A.V.Sorohod, Studes n the theory of random processes, Addson Wesley, 1965. [15] H. Toto 1961. A method of constructon of measures on functon spaces and ts applcatons to stochastc processes. Mem. Fac. Sc. Kyushu Unv. Ser. A 15, 178-19. [16] J.B. Walsh, An Introducton to Stochastc Partal Dfferental Equatons, In: École d Été de Probabltés de Sant-Flour XIV-1984, Sprnger-Verlag, ew Yor, 1986. [17] T. Yamada and S. Watanabe 1971. On the unqueness of solutons of stochastc dfferental equatons. J. Math. Kyoto Unv. 11, 155-167. 11