On A Priori Estimates for Rough PDEs

Transcription

1 On A Priori Etimate for Rough PDE Qi Feng and Samy Tindel Dedicated to Rodrigo Bañuelo on occaion of hi 60th birthday Abtract In thi note, we preent a new and imple method which allow to get a priori bound on rough partial differential equation. The technique i baed on a weak formulation of the equation and a rough verion of Gronwall lemma. The method i preented on a imple linear example, but might be generalized to a wide number of ituation. Keyword A priori etimate Rough Gonwall lemma Rough path Stochatic PDE 1 Introduction Thi paper propoe to review a recent method allowing to get a priori etimate for rough partial differential equation, taken from [6]. Our aim here i to how how to implement the technique on a imple example. Namely, we hall conider the following noiy heat equation on an interval Œ0; R d for > 0 and a patial dimenion d t u t.x/ D u t.x/ C ˇiu t.x/e i.x/dw i t ; (1.1) where tand for the Laplace operator, fe i I1g i an orthonormal bai of L.R d / and fˇii 1g i a family of coefficient atifying ome ummability condition (ee Hypothei.4 below). In Eq. (1.1), fw i I 1g i alo a family of noie, Q. Feng S. Tindel ( ) Department of Mathematic, Purdue Univerity, 150 N. Univerity Street, Wet Lafayette, IN 47907, USA tindel@purdue.edu Springer International Publihing AG 017 F. Baudoin, J. Peteron (ed.), Stochatic Analyi and Related Topic, Progre in Probability 7, DOI / _6 117

2 118 Q. Feng and S. Tindel interpreted a p-variation path with p <3, which can be lifted to a rough path w (ee Hypothei.3 for a more complete definition). The recent activity on exitence and uniquene reult for rough PDE ha been thriving. A lot of thi activity concern ituation which require renormalization technique and a way to handle pathwie product of ditribution [10, 1, 13]. Here we are concerned with a different context, for which the noie i mooth enough in pace, o that the olution of (1.1) i directly expected to be a function and the integral with repect to w are uual rough path integral. Thi ituation doe not require the whole regularity tructure machinery, and one advantage of thi reduced etting i that more information on the olution i available. We are concerned in thi paper about a priori etimate, which can be either een a a crucial tep in the proof of exitence of olution, or a a firt piece of valuable information about the olution. Furthermore, we believe that a priori etimate exhibit the core of the pathwie method for tochatic PDE, even though many more technical tep have to be performed in order to get exitence and uniquene reult. Let u ummarize ome of the (unrelated) approache leading to etimate of equation like (1.1). 1. The reference [, 11] handle tochatic PDE by conidering random flow (induced by a finite dimenional rough path) which change the tochatic PDE into a determinitic PDE with random coefficient. A priori bound are then potentially obtained by compoing bound on determinitic PDE and etimate on rough flow. Thi poibility ha not been fully exploited yet, and might lead to nontrivial conideration.. In [5, 9], a variant of the rough path theory i introduced in order to cope with PDE of the form (1.1), conidered in the mild ene. Thi involve ome lengthy and intricate conideration on twited increment of the form ıf O t D f t S t f,wheres deignate the heat emi-group and f i a generic L.R d /- valued function. However, thi formalim yield a priori etimate for (1.1), epecially when one conider related numerical cheme a in [4]. 3. For linear equation like (1.1), Feynman-Kac repreentation for the olution are available. Thi give raie to explicit moment computation for u t.x/, forafixed couple.t; x/ R C R d. Many cae of Gauian noie have been examined in thi context, and we refer to [3] for a ituation which i cloe to our, namely a rough noie in time which i mooth in pace. Let u highlight again the fact that we only recall here reult concerning mooth noie in pace. In cae like [10, 1, 13] where renormalization i needed, the mere exitence of moment for the renormalized olution i till an open problem (to the bet of our knowledge). With thee preliminary conideration in mind, the main point of the current paper i to how that the variational approach to rough PDE, introduced in [1, 6], provide a handy way to obtain L.R d / (and more generally L.R d /) etimate on the olution. The main advantage of thi new etting are the following: 1. The variational formulation i convenient at an algebraic and analytic level, when compared with the other method mentioned above.

3 On A Priori Etimate for Rough PDE 119. Unlike Feynman-Kac repreentation, the variational approach i not retricted to linear equation (though generalization require a nontrivial extra work). We hall illutrate thi point of view with the imple model (1.1), for which we hall deduce L -etimate in a detailed way. It hould be noticed that variational method have been conidered previouly in [15] for pathwie PDE driven by a fractional Brownian motion. With repect to thi reference, our computation are retricted to linear cae. However, [15] only conider fbm with a Hurt parameter H > 1, while we are concerned with a true rough cae (correponding to 1 3 < H 1 for fbm). Our article i tructured a follow: in Sect. we introduce ome notation and the variational method framework, and we alo preent our firt a priori etimate in Propoition.8. Thi etimate (adapted from [6, Theorem.5]) i valid for general linear equation, and will be uitable for our tochatic heat equation with multidimenional noie. Then in Sect. 3 we prove our main a priori bound, namely Theorem 3.5 and 3.9 for the olution of Eq. (1.1), both in L.R d / and L.R d / norm. Finally, Sect. 4 i devoted to the application of our abtract reult to equation driven by fractional Brownian motion. A firt example concern a bounded domain, which enable u to compare our reult with thoe of [15], while a econd example deal with the whole pace R d : Rough Variational Framework A mentioned above, our framework relie on a variational formulation of the heat equation, which i algebraically quite convenient. In thi ection we firt recall ome baic vocabulary about algebraic integration, then we give the main general reult needed for the rough heat equation (1.1)..1 Notion of Algebraic Integration Firt of all, let u recall the definition of the increment operator, denoted by ı. Ifg i a path defined on Œ0; T and ; t Œ0; T then we et ıg t WD g t g. Whenever g i a -index map defined on Œ0; T,wedefineıg ut WD g t g u g ut. The norm of the element g in the Banach pace E will be written a N ŒgI E. For two quantitie a and b the relation a. x b mean a c x b, for a contant c x depending on a multidimenional parameter x. In the equel, given an interval I we call control on I (and denote it by!) any continuou uperadditive map on I WD f.; t/ I W tg, that i, any continuou

4 10 Q. Feng and S. Tindel map! W I! Œ0; 1/ uch that, for all u t,!.; u/ C!.u; t/!.; t/: Given a control! on an interval I D Œa; b, we will ue the notation!.i/ WD!.a; b/. For a fixed time interval I,aparameterp >0, a Banach pace E and any continuou function g W I! E we define the norm N ŒgI V p 1.II E/ WD up.t i /P.I/ X i jıg ti t ic1 j p! 1 p ; where P.I/ denote the et of all partition of the interval I. In thi cae,! g.; t/ D N ŒgI V p 1.Œ; t I E/ p define a control on I. We denote by V p.ii E/ the et of continuou two-index map g W I I! E for which there exit a control! uch that jg t j!.; t/ 1 p for all ; t I. We alo define the pace V p ;loc.ii E/ of map g W I I! E uch that there exit a countable covering fi k g k of I atifying g V p.i ki E/ for any k. The following reult i often referred to a ewing lemma in the literature, and i at the core of our approach to generalized integration. Lemma.1 Fix an interval I, a Banach pace E and a parameter >1. Conider a function h W I 3! E uch that h Im ı and for every < u < t I, jh ut j!.; t/ ; (.1) 1 for ome control! on I. Then there exit a unique element ƒh V.II E/ uch that ı.ƒh/ D h and for every < t I, j.ƒh/ t jc!.; t/ ; (.) for ome univeral contant C. Our computation alo hinge on the following rough verion of Gronwall lemma, borrowed from [6, Lemma.7]. Lemma. Fix a time horizon T >0and let Q W Œ0; T! Œ0; 1/ be a path uch that for ome contant C; L >0, 1 and ome control! 1 ;! on Œ0; T, one ha ıq t C up 0rt Q r! 1.; t/ 1 C!.; t/; (.3)

5 On A Priori Etimate for Rough PDE 11 for every < t Œ0; T atifying! 1.; t/ L. Then it hold n o up 0tT Q t exp.c ;L! 1.0; T// Q 0 C up 0tT!.0; t/ exp. c ;L! 1.0; t// ; for a trictly poitive contant c ;L.. Linear Equation with Ditributional Drift In thi ection we hall firt generalize Eq. (1.1), and conider the following: dg t D.dt/ C ˇig t e i dw i t ; (.4) where i a ditributional-valued meaure. Before we give a rigorou meaning to thi equation, let u label our hypothei on the coefficient. We tart by a rough path aumption for each couple of component of the driving noie w: Hypothei.3 Let p Œ; 3/ be given. We aume that the family fw i I i 1g i uch that there exit increment w 1;i ; w ;ij atifying the two following propertie: (i) Algebraic condition: For each i; j 1 and 0 u t, Chen relation hold true: (ii) Analytic condition: For all i; j 1, we have ıw 1;i t D 0; and ıw ;ij ut D w 1;i u w1;j ut : (.5) N Œw 1;i I V p.œ; t / < 1; and N Œw;ij I V p=.œ; t / < 1: The rough variational etting introduced in [1, 6] ue the concept of cale. A cale i defined a a equence E n ; kk n nn 0 of Banach pace uch that E nc1 i continuouly embedded into E n. Beide, for n N 0 we denote by E n the topological dual of E n. For the heat equation (1.1), we will conider the cale E n D W n;1. Having the concept of cale in mind, the noie w hould alo fulfill the following hypothei a an infinite dimenional object: Hypothei.4 Recall that the cale E n i given by E n D W n;1. We aume that fˇii 1g i a family of poitive coefficient atifying P i1 ˇi < 1. Conider an orthonormal bai fe i I1g of L.R d /, compoed of bounded function. The noie w i uch that fw i I1g i a family of p-variation path with p <3, whoe firt and

6 1 Q. Feng and S. Tindel econd order increment w 1;i ; w ;ij are uch that! w 1 and! w below are two control on Œ0; :! w 1.; t/! p ˇi.1 Cje i j E1 / N Œw 1;i I V p.œ; t / (.6) 0 1! w.; ˇiˇjje i j E1 je j j E1 N Œw ;ij I V p=.œ; t / A p= : (.7) We can now give a more formal definition of olution to our Eq. (.4), in term of expanion of the increment up to a regularity order greater than 1: Definition.5 Let p Œ; 3/ andfixanintervali Œ0;.Letbeaditributional- valued meaure lying in V1 1.II E 1/. Apathg W I! E 0 i called olution (on I) of Eq. (.4) provided there exit q <3and g \ V q 3 ;loc.i; E 1 / uch that we have: ıg t.'/ D ˇig.e i '/w 1;i t C ı t.'/ C ˇiˇjg.e i e j '/w ;ij t C g \ t.'/; (.8) for every ; t I atifying < t and every ' E 1. Remark.6 On top of (.5), we will ue the following expreion for ıg t : ıg t.'/ D where g ] i a V p.e 1 / increment atifying: g ] t.'/ D ıg t.'/ ˇig.e i '/w 1;i t ˇig.e i '/w 1;i t C g ].'/; (.9) D ı t.'/ C ˇiˇjg.e i e j '/w ;ij t C g \ t.'/: (.10) Remark.7 Equation (.8) i expreed a an expanion along the increment of w i. However, according to [7, Theorem 4.10], a olution u of (.8) alo olve the following integral equation (which ha to be interpreted in the rough path ene in time and weak ene in pace): ıg t D.Œ; t// C ˇie i g r dw i r : (.11)

7 On A Priori Etimate for Rough PDE 13 Furthermore, a change of variable formula (ee [7, Propoition 5.6]) hold for g verifying (.11). Namely, for h C 3.R/ we have (till in the weak rough path ene): ıh.g/ t D h 0.g r /.dr/ C ˇie i h 0.g r / g r dw i r : (.1).3 A General Etimate for Linear Equation The following propoition give our firt a priori etimate for the olution to Eq. (.4). It hould be een a an adaptation of [6, Theorem.5] to our current context. Propoition.8 Let p Œ; 3/ and fix an interval I Œ0; T. Letw be a rough path verifying Hypothei.3 and.4. Conider a path V1 1.II E 1/ uch that for every ' E 1, there exit a control! verifying jı t.'/j!.; t/ k'k E1 : (.13) Let g be a olution on I of Eq. (.4), with the following additional hypothei: g i controlled over the whole interval I, that i we have g \ V q 3.II E 1 / for q <3. Moreover let St g D up t kg k E 0, and conider the following control:! I.; t/!.; t/! 1=p.; t/ C! =p.; t/ CS g w 1 w t! 1=p.; t/! =p.; t/c! 4=p.; t/ : w 1 w w (.14) Then there exit a contant L D L p >0(independent of I) uch that if! w 1.; t/ C! w.; t/ L; then for all ; t I uch that < t, we have: kg \ tk E 1. p! I.; t/: (.15) Proof Let! \.; t/ be a regular control uch that kg \ tk E 1! \.; t/ 3 q for any ; t I uch that < t. We divide thi proof in everal tep. Step 1: An Algebraic Identity Let ' E 1 be uch that k'k E3 1. Wefirthow that ıg \ ut.'/ D ˇig ] u.e i'/w 1;i ut C ˇiˇjıg u.e i e j '/w ;ij ut K 1 ut C K ut ; (.16)

8 14 Q. Feng and S. Tindel where g ] wa defined in (.10). Indeed, owing to (.8), we have g \ t D ıg t.'/ ˇig.e i '/w 1;i t ı t.'/ ˇiˇjg.e i e j '/w ;ij t : Applying ı on both ide of thi identity and recalling Chen relation (.5)awell a the fact that ıı D 0 we thu get ıg \ ut.'/ D ˇiıg u.e i '/w 1;i ut C 1 X ˇiˇjıg u.e i e j '/w ;ij ut ˇiˇjg.e i e j '/w 1;i u w1;j ut : Plugging relation (.10) again into thi identity, we end up with our claim (.16). Step : Bound for K 1 In order to bound the term g ] u.e i '/ in K 1,weinvoke decompoition (.10), which yield: and hence: g ] u.e i'/ D ı u.e i '/ C j;kd1 ˇjˇkg.e i e j e k '/w ;kl u C g\ u.e i'/; jg ] u.e i'/j 4!.; t/je i j E1 CSu g 3 ˇjˇkje i j E0 je j j E0 je k j E0! =p w;jk.; u/c!3=p \.; u/je i j E1 5 j'j E1 : j;kd1 Therefore, thank to our aumption (.7), we have: jg ] u.e i'/j h i!.; u/ C Su g!=p.; u/ C! 3=p w \.; u/ je i j E1 j'j E1 : (.17) Plugging thi identity into the definition of K 1, we have thu obtained: jk 1 ut jj'j E 1 h!.; u/ C S g u!=p w.; u/ C! 3=p \.; u/ i 1 X ˇije i j E1! 1=p w 1;i.u; t/ j'j E1 h!.; u/ C S g u!=p w.; u/ C! 3=p \.; u/ i! 1=p w 1.u; t/: (.18)

9 On A Priori Etimate for Rough PDE 15 Step 3: Bound for K and ıg \ The main term to treat for K i the increment ıg u. To thi aim, we reort to decompoition (.9). Thi yield: K ut D 1 X i;j;kd1 Furthermore, we have: jk 1 ut jsg t j'j E0 ˇiˇjˇk g.e i e j e k '/w 1;k u w;ij ut kd1 C! 0 ˇkje k j E0! 1=p.; w 1;k S g t j'j E0! 1=p w 1.; u/! =p w.u; t/: ˇiˇj g ].e i e j '/w ;ij ut K 1 ut C K ut : 1 ˇiˇjje i j E0 je j j E0! =p.u; t/ A w ;ij In order to handle K, we elaborate lightly on our etimate (.17) in order to get: h i 0 jkut jj'j X 1 E 1!.; u/ C Su g!=p.; u/ C! 3=p w \.; j'j E1 h i!.; u/ C Su g!=p.; u/ C! 3=p w \.; u/! =p.u; t/: w Hence, gathering our etimate on K 1 and K weendupwith: jk ut jj'j E 1 hs g t 1 ˇiˇjje i j E1 je j j E1! =p.u; t/ A w ;ij i! 1=p.; u/ C! =p.; u/ w 1 w C!.; u/ C! 3=p \.; u/! =p.u; t/: w (.19) We can now eaily conclude for the increment ıg \ : plugging (.18) and(.19) into (.16), we get: n ˇ ˇıg \ ut.'/ ˇ j'j E1!.; u/ C Su g!=p! 1=p C!.; u/ C St g C! 3=p \.; u/.; u/ w! 1=p.; u/ C! =p w 1! 1=p w 1.u; t/ C! =p w.u; t/.u; t/ w 1.; u/ w o : Otherwie tated, with our definition (.14) in mind, we have obtained:! =p w.u; t/ n o ˇ ˇıg \ ut.'/ ˇ j'j E1! I.; t/ C! 3=p \.; t/! 1=p.; t/ C! =p.; t/ : (.0) w 1 w Step 4: Concluion It i readily checked, thank to the fact that!,! w 1,! w and! \ are control, plu [8, Exercie 1.9], that! I i a control a well a! 3=p \.; t/.! 1=p.; t/ C! =p.; t//. One can thu apply Lemma.1 to relation (.0) w 1 w

10 16 Q. Feng and S. Tindel and get: n o ˇ ˇg \ t.'/ ˇ c p j'j E1! I.; t/ C! 3=p \.; t/! 1=p.; t/ C! =p.; t/ : w 1 w We now take I uch that c p.! 1=p.; t/ C! =p.; t// 1 w 1 w. We obtain: kg \ tk E 1 c p! I.; t/; which end our proof. ut Remark.9 In order to apply Propoition.8 to the heat equation (1.1), we hall conider a meaure defined by.œ0; t / D R t 0 u d. It i worth noting that for a noiy equation like (1.1), we cannot aume that u i properly defined. Thi i why we conider.œ0; t / a an element of E 1 and perform our computation with ditributional increment. 3 L and L Type Etimate Let u now go back to Eq. (1.1), for which we will derive ome a priori etimate in L.R d / and L.R d /. We tart by giving ome baic propertie of our linear heat equation. 3.1 Preliminary Conideration Let u begin by giving a precie meaning to Eq. (1.1), a a particular cae of rough PDE in the weak ene. Definition 3.1 Let w be a rough path atifying Hypothei.3 and.4. Conider the following equation: du t.x/ D 1 u t.x/ C ˇiu t.x/e i dw i t : (3.1) We interpret thi ytem a in Definition.5, with a meaure given by.œ; t// D u r dr: A mentioned in the introduction, we are only focuing here on a priori etimate for the heat equation, which are repreentative of the method at take without being too technical. To thi aim, we label the following aumption, which prevail until the end of the article:

11 On A Priori Etimate for Rough PDE 17 Hypothei 3. One can contruct a path u on Œ0; which olve (3.1) according to Definition 3.1. In addition, u can be obtaineda a limit of a equenceof function u ",whereu " olve: du " t.x/ D 1 u" t.x/ C 1 X ˇiu " t.x/e idw ";i t : (3.) In (3.), the family fw ";i t I ">0;i1giaequence of mooth function converging to w. Recalling our notation (.6) and (.7), we alo aume that: lim! w "!0 1 w 1;".0; / C! w w;".0; / D 0: Remark 3.3 Since we aume that u i obtained a a limit of moothed path u " (ee Hypothei 3.), all the remaining computation have to be undertood a follow: we firt derive our relation for u ", and we then take limit a "! 0. Thi tep will often be implicit for ake of conciene. With Hypothei 3. in hand, we now derive the equation followed by the path u a a firt tep toward L etimate. Propoition 3.4 Let u be the olution of Eq. (3.1) alluded to in Hypothei 3..We alo et f t Dku t k L C Then the following hold true: (i) Let be the E 1 -valued meaure defined a: Then we have:!.; t/ 3 ı t. / D jruj. /dr 0 kruk L dr C 1 kru r k L dr; and St f D up f : (3.3) t kuk L dr 3 provided the quantity above i finite. (ii) The quared path u admit the following repreentation: ıu t. / D ı t. /C 1 X ˇiu.e i /w 1;i t C 1 X.u r ru r /.r /dr: (3.4) jd1 kruk L dr C.t /Sf t ; (3.5) 4u. e ie j /ˇiˇjw ;ij t Cu ;\ t. /; (3.6)

12 18 Q. Feng and S. Tindel where i a generic tet function, and where u ;\ i an element of V q 3 for a certain q <3. (iii) The increment f atifie the following relation: for 0 < t we have ıf t D u.e i/ˇiw 1;i t C 4 jd1 u.e ie j /ˇiˇjw ;ij t C u ;\ t.1/; (3.7) where 1 deignate the function defined on R d and identically equal to 1. Proof With Remark 3.3 in mind, let u divide our proof in everal tep. Proof of (i) Similarlyto[6, Remark.6], and working in the cale E n D W n;1.r d /, we have 1 Z 1 t j.ı / t. /j kruk L drk k L 1C kruk L dr kuk L dr k kw 1;1 ; Invoking now Young inequality (namely AB A A; B with 1 C 1ˇ D 1) we get our claim (3.5). C Bˇ ˇ (3.8) for two poitive number Proof of (ii) According to Definition.5 and 3.1, the olution of Eq. (3.1) can be decompoed a: ıu t. / D ˇiu.e i /w 1;i t C ˇiˇiu.e i e j /w ;ij t Cı t. /Cu \ t. /: (3.9) A mentioned in Remark.7, u can alo be een a a olution to the integral equation (.11), for which the change of variable formula (.1) hold true. Applying thi relation (written in it weak form) to h.z/ D z, we obtain: ıu t. / D u r.u r /dr C o that an integration by part in the firt integral above yield: ıu t. / D jruj. / dr ˇi.u r ru r /.r /dr C u r.e i /dw i r ; ˇi u r.e i /dw i r : (3.10) We now expand the rough integral in (3.10) along the increment of w. We end up with relation (3.6), for a certain remainder u ;\ V q 3.E 1 /.

13 On A Priori Etimate for Rough PDE 19 Proof of (ii) Relation (3.7) i implyobtainedfrom (3.6) by conideringaequence of tet function f n I n 1g uch that lim n!1 n D 1 and lim n!1 r n D 0. ut 3. A Priori Etimate in L With Propoition 3.4 in hand, we can now derive the main etimate of thi ection. Theorem 3.5 Suppoe w fulfill Hypothei.3 and.4, and let u be the olution of Eq. (3.1) given in Hypothei 3..For0 < t,et:! 1.; t/ D! w 1.; t/ C! w.; t/ C! w 1.; t/! w.; t/ C! 4 w.; t/: (3.11) Then the following L norm etimate for the olution u hold true: S f D up ku r k L C 0t 0 kru r k L dr exp c p! 1.0; / ku 0 k L ; (3.1) where c p i a trictly poitive contant. Remark 3.6 Notice that ku r k and R t L 0 kru rk dr are poitive. Therefore relation (3.1) implie that both term are bounded from above. L Proof of Theorem 3.5 Recall that we have obtained the following decompoition in Propoition 3.4: ıu t. / D ı t. / C 1 X ˇiu.e i /w 1;i t C jd1 If we now et g D u and g D, we can recat (3.13)a: ıg t. / D ı g t. / C ˇi g.e i /w 1;i t C jd1 4u. e ie j /ˇiˇjw ;ij t C u ;\ t. /; (3.13) 4 g. e i e j /ˇiˇjw ;ij t C g \ t. /: Thi equation i of the ame form a (.8), and thu we can apply Propoition.8 directly. We get the following bound for g \ t, which i valid whenever! 1.; t/ C!.; t/ L p (recall that p i the regularity index of w): kg \ tk E 1 c p! I.; t/; or equivalently ku ;\ t k E 1 c p! I.; t/; (3.14)

14 130 Q. Feng and S. Tindel where the control! I i defined by:! I.; t/!.; t/! 1=p.; t/ C! =p.; t/ CS u w 1 w t! 1=p.; t/! =p.; t/ C! 4=p.; t/ ; w 1 w w (3.15) and where we recall that we have et: S u t D up t ju j E 0 D up ju j L : t Let u now go back to (3.13), and apply thi relation to D 1 (notice that the function 1 obviouly it in E 1 ). It i readily checked from (3.4)that: ı t.1/ D kruk L dr; and thu, with our notation (3.3) in mind, relation (3.13) become: ıf t D ˇiu.e i/w 1;i t C jd1 4u.e ie j /ˇiˇjw ;ij t C u ;\ t.1/ Therefore, bounding ku k E 0 by St f and invoking (3.14) in order to etimate u ;\ t.1/, we obtain: h i jıf t j! 1=p.; t/ C 4! =p.; t/ S f w 1 w t C c p! I.; t/; (3.16) where! I i given by (3.15). In order to cloe thi expreion, let u further bound the term! in the definition of! I. Namely, according to (3.5), we have!.; t/ 3 kruk L dr C.t /Sf t c S f t ; (3.17) where we recall that we are working on a time interval Œ0;. Plugging thi inequality into the definition of! I, we end up with:! I.; t/ c S f t! 1=p.; t/ C! =p.; t/ C! 1=p.; t/! =p.; t/ C! 4=p.; t/ : w 1 w w 1 w w Reporting the relation above into (3.16), we get jıf t j c S f t! 1=p.; t/ C! =p.; t/ C! 1=p.; t/! =p.; t/ C! 4=p.; t/ w 1 w w 1 w w c ;p S f t! 1.; t/; (3.18)

15 On A Priori Etimate for Rough PDE 131 where! 1 i the control introduced in (3.11). Recall again that inequality (3.18) i valid when! 1.; t/ C!.; t/ L p. It i thu alo atified when! 1.; t/ L p. We are now in a poition to directly apply our rough Gronwall Lemma. to (3.18), with Q D f, D 1=p and! D 0. It i readily checked that! 1 i a control, and hence: S f t exp c p! 1.0; / f 0 D exp c p! 1.0; / ku 0 k L ; (3.19) which end our proof. ut 3.3 L Type Etimate In thi part, we are going to derive ome L etimate for the olution of Eq. (3.1), generalizing the cae D. A the reader will notice, the method i the ame a for the L cae, but we include ome computational detail for convenience. Remark 3.7 We will handle the cae of L etimate for an even integer, inorder to have u 0 and u 0 in the computation below. However, notice that other value of can then be reached by imple interpolation method. We tart thi ection with an analogue of Propoition 3.4. Propoition 3.8 Let u be the olution of Eq. (3.1) alluded to in Hypothei 3., and conider an even integer. Wealoet `t Dku t k L C 0 r kru r k dr; and S`t D up `: u Then the following hold true: (i) Let be the E 1 -valued meaure defined a: 1/ ı t. / D. Then we have:!.; t/. 1/ 4 u r jruj. /dr provided the quantity above i finite. (ii) The path u admit the following repreentation : ıu t. / D ı t. / C 1 X t.u 1 r ru r /.r /dr (3.0) u r jru r j.t /S`t dr C ; (3.1) 4 ˇiu.e i /w 1;i t C jd1 u. e ie j /ˇiˇjw ;ij t C u ;\ t. / (3.)

16 13 Q. Feng and S. Tindel where i a generic tet function, and where u ;\ i an element of V q 3 for a certain q <3. (iii) The increment ` atifie the following relation: for 0 < t we have ı`t D u.e i/ˇiw 1;i t C jd1 u.e ie j /ˇiˇjw ;ij t C u ;\ t.1/; (3.3) where 1 deignate the function defined on R d and identically equal to 1. Proof With Remark 3.3 in mind and defined in (3.1), it i readily checked that: 1/ j.ı / t. /j. C u r jru r j dr u r jru r j drk k L 1 1= 1= ku r k L dr k k W 1;1: (3.4) Invoking now Young inequality a we did in the previou L cae, we get our claim (3.1). The proof of (3.) i imilar to the L cae, except that we apply the change of variable formula and relation (3.9)toh.z/ D z. We obtain: ıu t. / D u r.u 1 r /dr C o that an integration by part in the firt integral above yield: ˇi u r.e i /dw i r ; ıu t. / D. 1/ u r jruj. / dr C.u 1 r ru r /.r /dr ˇi u r.e i /dw i r : (3.5) We now expand the rough integral in (3.10) along the increment of w. We end up with relation (3.), for a certain remainder u ;\ V q=3.e 1 /. A in the L cae, relation (3.3) i imply obtained from (3.) by conidering a equence of tet function f n I n 1g uch that lim n!1 n D 1. ut With Propoition 3.8 in hand, we can now derive the announced etimate in L type pace. Theorem 3.9 Suppoe w fulfill Hypothei.3 and.4, and let u be the olution of Eq. (3.1) given in Hypothei 3..For0 < t,et:! 1.; t/ D! w 1.; t/ C! w.; t/ C! w 1.; t/! w.; t/ C! 4 w.; t/: (3.6)

17 On A Priori Etimate for Rough PDE 133 Then for any even integer, the following L norm etimate for the olution u hold true: ku r k L C u r jru r j dr exp c p! 1.0; / ku 0 k L ; (3.7) up 0t 0 where c p i a trictly poitive contant. Proof Recall that we have obtained the following decompoition in Propoition 3.8: ıu t. / D ı t. / C 1 X ˇiu.e i /w 1;i t C u. e ie j /ˇiˇjw ;ij t jd1 C u ;\ t. /: (3.8) If we now et g D u and g D, we can proceed a in Theorem 3.5 and recat (3.13)a: ıg t. / D ˇig.e i '/w 1;i t C ı g t.'/ C ˇiˇjg.e i e j '/w ;ij t C g \ t.'/; Thi equation i of the ame form a (.8), and thu we can apply Propoition.8 directly. We get the following bound for g \ t, which i valid whenever! 1.; t/ C!.; t/ L p; : kg \ tk E 1 c p! I.; t/; or equivalently ku ;\ t k E 1 c p! I.; t/; (3.9) where the control! I i defined by:! I.; t/!.; t/! 1=p.; t/ C! =p.; t/ C S u w 1 w t! 1=p.; t/! =p.; t/ C! 4=p.; t/ : w 1 w w (3.30) and where we recall that we have S u t D up t ju j E 0 D up ju j L : t Let u now go back to (3.13), and apply thi relation to D 1 (notice that the function 1 obviouly it in E 1 ). It i readily checked from (3.0)that: 1/ ı t.1/ D. u r jruj dr;

18 134 Q. Feng and S. Tindel and thu (3.13) become: ı`t D ˇiu.e i/w 1;i t C jd1 u.e ie j /ˇiˇjw ;ij t C u ;\ t.1/ Therefore, bounding ku k E 0 by S`t and invoking (3.9) in order to etimate u ;\ t.1/, we obtain: h i jı`t j! 1=p.; t/ C 4! =p.; t/ w 1 w S`t C! I.; t/; (3.31) where! I i given by (3.15). In order to cloe thi expreion, let u further bound the term! in the definition of! I. Namely, according to (3.1), we have!.; t/. 1/ 4 u r jru r j.t /S`t dr C 4 c ;p S`t ; which i the equivalent of relation (3.17) in our context. Starting from thi point, we can conclude exactly a in Theorem 3.5. ut 4 Application to Fractional Brownian Motion Thi ection i devoted to the application of our abtract reult of Sect. 3 to ome more concrete example of heat equation driven by an infinite dimenional fractional Brownian motion. Though our general analyi wa focued on equation in R d, we hall treat the cae of both bounded and unbounded domain. 4.1 Equation in Bounded Domain We firt conider the cae of an equation in a bounded domain D. Thi will enable u to compare our hypothei with the aumption contained in [15] for imilar ituation. Let u firt label the condition on our domain. Hypothei 4.1 In thi ection, we conider an open, bounded domain D with mooth and atifying the cone property. On uch a domain D, we wih to give condition which are cloe enough to the one produced in [15]. Thi i why we conider an operator C given a follow: Hypothei 4. In the remainder of the ection, C will tand for a linear, elfadjoint, poitive trace-cla operator on L.D/. Thi operator admit an orthonormal bai.e i / inc of eigenfunction, with correponding eigenvalue. i / inc.it

19 On A Priori Etimate for Rough PDE 135 alo admit an integral repreentation, whoe generating kernel i denoted a. Summarizing, for all i 0 and for almot every x D we have: Z Ce i.x/ D.x; y/e i.y/ dy D i e i.x/: (4.1) D We can now formulate our a priori etimate in thi context: Propoition 4.3 Let D R d be a domain fulfilling Hypothei 4.1, together with an operator C a in Hypothei 4.. On D, we conider the following equation: du t.x/ D 1 u t.x/ C i u t.x/e i.x/db i t ; (4.) where.b i t / tr C/ inc i a equence of one-dimenional, independent, identically ditributed fractional Brownian motion with Hurt parameter H. 1 3 ;1/, and 0 i a poitive parameter. For the definition of e i and i, we refer to Hypothei 4.. In addition, we uppoe that our operator C and it kernel atify the following condition: X A up k.x; /k L.D/ Ckr.x; /k L.D/ < 1; and i 1 < 1: (4.3) xd Then the reult from Theorem 3.5 and 3.9 apply. Proof It i well known (ee e.g. [8, Chap. 15]) that any finite dimenional fractional Brownian motion.b i / in can be lifted a a rough path. It i thu enough to prove condition (.6) and(.7). We hall focu on condition (.6), the other one being checked with the ame kind of argument. In order to verify (.6),imilarlyto[15], we tart by recating (4.1)a: Z e i.x/ D 1 i D Z.x; y/e i.y/ dy; and re i.x/ D 1 i r.x; y/e i.y/ dy: D Hence, invoking Cauchy-Schwarz inequality and relation (4.3), we obtain: je i j E1 1 i A ke i k L.D/ D 1 i A : (4.4) Now notice that (.6) i enured by the condition EŒ! 1=p w 1.0; / < 1,where i our time horizon. Furthermore, h i E! 1=p.0; / D w 1 i0 i.1 Cje i j E1 / E N Œw 1;i I V p.œ; t / ;

20 136 Q. Feng and S. Tindel and ince EŒN Œw 1;i I V p.œ; t / i uniformly bounded in i, we end up with h E! 1=p.0; / w 1 i c ;w 1 X 1 i ; which i a finite quantity according to our aumption (4.3). In concluion, Hypothei.3 and.4 are atified, and Theorem 3.5 and 3.9 hold true. ut Remark 4.4 With repect to [15], we have added here the aumption up kr.x; /k L.D/ < 1; xd which i an artifact of our variational approach. Thi being aid, let u recall that our method applie to rough ituation (compared to the cae H >1=dealt with in [15]). We alo believe that our method extend to non linear equation, with a noiy term of the form P 1 i.u t.x//e i.x/db i t for a mooth coefficient. 4. Equation in R d OnthewholepaceR d, choice of orthonormal bai of L are wide. For ake of concretene, we will tick here to a wavelet bai baed on Shannon wavelet, though a much more general etting can be found e.g. in [14]. Let u tart by defining the L bai alluded to above (we refer again to [14] for proof of general fact on wavelet). Lemma 4.5 Let W R! R be defined a.x/ D in.x 1=/.x 1=/ in.x 1=/ :.x 1=/ Then L.R/, and the following hold true: (i) Let u introduce a family of caled function f j;k I j 0; k Zg by: j;k.x/ D j x j k j : (4.5) Thi family i an orthonormal bai of L.R/. (ii) One can obtain an orthonormal bai of L.R d / by tenorizing the previou bai of L.R/. Namely, for all j 0 and for n D.n 1 ; ; n d /, we denote j;n.x/ D dj= x1 j n 1 ; ; x d j n d : j j

21 On A Priori Etimate for Rough PDE 137 Then f j;n.x/g. j;n/z dc1 readily checked that: i an orthonormal bai of L.R d /. In addition, it i j j;k j E1 jd ; (4.6) where we recall that we work in the cale E n D W n;1.r/. Remark 4.6 A completely correct verion of Lemma 4.5 hould include a o-called father wavelet. We omit thi tep for notational ake. Under the etting of Lemma 4.5, here i our example of tochatic heat equation on R d : Propoition 4.7 Conider the equation du t.x/ D 1 u t.x/ C X jd0 t ; nz d ˇj;n u t.x/ j;n.x/db j;n where fb j;n I j 0; n Z d g i a equence of one-dimenional, independent, identically ditributed fractional Brownian motion with Hurt parameter H. 1 3 ;1/, and fˇj;ni j 0; n Z d g i a family of poitive coefficient. We aume that Aˇ X jd0 nz d dj ˇj;n < 1: (4.7) Then the reult of Theorem 3.5 and 3.9 apply. Proof We proceed a for Propoition 4.3, and we are eaily reduced to how that EŒ! 1=p.0; / i a finite quantity. In our cae, we have w 1 h i E! 1=p.0; / D w 1 X ˇj;n jd0 nz d 1 Cj j;n j E1 E N Œw 1;j;n I V p.œ; t / : Moreover, the coefficient EŒN Œw 1;j;n I V p.œ; t / are uniformly bounded in j; n. Hence, owing to relation (4.6), we get: h i E! 1=p.0; / c w 1 ;w X jd0 nz d dj ˇj;n D c ;w Aˇ; where Aˇ i introduced in condition (4.7). Thi conclude our proof in a traightforward way. ut

22 138 Q. Feng and S. Tindel Reference 1. I. Bailleul, M. Gubinelli, Unbounded rough driver (015). arxiv: M. Caruana, P. Friz, H. Oberhauer, A (rough) pathwie approach to a cla of non-linear tochatic partial differential equation. Ann. Int. H. Poincaré Anal. Non Linéaire 8(1), 7 46 (011) 3. L. Chen, Y. Hu, K. Kalbai, D. Nualart, Intermittency for the tochatic heat equation driven by a rough time fractional Gauian noie (016). arxiv: A. Deya, A dicrete approach to rough parabolic equation. Electron. J. Probab. 16(54), (011) 5. A. Deya, M. Gubinelli, S. Tindel, Non-linear rough heat equation. Probab. Theory Relat. Field 153, (01) 6. A. Deya, M. Gubinelli, M. Hofmanova, S. Tindel, General a priori etimate for rough PDE with application to rough conervation law (016). arxiv: P. Friz, M. Hairer, A Coure on Rough Path (Springer, Berlin, 014) 8. P. Friz, N. Victoir, Multidimenional Dimenional Procee Seen a Rough Path (Cambridge Univerity Pre, Cambridge, 010) 9. M. Gubinelli, S. Tindel, Rough evolution equation. Ann. Probab. 38(1), 1 75 (010) 10. M. Gubinelli, P. Imkeller, N. Perkowki, Paracontrolled ditribution and ingular PDE. Forum Math. Pi 3(e6), 75 (015) 11. M. Gubinelli, S. Tindel, I. Torrecilla, Controlled vicoity olution of fully nonlinear rough PDE (014). arxiv: M. Hairer, Solving the KPZ equation. Ann. Math. 178(), (013) 13. M. Hairer, A theory of regularity tructure. Invent. Math. 198(), (014) 14. S. Mallat, A Wavelet Tour of Signal Proceing (Academic, London, 1998) 15. D. Nualart, P.-A. Vuillermot, Variational olution for partial differential equation driven by a fractional noie. J. Funct. Anal. 3, (006)