THE BORELL-EHRHARD GAME

Transcription

1 THE BORELL-EHRHARD GAME RAMON VAN HANDEL Abstrat. A preise desription of the onvexity of Gaussian measures is provided by sharp Brunn-Minkowski type inequalities due to Ehrhard and Borell. We show that these are manifestations of a game-theoreti mehanism: a minimax variational priniple for Brownian motion. As an appliation, we obtain a Gaussian improvement of Barthe s reverse Brasamp-Lieb inequality. 1. Introdution The onvexity properties of probability measures play an important role in various areas of probability theory, analysis, and geometry. They arise in a fundmental manner, for example, in the study of onentration phenomena [29, 2] and in funtional analysis and onvex geometry [12, 1]. Among the most deliate results in this area are the remarkable onvexity properties of Gaussian measures [14, 32, 28, 8, 1]. The aim of this paper is to shed some new light on the latter topi. Let γ n be the standard Gaussian measure on R n. The simplest expression of the onvexity of Gaussian measures is given by the log-onavity property: λ log(γ n (A + (1 λ log(γ n (B log(γ n (λa + (1 λb for all λ [, 1] and Borel sets A, B R n, where A + B := {x + y : x A, y B} denotes Minkowski addition. This inequality is easily dedued from the lassial Brunn-Minkowski inequality, whih is the analogous statement for Lebesgue measure. However, while the importane of log-onavity an hardly be overstated, we expet in the ase of Gaussian measures that onvexity should appear in a muh stronger form than an be explained by log-onavity alone. For example, the lassial isoperimetri inequality for Eulidean volume is an easy and fundamental onsequene of the Brunn-Minkowski inequality [21], but log-onavity fails to explain the analogous isoperimetri property of Gaussian measures [28]. A preise desription of the onvexity of Gaussian measures was developed in a remarkable paper by Ehrhard [14], who introdued the following sharp analogue of the Brunn-Minkowski inequality for Gaussian measures: λ (γ n (A + (1 λ (γ n (B (γ n (λa + (1 λb, where Φ(x := γ 1 ((, x]. This inequality beomes equality when A, B are parallel halfspaes, and is a strit improvement over log-onavity as the funtion log Φ is onave. It has numerous interesting and important impliations, inluding the isoperimetri property of Gaussian measures that arises as a speial ase [28, 32]. 2 Mathematis Subjet Classifiation. 6G15, 39B62, 52A4, 91A15. Key words and phrases. Gaussian measures; onvexity; Ehrhard inequality; stohasti games. Supported in part by NSF grant CAREER-DMS and by the ARO through PECASE award W911NF

2 2 RAMON VAN HANDEL Given the fundamental nature of Ehrhard s inequality, it is natural to seek other Gaussian analogues of the rih family of results that appear in the lassial Brunn- Minkowski theory (f. [21, 4] and the referenes therein. Progress in this diretion has remained relatively limited, however. Unlike the lassial Brunn-Minkowski inequality, whih is well understood from many different perspetives, only two approahes to Ehrhard s inequality are known. 1 Ehrhard s original proof [14], using a Gaussian analogue of Steiner symmetrization, is limited to the ase where the sets A, B are onvex; it was later extended by Lata la [27] to eliminate the onvexity assumption on one of the two sets. The long-standing problem of proving Ehrhard s inequality for arbitrary Borel sets was finally settled by Borell [8], who also introdued a number of signifiant generalizations of this inequality [9, 1]. Borell s elegant approah, using a nonlinear heat equation and the paraboli maximum priniple, relies on some surprising anellations (as will be explained below, ompliating efforts to identify how it an be applied in other settings. A more abstrat variant of Borell s approah is given in [5, 25, 24], but the mehanism that makes this approah work remains somewhat mysterious. Let us note, in addition, that unlike many other geometri inequalities (inluding the Gaussian isoperimetri inequality that extend to more general settings, Ehrhard s inequality appears to be uniquely Gaussian; see [26, 4.3] for some disussion on this point. The aim of this paper is to develop a new interpretation of Ehrhard s inequality: we will show that both Ehrhard s inequality and its generalizations due to Borell arise as manifestations of a stohasti game that appears to lie at the heart of these phenomena. This unexpeted game-theoreti mehanism provides new insight into the suess of earlier proofs, and allows us to identify new onvexity results for Gaussian measures. In partiular, we will develop a Gaussian improvement of Barthe s reverse Brasamp-Lieb inequality, addressing a question posed in [5] Borell s stohasti method. To motivate the ideas that will be introdued in the sequel, let us begin by realling a powerful approah, also due to Borell [7], for proving log-onavity of Gaussian measures. In order to show that γ n (or any other measure is log-onave, it is natural to seek a representation formula for log(γ n (A from whih the onavity property beomes evident. A fundamental representation of this type, the Gibbs variational priniple, dates bak to the earliest work on statistial mehanis [22]: { } log e f dγ n = sup f dµ H(µ γ n, µ where H(µ γ n denotes relative entropy and the supremum is taken over all probability measures µ. The log-onavity property ould be read off diretly from this formulation using displaement onvexity of relative entropy as developed in the theory of optimal transportation [37]. However, in the ase of Gaussian measures, a simpler approah beomes available by identifying γ n with the distribution of the value of a Brownian motion {W t } at time one. The advantage gained by this approah is that absolutely ontinuous hanges of measure of Brownian motion admit an expliit haraterization by Girsanov s theorem [33], whih gives rise to the 1 After this paper was ompleted, the author learned of reent work [34] where another rather deliate proof of Ehrhard s inequality is provided using the Ornstein-Uhlenbek semigroup.

3 THE BORELL-EHRHARD GAME 3 following reformulation of the Gibbs variational priniple for Gaussian measures: [ ( log e f dγ n = sup E f W 1 + α t dt 1 ] α t 2 dt, α 2 where the supremum is taken over all progressively measurable proesses α. This formula was originally obtained using PDE methods by Fleming [18]; the onnetion with the Gibbs variational priniple was developed by Boué and Dupuis [11]. It was observed by Borell in [7] that log-onavity of the Gaussian measure is an almost immediate onsequene of this identity. Let us illustrate this idea in its funtional (Prékopa-Leindler form. Let f, g, h be funtions suh that λ log(f(x + (1 λ log(g(y log(h(λx + (1 λy for all x, y, and denote by α f and α g the maximizing proesses when the above representation is applied to log f and log g, respetively. Then we have λ log f dγ n + (1 λ log g dγ n [ ( = λ E log f W 1 + [ ( + (1 λ E log g [ ( E log h W 1 + log h dγ n. α f t dt 1 2 ] α f t 2 dt W 1 + α g t dt 1 2 (λα f t + (1 λα g t dt 1 2 ] α g t 2 dt ] λα f t + (1 λα g t 2 dt Log-onavity follows readily by hoosing f = 1 A, g = 1 B, and h = 1 λa+(1 λb. The beauty of this stohasti approah is that it redues log-onavity of Gaussian measures to a trivial fat, viz. onvexity of the funtion x x 2. This idea has been further developed in [3, 31, 13] to prove various other inequalities, some of whih do not seem to be readily aessible by other methods. It is tempting to approah Ehrhard s inequality by seeking a Gaussian improvement of the Gibbs variational priniple. It is far from lear, however, why this should be possible. The Gibbs variational priniple is not a mysterious result: it simply expresses Fenhel duality for the onvex funtional f log e f dγ n. On the other hand, lassial results of Hardy, Littlewood, and Pólya [23, 3.16] imply that the funtional f Φ(f dγ n annot be onvex. In his proof of Ehrhard s inequality [8], Borell irumvents the lak of a representation formula by using partial differential equation methods. As a first step, he obtains a PDE for the transformation v f (t, x := (u f (t, x of the solution u f (t, x of the heat equation with initial ondition f (the latter arises naturally in this setting as the Markov semigroup of Brownian motion. It is not immediately obvious that the resulting nonlinear PDE, given in setion 2.2 below, possesses any useful onvexity properties. Instead, Borell onsiders diretly the desired ombination C(t, x, y := λv f (t, x + (1 λv g (t, y v h (t, λx + (1 λy, and observes that a fortuitous anellation ours: one an arrange the terms in the ombined PDEs for v f, v g, v h to obtain a paraboli PDE for C alone. This makes it possible to apply the paraboli maximum priniple to dedue nonpositivity of C, whih is essentially the statement of Ehrhard s inequality in its funtional form.

4 4 RAMON VAN HANDEL 1.2. The Borell-Ehrhard game. The main result of the present paper is a new stohasti representation formula that lies at the heart of Ehrhard s inequality, in diret analogy with Borell s stohasti approah to log-onavity. This priniple provides signifiant insight into the mehanism behind the onvexity properties of Gaussian measures, as well as a new tool to study suh properties. As was explained above, the lak of onvexity of f Φ(f dγ n prohibits us from obtaining a representation formula by a onvex duality argument. Instead, our main result shows that this funtional an be represented by a minimax variational priniple. An informal statement of our main result is as follows. Theorem (informal statement. For bounded and uniformly ontinuous f Φ(f dγ n = sup α [ ( inf E e 1 R t 2 βs 2ds α t, β t dt + e 1 R 1 2 βt 2dt f W 1 + β ] α t dt. This expression an be interpreted as the value of a zero-sum stohasti game between two players. The first player an apply a fore α t at time t to the underlying Brownian motion. The seond player annot affet the dynamis of the Brownian motion, but an instead hoose to end the game prematurely: her ontrol β t is the rate of termination of the game at time t (that is, the game ends prematurely in the interval [t, t + dt with probability β t 2 dt. The remarkable feature of this game is that the running ost α t, β t is not quadrati, as in the stohasti representation used to prove log-onavity, but rather linear in α. This reflets the fat that the transformation lies preisely at the border of where we an expet onvexity to appear: it linearizes the quadrati ost that arises from the Gibbs variational priniple. (It is pointed out in [13] that log-onavity of γ n an be strengthened in a different sense by exploiting uniform onvexity of the quadrati ost. As is typial in the theory of ontinuous-time games, it is essential to arefully define the information struture available to eah player in order for the above stohasti representation to be valid. In setion 2, we provide a preise formulation and proof of our main result. The essential observation behind the proof is that our stohasti game is losely onneted to Borell s PDE approah to Ehrhard s inequality: the nonlinear heat equation of Borell an be identified as the Bellman- Isaas equation [36, 19] for the value of our stohasti game. This observation leads not only to the above representation, but also reveals the reason behind the hidden onvexity that appears somewhat mysteriously in Borell s proof. With the above stohasti representation in hand, it is a simple exerise to dedue Ehrhard s inequality, and its generalizations due to Borell, in omplete analogy to the stohasti proof of log-onavity. This exerise is arried out in setion A Gaussian reverse Brasamp-Lieb inequality. As an illustration of the power of the stohasti approah, we will use it to obtain a Gaussian improvement of the reverse Brasamp-Lieb inequality of Barthe [3, 5]. Let us first reall Barthe s inequality in its Brunn-Minkowski form (see setion 4 for the funtional form. Let E 1,..., E k be linear subspaes of R n with dim(e i = n i. Denote by P i the orthogonal projetion on E i, and let λ 1,..., λ k be suh that λ 1 P 1 + +λ k P k = I n. Then Barthe s inequality states that for any Borel sets A i E i, we have λ 1 log(γ n1 (A λ k log(γ nk (A k log(γ n (λ 1 A λ k A k,

5 THE BORELL-EHRHARD GAME 5 where we identify γ ni with the standard Gaussian measure on E i. This is an extension of the log-onavity property where the sets A i may lie in lower-dimensional subspaes of the ambient spae (in whih ase log-onavity is a trivial statement. In view of Ehrhard s inequality, one might hope that it is possible to replae the logarithm by in Barthe s inequality to obtain a Gaussian improvement. However, this is ertainly impossible in general: if E 1,..., E k are orthogonal subspaes that span R n, then Barthe s inequality is in fat equality for all hoies of A i and no Gaussian improvement is possible (see setion 4.1. Nonetheless, it is possible to systematially improve Barthe s inequality in the Gaussian setting, as we will do in setion 4. To this end, define for every (, 1 the funtion (x := (x (. We will show in setion 4 that, under the same assumptions as in Barthe s inequality, λ 1 (γ n1 (A λ k (γ nk (A k (γ n (λ 1 A λ k A k for every (, 1. From this inequality, one an reover both Barthe s inequality ( and Ehrhard s inequality ( 1 as speial ases. The stohasti game approah was essential to disovering the orret formulation of this inequality Generalized means. While our main result sheds new light on the mehanism behind Ehrhard s inequality, there remains some residual mystery regarding the origin of this stohasti game. The stohasti representation used to prove log-onavity is entirely natural, as it arises simply as a speialization of the Gibbs variational priniple to the Brownian setting. It is unlear, however, whether there exists a natural minimax generalization of the Gibbs variational priniple that an provide an analogous explanation for the Borell-Ehrhard game. From another perspetive, however, there is nothing partiularly surprising about the stohasti representations that we enountered so far. To plae these results in a broader ontext, we an onsider the more general funtional f F 1 F (f dγ n for any stritly inreasing funtion F. Suh funtionals, alled generalized means, were studied by Hardy, Littlewood, and Pólya [23, hapter 3], who provide in partiular neessary and suffiient onditions for suh funtionals to be onvex. In setion 5, we will show that any onvex generalized mean admits a stohasti representation that is very similar to the speial ase F (x = e x, from whih onvexity an be immediately read off. Moreover, we will argue that essentially arbitrary hoie of F will admit a stohasti game representation, so the appearane of a game in the ase F (x = Φ(x is just one speifi example. Of ourse, it is a speial feature of this example that gave rise to Ehrhard s inequality; the potential utility of suh representations in other ontexts will depend on the problem at hand. 2. The Borell-Ehrhard game 2.1. Setting and main result. Let (Ω, F, {F t }, P be a probability spae with a omplete and right-ontinuous filtration, and let {W t } be a standard n-dimensional F t -Brownian motion. We denote by γ n the standard Gaussian measure on R n and by Φ(x := γ 1 ((, x]. Our main result is a variational priniple for Gaussian measures that will be expressed as a stohasti game for the Brownian motion W. As is often the ase in ontinuous time games, it is important to arefully define what information is available to eah player. Informally, we an view our game as the ontinuous time limit of a disrete time game where two players take turns

6 6 RAMON VAN HANDEL exerising some ontrol on the underlying Brownian motion. We denote the ontrols of the first and seond players at time t by α t and β t, respetively. As the seond player omes after the first, her ontrol may depend on the hoie of ontrol of the first player. Conversely, the ontrol of the first player may depend on the hoie of ontrol of the seond player in earlier turns. It is not entirely obvious how this information struture should be enoded when time is ontinuous. For our purposes, it will be onvenient to adopt an approah due to Elliott and Kalton [15, 19]. In this framework, the seond player may hoose any ontrol. Definition 2.1. A ontrol is a progressively measurable n-dimensional proess β = {β t } t [,1]. Denote by C the family of all ontrols suh that E[ β s 2 ds] <. On the other hand, the ation of the first player must expliitly aount for the fat that she has aess to the earlier hoie of ontrol of the seond player. To this end, we introdue the notion of an (Elliott-Kalton strategy. Definition 2.2. A strategy is a map α : C C suh that for every t [, 1] and β, β C suh that β s (ω = β s(ω for a.e. (s, ω [, t] Ω, we have α s (β(ω = α s (β (ω for a.e. (s, ω [, t] Ω. Denote by S the family of all strategies suh that sup{e[ α s(β 2 ds] : E[ β s 2 ds] R} < for all R <. In the Elliott-Kalton approah, the seond player hooses any ontrol β C, while the first player s ontrol α(β is defined by a strategy α S. The definition of a strategy ensures that the ontrol of the first player depends ausally on the ontrol of the seond player, thereby enoding the desired information struture. With these formalities out of the way, we an now formulate our main result. Theorem 2.3. Let f : R n R be bounded and uniformly ontinuous, and define [ R t J f [α, β] := E βs 2ds α t, β t dt + e 1 R 1 ] 1 2 βt 2dt f W 1 + α t dt e 1 2 for α, β C. Then Φ(f dγ n = sup α S inf J f [α(β, β] = inf β C sup α S β C J f [α(β, β]. The remainder of this setion is devoted to the proof of Theorem 2.3. The onnetion with geometri inequalities will be developed in setions 3 and 4 below. Remark 2.4. Let us emphasize that we defined a strategy α S as a map that assigns to every random proess β C another random proess α(β C, not as a mapping of the sample paths of β. In partiular, the proess α(β is progressively measurable by onstrution, so that no measurability issues arise in Theorem The Borell PDE. Throughout the proof, we will assume without loss of generality that f is bounded, smooth, and has bounded derivatives of all orders. One the result is proved in this ase, the onlusion is readily extended to funtions f that are only bounded and uniformly ontinuous (as the latter an be approximated in the uniform topology by smooth funtions with bounded derivatives by onvolution with a smooth ompatly supported kernel, f. [2, 8.2]. Define for (t, x [, 1] R n the funtion u(t, x := E[Φ(f(W 1 W t + x],

7 THE BORELL-EHRHARD GAME 7 so that u solves the heat equation Define u t + 1 u =, 2 u(1, x = Φ(f(x. v(t, x := (u(t, x. By the smoothness assumption on f and elementary properties of the heat equation, u and therefore v are bounded, smooth, and have bounded derivatives of all orders on [, 1] R n. Moreover, it is readily verified that v satisfies v t v 1 2 v v 2 =, v(1, x = f(x. This equation was introdued by Borell [8] in his study of the Ehrhard inequality. The following simple observation ontains the main idea behind the proof of Theorem 2.3: the nonlinear term in Borell s PDE admits a variational interpretation. Lemma 2.5. Let be a onstant suh that 2 sup x f(x. Then 1 2 v v 2 = sup inf { a + b, v + b 12 } v b 2, a R n b R n where the optimizer a = ( v v, b = v is a saddle point. Proof. Define for a, b R n the objetive Then it is readily verified that H(a, b := a + b, v + b 1 2 v b 2. H(a, b = 1 2 v v 2, H(a, b = 1 2 (2 v b + v v v 2. But note that as 2 f, we have 2 v by the definition of v. Therefore sup a inf b H(a, b sup and the proof is omplete. a H(a, b = 1 2 v v 2 = inf b H(a, b sup a inf H(a, b, b Lemma 2.5 reveals that the partial differential equation satisfied by v is none other than the Bellman-Isaas equation for the value of a stohasti game [19, 36]. We an now proeed along mostly standard lines to formalize this idea Upper bound. Fix 2 f, and onsider the stohasti differential equation dx β t = ( v(t, X β t v(t, X β t dt + β t dt + dw t, X β = for β C. As the funtion ( v v is smooth with bounded derivatives, this equation has a unique strong solution X β [33, Theorem 4.8]: that is, there is a progressively measurable map F : W W suh that X β = F [{ t β sds + W t }], where W denotes the spae of ontinuous paths with its anonial filtration. Define α t (β := ( v(t, X β t v(t, X β t. Then evidently α (β C (in fat, it is uniformly bounded and α depends ausally on β by the strong solution property. Thus we have shown that α S defines an Elliott-Kalton strategy in the sense of Definition 2.2.

8 8 RAMON VAN HANDEL Applying Itô s formula to the proess e 1 2 e 1 2 v(, + e e 1 2 R t βs 2 ds v(t, X β t gives R t βs 2ds α t (β + β t, β t dt + e 1 R 1 2 βt 2dt f(x β 1 = R t βs 2 ds v(t, X β t, dw t R t βs 2 ds{ v t (t, Xβ t v(t, Xβ t + α t (β + β t, v(t, X β t + β t 1 2 v(t, Xβ t β t 2 }dt. We now observe that the last integral in this expression is nonnegative by Borell s PDE and Lemma 2.5. Moreover, the Brownian integral is a martingale as v is bounded. Therefore, taking the expetation of this expression, we obtain [ ] R t v(, E βs 2ds α t (β + β t, β t dt + e 1 R 1 2 βt 2dt f(x β 1 e 1 2 = J f [α (β + β, β] for every β C. But evidently α (β := α (β + β defines another Elliott-Kalton strategy α S. We therefore readily obtain the upper bound in Theorem 2.3 Φ(f dγ n = v(, sup inf J f [α(β, β]. β C 2.4. Lower bound. For the proof of the lower bound, fix any α S. Given this strategy, our aim is to onstrut a ontrol β C that nearly minimizes J f [α(β, β]. We will do this by imitating the idea that our ontinuous game is the limit of disrete-time games, as was explained informally at the beginning of this setion. To this end, fix a time step δ = N 1 (N 1. For t [, δ, let β t := v(,. We now iteratively extend the definition of β as follows. Suppose that β has been defined on the interval [, kδ. Let βt k := β t 1 [,kδ (t, so that β k C. We define kδ β t := v kδ, W kδ + α s (β k ds for t [kδ, (k + 1δ. Iterating this proess N 1 times results in a ontrol β C that is defined for all times t [, 1] (we may arbitrarily hoose β 1 :=. Now define the proess X t := W t + t α S α s (β ds. As β t = βt k for all t [, kδ by onstrution, we have α s (β(ω = α s (β k (ω for a.e. (s, ω [, kδ Ω by the definition of an Elliott-Kalton strategy. In partiular, we have β t = v(kδ, X kδ for every t [kδ, (k + 1δ a.s. Applying Itô s formula to e 1 R t 2 βs 2ds v(t, X t as in the upper bound gives where Γ := e 1 2 J f [α(β, β] = v(, + E[Γ] } R t ds{ 1 βs 2 2 v(t, X t( v(t, X t 2 β t 2 + α t (β, v(t, X t +β t dt.

9 THE BORELL-EHRHARD GAME 9 As v is bounded and has bounded derivatives of all orders, we an estimate Γ C 1 (1 + α t (β v(t, X t + β t dt N 1 = C 1 N 1 C 2 (k+1δ k= kδ (k+1δ k= kδ (1 + α t (β v(t, X t v(kδ, X kδ dt (1 + α t (β (δ + X t X kδ dt for onstants C 1, C 2 that depend on f only. Note that for t (k + 1δ X t X kδ W t W kδ + [ t 1/2 δ α s (β ds] 2. kδ We an therefore estimate using Cauhy-Shwarz [ ] E[Γ] C 3 δ (1 + E α t (β 2 dt C 3 (K + 1 δ, where K := sup{e[ α t(β 2 dt] : β v } < by definition as α S and where C 3 depends only on f. We have therefore shown that inf J f [α(β, β ] J f [α(β, β] v(, + C 3 (K + 1 δ. β C As δ > and α S were arbitrary, we readily onlude that sup inf J f [α(β, β] v(, = Φ(f dγ n. β C α S 2.5. End of proof. Combining the upper and lower bound, we have shown Φ(f dγ n = sup inf J f [α(β, β]. β C It remains to prove the seond identity in Theorem 2.3. To this end, note that Φ(f dγ n = Φ( f dγ n = sup inf J f [α(β, β] β C as Φ( x = 1 Φ(x. But we an write sup inf J f [α(β, β] = inf β C α S sup α S β C α S α S ( J f [α(β, β] = inf sup α S β C J f [α(β, β]. As C is invariant under the transformation β β and S is invariant under the transformation α(β α( β, the seond identity in Theorem 2.3 follows. 3. The Ehrhard and Borell inequalities The aim of this short setion is to show that the lassial Gaussian Brunn- Minkowski inequality of Ehrhard [14, 8] and its generalizations due to Borell [9, 1] arise as immediate orollaries of Theorem 2.3. In setion 4 below, we will extend this approah to derive new geometri inequalities for Gaussian measures.

10 1 RAMON VAN HANDEL 3.1. Ehrhard s inequality. Ehrhard s inequality states that λ (γ n (A + (1 λ (γ n (B (γ n (λa + (1 λb for all Borel sets A, B R n and λ [, 1]. By approximating the indiator funtions of A and B by smooth funtions, it is routine to dedue this inequality from the following funtional form of the result (see [8] or setion 4.4 below. Corollary 3.1 ([14, 8]. Let λ [, 1], and let f, g, h be uniformly ontinuous funtions with values in [ε, 1 ε] for some ε >. Suppose that for all x, y R n λ (f(x + (1 λ (g(y (h(λx + (1 λy. Then λ f dγ n + (1 λ g dγ n h dγ n. Proof. Fix δ >, and hoose near-optimal α f, α g S and β h C suh that sup inf J Φ β C 1 (f[α(β, β] inf J Φ β C 1 (f[α f (β, β] + δ, α S sup inf J Φ β C 1 (g[α(β, β] inf J Φ β C 1 (g[α g (β, β] + δ, α S J (h[λα f (β h + (1 λα g (β h, β h ] inf β C J (h[λα f (β + (1 λα g (β, β] + δ. Then by Theorem 2.3 λ ( f dγ n + (1 λ Φ 1 ( g dγ n λ J (f[α f (β h, β h ] + (1 λ J (g[α g (β h, β h ] + 2δ J (h[λα f (β h + (1 λα g (β h, β h ] + 2δ ( h dγ n + 3δ, and the proof is ompleted by letting δ Borell s Gaussian Brunn-Minkowski inequalities. In [9], Borell proves a substantial generalization of Ehrhard s inequaliy: he shows that λ (γ n (A + µ (γ n (B (γ n (λa + µb holds for all Borel sets A, B R n if and only if λ + µ 1 and λ µ 1 (the neessity of the latter onditions is easily verified by expliit examples, see [9]. The dedution of this result from Theorem 2.3 requires only a minor modifiation of the proof of Corollary 3.1: it suffies to note that we do not need to hoose the same Brownian motion W in the variational problems for f and g. By hoosing instead two orrelated Brownian motions, we immediately reover Borell s result. Corollary 3.2 ([9]. Let λ, µ, and let f, g, h be uniformly ontinuous funtions with values in [ε, 1 ε] for some ε >. Suppose that for all x, y R n λ (f(x + µ (g(y (h(λx + µy. If λ + µ 1 and λ µ 1, then λ f dγ n + µ g dγ n h dγ n.

11 THE BORELL-EHRHARD GAME 11 Proof. Let ρ = (1 λ 2 µ 2 /2λµ. The assumptions λ + µ 1 and λ µ 1 guarantee that ρ [ 1, 1]. We an therefore define two standard n-dimensional Brownian motions {W t } and { W t } with quadrati ovariation W i, W j t = ρtδ ij. The point of this onstrution is that the proess { W t } defined as W t := λw t +µ W t is again a standard n-dimensional Brownian motion. Let J f, J f be defined analogously to J f in Theorem 2.3 where {W t } is replaed by { W t } and { W t }, respetively. The remainder of the proof is idential to that of Corollary 3.1, where J (g is replaed by J (g and J (h by J (h. Remark 3.3. We observe that it was essential for the suess of the proof of Corollary 3.2 that the game desribed by Theorem 2.3 is defined on a general probability spae: while the objetive funtion J f [α, β] depends only on a single Brownian motion {W t }, we allowed the ontrols α, β C to be adapted to a larger filtration {F t } that is not neessarily generated by the underlying Brownian motion alone. This freedom was used ruially in the proof of Corollary 3.2; here we an take F t to be (the augmentation of σ{w s, W s : s t}, but we annot ensure that the ontrol λα f (β h + µα g (β h will depend only on { W t }. The assumptions λ + µ 1 and λ µ 1 in Corollary 3.2 are preisely the onditions required for the existene of orrelated standard Brownian motions {W 1,t } and {W 2,t } suh that λw 1 +µw 2 is also a standard Brownian motion. Along idential lines, we immediately see that the inequality λ 1 (γ n (A λ k (γ n (A k (γ n (λ 1 A λ k A k holds for all Borel sets A 1,..., A k R n whenever there exist orrelated standard Brownian motions {W i,t }, i = 1,..., k suh that λ 1 W λ k W k is again a standard Brownian motion. The family of oeffiients λ 1,..., λ k for whih this is the ase is haraterized by [5, Lemma 3], and we reover in this manner the general Gaussian Brunn-Minkowski inequality of Borell [1]. Remark 3.4. We have stated Corollaries 3.1 and 3.2 for simpliity under the assumption that the funtions f, g, h are uniformly ontinuous and bounded away from zero and one. This ase ontains the main diffiulty of the problem: it is routine to derive from this the orresponding results for sets [8], and one an subsequently derive versions of Corollaries 3.1 and 3.2 where the funtions f, g, h are just Borel measurable with values in [, 1] as is explained in [28]. As these are standard results, we omit the details. However, in setion 4.4 below, we will work out in detail a diret approximation argument in the setting of Theorem 4.2 that ould also be applied here to dedue the measurable versions of Corollaries 3.1 and A Gaussian reverse Brasamp-Lieb inequality 4.1. Barthe s inequality. Both the lassial Brunn-Minkowski inequality and Ehrhard s inequality bound the measure of the Minkowski sum λa + (1 λb from below in terms of the measures of A and B. Therefore, when either A or B has measure zero, these inequalities neessarily beome trivial. Nonetheless, it is perfetly possible for λa + (1 λb to have positive measure even when A and B are, for example, ontained in lower-dimensional subspaes of R n. This phenomenon is aptured quantitatively by a signifiant generalization of the lassial Brunn-Minkowski inequality due to Barthe [3], whih we presently reall.

12 12 RAMON VAN HANDEL Fix λ 1,..., λ k, and let B 1,..., B k be linear maps B i : R n R ni suh that k λ i Bi B i = I n, B i Bi = I ni for all i. i=1 Note that Bi isometrially embeds Rni in the linear subspae E i = Im(Bi of Rn. Let f i : R ni R and h : R n R be funtions suh that λ 1 log(f 1 (x λ k log(f k (x k log(h(λ 1 B 1x λ k B kx k for all x i R ni. Then Barthe s inequality states that λ 1 log f 1 dγ n1 + + λ k log f k dγ nk log h dγ n (see [5, 31] for the formulation in terms of Gaussian rather than Lebesgue measure. When f i are taken to be indiator funtions of sets, this redues to the following generalization of the Brunn-Minkowski inequality: for any Borel sets A i E i λ 1 log(γ n1 (A λ k log(γ nk (A k log(γ n (λ 1 A λ k A k, where we impliitly identify γ ni with the standard Gaussian measure on E i. Remark 4.1. Barthe s inequality is also alled the reverse Brasamp-Lieb inequality. The lassial Brasamp-Lieb inequality is an analogous multilinear generalization of Hölder s inequality. Just as the Prékopa-Leindler inequality ould formally be viewed as a reverse form of Hölder s inequality, Barthe s inequality an be viewed as a reverse form of the Brasamp-Lieb inequality. Let us note that we have stated the inequality in its geometri form, whih is most natural for our purposes. The general form of the reverse Brasamp-Lieb inequality (for general matries B i an be dedued from the geometri form, see [6] and [31] for details. When n i = n and B i = I n for all i, Barthe s inequality redues to the Prékopa- Leindler inequality. However, we know that the latter is far from optimal for Gaussian measures: the sharp form of the Prékopa-Leindler inequality in the Gaussian ase is preisely Ehrhard s inequality (Corollary 3.1, where the logarithm is replaed by. It is therefore natural to ask whether there exists an analogous Gaussian improvement of Barthe s inequality. This question was raised in [5, 4.2]. We will show in setion 4.2 that there does in fat exist an interesting family of inequalities of this form, but the orret formulation of suh inequalities is not entirely obvious. Before we develop these inequalities, let us briefly disuss what sort of improvement ould reasonably be expeted. One might optimistially hope that as in the ase of Ehrhard s inequality, we may simply replae log by in Barthe s inequality to obtain the analogus Gaussian form. However, not only is this impossible, but in fat no improvement of Barthe s inequality is possible in general. To see why, onsider the ase where E 1 and E 2 are two orthogonal subspaes of R 2, whih fores λ 1 = λ 2 = 1. Suppose the inequality L(γ 1 (A 1 + L(γ 1 (A 2 L(γ 2 (A 1 + A 2 holds for a funtion L. As γ 2 (A 1 + A 2 = γ 1 (A 1 γ 1 (A 2 in this ase, we must have L(x + L(y L(xy for all x, y [, 1], whih is learly violated when L(x = (x (let x = y = 1 2. On the other hand, the above inequality holds with equality when L(x = log x.

13 THE BORELL-EHRHARD GAME 13 It follows that Barthe s inequality is already optimal in the orthogonal setting and annot be improved by any alternative hoie of funtion L. We have now onsidered two extreme ases. When E 1 = = E k = R n, Ehrhard s inequality is sharp and the optimal hoie of funtion is L =. On the other hand, when E 1,..., E k are orthogonal subspaes, Barthe s inequality is sharp and the optimal hoie of funtion is L = log. One an therefore not expet that any single hoie of funtion L an provide a systemati Gaussian refinement of Barthe s inequality: any general improvement requires the hoie of L to depend at least on the parameters λ i and B i. This feature is integral to the formulation of the Gaussian reverse Brasamp-Lieb inequalities that we will prove presently: we will introdue a family of inequalities that interpolate, in some sense, between the Ehrhard and Barthe inequalities; the best hoie of inequality within this family must depend on the parameters to whih it is applied A Gaussian refinement. In the remainder of this setion, we plae ourselves in the same setting as in the above formulation of Barthe s inequality: that is, we fix λ 1,..., λ k and let B 1,..., B k be linear maps B i : R n R ni suh that k λ i Bi B i = I n, B i Bi = I ni for all i. i=1 As before, we define the subspaes E i = Im(Bi. We also define the funtion (x := (x (, x [, 1] for (, 1. We will prove the following Gaussian form of Barthe s inequality. Theorem 4.2. Let (, 1, and let f 1,..., f k, h be Borel measurable funtions f i : R ni R, h : R n R with values in [, 1]. Suppose that λ 1 for all x i R ni. Then λ 1 (f 1 (x λ k (f k (x k (h(λ 1 B1x λ k Bkx k f 1 dγ n1 + + λ k f k dγ nk h dγ n. We immediately dedue the following generalization of Ehrhard s inequality. Corollary 4.3. For any (, 1 and Borel sets A i E i, i = 1,..., k, we have λ 1 (γ n1 (A λ k (γ nk (A k (γ n (λ 1 A λ k A k. Proof. Choose f i (x = 1 Ai (B i x and h(x = 1 λ 1A 1+ +λ k A k (x. It is instrutive to note that both Ehrhard s inequality and Barthe s generalized Brunn-Minkowski inequality arise as limiting ases of Corollary 4.3. Let us first reover Barthe s inequality. To this end, reall that Φ( y = A simple omputation shows that so that y e z2 /2 2π dz = (1 + o(1 e y2 /2 y 2π as y. (x 2 = 2 log x log log(1/x log 4π + o(1 as x, (x = Φ 1 (x 2 ( 2 2 log x + o(1 ( + = (x ( + (x as.

14 14 RAMON VAN HANDEL This implies, in partiular, that lim (x 2 log = log x. Thus Barthe s Brunn-Minkowski inequality is reovered as in Corollary 4.3. On the other hand, to reover Ehrhard s inequality, set n i = n and B i = I n for all i. This fores λ λ k = 1, so that Corollary 4.3 redues to λ 1 (γ n (A λ k (γ n (A k (γ n (λ 1 A λ k A k. Thus Ehrhard s inequality is reovered as 1 in Corollary 4.3. We have therefore seen that Corollary 4.3 is never worse than Barthe s Brunn- Minkowski inequality, and an be substantially better. For general parameters, one has the freedom to optimize over to obtain the best inequality in this family Proof of Theorem 4.2: smooth ase. The proof of Theorem 4.2 is based on a minor but ruial variation on Theorem 2.3. Before we turn to the proof, let us briefly disuss how this extension and the funtion Φ arise naturally when attempting to extend Ehrhard s inequality to the setting of Theorem 4.2. We begin by noting that the Gaussian measure γ ni on R ni an be obtained as a projetion of the Gaussian measure γ n. In partiular, as B i Bi = I n i by assumption, B i : R n R ni is a projetion, and we an write f i dγ ni = (f i B i dγ n. It is therefore natural to attempt to obtain a Gaussian analogue of Barthe s inequality by applying Theorem 2.3 to the funtions (f i B i. If we had [ R? t (f i B i dγ n = sup inf βs 2ds Bi B i α t (β, β t dt E α S β C + e 1 2 e 1 2 R 1 βt 2 dt ( f i (B i W 1 + ] B i α t (β dt, then we ould diretly repeat the proof of Corollary 3.1 to show that the analogue of Barthe s inequality holds even for the funtion rather than (whih is false as disussed above. However, Theorem 2.3 only gives α t (β, β t in the running ost, rather than Bi B iα t (β, β t as written in the above formula. This issue nearly resolves itself by inspetion of the proof of Theorem 2.3. We observe that the optimal strategy α that is defined in the proof of Theorem 2.3 is of the form α t (β v(t, X β t. As (f i B i = Bi ( f i B i, it follows readily that (f i B i, and therefore v(t, x, take values in the range of Bi. In partiular, B i B i α t (β = α t (β as B i is a projetion. It therefore follows that we may essentially insert the missing projetion B i B i for free in the statement of Theorem 2.3. The major problem that we have overlooked in the above disussion is that if f i, then it is not α (β but rather α (β := α (β+ (β that appears as the saddle point in the proof of Theorem 2.3. In the proof of Ehrhard s inequality, this regularization (whih arises in an essential manner in Lemma 2.5 in order for the game to be well-posed was ompletely irrelevant as it ould be absorbed into the definition of the optimal strategy α. In the present ase, however, the regularizing term is preisely the obstrution that prevents Theorem 4.2 from being valid for the funtion, as we annot assume that β takes values in the range of B i.

15 THE BORELL-EHRHARD GAME 15 Instead, we will see in the proof of Proposition 4.4 below that a simple appliation of the fundamental theorem of alulus allows us to transform the additional regularization term in the running ost into an additive onstant that depends only on the uniform upper bound on the funtion f i. The definition of the funtion simply arises by absorbing the uniform bound and additive onstant (note that applying to funtions f i with values in [, 1], as is done in Theorem 4.2, orresponds to applying to funtions with values in [, ]; in partiular, Theorem 4.2 would fail if we allowed the funtions f i to take values larger than one, even though the funtion ould naturally be extended to suh values. Having desribed the main ideas behind the proof of Theorem 4.2, we are ready to state the key extension of Theorem 2.3 that will be needed. Proposition 4.4. Let f : R m (, ] be bounded and uniformly ontinuous, let (, 1, and let B : R n R m be a linear map suh that BB = I m. Define [ J B, f [α, β] := E e 1 R t 2 βs 2ds B Bα t, β t dt + e 1 R βt 2dt f BW 1 + Bα t dt + Φ 1 ( 1 ] Bβ t dt 2 for α, β C. Then we have Φ (f dγ m = sup inf J B, f α S β C [α(β, β]. Proof. We begin by noting that, by the definition of, we an write Φ (f dγ m = Φ( ( + f B dγ n (, where we used that B is a projetion (so that γ m = γ n B 1. Define g := ( + f B. As f, we have g (. Following verbatim the proof of the upper bound of Theorem 2.3, we have Φ(g dγ n J g [α (β Φ 1 (β, β] for all β C, where α S is a strategy of the form α t (β = ( 1 2 Φ 1 ( v(t, X β t v(t, X β t, v(t, x = (E[Φ(g(W 1 W t + x] for a suitably defined random proess X β. The ruial observation at this point is that as g(x = B f(bx, we have v(t, x Im(B for all t, x. In partiular, the optimal strategy α satisfies B Bα (β = α (β for every β C. Therefore Φ(g dγ n sup inf J g[b Bα(β + 1 β C 2 Φ 1 (β, β]. α S On the other hand, the orresponding lower bound follows immediately from Theorem 2.3 (as strategies of the form B Bα(β+ 1 2 Φ 1 (β form a subset of all possible strategies S. Putting everything together, we have now shown that Φ (f dγ m = sup inf J g[b Bα(β + 1 β C 2 Φ 1 (β, β] (. α S

16 16 RAMON VAN HANDEL To omplete the proof, it suffies to note that J g [B Bα Φ 1 (β, β] ( [ = J B, 1 1 f [α, β] + ( E e 1 R t 2 βs 2ds β t 2 dt + e = J B, f [α, β], where we used the fundamental theorem of alulus. With Proposition 4.4 in hand, we immediately obtain: ] R 1 βt 2dt 1 Corollary 4.5. Theorem 4.2 is valid under the additional assumption that the funtions f 1,..., f k, h are uniformly ontinuous with values in [ε, 1] for some ε >. The proof is idential to that of Corollary 3.1, and we omit the details. Remark 4.6. Corollary 4.3 an be dedued diretly from Corollary 4.5 by introduing smooth approximations of the indiator funtions of the sets A 1,..., A k and λ 1 A λ k A k. Suh an argument is given in [8], and an be readily applied in the present setting. We therefore do not need the full strength of Theorem 4.2 to dedue Corollary 4.3. However, while the proof of Theorem 4.2 requires a bit more work, it yields a result that is potentially of broader utility Proof of Theorem 4.2: general ase. The important part Theorem 4.2 is already ontained in Corollary 4.5 above. The remaining arguments in the proof of Theorem 4.2 are tehnial: we must approximate the measurable funtions f 1,..., f k, h by uniformly ontinuous funtions so that Corollary 4.5 an be applied. The requisite approximation arguments are worked out in this setion. (Closely related approximation arguments an also be found in [13]. We begin by proving Theorem 4.2 in the ase that f 1,..., f k and h are uppersemiontinuous. The following lemma makes it possible to approximate uppersemiontinuous funtions by uniformly ontinuous funtions without violating the assumption of Theorem 4.2, so that Corollary 4.5 an be applied. Lemma 4.7. Let f 1,..., f k, h be upper-semiontinuous funtions f i : R ni R, h : R n R with values in [ε, 1] for some ε >. Let (, 1, and suppose that λ 1 (f 1 (x λ k (f k (x k (h(λ 1 B1x λ k Bkx k for all x i R ni. Then there exist for every s > uniformly ontinuous funtions f1 s,..., fk s, hs with values in [ε, 1] suh that λ 1 (f1 s (x λ k (fk(x s k (h s (λ 1 B1x λ k Bkx k for all x i R ni, and suh that fi s f i and h s h pointwise as s. Proof. Define f s i and h s by the sup-onvolutions (fi s (x := sup { (f i (y s 1 x y }, y R n i (h s (x := sup { y R n (h(y s 1 x y }. It is easily seen that fi s, hs take values in [ε, 1], and that (fi s and Φ 1 (h s are s 1 -Lipshitz; thus fi s and h s are ertainly uniformly ontinuous. We now laim that h s h as s. To see this, hoose for every s > a point y s suh that (h s (x (h(y s s 1 x y s + s.

17 THE BORELL-EHRHARD GAME 17 As h s ε and h 1, this evidently implies x y s s 2 s (ε for all s, so that y s x as s. But we an now estimate (h(x lim inf s lim sup s (h s (x lim sup s (h(y s (h(x, (h s (x where we have used that h is upper-semiontinuous in the last line. This shows that h s h pointwise as s, and fi s f i follows identially. Finally, note that λ 1 (f1 s (x λ k (fk(x s k = sup {λ 1 (f 1 (y λ k (f k (y k y 1,...,y k s 1 λ 1 x 1 y 1 s 1 λ k x k y k } sup { (h(λ 1 B1y λ k Bky k y 1,...,y k s 1 λ 1 B 1(x 1 y λ k B k(x k y k } (h s (λ 1 B 1x λ k B kx k, where we have used that B i z = z for z Rni and the triangle inequality. Using Lemma 4.7 and Corollary 4.5, we an now prove the following. Corollary 4.8. Theorem 4.2 is valid under the additional assumption that the funtions f 1,..., f k, h are upper-semiontinuous with values in [, 1]. Proof. We first approximate f 1,..., f k, h by funtions that are bounded away from zero. To this end, fix ε (, 1 and let δ := max i Φ (λ i (ε. Define the uppersemiontinuous funtions h := h δ and f i := f i ε for all i. We laim that λ 1 ( f 1 (x λ k ( f k (x k ( h(λ 1 B1x λ k Bkx k. Indeed, if f i (x i > ε for all i this follows from the assumption of Theorem 4.2, while if f i (x i ε for some i the left-hand side is at most (δ. Applying Lemma 4.7, we an find uniformly ontinuous funtions f 1 s,..., f k s, h s with values in [ε, 1] suh that f i s f i and h s h pointwise as s and λ 1 ( f 1 s (x λ k ( f k(x s k ( h s (λ 1 B1x λ k Bkx k for every s >. Corollary 4.5 implies λ 1 f 1 s dγ n1 + + λ k f k s dγ nk h s dγ n. The onlusion follows using dominated onvergene as s and ε. We an now omplete the proof of Theorem 4.2. Proof of Theorem 4.2. Let f 1,..., f k be upper-semiontinuous funtions with ompat support and with values in [, 1] suh that f i f i for all i. Define h by ( h(x := sup {λ 1 λ 1B1 x1+ +λ kbk x ( f 1 (x λ k ( f k (x k }. k=x

18 18 RAMON VAN HANDEL Then h h by onstrution, and h is also upper-semiontinuous [35, Prop. 1.27]. Moreover, the upper-semiontinuous funtions f 1,..., f k and h learly satisfy the assumptions of Theorem 4.2. Therefore, Corollary 4.8 implies λ 1 f 1 dγ n1 + + λ k f k dγ nk h dγ n. The onlusion now follows by taking the supremum on the left-hand side over all ompatly supported upper-semiontinuous funtions f i f i [2, Prop. 7.14]. 5. Generalized means Unlike the logarithmi funtional f log e f dγ n, whose stohasti representation has a natural interpretation through the Gibbs variational priniple, the emergene of a stohasti game representation for f Φ(f dγ n may appear rather unexpeted. To provide some further insight into suh representations, we aim in this setion to plae the result of Theorem 2.3 in a broader ontext. Throughout this setion, let I R be a ompat interval, and let F : I R be a smooth funtion that is stritly inreasing F >. Following Hardy, Littlewood, and Pólya [23, hapter 3], we define the generalized mean M F as M F (f := F 1 F (f dγ n for any measurable funtion f : R n I. We will argue below that the generalized mean M F admits a stohasti representation for any suffiiently regular funtion F : from this perspetive, there is nothing partiularly speial about the speifi ases F (x = e x and F (x = Φ(x that we enountered so far. Of ourse, the potential utility of suh stohasti representations in other settings depends on the problem at hand. For example, to establish Brunn-Minkowski type inequalities, we ruially exploited a speial feature of the funtions F (x = e x and F (x = Φ(x: in both ases, the running ost in the stohasti representation proves to be a onave funtion of the strategy that is being maximized over. While suh strutural features of the representation are speifi to partiular hoies of F, the existene of a stohasti representation is not anything speial in its own right. In their study of generalized means, Hardy, Littlewood, and Pólya [23, 3.16] obtained neessary and suffiient onditions for f M F (f to be a onvex funtional. In setion 5.1, we will show that stohasti representations provide an interesting perspetive on this haraterization: the onditions of Hardy, Littlewood, and Pólya are preisely those that are needed to obtain a stohasti representation for M F involving only a supremum (as in the ase F (x = e x. In partiular, we an state a very general expression for the stohasti representation in this setting, despite that the Fenhel transform of M F (and therefore the natural analogue of the Gibbs variational priniple rarely admits a tratable expression. For generalized means that are not onvex, we will outline in setion 5.2 how one an obtain in this ase a stohasti game representation of M F under essentially no assumptions on F. As the expliit expressions that define suh games for general F do not provide muh insight, we do not state a general theorem, but rather illustrate by means of an example how easily suh representations an be obtained in pratie The onvex ase. The following result due to Hardy, Littlewood, and Pólya haraterizes preisely when the funtional f M F (f is onvex.

19 THE BORELL-EHRHARD GAME 19 Theorem 5.1 ([23, 3.16]. The generalized mean funtional f M F (f is onvex if and only if the funtion F is onvex and the funtion F /F is onave. Proof. The following fats are expliitly stated and proved in [23, 3.16]: Convexity of F is neessary for M F to be onvex. If F is stritly onvex F >, then onavity of F /F is neessary and suffiient for M F to be onvex. For ompleteness, we spell out what happens when F is onvex but fails to be stritly onvex. We should onsider two separate ases: If F vanishes everywhere in I, then F is linear and onvexity of M F is trivial (note that in this ase F /F + is learly onave. If F vanishes at some point but not everywhere in I, then F /F must blow up to + near that point as we assumed that F > and that F is smooth. This implies there is a subinterval J I on whih F > but where F /F fails to be onave, so onvexity of M F must fail. We have therefore established all possible ases of Theorem 5.1. When F (x = e x, the ondition of Theorem 5.1 is evidently satisfied; in this ase, onvexity of M F is simply the statement of Hölder s inequality. On the other hand, when F (x = Φ(x, the ondition for onvexity fails to be satisfied on any interval I. In partiular, while log-onavity ould formally be viewed as a reverse form of Hölder s inequality, there annot exist a Gaussian improvement of Hölder s inequality that is analogous to Ehrhard s improvement of log-onavity. Similarly, in ontrast to the Gaussian improvement of the reverse Brasamp-Lieb inequality of Theorem 4.2, there annot exist an analogous Gaussian improvement of the diret Brasamp-Lieb inequality (of whih Hölder s inequality is a speial ase. Remark 5.2. The proof of Theorem 5.1 shows that unless F is linear, onvexity of the funtional M F requires that F is stritly onvex F > everywhere in I. We will therefore assume the latter without loss of generality in our development of stohasti representations for onvex generalized means. The relevane of the onditions of Theorem 5.1 is far from obvious at first sight. We will presently see that these onditions arise in a very natural manner when we attempt to obtain a stohasti representation for M F. We begin by developing the argument of setion 2.2 in the present setting. Let f : R n I be a Lipshitz funtion and define for (t, x [, 1] R n u(t, x := E[F (f(w 1 W t + x], so that u solves the heat equation. Define v(t, x := F 1 (u(t, x. Note that as F is smooth and F >, the funtion F 1 is smooth by the inverse funtion theorem. Therefore, by elementary properties of the heat equation, v takes values in I, is smooth and has bounded derivatives of all orders on [, 1 ε] R n for every ε >, and v(t, x f(x uniformly in x as t 1. Using u t = F (v v t, u = F (v v + F (v v 2