Semilinear Kolmogorov equations and applications to stochastic optimal control

Transcription

1 Semilinear Kolmogorov equaions and applicaions o sochasic opimal conrol Federica Masiero 1 Advisor: Prof. Marco Fuhrman 2 1 Diparimeno di Maemaica, Universià degli sudi di Milano, Via Saldini 5, 2133 Milano, Ialy; fmasiero@ma.unimi.i 2 Diparimeno di Maemaica, Poliecnico di Milano, piazza Leonardo da Vinci 32, 2133 Milano, Ialy; marco.fuhrman@polimi.i

2 Acknowledgemens I would like o hank Gianmario Tessiore for his consan ineres in his hesis, Alessandra Lunardi, Fauso Gozzi and Robero Triggiani for encouragemen and ineresing suggesions. I wish o hank also Giuseppe Da Prao for hospialiy a he Scuola Normale Superiore in Pisa, suggesions and helpful discussions. I wish o hank my advisor, M.Fuhrman: I m indebed o him for his help and encouragemen in hese las years of my Ph.D. programm, while working for his hesis. Finally, I hank all people who encouraged me in he period of my Ph.D. program. 2

3 Conens 1 Inroducion Plan of he work and main resuls Originaliy of he work Preliminaries and noaions Trace class and Hilber Schmid operaors Gaussian measures in Hilber spaces Reproducing kernel Real and H-valued maringales Q-Wiener process Sochasic inegral wih respec o Q-Wiener processes Cylindrical Wiener processes and sochasic inegral Space ime whie noise Spaces of coninuous and differeniable funcions Noaions on spaces of funcions Regularizaion of coninuous funcions The class G G 33 5 The semilinear backward Kolmogorov equaion The sochasic differenial equaion Kolmogorov equaions in Hilber spaces Special case: he Ornsein-Uhlenbeck semigroup The finie dimensional case A perurbed Ornsein-Uhlenbeck process Applicaion of he Bismu-Elworhy-Xe formula Sochasic opimal conrol problems Formulaion of sochasic opimal conrol problems Applicaions o a class of opimal conrol problems: srong formulaion The fundamenal relaion for he opimal conrol problem Applicaion of he Bismu-Elworhy-Xe formula: a conrol problem Soluion of he closed loop equaion: a case of addiive noise Applicaions o a class of opimal conrol problems: weak formulaion Sochasic opimal conrol for some specific models The conrolled nonlinear hea equaion The conrolled wave equaion Sochasic delay equaions

4 1 Inroducion This hesis is devoed o he sudy of semilinear parabolic differenial equaions in Hilber spaces, in paricular Hamilon Jacobi Bellman equaions, and applicaions o sochasic conrol problems. The firs par of he hesis concerns a class of semilinear parabolic differenial equaions in a separable Hilber space H, namely we sudy he following equaion: { u, x) = A u, x) + ψ, x, u, x), G, ) u, x) ), ut, x) = ϕ x). [, T ], x H 1.1) We ofen refer o his equaion as semilinear backward Komogorov equaion. The linear operaor A is formally defined by A f x) = 1 2 T race G, x) G, x) 2 f x) ) + Ax, f x) H + F, x), f x) H, where G is he derivaive in he direcions seleced by G. More precisely, le f : H R and G : H L Ξ, H): we define G f x + sg x) h) f x) f x; h) = lim, s R. s s The operaor A arises as he generaor of an appropriae Markov process X in H. X is consruc as follows: consider a sochasic evoluion equaion { dx τ) = [AX τ) + F τ, X τ))] dτ + G τ, X τ)) dw τ), τ [, T ] 1.2) X ) = x. A is he generaor of a srongly coninuous semigroup e τa, τ, in H, F is a map from [, T ] H in H, W is a cylindrical Wiener process in anoher separable Hilber space Ξ and G is a map defined in [, T ] H and aking values in L Ξ, H). Under suiable assumpions on he coefficiens, equaion 1.2) admis a unique soluion see [15]), such ha, P-a.s., X τ,, x) = e τa x + e τ s)a F s, X s)) ds + e τ s)a G s, X s)) dw s), 1.3) for arbirary τ [, T ]. X τ,, x) sands for he value of he process a ime τ, wih iniial condiion x given a ime. We can define he ransiion semigroup on coninuous and bounded funcions ϕ : H R as P,τ [ϕ] x) = Eϕ X τ,, x)). If ϕ is sufficienly regular hen u, x) = P,T [ϕ] x) is a classical soluion of he Kolmogorov equaion { u, x) + A u, x) =, [, T ], x H 1.4) ut, x) = ϕ x). When ϕ is no regular, he funcion u, x) = P,T [ϕ] x) can be regarded as a generalized mild) soluion of he linear backward Kolmogorov equaion 1.4). For a deailed sudy of equaion 1.4) we refer o he book [15]. We look for mild soluions of he semilinear backward Kolmogorov equaion 1.1), ha is funcions u, x) saisfying a sor of variaion of consans formula, namely a mild soluion has o saisfy u, x) = P,T [ϕ] x) ] P,s [ψs,, u s, ), Gs, ) u s, ) x) ds, [, T ], x H. 4

5 We noice ha his formula is meaningful if u has only he firs derivaive G, ) u, x), and provided ϕ and ψ saisfy some growh and measurabiliy condiions. In he case of G consan, he G-gradien has already been used, e.g. by E. Priola and L. Gross, see he discussion in [17], chaper.3. We consider he G gradien also in he case of G no necessarily consan, anyway he main novely is he use of he G gradien in he definiion of mild soluions for he semilinear backward Kolmogorov equaion 1.1). Semilinear Kolmogorov equaions of a more general form han equaion 1.1) have already been sudied, namely equaions of he form { u, x) = A u, x) + ψ, x, u, x), u, x)), ut, x) = ϕ x), [, T ], x H 1.5) see e.g. [6] and [28], in he case of G consan. In his case, in order o find mild soluions o equaion 1.5), i is required ha, for every bounded and coninuous funcion φ, P,τ [φ] is differeniable, and he direcional derivaive saisfies P,τ [φ] x) h c τ ) α sup φ x) h, 1.6) x H for some α, 1) and for every h H. We sudy similar condiions which allow o solve equaion 1.1) in mild sense only wih assumpions on he G-derivaive of he semigroup. We briefly gives a moivaion of his fac. Consider a parabolic differenial equaion of he form { u, x) = A u, x) + ψ, x, G, ) u, x) ), ut, x) = ϕ x). [, T ], x H The mild soluion o his equaion has o saisfy ] u, x) = P,T [ϕ] x) P,s [ψs,, Gs, ) u s, ) x) ds. We formally ake he G-derivaive on boh sides: ] G, ) u, x) = G, ) P,T [ϕ] x) G, ) P,s [ψs,, u s, ), Gs, ) u s, ) x) ds. I urns ou ha his is a problem of finding a fixed poin G u, and o solve i we need assumpions on he G-derivaive of P,τ [φ] Namely, we make he following fundamenal assumpion: we require ha for every bounded and coninuous funcion φ, P,τ [φ] is G-differeniable, and he G-direcional derivaive saisfies G, ) P,τ [φ] x) ξ c τ ) α sup φ x) ξ, 1.7) x H for some α, 1) and for every ξ Ξ. Or, more generally, we require ha for every coninuous funcion φ such ha sup x H < for some k >, he G-direcional derivaive of he semigroup saisfies φx) 1+ x 2 ) k/2 G, ) P,τ [φ] x) ξ c τ ) α 1 + x 2) k/2 sup x H φ x) 1 + x 2) ξ, 1.8a) k/2 5

6 for some α, 1) and for every ξ Ξ. We consider he case of he Ornsein Uhlenbeck process, ha is X is a soluion o equaion { dx τ) = AX τ) dτ + GdW τ), τ [, T ] X ) = x. This is nohing else han equaion 1.2) wih F = and G consan. In his case, we are able o relae he assumpion on he G-derivaive wih properies of A and G. We prove ha P,τ [φ] is G-differeniable if e A G Ξ) Q 1/2 H). Moreover condiion 1.7) holds if for some < α < 1 and c > he operaor norm saisfies Q 1/2 e A G c α, for < T. On he oher side, for he Ornsein Uhlenbeck semigroup, differeniabiliy of P,τ [φ] is he same as o ask ha he semigroup is srong Feller, see e.g. [15]. I has been also proved ha condiion 1.6) holds rue if and only if e A H) Q 1/2 H), and Q 1/2 e A c α, for < T. We give examples of Ornsein Uhlenbeck processes when i is clear ha condiion 1.7) is less resricive han condiion 1.6). For he Ornsein Uhlenbeck process he assumpion 1.6) on he explosion for small imes of he derivaive has been also relaed o he null conrollabiliy of he sysem { y ) = Ay ) + Gu ), 1.9) y ) = x, wih he iniial sae x in H, and o some growh condiions of he energy o seer x yo in ime, see e.g. [15]. If x H is null conrollable, he energy o seer an iniial sae x o in ime is given by { 1/2 } E C, x) = min u s ds) 2 : y ) = x, y ) =, and i urns ou ha E C, x) = Q 1/2 e A x. So asking e A H) Q 1/2 H) is equivalen o require null conrollabiliy of 1.9), and asking e A c α is equivalen o ask ha he energy o seer x o in ime blows up like Q 1/2 α. Insead, our assumpions 1.7) on he explosion for small imes of he G-derivaive can be relaed o he null conrollabiliy of he sysem { y ) = Ay ) + Gu ), 1.1) y ) = x, wih he iniial sae x belonging o Im G H, and o some growh condiions of he energy o seer x yo in ime. If x Im G, namely x = Gξ, E C, x) = Q 1/2 e A Gξ. 6

7 So e A G Ξ) Q 1/2 H) is equivalen o null conrollabiliy of 1.1) wih he iniial sae in Im G, and Q 1/2 e A G c α is equivalen o ask ha he energy o seer x Im G o in ime blows up like α. In he finie dimensional case hese condiions are always verified and he Kolmogorov equaion 1.1) can be solved, in he generalized sense we have menioned above, wihou any non degeneracy assumpions on G. Namely we do no have o ask ha A is hypoellipic in he sense of Hormander, see [32]. Indeed requiring ha A is hypoellipic is he same as requiring ha he conrolled sysem 1.9) is null conrollable: we only ask null conrollabiliy for an iniial sae in Im G. Anoher case when we verify condiion 1.7) is when we can apply he Bismu-Elworhy-Xe formula, see e.g. [4] and [21]. In his case also 1.6) is saisfied; anyway we apply his resul o solve parabolic differenial equaions of he form of 1.1) wih G no consan. Nex we consider applicaions o opimal conrol problems. We ake a conrolled sochasic evoluion equaion, dx u τ) = [AX u τ) + F τ, X u τ)) + C τ, X u τ)) u τ)] dτ + G τ, X u τ)) dw τ), τ [, T ] X u ) = x, 1.11) where we require C τ, X u τ)) = G τ, X u τ)) R τ, X u τ)). This special srucure of he conrol erm we need in equaion 1.11) is a resricion, however i arises from concree models, such as conrolled wave equaions. Due o he special form of he conrol erm he Hamilon Jacobi Bellman equaion urns ou o be { v, x) = A v, x) + ψ, x, G, ) v, x) ), vt, x) = ϕ x), [, T ], x H 1.12) and so i has he form of 1.1), wih he nonlinear par depending on, x and on he G-derivaive of v. In he case of addiive noise, ha is G consan, Hamilon Jacobi Bellman equaions have already been sudied in differen ways. In he book [2], equaion 1.12) is reaed in he case when F = and G saisfies T racegg < +, moreover R equals he ideniy. Under some regulariy condiions he auhors find a unique classical soluion, by using an analyic approach, and solve he opimal conrol problem relaed. In [6] and [7] i is assumed ha A is self adjoin and C and G equal he ideniy. This assumpion on G is a nondegeneracy assumpion. Exisence and uniqueness of a mild soluion of he Hamilon Jacobi Bellman equaion is proved in a space of coninuous funcions wih a derivaive ha blows up as ends o. In [28] and [29] he resuls in he papers [6] and [7] are exended o more general siuaions, anyway G is required o be consan. Some general Hamilon Jacobi Bellman equaions arising in opimal conrol problems for reacion diffusion equaions have been sudied in he monograph [11]. A complee probabilisic approach in he case of G non consan is given in he paper [25]. A semilinear backward Kolmogorov equaion like 1.1) is sudied wihou any nondegeneracy assumpion on G, and a unique Gaeaux differeniable mild soluion o 1.1) is found by an approach based on backward sochasic differenial equaions. More regulariy on he final daum ϕ and on he Hamilonian ψ is asked, on he conrary no smoohing propery for he semigroup is needed. We also menion he noion of viscosiy soluion in relaion o he Hamilon Jacobi Bellman equaion: his heory has been developed by several auhors, and a collecion of main resuls is [12]. For wha concerns he infinie dimensional case we cie he fundamenal papers [13], [35] and [37]. In general in his approach i does no follows a feedback law for he opimal conrol. We menion also he papers [3] and [36] which deal wih he conrol of he Zakai equaion. 7

8 Recenly Hamilon Jacobi Bellman equaions have been also considered in L 2 H, µ), where µ is a probabiliy measure on H, see e.g. [27]. We formulae he conrol problem a firs in he srong sense: we consider a cos J, x, u) = E g s, X u s), u s)) ds + Eφ X u T )), wih g and φ saisfying some measurabiliy and growh condiions in x. This cos J has o be minimized over all adaped conrols u aking values in a preassigned bounded se. We define he hamilonian ψ as follows: ψ, x, q) = inf {g, x, u) + qu : u U}, ψ, x, p) = ψ, x, pr τ, x)). We prove he fundamenal relaion for he cos J, ha is J, x, u) = v, x) + E { ψ s,, G v s, ) ) X u s)) 1.13) + G v s, X u s)) R s, X u s)) u s) + g s, X u s), u s)) } ds. By he definiion of ψ, we see ha, for every u U, J, x, u) v, x), for every [, T ] and x H, and so also he value funcion saisfies J, x) v, x). In he case of addiive noise he fundamenal relaion has been already prooved by smoohing he coefficiens and hen applying Io s formula, see [28]. The fundamenal relaion allows us o say ha for every admissible conrols u. We se J, x, u) v, x) 1.14) Γ τ, x, q) = {u U : g τ, x, u) + qu = ψ, x, q)}. In 1.14), equaliy holds if and only if u τ) belongs o a measurable selecion Γ τ, X u τ), G v τ, X u τ)) R τ, X u τ)) ) of Γ. Wih u saisfying his feedback law, we can wrie he closed loop equaion dx τ) = [AX τ) + F τ, X τ))] dτ + G τ, X τ)) dw τ) +G τ, X τ)) R τ, X τ)) Γ τ, X τ), G v τ, X τ)) R τ, X τ)) ) dτ, τ [, T ] X ) = x. 1.15) In he case of addiive noise and wih A generaor of a compac semigroup we solve he closed loop equaion, and we find an opimal pair u τ), X τ)). If G = εi, a similar problem of finding he soluion of he closed loop equaion has already been solved in he paper [6]. We also rea he conrol problem in he weak sense, see e. g. [22]; in his formulaion we are able o perform he sinhesys of he opimal conrol, by solving in he weak sense he closed loop equaion: in he weak sense he conrol problem can be compleely solved. We consider applicaions o specific conrolled models: a conrolled semilinear hea equaion and a conrolled wave equaion. In absrac formulaion, he conrolled wave equaion refers o a perurbed Ornsein Uhlenbeck process. In order o verify 1.7), we consider he associaed conrolled linear sysem of he form 1.1). I is well known ha in he case of he wave equaion 1.9) is conrollable, bu o seer a general sae x o requires an energy growing oo fas as goes o. Wih our resricion on he iniial sae x he energy goes like 1/ and his implies our hypohesis 1.7) on he G-derivaive of he semigroup wih α = 1 2, while i does no exis α, 1) such ha 1.6) is verified. So his is a case when also he derivaive exiss, bu i is no saisfy condiion of explosion for small ime necessary o apply known resuls in solving he relaed Hamilon Jacobi Bellman equaion 1.12). 8

9 1.1 Plan of he work and main resuls This hesis is organized as follows: in chaper 2 we inroduce sandard analyical and probabilisic conceps ha we will use in he following, and ha are fundamenal in he infinie dimension approach. In Chaper 3 we inroduce some noaions on spaces of coninuous and differeniable funcion, nex we remember some resuls on approximaion of funcions in infinie dimensions and we prove some furher resuls of approximaion. These resuls will be used in chaper 6, o prove he fundamenal relaion for he opimal conrol problem. In Chaper 4 we define he class of G differeniable funcions and we sudy some properies of his class. Chapers 5 and 6 are he core of he hesis: in chaper 5 we sudy he semilinear Kolmogorov equaion 1.1): we inroduce he concep of mild soluion and we prove an exisence and uniqueness resul. We apply hese resuls o he Ornsein Uhlenbeck semigroup and we deduce some resul of exisence and uniqueness of mild soluions of semilinear Kolmogorov equaions in finie dimensions, even when he second order differenial operaor is no hypoellipic. Chaper 6 is devoed o he sudy of he sochasic opimal conrol problem by means of he Hamilon Jacobi Bellman equaion. We sudy sochasic opimal conrol problems of he form 1.11) such ha he Hamilon Jacobi Bellman equaion has he form of he semilinear Kolmogorov equaion 1.1). We formulae he conrol problems in he srong and weak sense. In chaper 7 we apply our resuls o a conrolled hea equaion and o a conrolled wave equaion. The main resuls of he hesis are conained in heorem 5.15, on he exisence and uniqueness of a mild soluion for he semilinear Kolmogororv equaion 1.1), in heorem 5.2, where we apply resuls on exisence and uniqueness o he case of a Ornsein Uhlenbeck process; in heorem 6.15 and heorem 6.19, which sum up he resuls on he soluion of he conrol problem, respecively in he srong and weak formulaion. There are resuls also in chaper 7, which deals wih applicaions o specific conrolled models. Resuls concernig conrollabiliy of a conrolled hea equaion are summed up in heorems 7.3 and 7.3. Some ineresing resuls are achieved in relaion o he conrolled wave equaion, and are summed up in he heorems 7.9 and Originaliy of he work As we have menioned above, chaper 5 and chaper 6 are he core of he hesis, and conain mos of he original resuls. Chaper 2 and 3 are inroducive and conain well known facs, apar from some approximaion resuls ha we prove mainly in lemma 3.6 and in proposiion 3.7. In chaper 4 we inroduce he class of G-differeniable funcions: he originaliy of his class of funcions is in he applicaions o he soluion of semilinear Kolmogorov equaions. Chaper 7 is abou applicaions, and conains some new resuls. We do no usually remember he proofs of known facs, we only wrie down proofs of original resuls, or proofs of facs where somehing new has been inroduced. 9

10 2 Preliminaries and noaions We inroduce some noaions and some facs abou race class and Hilber Schmid operaors, Gaussian measures, cylindrical Wiener processes and sochasic inegraion in infinie dimensional Hilber spaces. 2.1 Trace class and Hilber Schmid operaors In he following he norm of an elemen x in a Banach space E will be denoed by x E or simply x, if no confusion is possible. If F is anoher Banach space, LE, F ) denoes he space of bounded linear operaors from E o F, endowed wih he usual operaor norm. If E is a Banach space we denoe by E is dual space. Wih he leers Ξ, H, K, we will always denoe Hilber spaces, wih scalar produc,. All Hilber spaces are assumed o be real and separable. We will work also wih he ses of race class and Hilber-Schmid operaors from Ξ o K. We refer mainly o [2]. An operaor Q L Ξ, K) is said o be nuclear or of race class if here exis wo sequences {k j } K and {ξ j } Ξ such ha k j ξ j < 2.1) and Q has he represenaion Qx = j=1 k j ξ j, x, x Ξ. j=1 2.2a) Noe ha he sequence {ξ j } Ξ is idenified wih a sequence of operaors in he dual space Ξ. So, more generally, he space of race class operaors acing among wo Banach spaces E and F can be defined: le T LE, F ). T is said o be nuclear or of race class if here exis wo sequences {f j } F and {e j } E such ha and T has he represenaion T x = f j e j < j=1 f j e j x), x E. j=1 In he following we will work wih race class operaors acing among Hilber spaces. If condiion 2.1) holds, hen he series in 2.2a) is norm convergen, moreover Q is compac. The space of all race class operaors from Ξ o K, endowed wih he norm Q 1 = inf k j ξ j : Qx = k j ξ j, x j=1 is a Banach space, see [2], ha we denoe by L 1 Ξ, K). Le K be a separable Hilber space and le {k j } K be a complee orhonormal sysem in K. We denoe L 1 K, K) by L 1 K). If Q L 1 K) we inroduce he race of Q: Tr Q = j=1 T k j, k j. j=1 1

11 I can be proved ha if Q L 1 K) hen Tr Q is well defined and i is independen on he choice of he basis {k j }. Moreover Tr Q Q 1. If Q is a nonnegaive operaor in L K), i is nuclear if and only if here exiss an orhonormal basis {e j } K such ha T e j, e j <, j=1 and in his case Tr Q = Q 1. We briefly inroduce he space of Hilber Schmid operaors acing beween wo separable Hilber spaces Ξ and K. Le {ξ j } be a complee orhonormal basis in Ξ. An operaor S L Ξ, K) is a Hilber Schmid operaor if T e j 2 <. We inroduce he norm j=1 T 2 = T e j 2 j=1 1/2. 2.3) I urns ou ha T 2 = T 2. The space L 2 Ξ, K) of Hilber Schmid operaors, endowed wih he norm 2.3), is iself a separable Hilber space, wih scalar produc T, S = T e j, Se j, T, S L 2 Ξ, K). j=1 2.2 Gaussian measures in Hilber spaces We wan o recall some basic facs on Gaussian measures in Hilber spaces. We refer mainly o [5]. In he following H is a separable real Hilber space and by B H) we denoe he Borel σ-field in H. We se L + H) and L + 1 H) respecively he subse of L H) and L 1 H) consising of all nonnegaive symmeric operaors. Le H = R n and le Q be an n n posiive definie marix. We remember ha in R n a non degenerae Gaussian measure is defined by 1 N a,q dx) = 2π) n/2 de Q) exp 1 Q 1 x a), x a) ) dx, 2 where a R n is he mean of he Gaussian measure and Q is he covariance operaor. If a =, N Q dx) sands for N a,q dx). The Fourier ransform of N a,q is given by N a,q h) := e R i h,x N a,q dx) = e i a,h 1 2 Qh,h, h R n. 2.4) n A general gaussian disribuion on R n is an image of a non degenerae Gaussian disribuion under a linear ransformaion. Is Fourier ransform is of he he form 2.4), and is deermined by he mean a R n and by a nonnegaive symmeric marix Q. The Fourier ransform fully defines he measure, indeed if a R n and Q L + R n ), and if µ is a probabiliy measure on R n, B R n )) such ha e R i h,x µ dx) = e i a,h 1 2 Qh,h, h R n, n 11

12 hen µ = N a,q. Le H be an arbirary separable Hilber space: if a H and Q L + 1 H), hen here exiss a unique probabiliy measure µ on H, B H)) such ha e i h,x µ dx) = e i a,h 1 2 Qh,h, h H. H We se µ = N a,q. Consider a random variable X : Ω, F, P) H. The law of X, denoed by L X), is given by he following probabiliy measure on H, B H)) L X) B) = P X 1 B) ), B B H). A random variable X aking values in H is said o be Gaussian if is law L X) is a Gaussian measure. A random variable X wih values in H is Gaussian if and only if for every h H he real random variable h, X is a real Gaussian random variable. We lis some resuls on inegrals wih respec o Gaussian measures: xn a,q dx) = a, H H H H x a, y x a, z N a,q dx) = Qy, z, x a 2 N a,q dx) = Tr Q e h,x N a,q dx) = e i a,h Qh,h. Nex we invesigae wheher wo Gaussian measures N Q and N a,q are equivalen or singular. If H is finie dimensional, ha is H = R n, wo non degenerae Gaussian measures N Q and N a,q are equivalen and we have dn a,q dn Q x) = e 1 2 Q 1/2 a 2 + Q 1/2 a,q 1/2 x, x R n. In he infinie dimensional case he wo Gaussian measures N Q and N a,q can be eiher equivalen or singular, more precisely i happens ha if a Q 1/2 H) hen N Q and N a,q are equivalen, if a / Q 1/2 H) hen N Q and N a,q are singular. Moreover, if he wo Gaussian measures N Q and N a,q are equivalen, he Cameron Marin formula for he Radon Nikodym derivaive holds: dn a,q dn Q x) = e 1 2 Q 1/2 a 2 + Q 1/2 a,q 1/2 x, x H. 2.5) In he previous formula by Q 1/2 we denoe he pseudoinverse of Q 1/2. In wha follows we will ofen mee inverses of operaors which are no one-o-one. Le Ξ and K be wo Hilber spaces and le O L Ξ, K). Then Ξ = ker O is a closed subspace of Ξ. Le Ξ 1 be he orhogonal 12

13 complemen of Ξ in Ξ: Ξ 1 is closed, oo. Denoe by O 1 he resricion of O o Ξ 1 : O 1 is one-o-one and Im O 1 = Im O. For k Im O, we define O 1 by seing O 1 k) := O 1 1 k). The operaor O 1 : Im O Ξ is called he pseudoinverse of O. O 1 is linear and closed bu in general no coninuous. Noe ha if k Im O, Reproducing kernel inf { ξ : O ξ) = k} = O 1 1 k). We are given an operaor Q L + 1 H). Le e k) be a complee orhonormal basis in H and le λ k ) be a sequence of nonnegaive numbers such ha Q admis he decomposiion Qe k = λ k e k. In virue of his decomposiion, i can be seen ha Q 1/2 H) is a proper subspace of H, which is called he reproducing kernel of he Gaussian measure N Q. For simpliciy we assume ha ker {Q} =, ha is equivalen o he requiremen ha λ k > for all k. In his case Q 1/2 H) is dense in H. We denoe by L 2 H, N Q ) he space L 2 H, B H), N Q ) of funcions defined in H, aking values in R and measurable wih respec o he Borel σ field, and square inegrable wih respec o he gaussian measure N Q. We build an isomorphism W from H ino L 2 H, N Q ). A firs we define an applicaion for f in he subspace Q 1/2 H): if f Q 1/2 H) hen W f L 2 H, N Q ) is well defined by seing W f x) = Q 1/2 f, x, x H. Moreover, by H H Q 1/2 2 f, x NQ dx) = Q 1/2 f, x Q 1/2 g, x N Q dx) = Q 1/2 f, Q 1/2 f = f 2, Q 1/2 f, Q 1/2 g = f, g we can see ha W is an isomery for he norm of H from Q 1/2 H) in L 2 H, N Q ). By densiy of Q 1/2 H) in H, i can be uniquely exended o all H. The applicaion W is called he whie noise funcion. If ker Q), hen W can be defined in he same way for any f H = Q 1/2 H). 2.3 Real and H-valued maringales Le Ω, F, P) be a complee probabiliy space. A collecion of H valued random variables {X ), } is called a sochasic process; we say ha he process is coninuous if for almos all ω Ω he funcion X ) ω) : [, + ) H is coninuous. Le {F, } be a filraion in Ω, F, P). We say ha a filraion {F, } saisfies he usual condiions if: {F, } is complee, ha is F, and so every F, conains he measurable ses of null P-measure; {F, } is coninuous on he righ, ha is for every F = F +, where by definiion F + = s> F s. In he following we will always consider filraions saisfying he usual condiions. We recall some basic definiions on sochasic processes, in paricular of maringales. We refer mainly o [33] and [45]. We lis some definiions concerning measurabiliy of sochasic processes. 13

14 Definiion A sochasic process {X ), } is adaped o he filraion {F, } if for every X ) : Ω H is measurable wih respec o F. 2. A sochasic process {X ), } is said o be measurable on [, T ] if i is measurable as an applicaion from Ω [, T ] equipped wih he σ-field generaed by F T B [, T ]), ino H, B H)). 3. A sochasic process {X ), } is said o be progressively measurable if i is measurable on every inerval [, ],. 4. A sochasic process {X ), } is said o be predicable if i is measurable wih respec o he σ algebra generaed by ses of he form s, ] F, wih s < and F F s, and by {} F wih F F. We inroduce he concep of maringale. Definiion 2.2 An H-valued sochasic process {X ), [, T ]} is a maringale if E X ) < for every [, T ] and E [X ) F s ] = X s), for every > s,, s [, T ]. Definiion 2.3 An H-valued sochasic process {X ), [, T ]} is a local maringale if here exiss an increasing sequence of sopping imes {T n }, T n, such ha he sopped process {X n ) = X T n ), [, T ]} is a maringale. Definiion 2.4 An H-valued sochasic process {X ), [, T ]} is a semimaringale if i can be wrien as he sum of a local maringale and of a process of bounded variaion. We work in he framework of [33]: le M loc c be he linear space of all coninuous local maringales X such ha X =. Endow M loc c wih he opology of he uniform convergence in probabiliy. More precisely le ρ be he disance defined by ρ X, Y ) = E [ sup [,T ] X ) Y ) sup [,T ] X ) Y ) 2 for all X, Y M loc c. A sequence {X n } M loc c is a Cauchy sequence if and only if for any ε >, P sup X n ) X m ) > ε) ends o as n, m. Wih respec o he opology of he uniform convergence in probabiliy M loc c is complee, see [33], heorem 1.4, pp 142. Moreover le M c be he linear space of all coninuous square inegrable maringales X, endowed wih he norm [ ] 1 X = E sup X ) 2 2. [,T ] I urns ou ha M c is a Banach space, and M c is a dense subse of M loc c. In wha follows we consider real valued sochasic processes. We wan o give noions ha relae convergence of sequences of local maringales o he behavior of heir quadraic variaion. We inroduce he quadraic variaion of a coninuous sochasic process {X ), [, T ]}. Consider a pariion of he inerval [, T ] : = < 1 <... < n = T. Define a coninuous process ] 1 2, X = n X i ) X i 1 ) 2. i=1 14

15 Assume ha, as, here exiss he limi in probabiliy of X uniformly for [, T ] and his limi is independen on he choice of he pariion. The limi in probabiliy of X, as, is called he quadraic variaion of X and i is denoed by X. The quadraic variaion is no well defined for any coninuous sochasic process, he naural class of processes where quadraic variaion is well defined is he one of coninuous semimaringales. We begin by collecing some facs abou quadraic variaion of local maringales. If X is a coninuous local maringale, hen here exiss a coninuous increasing process denoed X such ha X converges uniformly in probabiliy o X. Moreover he quadraic variaion of a local maringale can be characerized as follows: Theorem 2.5 Le {X ), [, T ]} be a coninuous local maringale. A process {A ), [, T ]} coincides wih he quadraic variaion of X if and only if A is coninuous, adaped, increasing and saisfies A ) = ; he process { X 2 ) A ), [, T ] } is a local maringale. Now we relae convergence of quadraic variaion o convergence of sequences of maringales and local maringales. A sequence of coninuous square inegrable maringales X n converges o X in M c if and only if he quadraic variaion X n X T converges o in L 1 Ω). A sequence of coninuous local maringales X n converges o X in M loc c if and only if he quadraic variaion X n X T converges o in probabiliy. Nex i is possible o define he join quadraic variaion of wo local maringales X and Y. Se X, Y = n X i ) X i 1 )) Y i ) Y i 1 )), i=1 and define he join quadraic variaion of X and Y, X, Y, as he uniform limi in probabiliy of X, Y as. I is immediae o see ha X, Y = 1 4 X + Y, X + Y X Y, X Y ). Similarly i can be defined he quadraic variaion of a coninuous semimaringale and he join quadraic variaion beween wo coninuous semimaringales. We wan o underline ha if a process A is of bounded variaion and if X is a coninuous local maringale, hen A = and A, X =. So if M and N are wo coninuous semimaringales, which admi he following decomposiion ino a coninuous local maringale and a coninuous process of bounded variaion respecively, M = X + A, N = Y + B, hen he join quadraic variaion of he semimaringales is given by he join quadraic variaion of he local maringale par: M, N = X, Y. 2.4 Q-Wiener process Le Ω, F, F, P) be a filered probabiliy space. We recall ha a real sochasic process {W ), }, adaped o he filraion {F, }, is a real F )-Wiener process if 1. W ) = P-a.s. and W has coninuous rajecories; 2. for s <, W ) W s) is a real Gaussian random variable wih law N s ; 15

16 3. for s he incremen W ) W s) is independen from F s. In many cases, as a filraion, we will ake he naural filraion of W, augmened wih he family N of P null ses of F: F = σ W s), s [, ]) N. The filraion F ) saisfies he usual condiions. In his case poin 3 means ha W has independen incremens. We wan o inroduce he concep of Ξ-valued Wiener process, where Ξ is a real separable Hilber space. Le Q L + 1 Ξ). A sochasic process {W ), } defined on a filered probabiliy space Ω, F, F, P) and adaped o he filraion F ), aking values in Ξ is a Q- Wiener process if 1. W ) = P-a.s. and W has coninuous rajecories P-a.s.; 2. for s <, L W ) W s)) = N s)q ; 3. for s he incremen W ) W s) is independen from F s. I urns ou ha E W ), a Ξ W s), b Ξ = s) Qa, b Ξ, a, b Ξ. A Ξ valued Q-Wiener process can be buil by saring from a sequence of independen sandard real Wiener processes. Indeed le {e k } be a complee orhonormal sysem in Ξ and le {λ k } be a summable sequence of nonnegaive numbers; consider a sequence {β k } of independen sandard real Wiener processes, hen W ) = λk β k ) e k 2.6) k=1 is a Q-Wiener process, where Q is defined by Qe k = k=1 λ ke k. The series in 2.6) converges in L 2 Ω, F, P). Viceversa, given a Q- Wiener process W, le {e k } be a complee orhonormal sysem of eigenvecors of Q and le {λ k } be he corresponding eigenvalues. If λ k > hen β k ) = λ k ) 1 2 W ), ek Ξ,, is a one dimensional sandard Wiener process and for k = 1, 2,... he processes {β k ), } are independen. 2.5 Sochasic inegral wih respec o Q-Wiener processes Le {W ), } be a Q-Wiener process in Ξ. Se Ξ = Q 1/2 Ξ): Ξ is a separable Hilber space endowed wih he scalar produc u, v Ξ = Q 1/2 u, Q 1/2 v. We wan o inroduce he Ξ sochasic inegral Ψ s) dw s), [, T ], 2.7) where Ψ s) is a predicable process aking values in L 2 Ξ, H), he space of Hilber-Schmid operaors acing beween he real and separable Hilber spaces Ξ and H. We underline ha if Φ L 2 Ξ, H), hen Φ L2 Ξ,H) = Tr ΦQΦ ), 16

17 where Tr denoes he race in Ξ. Moreover, if {Φ ), } is a predicable process aking values in L 2 Ξ, H), hen we define he norms Φ = { E Φ s) 2 L 2 Ξ,H) ds } 1 2 = { E } 1 Tr Φ s) QΦ 2 s)) ds. 2.8) I is well known ha also in he finie dimensional case he consrucion of he sochasic inegral in 2.7) is no sraighforward since he Wiener process is no of finie variaion. We describe he main seps in he consrucion of he sochasic inegral. Le = < 1 <... < n = T be a pariion of he inerval [, T ]. We call elemenary process a process Ψ of he form n 1 Ψ ) = Ψ k ) χ k, k+1 ], k= where Ψ k ) are F k -measurable random variables aking a finie number of values in L 2 Ξ, H). For an elemenary process Ψ he sochasic inegral is defined by n 1 Ψ s) dw s) = Ψ k ) [W k+1 ) W k )]. k= I urns ou ha if a process Ψ is elemenary and Ψ T <, hen he sochasic inegral Ψ s) dw s) is a coninuous, square inegrable H-valued maringale on [, T ] and he so called Io isomery holds: E Ψ s) dw s) 2 = { E Φ s) 2 L 2 Ξ,H) ds }. 2.9) So, by 2.9), he sochasic inegral is an isomery of he space of elemenary processes endowed wih he norm T defined in 2.8) ino he space of coninuous square inegrable H-valued maringales. The furher sep is o build he sochasic inegral for progressively measurable processes Ψ, aking values in L 2 Ξ, H) and saisfying Ψ T <. This is done by a densiy argumen, since he elemenary processes are dense in he space of progressively measurable processes Ψ, aking values in L 2 Ξ, H) and saisfying Ψ T <. Also i can be seen ha he Io isomery sill holds rue. An exension of he sochasic inegral allows o inegrae predicable processes in L 2 Ξ, H) which only saisfy ) P Φ s) 2 L 2 Ξ,H) ds < = ) In his case he inegral can be defined by mean of a localizaion procedure, and i urns ou o be a local maringale; Io isomery does no necessarily holds rue, anyway E Ψ s) dw s) 2 E Φ s) 2 L 2 Ξ,H) ds. We remember he Burkholder-Davis-Gundy inequaliy, which holds also for inegrands saisfying 2.1): for every p > here exiss a consan c p > such ha for [, T ] E sup u u Ψ s) dw s) 2 ) p/2 c p E Φ s) 2 L 2 Ξ,H) ds. 2.11) 17

18 Since in is general definiion he Io inegral wih respec o a Q-Wiener process is a local maringale, we can evaluae he join quadraic variaion of wo Io inegrals. Le Φ 1 s) and Φ 2 s) be wo progressively measurable processes aking values in L 2 Ξ, H) and saisfying 2.1), and consider he join quadraic variaion of he local maringales Φ 2 s) dw s). Then Φ 1 s) dw s), Φ 2 s) dw s) = For more deails on sochasic inegraion we remind o [46]. We also remember he so called Io s formula: le Φ 1 s) dw s) and Φ 1 s) Q 1/2) Φ 2 s) Q 1/2) ds. {W ), } a Q-Wiener process aking values in Ξ, {Ψ ), T } a process wih values in L 2 Ξ, H) saisfying ) P Ψ s) 2 L 2 Ξ,H) d < = 1, {ψ ), T } an H-valued progressively measurable process wih rajecories Bochner inegrable P-a.s.; Z an F -measurable H-valued random variable. Under hese assumpions he following process is well defined: Z ) = Z + ψ s) ds + Ψ s) dw s), T. This inegral definiion for he process {Z ), T } gives a precise sense o he differenial noaion, which is ofen used while dealing wih sochasic differenial equaions: dz ) = ψ s) ds + Ψ s) dw s), T and Z ) = Z. Le F : [, T ] H R be a funcions uniformly coninuous on bounded subses of [, T ] H ogeher wih is parial derivaives F, x F and x,x F. The Io s formula for he process {F, Z )), T } holds: F, Z )) = F, Z ) + x F s, Z s)), Ψ s) dw s) H 2.12) + { F s, Z s)) + x F s, Z s)), ψ s) H + 1 [Ψ 2 Tr s) Q 1/2) x,x F s, Z s)) Ψ s) Q 1/2)]} ds 18

19 2.6 Cylindrical Wiener processes and sochasic inegral We have presened Q-Wiener processes when he covariance operaor Q is of race class, and we have inroduced sochasic inegraion wih respec o such processes. In wha follows we will deal wih sochasic inegrals wih respec o Q-Wiener processes where Tr Q = +. Namely we will rea Q-Wiener processes where Q = I. We refer mainly o [15]. A cylindrical Wiener process W ), on Ξ is a process such ha for every ξ Ξ W ξ ) = β k ) ξ, ξ k Ξ, k=1 where β k ), k 1, are real independen sandard Wiener processes and ξ k ) k 1 is a complee orhonormal sysem in Ξ. Le Ξ 1 be an arbirary Hilber space such ha Ξ is coninuously embedded in Ξ 1 and he embedding is Hilber Schmid wih image dense in Ξ 1 : i : Ξ Ξ 1. Le Q 1 = ii. Q 1 is a posiive, symmeric race class operaor. Consider η k ) k 1 a complee orhonormal sysem in Ξ 1, and a sequence of posiive real numbers λ k ) k 1 such ha Q 1 η k = λ k η k. As a complee orhonormal sysem in Ξ fix now ξ k = i η k λk. We can consider a Q 1 -Wiener process W 1 ), aking values in Ξ 1. I urns ou ha Moreover W 1 ) = λk β k ) η k. k=1 β k ) = W 1 ), η k Ξ1, λk and for every η Ξ 1, W 1 ), η Ξ1 = W i η ). The following sep is o define he sochasic inegral of a predicable process ψ L 2 Ω [, T ] ; L 2 Ξ, H)) wih respec o a cylindrical Wiener process W. Since Im ))Q 1 ) 1/2 = Ξ, ψ can be regarded as a predicable process in L 2 Ω [, T ] ; L 2 Im Q 1 ) 1/2, H. So i is well defined he sochasic inegral of ψ wih respec o he Q 1 -Wiener process W 1 associaed o W in he above consrucion. For every [, T ], we se Ψ s) dw s) := Ψ s) dw 1 s). I urns ou ha he definiion is independen on he choice of he space Ξ 1 and on he process W 1 associaed o W. 19

20 2.7 Space ime whie noise We briefly inroduce whie noise. We refer mainly o [49] and [5]. Le W be a Gaussian random field on [, T ] [, 1], B [, T ] [, 1]), d dx) ha is: for every A [, T ] [, 1], W A) is a gaussian random variable wih law N, ) [,T ] [,1] χ A, x) ddx where χ A is he indicaor funcion of he se A; if A, B [, T ] [, 1] and A B =, hen W A) and W B) are independen and W A B) = W A) + W B). We wan o recall some properies abou Wiener and Io s inegral wih respec o space-ime Gaussian random fields. Consider W, a Gaussian random field on [, T ] [, 1], B [, T ] [, 1]), d dx). We say ha f s, x) is a simple funcion if here exis recangles A 1 = s 1, 1 ] x 1, y 1 ],..., A k = s k, k ] x k, y k ] and real numbers a 1,,..., a k such ha f s, x) = k a i I Ai s, x). i=1 If f is a simple funcion i is well defined he inegral [,T ] [,1] f s, x) dw s, x) := k a i W A i ) 2.13) This definiion is well posed: i is independen on he represenaion of f. If he A i, for i = 1,..., k are disjoin, i urns ou ha E f s, x) dw s, x) =, [,T ] [,1] 2 E f s, x) dw s, x)) = [,T ] [,1] i=1 [,T ] [,1] f 2 s, x) dsdx. 2.14) By he densiy of simple funcions in L 2 [, T ] [, 1]) his Wiener s ype inegral can be exended o all funcions in L 2 [, T ] [, 1]). For every f L 2 [, T ] [, 1]), he random variable f s, x) dw s, x) is Gaussian. If f, g L 2 [, T ] [, 1]), [,T ] [,1] E f s, x) dw s, x) [,T ] [,1] [,T ] [,1] ) g s, x) dw s, x) = The following proposiion holds, see [49], proposiion A.4. [,T ] [,1] f s, x) g s, x) dsdx. Proposiion 2.6 Le k L 2 [, T ]) and h L 2 [, 1]) such ha σ 2 = h2 x) dx. Define B = 1 h x) dw s, x), T. Then {B, } is a Brownian moion wih EB 2 = σ 2 and such ha we have almos surely 1 h x) k s) dw s, x) = k s) db s. 2

21 The above consrucion can be exended o predicable processes {f, ), T } wih values in L 2 [, 1]) and saisfying E f 2 s, x) dsdx = E f s, ) 2 L 2 [,1]) ds <. 2.15) [,T ] [,1] [,T ] The Io inegral is defined firs for simple processes in he obvious way and exended due o he isomeric formula 2 E f s, x) dw s, x)) = f 2 s, x) dsdx. [,T ] [,1] [,T ] [,1] I urns ou ha 1 f s, x) dw s, x) = f s, ), dw s, ) L 2 [,1]). 2.16) where {W, ), T } is an appropriae cylindrical Wiener process aking values in L 2 [, 1]). Again he validiy of equaliy 2.16) can be verified a firs for simple processes and deduced for he whole class of processes by densiy. We define a space ime whie noise as Ẇ, x) = dw τ, ξ). [,] [,x] 21

22 3 Spaces of coninuous and differeniable funcions In his chaper a firs we inroduce some noaions on ses and spaces of coninuous and differeniable funcions. Nex we perform he approximaion of uniformly coninuous funcions wih Freche differeniable funcions. The main ool for his approximaion is he inf-sup convoluion of a funcion inroduced by Lasry and Lions in he paper [34]. We perform he approximaion of a uniform coninuous funcion by avoiding he hypohesis of boundedness. These resuls will be used in paragraph 6.2.1, lemmas 6.7 and Noaions on spaces of funcions We inroduce he following spaces: C b H) is he space of mappings f : H R which are coninuous and bounded; UC b H) is he space of mappings f : H R which are uniformly coninuous and bounded; We se as usual f = sup f x). x H C b H) and UC b H), endowed wih he norm are Banach spaces. In he following we will denoe by C H) he se of coninuous funcions f : H R. C,1 b H) is he subspace of UC b H) of all lipschiz funcions. Se [f] 1 := C,1 b H) endowed wih he norm sup x,y H, x y f x) f y). x y is a Banach space. f C,1 b = f + [f] 1 In he following we will denoe by C,1 H) he se of lipschiz coninuous funcions f : H R: here exiss L > such ha, for every x, y H, f x) f y) L x y. We will also deal wih spaces of coninuous funcion wih polynomial growh, ha is funcions fx) f : H R such ha sup x X < + for some k >. 1+ x 2 H) k/2 C k H) is he se of coninuous funcions f : H R such ha sup x X for k >. UC k H) is he se of uniformly coninuous funcions f : H R such ha sup x X + for k >. UC k H) and C k H), endowed wih he norm are Banach spaces. f Ck = sup x H f x) ) k/2, 1 + x 2 H fx) 1+ x 2 H) k/2 < + fx) 1+ x 2 H) k/2 < 22

23 Le now f : H R. The direcional derivaive a a poin x H in a direcion h H is defined as f x + sh) f x) f x; h) = lim, s R s s The funcion f is Gaeaux differeniable a a poin x H if f admis he direcional derivaive in every direcions h H and here exiss an operaor, he gradien f x) L H, R), such ha f x; h) = f x) h. The funcion f is Gaeaux differeniable on X if i is Gaeaux differeniable a every poin x X. Following [25] we say ha a funcion f : H R belongs o he class G 1 H) if f is coninuous and Gaeaux differeniable on H, and he gradien f : H L H, R) is srongly coninuous, ha is for every direcions h H he map f ) h : H R is coninuous. The las condiion is weaker han requiring f o be coninuous in norm opology. If his happens, hen clearly f : H R is Freche differeniable, ha is for every x H he gradien is such ha f x + sh) f x) lim f x) h =, s s uniformly for h in bounded subses of H. We define he following spaces of differeniable funcions: Cb 1 H) is he space of coninuous and bounded mappings f : H R which are Freche differeniable on H wih a coninuous and bounded derivaive f; UCb 1 H) is he space of uniformly coninuous and bounded mappings f : H R which are Freche differeniable on H wih a uniformly coninuous and bounded derivaive f. We se f 1 = sup f x) + sup f x). x H x H Cb 1 H) and UC1 b H), endowed wih he norm 1 are Banach spaces. We will also denoe by C 1 k H) he subse of C k H) of Freche differeniable funcions. C 1,1 b H) is he space of all funcions f Cb 1 H) such ha f is lipschiz. We se f C 1,1 b [f] 1,1 := sup x,y H, x y C 1,1 b H) endowed wih he norm C 1,1 b f x) f y). x y = sup f x) + sup f x) + [f] 1,1. x H x H is a Banach space. We will also denoe by C 1,1 H) he se of coninuous and Freche differeniable funcions, wih a lipschiz Freche derivaive. Cb 2 H) is he space of coninuous and bounded mappings f : H R which are wice Freche differeniable on H wih a coninuous and bounded second derivaive 2 f; 23

24 UCb 2 H) is he space of uniformly coninuous and bounded mappings f : H R which are wice Freche differeniable on H wih a coninuous and bounded second derivaive 2 f. We se f 2 = sup f x) + sup f x) + sup 2 f x). x H x H x H Cb 2 H) and UC2 b H), endowed wih he norm 2, are Banach spaces. In he following i is necessary o generalize he definiion of G 1 H) o funcions depending on several variables. Le W and Y be wo oher Hilber spaces. G,1,1 W, H, Y ) is he class of coninuous funcions f : W H Y R which are Gaeaux differeniable wih respec o x H and y Y on W H Y, and x f : W H Y L H, R) and y f : W H Y L Y, R) are srongly coninuous. 3.2 Regularizaion of coninuous funcions If dim H = n, a funcion φ C b H) can be uniformly approximaed by is convoluion wih a smooh funcion. More precisely, le ρ Cb H) be a nonnegaive funcion wih compac suppor in he uniary ball, such ha ρ x) dx = 1. Se, for < < 1, H φ x) = n H ) x y ρ φ y) dy. φ Cb H) and as, φ φ. If H is infinie dimensional, hen in he paper [44] by Pesza and Zabczyk somehing similar has been done, in order o approximae funcions in C b H). Le φ C b H). For every n N, le ρ n Cb R n ) be a nonnegaive funcion wih compac suppor conained in he ball of radius 1 n R and such ha ρ n x) dx = 1. Le {e k } k N n be a complee orhonormal sysem in H and, for every n N, le Q n : H e 1,..., e n be he orhogonal projecion on he linear space generaed by e 1,..., e n. We idenify e 1,..., e n wih R n. Se n ) φ n x) = ρ R n y Q n x) φ y i e i dy, n i=1 where for every k N, y k = y, e k H. I urns ou ha φ n Cb L > and C > such ha H). Moreover if here exis φ x) φ y) L x y, φ x) C 1 + x ), for every x, y H hen for every k N φ n x) φ n y) L x y, φ n x) C 1 + x ), for every x, y H. For every x H, lim φ n x) φ x) =. n 24

25 So φ n ) n is a poinwise approximaion of φ, preserving he lipschiz consan and he linear growh of φ. Bu his approximaion is no uniform. We need smooh approximaions in he norm of he uniform convergence even when H is infinie dimensional. In he following, for every φ UC b H) we denoe by ω φ ) he uniform coninuiy modulus of φ: ω φ ) = sup { φ x) φ y) : x y }. We wan o approximae uniformly coninuous funcions wih Freche differeniable funcions. We recall he resuls inroduced by Lasry and Lions in he paper [34]: he approximaion is based on he inf-sup convoluions: if φ UC b H), i can be uniformly approximaed by funcions in H) by mean of is inf-sup convoluion. The resuls on his approximaion procedure have been inroduced in [34]. In wha follows we ofen cie he book [17], where hey are reaed wih more deails on he proofs. Le φ UC b H). We can define, see e.g. [17] and [34], he inf-sup convoluions by seing, for < 1, UC 1,1 b and he sup-convoluion by U φ x) = inf y H V φ x) = sup y H { { φ y) + x y 2 H 2 φ y) x y 2 H 2 } }, 3.1). 3.2) U and V enjoy he properies summarized in he following proposiion, see [17], proposiion C.3.2. Proposiion 3.1 The following saemens hold: 1. if φ UC b H), hen U φ and V φ UC b H); ω Uφ ω φ and ω Vφ ω φ ; moreover U φ x) φ x) V φ x) for all x H; 2. he semigroup propery holds: for, s, U +s φ = U U s φ and V +s φ = V V s φ; 3. if φ, ψ UC b H), hen U φ U ψ φ ψ, V φ V ψ φ ψ ; 4. if φ UC b H), hen ) U φ x) φ x) ω φ 2 φ, ) V φ x) φ x) ω φ 2 φ. In paricular i follows ha U φ and V φ approximae φ in UC b H). Bu i is well known ha in general U φ and V φ are no differeniable. In order o find a uniform approximaion which is differeniable, he noion of inf-sup convoluion has been inroduced. 25

26 Definiion 3.2 Le φ UC b H). The inf-sup convoluion of φ is denoed by φ and is defined by { [ ] } φ x) = V /2 U φ x) = sup z H inf y H φ y) + z y 2 H 2 x z 2 H. 3.3) This definiion makes sense even if φ is no bounded. Moreover, if φ is bounded, φ UC 1,1 b H) and i approximaes φ in UC b H). We underline ha if H is infinie dimensional his is he bes approximaion we can achieve, indeed in 1973 A. S. Nemirowski and S. M. Semenov proved ha UCb 2 H) is no dense in UC b H), whereas UC 1,1 b H) is, see [4]. Moreover, by he inf-sup convoluions, we are given an explici represenaion of he approximaing funcions, which are ranslaion invarian and order preserving, ha is if φ x) ψ x) for all x H, hen by he definiion φ x) ψ x) for all x H. Moreover, see [17], heorem 2.2.1, he following properies hold rue: Proposiion 3.3 If φ UC b H), hen φ UC 1,1 b H) and, for [, 1] ) φ φ, φ φ ω φ 2 φ [φ ] 1 2 φ, [φ ] 1,1 1 By he second inequaliy we ge ha, as ends o, φ converges o φ uniformly, so in paricular we conclude ha UC 1,1 b H) is dense in UC b H). The proof of hese resuls is based on properies of semiconcave and semiconvex funcions. A funcion φ : H R is semiconvex if here exiss K > such ha he mapping x φ x) + K 2 x 2 is convex, and i is semiconcave if here exiss K > such ha he mapping x φ x) K 2 x 2 is concave. If a funcion φ in C,1 b H) is boh K-semiconcave and K-semiconvex, hen φ C 1,1 b H) and [φ ] 1,1 K, see [17], proposiion C.2.1. We now sae he following resul. Lemma 3.4 Le φ : [, T ] H R a bounded funcion such ha for every a [, T ], φ a, ) is uniformly coninuous in x uniformly wih respec o a [, T ]. Then, for every, 1] here exiss φ : [, T ] H R such ha for every a [, T ], φ a, ) UC 1,1 b and φ a, ) φ a, ), for every < 1, and uniformly in a [, T ]. lim sup φ a, x) φ a, x) =, x H 26

27 Proof. Since by our assumpions for every a [, T ] φ a, ) UC b H), hen, by proposiion 3.3, here exiss funcions φ a, )) < 1 UC 1,1 b H) such ha, as, sup φ a, x) φ a, x). x H Since φ is uniformly coninuous in x uniformly wih respec o a [, T ], he convergence is uniform in a. We now consider approximaion of coninuous funcions wih polynomial growh, more precisely le k > be such ha The following lemma holds rue: sup x H φ x) 1 + x 2) k/2 <. Lemma 3.5 Le φ C k H) and define φ x) =, x H. Assume also ha φ 1+ x 2 ) k/2 UC b H). Then, for every, 1] here exiss φ C k H) such ha φ C 1 H) and φx) φ C k φ C k, lim φ φ C k =. Proof. Since ) by our assumpions φ UC b H), hen, by proposiion 3.3, here exiss funcions φ < 1 UC1,1 b H) such ha, as, φ φ in UC b H). We pu φ x) = φ x) 1 + x 2) k/2 : we claim ha as ends o, φ converges o φ in C k H), indeed Moreover φ x) φ x) φ φ Ck = sup x H 1 + x 2) k/2 = sup φ x) φ x) as. φ C k = sup x H φ x) 1 + x 2) k/2 = sup x H φ x) x H sup φ x) = sup x H x H φ x) 1 + x 2) k/2 = φ C k. So by he previous lemma we have performed he approximaion of funcions in C k H). We make an exension of he previous lemma when φ depends also on a [, T ], for some T >. Lemma 3.6 Le φ : [, T ] H R such ha for every a [, T ], φ a, ) C k H) and define φa,x) φ a, x) =, a [, T ], x H. Assume also ha φ is uniformly coninuous in x 1+ x 2 ) k/2 uniformly wih respec o a [, T ]. Then, for every, 1] here exiss φ : [, T ] H R such ha for every a [, T ], φ a, ) C k H) C 1 H) and sup x H φ a, x) 1 + x 2) k/2 sup x H φ a, x) 1 + x 2), for every < 1, k/2 and lim sup φ a, x) φ a, x) x H 1 + x 2) =, k/2 uniformly in a [, T ]. 27