TRANSPORTATION COST-INFORMATION INEQUALITY FOR STOCHASTIC WAVE EQUATION

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1 TRANSPORTATION COST-INFORMATION INEQUALITY FOR STOCHASTIC WAVE EQUATION YUMENG LI AND XINYU WANG arxiv: v [math.pr] 4 Jan 19 Abstract: In this paper, we prove a Talagrand s T transportation cost-information inequality for the law of a stochastic wave equation in spatial dimension d 3 driven by the Gaussian random field, white in time and correlated in space, on the continuous paths space with respect to the uniform topology. Keyword: Stochastic wave equation; Girsanov transformation; Transportation costinformation inequality. MSC: 6H15; 6H. 1. Introduction ThepurposeofthispaperistostudytheTalagrand st transportationcost-information inequality for the following stochastic wave equation in spatial dimension d 3: $ B & upt,xq b`upt,xq `F 9 pt,xq, Bt up,xq ν 1 pxq, (1) % B up,xq ν Bt pxq for all pt,xq P r,t s ˆR 3, where the coefficient b : R Ñ R is Lipschitz continuous, the term u denotes the Laplacian of u in the x-variable and the process F 9 is the formal derivative of a Gaussian random field, white in time and correlated in space. We recall that a random field solution to (1) is a family of random variables tupt,xq,t P R`,x P R 3 u such that pt,xq ÞÑ upt,xq from R` ˆ R 3 into L pωq is continuous and solves an integral form of (1), see Section for details. It is known that random field solutions have been shown to exist when d P t1,,3u, see [8]. In spatial dimension 1, a solution to the non-linear wave equation driven by space-time white noise was given in [7] by using Walsh s martingale measure stochastic integral. In dimensions or higher, there is no function-valued solution with space-time white noise, some spatial correlation is needed. A necessary and sufficient condition on the spatial correlation for existence of a random field solution was given in [1]. Since the fundamental solution in spatial dimension d 3 is not a function, this required an extension of Walsh s martingale measure stochastic integral to integrands that are Schwartz distributions, the existence of a random field solution to (1) is given in [8]. Hölder continuity of the solution was established in [1]. In spatial dimensional d ě 4, since thefundamental solutionofthewave equationisnotameasure, but aschwarz distribution that is a derivative of some order of a measure, the methods used in dimension 3 do not apply to higher dimensions, see [1] for the study of the solutions. 1

2 YUMENG LI AND XINYU WANG Transportation cost-information inequalities have been recently deeply studied, especially for their connection with the concentration of measure phenomenon, log-sobolev inequality, Poincaré inequalities and Hamilton-Jacobi s equation, see [1, 3, 5, 15, 18,, 4, 6]. Let us recall the transportation inequality. Let pe, dq be a metric space equipped with σ-field B such that dp, q is B ˆB measurable. Given p ě 1 and two probability measures µ and ν on E, the Wasserstein distance is defined by j1 W p,d pµ,νq : inf d p p px,yqπpdx,dyq, π where the infimum is taken over all the probability measures π on E ˆE with marginal distributions µ and ν. The relative entropy of ν with respect to (w.r.t. for short) µ is defined as " ş log dνdν, if ν! µ; Hpν µq : dµ () `8, otherwise. Definition 1.1. The probability measure µ is said to satisfy the transportation costinformation inequality T p pcq on pe,dq if there exists a constant C ą such that for any probability measure ν on E, W p,d pµ,νq ď a CHpν µq. Recently, the problem of transportation inequalities and their applications to diffusion processes have been widely studied. The T pcq inequality, first established by M. Talagrand [4] for the Gaussian measure with the sharp constant C. The approach of M. Talagrand is generalized by D. Feyel and A.S. Üstünel [14] on the abstract Wiener space with respect to Cameron-Martin distance using the Girsanov theorem. With regard to the paths of finite stochastic differential equation (SDE for short), by means of Girsanov transformation and the martingale representation theorem, the T pcq w.r.t. thel andthecameron-martindistanceswereestablishedbyh.djelloutet al. [13]; the T pcq w.r.t. the uniform metric was obtained by [5, 8]. J. Bao et al. [] established the T pcq w.r.t. both the uniform and the L distances on the path space for the segment process associated to a class of neutral function stochastic differential equations. B. Saussereau [3] studied the T pcq for SDE driven by a fractional Brownian motion, and S. Riedel [] extended this result to the law of SDE driven by general Gaussian processes by using Lyons rough paths theory. S. Pal [1] proved that probability laws of certain multidimensional semimartingales which includes time-inhomogenous diffusions, satisfy quadratic transportation cost inequality under the uniform metric. Those, in particular, imply some results about concentration of boundary local time of reflected Brownian motions. Motivated by the source of the noise modeled by the random terms in partial differential equations, which include physical noise (such as thermal noise), the stochastic partial differential equations have been studied in many literatures in past thirty years. For the stochastic reaction-diffusion equation, L. Wu and Z. Zhang [9] studied the T pcq w.r.t. L -metric by Galerkin s approximation. By Girsanov s transformation, B. Boufoussi and S. Hajji [6] obtained the T pcq w.r.t. L -metric for the stochastic

3 TRANSPORTATION COST-INFORMATION INEQUALITIES FOR STOCHASTIC WAVE EQUATION3 heat equations driven by space-time white noise and driven by fractional noise. Those results are established for the stochastic parabolic equations. However, the hyperbolic case is much more complicated, one difficulty comes from the more complicated stochastic integral, another one comes from the lack of good regularity properties of the fundamental solutions. See [9] for the study of the stochastic wave equation. In this paper, we shall study the Talagrand s T -transportation inequality for the law of a stochastic wave equation (1) on the continuous paths space with respect to the uniform metric. The rest of this paper is organized as follows. In Section, we first give the properties of the Eq. (1), and then state the main result of this paper. In Section 3, we shall prove the main result.. Framework and main result.1. Framework. For any d ě 1, let SpR d`1 q be the space of Schwartz functions, all of whose derivatives are rapidly decreasing. F pf pϕq,ϕ P SpR d`1 qq is a Gaussian process defined on some probability space with zero mean and covariance functional EpF pϕqf pψqq Jpϕ,ψq : ds dx dyϕps,xqfpx yqψps,yq, (3) R d R d R` where f : R d Ñ R` continuous on R d ztu. According to [8], there are some requirements on f. As a covariance functional of a Gaussian process, the function Jp, q should be non-negative definite, this implies that f is symmetric (fpxq fp xq for all x P R d ), and is equivalent to the existence of a non-negative tempered measure µ on R d, whose Fourier transform is f. More precisely, for any ϕ P SpR d q, let Fϕ be the Fourier transform of ϕ: Fϕpξq : expp iπξ xqϕpxqdx. R d The relationship between µ and f is, by definition of the Fourier transform on the space S 1 pr d q of tempered distributions, that is for all ϕ P SpR d q, fpxqϕpxqdx Fϕpξqµpdξq. R d R d Elementary properties of Fourier transform show that for all ϕ,ψ P SpR d q, xϕ,ψy H : dx dyϕpxqfpx yqψpyq R d R d (4) µpdξqfϕpξqfψpξq. R d Here z is the complex conjugate of z. According to [11], the Gaussian process F with covariance (3) can be extended to a martingale measure M M t paq, t ě, A P B b pr 3 q (, where B b pr 3 q denotes the collection of all bounded Borel measurable sets in R 3.

4 4 YUMENG LI AND XINYU WANG Foreacht ě,denotebyf t theσ-fieldgeneratedbytherandomvariables tm s paq,s P r,ts,a P BpR 3 qu, that is F t : σtm s paq,s P r,ts,a P BpR 3 qu. Let H be the Hilbert space obtained by the completion of SpR 3 q with the inner product x, y H defined by (4), and denote by } } H the induced norm. Let H T : L pr,t s;hq and consider the usual L -norm } } HT on this space. Then H T is a Hilbert space with the inner product xψ 1,ψ y HT : T xψ 1 ptq,ψ ptqy H dt, ϕ,ψ P H T. (5) Let D be an open set D in Euclidean space R d for d ě 1. For any integer n ě 1, let C n pdq be the space of all continuous functions from D to R, whose derivatives up to order n are also continuous. For any δ P p,1q, let C δ pdq be the space of all Hölder continuous functions of degree δ, with the Hölder norm gpxq gpyq }g} δ : sup ă P C δ pdq; x y x y δ and let C Lip pdq be the space of all Lipschitz continuous functions, with the norm gpxq gpyq }f} Lip : sup ă P C Lip pdq. x y x y Hypothesis (H): (H.1) There exists a constant K ą such that bpxq bpyq ď K x P R. (6) (H.) The function f given in (3) can be expressed by fpxq ϕpxq x β,x P R 3 ztu, with β Ps,r. Here the functions ϕ and ϕ are bounded, ă ϕ P C 1 pr 3 q, ϕ P C δ pr 3 q with δ Ps,1s. (H.3) The initial values ν 1,ν are bounded, ν 1 P C pr 3 q, ν 1 is bounded, ν 1 and ν are Hölder continuous with degrees γ 1,γ Ps,1s, respectively. We remark that the hypothesis (H.) implies that, for any T ą, MpT q : sup FGptqpξq µpdξq ă 8, (7) tpr,t s R 3 see [9]. By Walsh s theory of stochastic integration with respect to (w.r.t. for short) martingale measures, for any t ě and h P H, the stochastic integral t B t phq : hpyqmpds,dyq R d is well defined, and " t * Bt k : e k pyqmpds,dyq; k ě 1 R d

5 TRANSPORTATION COST-INFORMATION INEQUALITIES FOR STOCHASTIC WAVE EQUATION5 defines asequence of independent standardwiener processes, here te k u kě1 is a complete orthonormal system of the Hilbert space H. Thus, B t : ř kě1 Bk t e k is a cylindrical Wiener process on H. See [7]. According to Dalang and Sanz-Solé [1], under hypothesis (H), Eq. (1) admits a unique solution u: upt,xq wpt,xq ` ÿ t xgpt s,x q,e k p qy H dbs k kě1 t (8) ` Gpt s,x,yqbpups,yqqdyds, R 3 where wpt,xq : d dt Gpt,x,yqν 1pyq `Gpt,x,yqν pyq, with Gpt,x,yq 1 4πt σ tpx yq, σ t is the uniform surface measure (with total mass 4πt ) on the sphere of radius t. Furthermore, for any p P r, 8r, sup E r upt,xq p s ă `8. (9) pt,xqpr,t sˆr 3 See Dalang and Sanz-Solé [1] or Hu et al [16] for details... Main results. Let Cpr,T s ˆ R 3 q be the space of all continuous functions from r,t s ˆR 3 to R, endowed with the uniform norm }f} 8 : sup fpt,xq. pt,xqpr,t sˆr 3 For initial function ν : pν 1,ν q satisfied (H.3), let P ν be the law of tupt,xq, pt,xq P r,t s ˆR 3 u on Cpr,T s ˆR 3 q with initial value up,xq ν 1 pxq and B Bt up,xq ν pxq. Recall the constants K,MpT q given by (6) and (7) respectively. In this paper, we establish the following result: Theorem.1. Under Hypothesis (H), there exists a constant CpT,Kq : TMpT qe T4 K such that the probability measure P ν satisfies T pcq on the space Cpr,T sˆr 3 q endowed with the uniform norm } } 8. As indicated in [3], many interesting consequences can be derived from Theorem.1, see also Corollary 5.11 of [13]. Corollary.. Under Hypothesis (H), we have for any T ą, the following statements hold for the constant CpT,Kq TMpT qe T4 K. (a) For any Lipschitzian function U on Cpr,T sˆr 3 q with respect to uniform norm, we have E Pν exp `U E U Pν ď e CpT,Kq }U} Lip. (b) (Inequality of Hoeffding type) For any V : R Ñ R such that }V } Lip ă 8, we have that for any r ě, ˆ1 T 1 T j r P }V puptqq} 8 dt E }V puptqq} 8 dt ą r ď exp. T T CpT,Kq}V } Lip

6 6 YUMENG LI AND XINYU WANG 3. The proof 3.1. An important lemma. This lemma is an analogue of the result [13, Theorem 5.6] for the space-colored time-white noise instead of the finite-dimensional Brownian motion. Lemma 3.1. [17, Lemma 6.] For every probability measure Q! P ν on the space L pr,t s ˆR 3 ;Rq, there exists an adapted Q-a.s. h thps,xq, ps,xq P r,t s ˆR 3 u such that }h} HT ă 8, Q-a.s., and the function F r : L pr,t s ˆR 3 ;Rq Ñ L pωq defined by t F pφq : F pφq φps,xqhps,xqdxds, (1) R 3 is a space-colored time-white noise with the spectral density f with respect to the measure Q. The Randon-Nikodym derivative is given by ˆ dq T exp hps,xqf pds,dxq dp ν ˇˇˇFT 1 R 3 and the relative entropy is given by 3.. The proof of Theorem.1. T }hpsq} H ds, (11) HpQ P ν q 1 EQ }h} H T. (1) Proof. It is enough to prove the result for any probability measure Q on Cpr,T s ˆR 3 q such that Q! P ν and HpQ P ν q ă 8. Assume that the Randon-Nikodym derivative is given by ˆ dq T exp dp ν ˇˇˇFT and the relative entropy is R 3 hps,xqf pds,dxq 1 T }hpsq} Hds, (13) HpQ P ν q 1 EQ }h} H T. (14) Let pω,f, r Pq be a complete probability space on which F is space-colored time-white noise with the spectral density f with respect to the measure Q. Let F t F F t σpf ps,aq,s ď Ă R 3 q r P completion by r P. Let u t pνq be the unique solution of (1) with initial condition ν pν 1,ν q. Then the law of u pνq is P ν. Consider rq : dq dp ν pu pνqq rp. Then HpQ P ν q Hp r Q r Pq 1 EQ }h} H T For a complete orthonormal system te k u kě1 of the Hilbert space H, let " t * Bt k : e k pyqf pds,dyq; k ě 1. R 3

7 TRANSPORTATION COST-INFORMATION INEQUALITIES FOR STOCHASTIC WAVE EQUATION7 Then B t : ř kě1 Bk t e k is a cylindrical Wiener process on H under P, r and ř kě1 pbk t ` xh,e k y H qe k is a cylindrical Wiener process on H under Q r by Lemma 3.1. According to Lemma 3.1, we couple pp,qq as the law of a process pu,vq under Q t upt,xq wpt,xq ` Gpt s,x,yqfpds,dyq r R 3 t ` Gpt s,x,yqbpups,yqqdyds (15) R 3 ` ÿ t xgpt s,x q,e k p qy H xhps, q,e k p qy H ds, and kě1 t vpt,xq wpt,xq ` Gpt s,x,yqfpds,dyq r R 3 t ` Gpt s,x,yqbpvps,yqqdyds. R 3 By the definition of the Wasserstein distance, «W,} } 8 pq,pq ď E r Q sup upt,xq vpt,xq pt,xqpr,t sˆr 3 In view of (1) and (17), it remains to prove that «ff E r Q sup upt,xq vpt,xq pt,xqpr,t sˆr 3 ff (16). (17) ď CE rq r}h} H T s. (18) From (8), (15) and (16), we can represent upt,xq vpt,xq as t upt,xq vpt,xq Gpt s,x,yqrbpups,yqq bpvps,yqqsdyds R 3 ` ÿ t xgpt s,x q,e k p qy H xhps, q,e k p qy H ds For every t P r,t s, define kě1 :I 1 pt,xq `I pt,xq. ηptq sup ups,xq vps,xq. ps,xqpr,tsˆr 3 By using the elementary inequality pa `bq ď pa `b q, we have (19) upt,xq vpt,xq ďi1 pt,xq `I pt,xq. () By the Cauchy-Schwarz inequality with with respect to the finite measure Gpt s, x, yqdyds on r,t s ˆR 3, with total measure t {, and by the Lipschitz continuity of b, we obtain

8 8 YUMENG LI AND XINYU WANG that for any t ď T, t I 1 pt,xq ď K ď t K t K ď T3 K t t Gpt s,x,yqdyds R 3 t t R 3 Gpt s,x,yqηpsqdyds pt sqηpsqds ηpsqds, R 3 Gpt s,x,yq ups,yq vps,yq dyds where ş Gpt s,x,yqdy t s is used in the last second line. Let us estimate the R 3 second term. By the Cauchy-Schwarz inequality and (7), we have for any t ď T, I pt,xq ď t ď TMpT q }Gpt s,x, q} Hds ˆ t }hpsq} Hds Putting (), (1) and () together, we have for any t ď T, ηptq ď T3 K Using the Gronwall s inequality, we obtain that The proof is complete. t t (1) }hpsq} H ds. () ηpsqds `TMpT q}h} H T. E rq rηpt qs ď TMpT qe T4 K E rq r}h} H T s. Acknowledgements. The authors are grateful to the anonymous referees for comments and corrections. References [1] Bakry, D., Gentil, I., Ledoux, M.: Analysis and geometry of Markov diffusion operators. Fundamental Principles of Mathematical Sciences, 348. Springer, 14. [] Bao, J., Wang, F. Y., Yuan, C.: Transportation cost inequalities for neutral functional stochastic equations. Z.. Anal. Anwend., 3(4), (13) [3] Bobkov, S., Gentil, I., Ledoux, M.: Hypercontractivity of Hamilton-Jacobi equations. J. Math. Pures Appl., 8(7), (1) [4] Bobkov, S. G., Götze, F.: Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal., 163(1), 1-8 (1999) [5] Boucheron, S., Lugosi, G., Massart, P.: Concentration inequalities. A nonasymptotic theory of independence. Oxford University Press, Oxford, 13. [6] Boufoussi, B., Hajji, S.: Transportation inequalities for stochastic heat equations. Statist. Probab. Lett., 139, (18) [7] Cairoli, R., Walsh, J. B.: Martingale Representations and Holomorphic Processes. Ann. Probab. 5(4), (1977)

9 TRANSPORTATION COST-INFORMATION INEQUALITIES FOR STOCHASTIC WAVE EQUATION9 [8] Dalang, R.: Extending martingale measure stochastic integral with applications to spatially homogeneous S.P.D.E s. Electron. J. Probab,, 4(6), 1-9 (1999) [9] Dalang, R.: The stochastic wave equation. A minicourse on stochastic partial differential equations, 39-71, Lecture Notes in Math., 196, Springer, Berlin, 9. [1] Dalang, R., Frangos, N.: The stochastic wave equation in two spatial dimensions. Ann. Probab., 6(1), (1998) [11] Dalang, R., Mueller, C.: Some non-linear S.P.D.E s that are second order in time. Electron. J. Probab. 8(1), 1-1 (3) [1] Dalang, R., Sanz-Solé, M.: Hölder-Sobolev regularity of solution to stochastic wave equation in dimention three. Mem. Am. Math. Soc., 199 (9) [13] Djellout, H., Guillin, A., Wu, L.: Transportation cost-information inequalities and applications to random dynamical systems and diffusions. Ann. Probab. 3(3), 7-73 (4) [14] Feyel D., Üstünel A.S.: Monge-KantorovitchMeasure Transportation and Monge-Ampère Equation on Wiener Space, Probab. Th. Rel. Fields, 18 (3), , (4) [15] Gozlan, N.: Transport inequalities and concentration of measure. ESAIM: Proceedings and Surveys, 51(89), 1-3 (15) [16] Hu, Y., Huang, J., Nualart, D.: On Hölder continuity of the solution of stochastic wave equations in dimension three. Stoch. Partial Differ. Equ. Anal. Comput. (3), (14) [17] Khoshnevisan, D., Sarantsev, A.: Talagrand concentration inequalities for stochastic partial differential equations, 1-6. Arxiv.org/abs/ v1 (17) [18] Ledoux, M.: The concentration of measure phenomenon. Mathematical Surveys and Monographs, 89. American Mathematical Society, Providence, RI, 1. [19] Ortiz-López, V., Sanz-Solé, M.: A Laplace principle for a stochastic wave equation in spatial dimension three. Stochastic Analysis 1, 31-49, Springer, Heidelberg, 11. [] Otto, F., Villani, C: Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal., 173(), () [1] Pal, S: Concentration for multidimensional diffusions and their boundary local times. Probab. Theory Related Fields, 154(1-), 5-54 (1) [] Riedel, S.: Transportation-cost inequalities for diffusions driven by Gaussian processes. Electron. J. Probab.,, 1-6 (17) [3] Saussereau, B.: Transportation inequalities for stochastic differential equations driven by a fractional Brownian motion. Bernoulli, 18(1), 1-3 (1) [4] Talagrand, M.: Transportation cost for Gaussian and other product measures. Geom. Funct. Anal., (1996) [5] Üstünel, A.S.: Transportation cost inequalities for diffusions, Stochastic Analysis and Related Topics, Springer Proceedings in Mathematics and Statistics, 3-14 (1) [6] Villani, C.: Optimal Transport: Old and New. A series of comprehensive studies in mathematics, Volume 338, Springer Verlag (9) [7] Walsh, J.: An introduction to stochasticpartialdifferential equations.écoled étéde Probabilités St Flour XIV, Lect Notes Math, 118, Springer, [8] Wu, L., Zhang, Z.: Talagrand s T -transportation inequality w.r.t. a uniform metric for diffusions. Acta Math. Appl. Sinica, English Series, (3), (4) [9] Wu, L., Zhang, Z.: Talagrand s T -transportation inequality and log-sobolev inequality for dissipative SPDEs and applications to reaction-diffusion equations. Chinese Ann. Math., Series B, 7(3), 43-6 (6)

10 1 YUMENG LI AND XINYU WANG Yumeng Li, School of Statistics and Mathematics, Zhongnan University of Economics and Law, 4373, P. R. China. address: XinYu Wang, School of Mathematics and Statistics, Huazhong University of Science and Technology, 4373, P. R. China. address: wang xin