A DYNAMIC PROGRAMMING APPROACH TO THE PARISI FUNCTIONAL

A DYNAMIC PROGRAMMING APPROACH TO THE PARISI FUNCTIONAL AUKOSH JAGANNATH AND IAN TOBASCO Absrac. G. Parisi prediced an imporan variaional formula for he hermodynamic limi of he inensive free energy for a class of mean field spin glasses. In his paper, we presen an elemenary approach o he sudy of he Parisi funcional using sochasic dynamic programing and semi-linear PDE. We give a derivaion of imporan properies of he Parisi PDE avoiding he use of Ruelle Probabiliy Cascades and Cole-Hopf ransformaions. As an applicaion, we give a simple proof of he sric convexiy of he Parisi funcional, which was recenly proved by Auffinger and Chen in []. 1. Inroducion Consider he mixed p-spin glass model on he hypercube Σ N = { 1, 1} N, which is given by he Hamilonian H N (σ = H N(σ + h σ i i where H N is he cenered gaussian process on Σ N wih covariance EH N(σ 1 H N(σ = Nξ((σ 1, σ /N. The parameer ξ saisfies ξ( = p 1 β p p where we assume here is a posiive ɛ such ha ξ(1+ɛ <, and h is a non-negaive real number. I was prediced by Parisi [13], and laer proved rigorously by Talagrand [19], and Panchenko [16], ha he hermodynamic limi of he inensive free energy is given by lim N 1 N log σ Σ N e HN (σ = inf P(µ; ξ, h a.s. µ Pr[,1] Here Pr([, 1] is he space of probabiliy measures on [, 1], and he Parisi funcional, P, is given by P(µ; ξ, h = u µ (, h 1 ξ (µ[, ] d, where u µ solves he Parisi PDE: ( u µ (, x + ξ ( xx u µ (, x + µ [, ] ( x u µ (, x = (, x (, 1 R u µ (1, x = log cosh(x. In he case ha µ has finiely many aoms, he exisence of a soluion of he Parisi PDE and is regulariy properies are commonly proved using he Cole-Hopf ransformaion and Ruelle Probabiliy Cascades. A coninuiy argumen is hen used o exend he definiion of u µ o general µ and o prove corresponding regulariy properies. Such approaches do no address he quesion of uniqueness of soluions. See [1, 15, 1, ] for a summary of hese resuls. 1 Mahemaics Subjec Classificaion. Primary: 6K35, 8B44, 8D3, 49N9; Secondary: 35Q8, 35K58, 49S5. Key words and phrases. Parisi formula; Sherringon-Kirkparick model; Dynamic Programming. 1 ˆ 1

In his noe, we presen a differen approach. In Secion, we prove he exisence, uniquness, and regulariy of he Parisi PDE using sandard argumens from semi-linear parabolic PDEs. Theorem 1. The Parisi PDE admis a unique weak soluion which is coninuous, differeniable in ime a coninuiy poins of µ, and smooh in space. See Secion for he precise saemen of his resul, and in paricular for he definiion of weak soluion. Due o he non-lineariy of he Parisi PDE, low regulariy of he coefficiens, loss of uniform ellipiciy a =, and unboundedness of he iniial daa, he proof of Theorem 1 requires he careful applicaion of many differen (hough relaively sandard argumens in andem. The presenaion of a PDE driven approach o he sudy of his funcional is no only of ineres o expers in he field of spin glasses, bu may also be of ineres o praciioners of he Calculus of Variaions, PDEs, and Sochasic Opimal Conrol. There are many imporan, purely analyical quesions surrounding his funcional ha mus be addressed before furher progress on quesions in spin glasses can be made. See [1,, 18] for a discussion. Some of hese quesions are hough o be inracable o he mehods currenly used in he spin glass lieraure bu appear o be wellsuied o he echniques of he aforemenioned fields; as such i is imporan o presen he sudy of his funcional in a language ha is boh basic and palaable o heir praciioners. Besides is inrinsic ineres, he preceding heorem has useful applicaions o he sudy of he Parisi funcional. Afer proving he exisence of a sufficienly regular soluion o he above PDE, we can use elemenary argumens from sochasic analysis o prove many imporan and basic properies of his funcional, such as fine esimaes on he soluion of he Parisi PDE and he sric convexiy of he Parisi funcional iself. As a firs applicaion of his ype, we furher develop he well-posedness heory of he Parisi PDE by quaniaively proving he coninuiy of he soluion in he measure µ. We also prove sharp bounds on some of he derivaives of he soluion. Such bounds are imporan o he proofs of many imporan resuls regarding he Parisi funcional, see for example Talagrand s proof of he Parisi formula in [1] and also [1,, 18]. They were previously proved using manipulaions of he Cole-Hopf ransformaion and Ruelle Probabiliy Cascades [1]. This is presened in Secion.4. As a furher demonsraion how Theorem 1 can be combined wih mehods from sochasic opimal conrol, we presen a simple proof of he sric convexiy of he Parisi funcional. As background, recall he predicion by Parisi [13] ha he minimizer of he Parisi funcional should be unique and should serve he role of he order parameer in hese sysems. The quesion of he sric convexiy of P was firs posed by Panchenko in [14] as a way o prove his uniqueness. I was sudied by Panchenko [14], Talagrand [18, 19], Bovier and Klimovsky [4], and Chen [5], and finally resolved by Auffinger and Chen in heir fundamenal work []. The work of Auffinger and Chen resed on a variaional represenaion of he log-momen generaing funcional of Brownian moion [3, 7], which hey combine wih approximaion argumens o give a variaional represenaion for he soluion of he Parisi PDE. We noe here ha an early version of his variaional represenaion appeared in [4], where i is shown, using he heory of viscosiy soluions, o hold when he coefficien µ[, ] is piecewise coninuous wih finiely many jumps. Since he Parisi PDE is a Hamilon-Jacobi-Bellman equaion, i is naural o obain he desired variaional represenaion for is soluion as an applicaion of he dynamic programming principle from sochasic opimal conrol heory. The required argumens are elemenary, and are commonly used in sudying nonlinear parabolic PDEs of he ype seen above. We prove he variaional represenaion in Secion 3, and hen deduce from i he sric convexiy of he Parisi funcional in Secion 4. Theorem. The funcional P(µ; ξ, h is sricly convex for all choices of ξ and h. The variaional represenaion which was discussed above is given in Lemma 18. From his i follows immediaely ha one has he following represenaion for he Parisi Formula.

Proposiion 3. The Parisi Formula has he represenaion 1 lim N N log [ (ˆ 1 e HN (σ = inf sup E log cosh ξ (sµ[, s]α s ds + µ Pr([,1] σ Σ α A N 1 ˆ 1 ξ (sµ[, s] ( αs + s ] ds ξ (sdw s + h where A consiss of all bounded processes on [, 1] ha are progressively measurable wih respec o he filraion of Brownian moion. Acknowledgemens We would like o hank Anonio Auffinger for encouraging he preparaion of his paper, and asking A.J. if one could prove sric convexiy using dynamic programming echniques. We would also like o hank Anon Klimovsky for bringing our aenion o [4]. Finally, we would like o hank our advisors G. Ben Arous and R.V. Kohn for heir suppor. This research was conduced while A.J. was suppored by an NSF Graduae Research Fellowship DGE-813964, NSF Gran DMS-19165, and NSF gran OISE-73136, and while I.T. was suppored by an NSF Graduae Research Fellowship DGE-813964, NSF Gran OISE-96714, and NSF Gran DMS-1311833.. Well-posedness of he Parisi PDE Le u : [, 1] R R be a coninuous funcion wih essenially bounded weak derivaive x u. We call u a weak soluion of he Parisi PDE if i saisfies ˆ 1 ˆ = u φ + ξ ( ( ˆ u xx φ + µ [, ] ( x u φ dxd + φ (1, x log cosh x dx R R for every φ Cc ((, 1] R. We now sae he precise version of Theorem 1 from he inroducion. Theorem 4. There exiss a unique weak soluion u o he Parisi PDE. The soluion u has higher regulariy: j xu C b ([, 1] R for j 1 j xu L ([, 1] R for j. For all j 1, he derivaive xu j is a weak soluion o { ( xu j + ξ ( xx xu j + µ [, ] x j ( x u = xu j (1, x = dj log cosh x dx j (, x (, 1 R. x R Remark 5. The soluion described in [1] can be shown o be a weak soluion of he Parisi PDE, using he approximaion mehods developed here. I was also shown in [1] ha his soluion has he higher regulariy described above. Remark 6. The reader may noice ha he essenial boundedness of x u is no sricly necessary o make sense of he definiion of weak soluions. I is used in he proof of uniqueness in an essenial way, however we do no claim ha his proof is opimal by any means. Coninous dependence is proved in Secion.4. We begin he proof of Theorem 4. Afer performing he ime change s ( = 1 (ξ (1 ξ ( and exending he ime-changed CDF µ [, s 1 ( ] by zero, we are led o consider he semi-linear parabolic PDE { u u = m ( u x (, x R + R (1 u (, x = g (x x R 3

where g (x = log cosh x and m ( = µ [, s 1 ( ] 1 (ξ (1 ξ (/. We carry over he definiion of weak soluion from before: a coninuous funcion u : [, R R wih essenially bounded weak derivaive x u is a weak soluion o (1 if i saisfies ˆ ˆ ˆ = u φ + u xx φ + m( ( x u φ dxd + φ (, x g(x dx R for every φ Cc ([, R. Evidenly, he exisence, uniqueness, and regulariy heory of weak soluions o he Parisi PDE is capured by ha of (1. Our proof of he well-posedness of (1 boils down o he sudy of a cerain fixed poin equaion, which we inroduce now. Le e be he hea semigroup on R, i.e. ( e h (x = 1 ˆ 4π Then, u weakly solves (1 if and only if u saisfies ( u ( = e g + R R e x y 4 h (y dy. e ( s m (s u x (s ds. This is an applicaion of Duhamel s principle (see e.g. [6, Ch. ]. For compleeness, we presen his in Proposiion 4. In Secions.1-.3 below, we prove he exisence, uniqueness, and regulariy of fixed poins of ( on a cerain complee meric space. The properies of g and m we will be using are ha g L and dj dx j g L L for j m is a monoonic funcion of ime alone and m 1. These properies will inform our choice of space on which o sudy (. The exac bound on m does no maer, bu we include i for convenience. Once Theorem 4 is esablished, one can give a quick proof of he final componen of wellposedness, namely he coninuiy of he map from µ o he corresponding soluion of he Parisi PDE, using sandard SDE echniques. This is in Secion.4. The noaion c denoes an inequaliy ha is rue up o a universal consan ha depends only on c. Throughou he proofs below, we will use wo elemenary esimaes for he hea kernel which we record here: (3 e L p L p 1 and xe L p L p 1..1. Exisence of a fixed poin. We prove exisence of a fixed poin o (. Firs we show here exiss a soluion for shor-imes < T, hen by using an a priori esimae we prove a soluion exiss for all ime. Shor-ime exisence comes via a conracion mapping argumen. Define he Banach space X = {ψ L (R} {ψ x L (R} { ψ xx L (R } wih he norm ψ X = ψ ψ x ψ xx, and for each T > define he complee meric space X h T = { e h + φ : φ L ([, T ] ; X } { φ x L ([,T ] R h, φ xx L ([,T ];L (R h } wih he disance d X h T (u, v = u v L ([,T ];X. The symbol h in he definiion of he space refers o he iniial daa, which is assumed o saisfy h L and h L. 4

Given u XT h define he map (4 A [u] = e h + e ( s m (s u x (s ds. Lemma 7. (shor-ime exisence Le (5 T (h = min {1, [ C ( h + h ] } where C R + is a universal consan. Then for all T (, T, (self-map A : X h T Xh T (sric conracion There exiss α < 1 such ha d X h T (A [u], A [v] α d X h T (u, v, u, v X h T. Therefore for every T < T (h here exiss u X h T Proof. Firs we prove A is a self-map. Le u X h T saisfying u = A [u]. and call ψ = A [u] e h = Noe ha ψ x = ψ xx = e ( s m (s u x (s ds. x e ( s mu x(s ds x e ( s mu x u xx (s ds. The esimaes in (3 and he definiion of XT h imply he bounds ψ L ([,T ] R T h ψ x L ([,T ] R T 1/ h ψ xx L ([,T ];L (R T 1/ h h. Therefore here is a universal consan C R + such ha A : XT h Xh T whenever T T (h = ( C h. Now we prove A is a sric conracion. Le u, v X h T and call D = A [u] A [v] = e ( s m (s ( u x (s v x (s ds. The esimaes in (3 and he definiion of XT h give { d X h (A [u], A [v] C max T h, T 1/ h, T 1/ ( h + h } d T X h (u, v T where C R + is a universal consan. Therefore, if T 1 (h = min {1, [ C ( h + h ] } hen A is a sric conracion on X h T for all T < T T 1. Since T 1 T we may ake T = T 1. 5

To prove he exisence of a global-in-ime soluion o ( we will work in he space X T = { e g + φ : φ L ([, T ] ; X } defined for each T R +. Noe X g T X T so ha by Lemma 7, if we ake T < T (g hen here exiss u X T saisfying he fixed poin equaion (. To exend u o all of ime we require he following a priori esimaes. Lemma 8. (a priori esimaes Le T R + and assume u X T saisfies (. Then u x L ([,T ] R g u xx L ([,T ];L (R g exp ( g T. Proof. The esimae on u x is derived by he maximum principle. By Corollary 11 (see below we have ± u x (, x u x (, x = m (± u x x u x (, x, (, x (, T R and by assumpion u x is bounded. Now he usual proof of he maximum principle for linear parabolic PDE in unbounded domains goes hrough [1]. For he esimae on u xx observe ha u x = e g + e ( s mu x u xx (s ds, so by a sandard energy esimae (see Lemma 1 below we have for almos every T u xx L (R ( u x L ([,T ] R u xx L (R (sds + g. The desired resul follows from Gronwall s inequaliy [6] and he a priori bound on u x. Corollary 9. (global exisence For each T R +, here exiss u T X T saisfying (. The soluions {u T } T R+ so produced agree on heir common domains. Proof. Define he maximal ime of exisence T M o be he supremum over T R + such ha here exiss u T X T saisfying (. If T M < hen by Lemma 7 we mus have lim sup T T M (u T x L ([,T ] R + (u T xx L ([,T ];L (R =, oherwise we could consruc a soluion exending for imes beyond T M. Therefore by Lemma 8 we mus have T M =. A quick applicaion of Lemma 13 shows ha u T = u T for T T... Regulariy of fixed poins. One proves he higher regulariy of he fixed poin u by a parabolic boosrapping procedure. Lemma 1. (higher regulariy Assume u X T saisfies (. Then u saisfies j xu L ( [, T ] ; L (R L (R for j u L ([, T ] R and j xu L ( [, T ] ; L (R L (R for j 1. Proof. Le us describe he firs sep of he argumen. Since u X T we have u x L x and u xx L L x. Our goal will be o deduce u xx L x and u xxx L L x. I will be imporan o noe we are working on he finie-ime domain [, T ] R, so ha in paricular L L x L x. Sar by wriing u x = e g + e ( s mu x u xx (s ds, 6

hen by Lemma 1 we ge u xxx L x. Since mu x u xx L L x, g L and u xx = e g + x e ( s mu x u xx (s ds, we conclude ha u xx L x. Here we have used ha xe ( s ds : L L x L x which follows from (3. Now x (mu x u xx = m ( u xx + u x u xxx L x, so ha u xx = e g + e ( s m ( u xx + u x u xxx ds and finally we conclude u xxx L L x using Lemma 1 again. The res of he esimaes on j xu are proved in he same way; he j xu esimaes follow easily. There is a sense in which he weak soluion u is a classical soluion. Corollary 11. Le u X T saisfy (. Then for all j we have xu j exiss poinwise and is coninuous he lef/righ derivaives ± j xu exis poinwise, and xu j exiss a coninuiy poins of m Moreover, we have ha ± j xu (, x xu j (, x = m (± x j [ ] u x (, x, (, x (, T R. For compleeness, we record he energy esimae which was used above. The proof is sandard (see [6] and is omied. Lemma 1. Le h be weakly differeniable wih h L and le f L ([, T ] R. Then saisfies ψ( = e h + e ( s f(s ds ψ x L ([,T ];L (R + ψ xx L ([,T ] R f L ([,T ] R + h L (R..3. Uniqueness of fixed poins. Since we used a conracion mapping argumen o consruc fixed poins for (4 in he spaces XT h, we have implicily demonsraed a uniqueness heorem here. The following resul achieves uniqueness wihou menion of he second derivaive u xx. Lemma 13. Assume u, v : [, T ] R R are weakly differeniable and ha u x, v x are essenially bounded. Then if u, v saisfy he fixed poin equaion (, i follows u = v. Proof. In he following, C denoes a universal consan which may change from line o line. Le d = u v, hen by assumpion we have Therefore d ( = d x ( = e ( s m (s (u x + v x d x (s ds, T. x e ( s m(s(u x + v x d x (s ds, T. Using he second hea kernel esimae in (3, we conclude he conracive esimae d x L ([,] R C u x + v x L ([,T ] R 1 s d x (s L (dx ds for all T. I now follows from an ieraive argumen ha d x =, and hence ha d =. To see his noe ha if d x = on [, 1 ] R, hen by he conracive esimae above, d x L ([ 1,] R C u x + v x L ([,T ] R 1 d x L ([ 1,] R 7

for all [ 1, T ]. Therefore d x = on [, 1 + ɛ] where ɛ depends only on he L bounds on u x, v x. This complees he proof..4. Coninuous dependence of soluions. For convenience we merize he weak opology on he space of probabiliy measures on he inerval Pr [, 1] wih he meric d (µ, ν = ˆ 1 µ [, s] ν [, s] ds. Lemma 14. Le µ, ν Pr[, 1] and u, v be he corresponding soluions o he Parisi PDE. Then u v ξ (1 d(µ, ν u x v x exp ( ξ (1 ξ ( ξ (1 d(µ, ν. Remark 15. The firs inequaliy is originally due o Guerra [11]. Proof. Le u, v solve he Parisi PDE weakly, hen w = u v solves ( {w + ξ wxx + µ[, ] (u x + v x w x + (µ[, ] ν[, ] vx = (, x (, 1 R w (1, x = x R weakly. Since u x, v x are Lipschiz in space uniformly in ime and bounded in ime, we can solve he SDE dx = ξ ( µ [, ] u x + v x (, X d + ξ (dw. Furhermore, as w weakly solves he above PDE and has he same regulariy as u and v, we can wrie (ˆ 1 1 w (, x = E X=x ξ (s (µ [, s] ν [, s] vx (s, X s ds by Proposiion. Therefore w ξ (1 d (µ, ν since ξ is non-decreasing and v x 1 by Lemma 16. Differeniaing he PDE for w in x, one finds by similar argumens o Proposiion ha w x has he represenaion (ˆ 1 w x (, x = E X=x E (, s ξ (s (µ [, s] ν [, s] v x v xx (s, X s ds where (ˆ s E (, s = exp ξ (τ µ [, τ] v xx + u xx (τ, X τ dτ. Using ha v x 1 and u xx v xx 1 from Lemma 16, and since ξ is non-decreasing, w x e ξ (1 ξ ( ξ (1 d (ν, µ. Lemma 16. The soluion u o he Parisi PDE saisfies u x < 1 and < u xx 1. Remark 17. The Auffinger-Chen SDE and he corresponding Iô s formula s for u x and u xx used in he proof below were firs proved in [] using approximaion argumens. 8

Proof. Using he PDEs for u x, u xx given in Theorem 4, along wih Proposiion, we can wrie u x (, x = E X=x (anh X 1 ( u xx (, x = E X=x sech X 1 + where X solves he Auffinger-Chen SDE ˆ 1 ξ (s µ [, s] u xx (s, X s ds dx = ξ ( µ [, ] u x (, X d + ξ (dw. The firs equaliy immediaely implies he bound on u x, and he second equaliy implies u xx >. Then by a rearrangemen one finds (ˆ 1 u xx (, x = 1 µ[, u x (, x E X=x u x (s, X s dµ (s and u xx 1 follows. 3. A variaional formulaion for he Parisi PDE In his secion we use he mehods of dynamic programming (see e.g. [8] o give a new proof of he variaional formula for he soluion of he Parisi PDE. Lemma 18. Le u µ solve he Parisi PDE as above and define he class A of processes α s on [, 1] ha are bounded and progressively measurable wih respec o Brownian moion. Then [ (6 u µ (, x = sup E X α =x 1 ˆ 1 ] ξ (sµ[, s]α α A sds + log cosh(x1 α where X α s solves he SDE (7 dx α s = ξ (sµ[, s]α s ds + ξ (sdw s wih iniial daa X α = x. Furhermore, he opimal conrol saisfies µ[, s]α s = µ[, s]u x (s, X s a.s. where X s solves he Auffinger-Chen SDE wih he same iniial daa: dx s = ξ (sµ[, s] x u(s, X s ds + ξ (sdw s. Remark 19. This formula was firs proved by Auffinger and Chen in []. Taking advanage of he Cole-Hopf represenaion in he case of aomic µ, hey prove he lower bound for every α using Girsanov s lemma and Jensen s inequaliy. They hen verify ha heir opimal conrol achieves he supremum, by an applicaion of Iô s lemma. The uniqueness follows from a convexiy argumen. In conras, we recognize he Parisi PDE as a specific Hamilon-Jacobi-Bellman equaion. I is well-known ha he soluion of such an equaion can be seen as he value funcion of a sochasic opimal conrol problem. As such, his represenaion can be obained by a exbook applicaion of he verificaion argumen. This argumen simulaneously gives he variaional represenaion and a characerizaion of he opimizer. We also noe ha he argumen presened here is more flexible, as is evidenced by replacing he nonlineariy u x wih F (u x in he Parisi PDE, where F is smooh, sricly convex, and has super linear growh. In paricular, observe ha one canno use he Cole-Hopf ransformaion on he resuling PDE, bu he argumens of his paper follow hrough muais muandis. 9

Proof. Le u solve he Parisi PDE. Noice ha he nonlineariy is convex, so if we le (8 hen by he Legendre ransform we have ξ ( µ [, ] ( xu L (, λ = ξ ( µ [, ] λ f(, λ = ξ ( µ [, ] λ, = ξ { (µ[, ] sup λ / + λ x u } = sup {L (, λ + f (, λ x u}. λ R λ R Therefore, we can wrie he Parisi PDE as a Hamilon-Jacobi-Bellman equaion: = u + ξ ( xxu + sup {L (, λ + f (, λ x u}. λ R Since α s in A is bounded and progressively measurable, we can consider he process, X α, which solves he SDE dx α = f(s, α s ds + ξ (dw wih iniial daa X α Noice ha u is a (weak sub-soluion o = x. This process has corresponding infiniesimal generaor L(, α = 1 ξ ( xx + f(, α x. u + L(, αu + L(, α wih he regulariy obained in Theorem 4. I follows from Iô s lemma (Proposiion ha [ˆ 1 ] u(, x sup E x L (s, α s ds + log cosh (X1 α. α A The resul now follows upon observing ha he conrol u x (s, X s achieves equaliy in he above since i achieves equaliy in he Legendre ransform. Tha his conrol is in he class A can be seen by an applicaion of he parabolic maximum principle (Lemma 16. Uniqueness follows from he fac ha λ achieves equaliy in he Legendre ransform if and only if ξ (µ[, ]λ = ξ (µ[, ]u x. Applying his represenaion o he Parisi formula gives Proposiion 3. 4. Sric convexiy As an applicaion of he above ideas, we give a simple proof of sric convexiy of P. Theorem. The Parisi Funcional is sricly convex. Proof. We will prove µ u µ (, h is sricly convex. Then P(µ = u µ (, h 1 ˆ 1 ξ (µ[, ]s ds will be he sum of a sricly convex and a linear funcional, so P will be sricly convex. Recall [ˆ 1 ] u µ (, h = sup E h ξ (sµ[, s] α s α A ds + log cosh (Xα 1. 1

Fix disinc µ, ν Pr [, 1] and le µ θ = θµ + (1 θ ν, θ (, 1. Le α θ be he opimal conrol for he Parisi PDE associaed o µ θ, so ha [ˆ 1 ( α u µθ (, h = E h ξ θ ( ] (sµ θ [, s] s ds + log cosh X1 αθ. Consider he auxiliary processes Y αθ and Z αθ given by solving dy = ξ (µ[, ]α θ d + ξ (dw and dz = ξ (ν[, ]α θ d + ξ (dw wih iniial daa Y = Z = h, and noe ha X αθ = θy αθ + (1 θz αθ. By he lemma below, P (Y 1 Z 1 >. Therefore by he sric convexiy of log cosh and he variaional represenaion (6, [ˆ 1 ( α u µθ (, h = E h ξ θ ( ] (s µ θ [, s] s ds + log cosh X1 αθ ( [ˆ 1 ( α < θ E h ξ θ ( ] (s µ [, s] s ds + log cosh Y1 αθ ( [ˆ 1 ( α + (1 θ E h ξ θ ( ] (s ν [, s] s ds + log cosh Z1 αθ as desired. θu µ (, h + (1 θu ν (, h Lemma 1. Le Y and Z be as above. Then P (Y 1 Z 1 >. Proof. I suffices o show ha By definiion we have Y 1 Z 1 = V ar(y 1 Z 1 >. ˆ 1 ξ (s(µ[, s] ν[, s]α θ s ds. Observe ha by he PDE for u x in Theorem 4 and Iô s lemma (see Proposiion, he opimal conrol α θ = (u µθ x is a maringale, α θ α θ = ξ (su xx (s, X s dw s. Therefore if we call s = ξ (s(µ[, s] ν[, s], (ˆ 1 ˆ V ar(y 1 Z 1 = E h s (αs θ αds θ = s K(s, dsd [,1] where s K(s, = E h [(α θ s α θ ( ] α θ α θ. Now since s L [, 1], i suffices o show ha K(s, is posiive definie. We have [ˆ s ] K(s, = E h ξ (su xx (s, Xs αθ dw s ξ (u xx (, X αθ dw = ξ ( E h u xx(, X αθ d = p ( s = p( p(s 11

where p(s = ˆ s ξ ( E h u xx(, X αθ d. By he maximum principle (Lemma 16, u xx >, so ha p( is sricly increasing. Since his kernel corresponds o a monoonic ime change of a Brownian moion, i is posiive definie. 5. Appendix We will say ha a funcion f : [, R R wih a mos linear growh if i saisfies an inequaliy of he form f(, x T 1 + x T R +, (, x [, T ] R. We will say he same in he case ha f : R R wih he obvious modificaions. In he following we fix a probabiliy space (Ω, F, P and le W be a sandard brownian moion wih respec o P. Le F be he filraion corresponding o W. To make his paper self-conained, we presen a version of Iô s lemma in a lower regulariy seing. The argumen is a modificaion of [17, Corr. 4..]. Proposiion. Le a, b : [, T ] (Ω, F, P R be be bounded and progressively measurable wih respec o F and le a. Le X solve dx = a(dw + b(d wih iniial daa X = x. Le L = 1 a(, ω + b(, ω x. Finally assume ha we have u saisfying: (1 u C([, T ] R wih a mos linear growh. ( u x, u xx C b ([, T ] R (3 u is weakly differeniable in wih essenially bounded weak derivaive u, and which has a represenaive ha is Lipschiz in x uniformly in. Then u saisfies Io s lemma: u(, X u(s, X s = s ( + L u(s, X s ds + s u x (s, X s a(s dw s Remark 3. This resul is applied hroughou he paper o he soluion u from Theorem 1 and is spaial derivaives. We noe here ha, given he regulariy in Theorem 4, he weak derivaives j xu, j, have represenaives saisfying he above Lipschiz propery. Proof. To prove his, we will smooh u by a sandard mollificaion-in-ime procedure and apply Iô s lemma. Wihou loss of generaliy, assume T = 1 and s =. Exend u o all of space-ime by { u(, x < u(, x = u(1, x > 1. Abusing noaion, we call he exension u and noe ha i saisfies each of he assumpions above. Le φ(y Cc ( 1, 1 wih φ 1 and φ = 1, and define φ ɛ (s = φ(s/ɛ/ɛ. Define he ime-mollified version of u as ˆ u ɛ (, x = φ ɛ (su( s, xds. R 1

Since u ɛ C 1, has bounded derivaives, and grows a mos linearly, Io s lemma implies ha sup x R u ɛ (, X u ɛ (, x = = ( + L u ɛ (s, X s ds + u ɛ (s, X s ds + = A ɛ + B ɛ + C ɛ Lu ɛ (s, X s ds u ɛ x(s, X s adw s u ɛ x(s, X s adw s for all ɛ >. Since hese quaniies are well-defined a ɛ =, i suffices o show heir convergence. Firs we show he lef-hand side converges. Noe u is Lipschiz wih consan u. Therefore, ˆ sup u ɛ (, x u(, x = sup sup φ(y (u( ɛy, x u(, x dy u ɛ. x R Thus u ɛ (, X u(, X uniformly P -a.s. Now we consider he righ-hand side. For A ɛ, noe ha since u is Lipschiz in x uniformly in, by an applicaion of Lebesgue s differeniaion heorem, we have ha u ɛ u for all x, Lebesgue-a.s. in. Thus by he bounded convergence heorem, we have ha sup [,1] u ɛ (s, X s ds u (s, X s ds ˆ 1 u ɛ (s, X s u (s, X s ds. Thus, A ɛ A uniformly P -a.s. The convergence for B ɛ follows from a similar argumen. Since u x, u xx C b, commuing derivaives wih mollificaion shows ha u ɛ x and u ɛ xx converge o u x and u xx poinwise. Then, he bounded convergence heorem implies ha B ɛ B uniformly P -a.s. jus as before. Now we prove uniform a.s. convergence of C ɛ o C. Combining he above argumens proves ha C ɛ is uniformly a.s. convergen, so i suffices o check is convergence o C in probabiliy. By Doob s inequaliy and Io s isomery, P ( sup [,1] u ɛ x adws u x adws η a 1 η where he las convergence is again by he bounded convergence heorem. ˆ 1 E u ɛ x u x We finish wih a discussion of Duhamel s principle, which jusifies he inroducion of he fixed poin equaion ( in he proof of Theorem 4. Noe ha since our weak soluions saisfy x u L by definiion, hey have a mos linear growh. Proposiion 4. Suppose ha u, f : [, R R, g : R R have a mos linear growh. Assume ha f is Borel measurable, and ha u and g are coninuous. Then ˆ ˆ ˆ (9 = u φ + u xx φ + fφ dxd + φ (, x g (x dx φ Cc ([, R if and only if R (1 u ( = e g + R e ( s f (s ds [,. Remark 5. Alhough he assumpion of linear growh is no opimal, i will be sufficien for our applicaion. Implici here is a uniqueness heorem for weak soluions of he hea equaion wih a mos linear growh. Recall ha even classical soluions fail o be unique wihou cerain growh condiions a x = (see e.g. [1, Ch. 7]. 13

Proof. Tha u saisfies (9 if i saisfies (1 is clear in he case ha f, g are smooh and compacly suppored. Then, a cuoff and mollificaion argumen upgrades he resul o he given class. In he oher direcion, suppose ha u saisfies (9. Define he funcion Θ (, x = u (, x [ e g ( ] [ ] (x e ( s f (s, (x ds, which is coninuous and saisfies Θ (, =. By a similar argumen as above, Θ saisfies he hea equaion in he sense of disribuions on R + R. Since he hea operaor is hypoellipic, i follows ha Θ is a classical soluion [9]. By is definiion, Θ grows a mos linearly since he same is rue for u, f, and g. By he maximum principle for he hea equaion in unbounded domains [1], we conclude ha Θ =. References [1] Anonio Auffinger and Wei-Kuo Chen. On properies of Parisi measures. Probabiliy Theory and Relaed Fields, o appear, March 13. [] Anonio Auffinger and Wei-Kuo Chen. The Parisi formula has a unique minimizer. ArXiv e-prins, February 14. [3] Michelle Boué and Paul Dupuis. A variaional represenaion for cerain funcionals of brownian moion. The Annals of Probabiliy, 6(4:1641 1659, 1 1998. [4] Anon Bovier and Anon Klimovsky. The aizenman-sims-sarr and guerras schemes for he sk model wih mulidimensional spins. Elecronic Journal of Probabiliy, 14:161 41, 8. [5] Wei-Kuo Chen. Parial resuls on he convexiy of he parisi funcional. Proc. Amer. Mah. Soc., o appear. [6] Lawrence C. Evans. Parial differenial equaions, volume 19 of Graduae Sudies in Mahemaics. American Mahemaical Sociey, Providence, RI, second ediion, 1. [7] Wendell H. Fleming. Exi probabiliies and opimal sochasic conrol. Appl. Mah. Opim., 4(4:39 346, 1977/78. [8] Wendell H. Fleming and Raymond W. Rishel. Deerminisic and sochasic opimal conrol. Springer-Verlag, Berlin-New York, 1975. Applicaions of Mahemaics, No. 1. [9] Gerald B Folland. Inroducion o parial differenial equaions. Princeon Universiy Press, 1995. [1] Avner Friedman. Parial differenial equaions of parabolic ype. Courier Corporaion, 13. [11] Francesco Guerra. Sum rules for he free energy in he mean field spin glass model. In Mahemaical physics in mahemaics and physics (Siena,, volume 3 of Fields Ins. Commun., pages 161 17. Amer. Mah. Soc., Providence, RI, 1. [1] Friz John. Parial differenial equaions, volume 1 of Applied Mahemaical Sciences. Springer-Verlag, New York, 198. [13] Marc Mézard, Giorgio Parisi, and Miguel Angel Virasoro. Spin glass heory and beyond, volume 9. World scienific Singapore, 1987. [14] Dmiriy Panchenko. A quesion abou he Parisi funcional. Elecron. Commun. Probab., 1:no. 16, 155 166, 5. [15] Dmiry Panchenko. The Sherringon-Kirkparick model. Springer, 13. [16] Dmiry Panchenko. The Parisi formula for mixed p-spin models. Ann. Probab., 4(3:946 958, 14. [17] Daniel W. Sroock and S. R. Srinivasa Varadhan. Mulidimensional diffussion processes, volume 33. Springer Science & Business Media, 1979. [18] Michel Talagrand. Parisi measures. Journal of Funcional Analysis, 31(:69 86, 6. [19] Michel Talagrand. The Parisi formula. Ann. Mah. (, 163(1:1 63, 6. [] Michel Talagrand. Mean field models for spin glasses. Volume I, volume 54 of Ergebnisse der Mahemaik und ihrer Grenzgebiee. 3. Folge. A Series of Modern Surveys in Mahemaics [Resuls in Mahemaics and Relaed Areas. 3rd Series. A Series of Modern Surveys in Mahemaics]. Springer-Verlag, Berlin, 11. Basic examples. [1] Michel Talagrand. Mean field models for spin glasses. Volume II, volume 55 of Ergebnisse der Mahemaik und ihrer Grenzgebiee. 3. Folge. A Series of Modern Surveys in Mahemaics [Resuls in Mahemaics and Relaed Areas. 3rd Series. A Series of Modern Surveys in Mahemaics]. Springer, Heidelberg, 11. Advanced replicasymmery and low emperaure. E-mail address: aukosh@cims.nyu.edu E-mail address: obasco@cims.nyu.edu 14

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