Bridging Schrödinger and Bass: A Semimartingale Optimal Transport Problem with Diffusion Control

Bridging Sc hr ¨ o dinger and Bass: A Semimartingale Optimal T ransp ort Problem with Diﬀusion Con trol Pierre HENR Y-LABORDERE ∗ Gr ´ egoire LOEPER † Othmane MAZHAR ‡ Huy ˆ en PHAM § Nizar TOUZI ¶ Marc h 31, 2026 Abstract W e study a semimartingale optimal transport problem interpolating b etw een the Schr ¨ odinger bridge and the stretched Bro wnian motion asso ciated with the Bass solution of the Skorokhod em b edding problem. The cost combi nes an entrop y term on the drift with a quadratic p enalization of the diﬀusion co eﬃcien t, leading to a sto c hastic con trol problem ov er drift and volatilit y . W e establish a comple te dualit y th eory for this problem, despite the lac k of co ercivit y in the diﬀusion comp onen t. In particular, we prov e strong dualit y and dual attainment, and derive an equiv alen t reduced dual formulation in terms of a v ariational problem o ver terminal p oten tials. Optimal solutions are characterized by a coupled Schr ¨ odinger–Bass bridge sys- tem, inv olving a backw ard heat p oten tial and a transp ort map giv en by the gradien t of a β -con vex function. This system interpolates b et ween the classical Schr ¨ odinger system and the Bass martingale transp ort. Our results furnish a uniﬁed framew ork encompassing entropic and martingale optimal transp ort, and yield a v ariational foundation for data-driven diﬀusion mo dels. Keyw ords : Opt imal transp ort along Itˆ o pro cesses, Sc hr ¨ odinger bridge, Bass martingale, dual p otentials, Sc hr ¨ odinger-Bridge-Bass system. MSC Classiﬁcation : Primary: 49Q22, 60H30. Secondary: 93E20, 60J60, 60G44. ∗ QubeR T. Email: phl@hotmail.com † BNPP and Monash Univ ersity . Email: gregoire.lo eper@bnpparibas.com ‡ LPSM, Univ ersit´ e P aris Cit´ e. This author w as supp orted b y the BNP-Paribas Chair “F utures of Quan titative Finance” . Email: othmane.xx90@gmail.com § Ecole Polytec hnique, CMAP . This author is supported by the Chair “Financial Risks” , by FiME (Laboratory of Finance and Energy Mark ets), and the EDF–CACIB Chair “Finance and Sustainable Dev elopment” . Email: huy en.pham@p olytec hnique.edu ¶ NYU T ando n Sc ho ol of Engineering. This author is partially supported b y NSF grant #DMS- 2508581. Email: nizar.touzi@n yu.edu 1 Con ten ts 1 In tro duction 2 2 Sc hr ¨ odinger bridge Bass optimal transport 4 2.1 Problem form ulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 A ﬁrst dual problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 HJB equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 Dual p oten tial maps and reduced dual problem . . . . . . . . . . . . . . . 7 2.5 A relaxed dual form ulation . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Main results 8 4 Pro of of the ﬁrst dualit y results 9 5 Reduced dual v eriﬁcation 14 6 Dual attainment 20 7 Primal Attainmen t 29 1 In tro duction Classical Sc hr ¨ odinger bridges and martingale optimal transp ort pro vide tw o fundamen tal constructions of optimal couplings of probabilit y measures via contin uous-time sto c has- tic pro cesses. In the Schr ¨ odinger problem, one ﬁxes a reference Brownian motion and prescribes initial and terminal distributions, then minimises the relativ e entrop y of the la w of the pro cess with resp ect to the reference path measure. This leads to a sto chastic con trol problem with controlled drift and ﬁxed diﬀusion co eﬃcien t, admitting a well- p osed dual formulati on and a c haracterisation in terms of the Sc hr ¨ odinger system, see [9], [4], [10]. In con trast, martingale optimal transp ort imposes a martingale constraint and allo ws for non-trivial con trol of the diﬀusion co eﬃcient. The Bass problem, whose optimizer is the stretc hed Brow nian motion, consists in prescribing marginal la ws of a con tinuous martingale while minimising a deviation from Bro wnian motion; see [5, 3, 1]. This form ulation is closely related to m artingale Benamou–Brenier transport [2] and pla ys a central role in mo del-independent ﬁnance. In contrast with the Schr ¨ odinger problem, it exhibits degeneracies and lac ks a regular dual structure. The aim of this pap er is to introduce and analyse a class of optimal semimartingale transp ort problems which in terp olate b et ween these tw o regimes. Giv en µ 0 , µ T ∈ P 2 ( R d ), w e consider Itˆ o pro cesses X = ( X t ) 0 ≤ t ≤ T with X 0 ∼ µ 0 , X T ∼ µ T , and w e minimise a functional whic h p enalises b oth the drift and the v olatilit y of X , w eighte d b y a parameter β > 0. W e denote the resulting v alue by SBB( µ 0 , µ T ) and refer to this optimisation problem as the Sc hr ¨ odinger–Bridge–Bass (SBB) problem. As β → ∞ , one reco vers the Sc hr ¨ odinger bridge, while as β → 0, the problem reduces to a Bass-t yp e martingale 2 transp ort. The SBB problem therefore provides a uniﬁed sto c hastic cont rol form ulation of entropic and martingale transp ort. The main diﬃculty in the analysis stems from the lac k of co ercivit y in the diﬀusion v ariable. In con trast with the framew ork of con trolled diﬀusions studied in [12], [6], the cost functional do es not p enalise large diﬀusion co eﬃcients in a co erciv e manner, and standard compactness and duality argumen ts do not apply directly . Our ﬁrst main result establishes strong dualit y for the SBB problem. W e prov e that the primal sto chastic control problem admits a minimiser and that its v alue co- incides with that of a dual problem o ver functions v ∈ C 1 , 2 solving a fully nonlinear Hamilton–Jacobi–Bellman equation under a curv ature constrain t D 2 v < β I d . Moreov er, the HJB equation admits an explicit representation in terms of a terminal p oten tial ϕ through a quadratic inf–con volution op erator, leading to an equiv alen t static dual for- m ulation of Donsk er–V aradhan type. W e show that the static dual problem attain its suprem um o ver a suitable relaxed space. Our second main result is a complete c haracterisation of solution to SBB. W e sho w that the optimal SBB bridge is enco ded by a triplet ( h, ν, Y ) solving a coupled system, whic h w e call the Sc hr ¨ odinger–Bass bridge system. The functi ons h and ν solv e bac kw ard and forwar d heat equations, resp ectiv ely , while the map Y is the gradien t of a quadratic inf–con volution transform. The marginal la ws ( µ t ) t of the optimal pro cess ( X t ) t satisfy Y t # µ t = h t ν t , i.e. µ t = X t #( h t ν t ) , 0 ≤ t ≤ T , where X t = Y − 1 t is the inv erse of the homeomorphism Y t , and given b y the gradient of a conv ex function: X t ( y ) = ∇ y  | y | 2 2 + 1 β log h t ( y )  = y + 1 β ∇ y log h t ( y ) , hence in terms of the score of the potential densit y function h t . This system reduces to the classical Schr ¨ odinger system as β → ∞ and to the Bass martingale transp ort system as β → 0. Finally , w e identify the structure of the optimal semimartingale. The optimal drift and diﬀusion co eﬃcien ts admit explicit feedback represen tations in terms of ( h, Y ). Moreo ver, after a c hange of v ariable: Y t = Y t ( X t ), 0 ≤ t ≤ T , the pro cess Y is a Sc hr ¨ odinger bridge diﬀusion, and b ecomes a Brownian motion under an equiv alen t c hange of measure. This yields a stretc hed Bro wnian represen tation of the optimal SBB bridge, extending the Bass construction from martingales to general semimartingales. The pap er is organised as follows. Section 2 introduces the SBB problem and deriv es a ﬁrst dual formulati on. Section 3 states the main results. Sections 4 and 5 establish dualit y and analyse the reduced dual problem. Sections 6 and 7 prov e dual and primal attainmen t and derive the structure of optimal solutions. 1 1 During the ﬁnal stage of the p reparation of this pap er, G. P ammer brought to our atten tion his work in [8] with M. Hasenbic hler and S. Thonhauser, motiv ated by early presen tations of the results rep orted here, in whic h they analyze the same problem through the lens of w eak optimal transp ort. 3 2 Sc hr ¨ odinger bridge Bass optimal transp ort 2.1 Problem form ulation Let T > 0 b e some ﬁnite maturity and let Ω = C ([0 , T ] , R d ) b e the canonical space equipp ed with its canonical ﬁltration F = ( F t ) t and canonical pro cess X , i.e. X t ( ω ) = ω ( t ) for all ω ∈ Ω, t ∈ [0 , T ]. W e denote b y P the set of probabilit y measures P on Ω under which X has the diﬀusion decomp osition X t = X 0 + Z t 0 α P s d s + Z t 0 σ P s d W P s , t ∈ [0 , T ] , P − a.s. for some d -dimensional P − Bro wnian motion W P , and some characte ristics ν P = ( α P , σ P ) that are F -progressively measurable pro cesses v alued in R d , S d + , satisfying R T 0 | α P t | d t + R T 0 | σ P t | 2 d t < ∞ , P -a.s. Giv en tw o distributions µ 0 , µ T ∈ P 2 ( R d ), we in tro duce the subset of transport plans P ( µ 0 , µ T ) =  P ∈ P : P ◦ X − 1 0 = µ 0 , and P ◦ X − 1 T = µ T  . Giv en a parameter β > 0, and denoting by I d the iden tity matrix in R d × d , w e intr o duce the cost function c ( a, b ) := 1 2 | a | 2 + β 2 | b − I d | 2 , a ∈ R d , b ∈ S d + . Our ob jectiv e in this paper is to derive a full ch aracterization of a critical transp ort plan obtained through the following optimal transp ort problem on the Wiener space: SBB( µ 0 , µ T ) := inf P ∈P ( µ 0 ,µ T ) J 0 ( P ) , with J 0 ( P ) := E P h Z T 0 c ( ν P t )d t i , Here the SBB acron ym stands for Shr ¨ odinger Bridge Bass , and is justiﬁed b y the t wo extreme cases β → 0 and β → ∞ : • F ormally , when β goes to inﬁnity , we constrain to those transp ort plans in the subset P 0 ( µ 0 , µ T ) := { P ∈ P ( µ 0 , µ T ) : σ P = I d on [0 , T ] } ; the problem SBB reduces to the classical Schr ¨ odinger bridge problem whic h aims to ﬁnd the closest transport plan P ∈ P 0 ( µ 0 , µ T ) to the Wiener measure in the sense of the relative entrop y (Kullbac k-Leibler) distance, see, e.g. [9], SB( µ 0 , µ T ) := inf P ∈P 0 ( µ 0 ,µ T ) E P h 1 2 Z T 0 | α P t | 2 d t i . • In the other extreme case, b y dividing the criterion J by β , and sending β to zero, we formally constrain the drift co eﬃcien t to b e zero, and then we are lo oking for a Brownian martingale whic h is closest to the Brownian motion according to the quadratic norm, under marginal constrain ts. This martingale transp ort 4 problem was studied in Backhoﬀ-V eraguas, Beiglb ¨ oc k, Huesmann & K ¨ allblad [2], Bac khoﬀ-V eraguas, Sc hacherma y er & Tsc hilderer [3], who named its solution as the Str etche d Br ownian motion and whic h is intimately related to the Bass martingale with degenerate initial la w. The latter w as motiv at ed b y calibration problems in ﬁnancial engineering, and w as ﬁrst studied b y Conze & Henry-Labord` ere [5] and Acciaio, Pammer & Marini [1]. The solution (when it exists) ˆ P of SBB( µ 0 , µ T ) is called the Schr ¨ odinger Bridge Bass (SBB) transp ort plan. W e ﬁrst start with some initial consi derations whic h will underpin the subsequent analysis of the problem SBB. Remark 2.1. The line ar c oupling P π lin ∈ P ( µ 0 , µ T ) is deﬁned for all static coupling π ∈ Π( µ 0 , µ T ) as follows. Let ( Y 0 , Y T ) b e a random v ector on some probabilit y space, with la w π , and let P π lin := La w( Y ) b e the law of the linear in terp olation Y t := T − t T Y 0 + t T Y T , t ∈ [0 , T ]. Back to our notations on the canonical space, w e observe that dX t = α P π lin t d t + 0 d W t , P π lin − a.s. with corresp onding α P π lin t := X T − X 0 T = X t − X 0 t satisfying J 0 ( P π lin ) = 1 2 E R T 0 | α P π lin t | 2 dt = 1 2 T − 1 E | X T − X 0 | 2 < ∞ . Consequen tly P π lin ∈ P ( µ 0 , µ T ). Lemma 2.2. SBB( µ 0 , µ T ) ∈ (0 , ∞ ) , and E P [sup t ≤ T | X t | 2 ] < ∞ for every P ∈ P ( µ 0 , µ T ) with J 0 ( P ) < ∞ . Pr o of. By Remark 2.1, w e hav e SBB( µ 0 , µ T ) ≤ J 0 ( P lin ) < ∞ . Next, for P ∈ P ( µ 0 , µ T ) with J 0 ( P ) < ∞ , it follo ws from the BDG inequality that E P [sup t ≤ T | X t | 2 ] ≤ C (1 + E P [ | X 0 | 2 ] + E P R T 0  | α P t | 2 + | σ P t | 2  d t ) ≤ C ′ (1 + E P [ | X 0 | 2 ] + J 0 ( P )) < ∞ , for some constan ts C, C ′ > 0. ⊔ ⊓ 2.2 A ﬁrst dual problem The problem SBB i s a semimartingale optimal transport problem in the sense of [12], see also [6]. Notice ho wev er that our cost function does not satisfy the coercivity condition in b that is required in [12]. Despite this, we shall justify that the linear programming dualit y still holds in our setting and w e ma y then express the constrained con trol problem SBB in terms of the p enalized unconstrained con trol problem. Let w := 1 + | . | 2 and let C w := { ψ ∈ C 0 ( R d , R ) : | ψ w | ∞ < ∞} b e the collection of all con tinuous functions with quadratic growth on R d . As µ T ∈ P 2 ( R d ), we hav e P ( µ 0 , µ T ) =  P ∈ P ( µ 0 ) : E P [ ψ ( X T )] = µ T ( ψ ) for all ψ ∈ C w  , with P ( µ 0 ) := ∪ ν ∈P 2 ( R d ) P ( µ 0 , ν ). W e may then rewrite our problem as the p enalized con trol problem SBB( µ 0 , µ T ) = inf P ∈P 0 ( µ 0 ) sup ψ ∈C w µ T ( ψ ) + J ψ ( P ) , J ψ ( P ) := E P h − ψ ( X T ) + Z T 0 c ( ν P t )d t i . 5 F ollow ing the standard terminology in optimal transp ort theory , we call the p enal- ization maps ψ p otential functions . By the trivial inequality inf P ∈P 0 ( µ 0 ) sup ψ ∈C w ≥ sup ψ ∈C w inf P ∈P 0 ( µ 0 ) , this pro vides the weak dualit y inequalit y SBB( µ 0 , µ T ) ≥ sup ψ ∈C w µ T ( ψ ) + inf P ∈P 0 ( µ 0 ) J ψ ( P ) . (2.4) W e next rewrite the last minimization on the right hand side as a standard sto chast ic con trol problem by expressing it as the in tegration of the v alue function of a standard sto c hastic control problem with respect to µ 0 . T o do this, we denote for all P ∈ P 0 ( µ 0 ) b y { P x } x ∈ R d a regular conditional la w of P given X 0 , and we write b y the to wer property that J ψ ( P ) = R J ψ ( P x ) µ 0 (d x ) ≥ R inf P ∈P 0 ( δ x ) J ψ ( P x ) µ 0 (d x ). Combining with (2.4), we see that SBB( µ 0 , µ T ) ≥ V ( µ 0 , µ T ) := sup ψ ∈C w µ T ( ψ ) − µ 0 ( V ψ 0 ) , (2.5) where V ψ 0 is the v alue function of a standard stochastic control problem: V ψ 0 ( x ) := sup P ∈P 0 ( δ x ) − J ψ ( P ) = sup P ∈P 0 ( δ x ) E P h ψ ( X T ) − Z T 0 c ( ν P t ) d t i , x ∈ R d . (2.6) The penalized optimization problem V ( µ 0 , µ T ) is our ﬁrst dual problem. Our primary ob jective in this pap er is to prov e that there is no duality gap in (2.5). In addition, w e shall prov e existence for both primal and dual problems, together with a complete c haracterization of the corresp onding solution which exhibits a p erfect interpolation b et we en the w ell-known Shr ¨ odinger bridge system and the Bass-Brenier transport map. 2.3 HJB equation W e denote b y Q T := [0 , T ) × R d the time-space parabolic domain, and Q T := [0 , T ] × R d its closure. By standard optimal control theory , the HJB equation corresponding to the problem V ψ 0 is: ∂ t v + H ( D v , D 2 v ) = 0 on Q T , and v T = ψ on R d , (2.7) where the Hamiltonian H is giv en by H ( p, A ) := sup a ∈ R d ,b ∈ R d × d n a · p + 1 2 bb ⊺ : A − c ( a, b ) o = 1 2 | p | 2 + β 2 I d : [ β ( β I d − A ) − 1 − I d ] on dom( H ) := { ( p, A ) : A < β I d } , and H = ∞ outside its domain dom( H ). F or later use, w e record that the maximizers of the Hamiltonian are giv en b y: b a ( p ) = p, b b ( A ) = β ( β I d − A ) − 1 , ( p, A ) ∈ dom( H ) . 6 Notice that dom( H ) implies formally that D 2 v < β I d , and w e then exp ect that the solution of the HJB equation (2.7) exhibits a b oundary la yer in the sense that lim t ↗ T v t − β 2 | . | 2 is concav e. F or this reason, it turns out that the set of p oten tial maps can b e reduced to the subset of all such β − c onc ave maps: C conc w := C w ∩ C conc where C conc :=  ϕ ∈ C 0 ( R d ) : ϕ is β − concav e  . Similarly , we say that ϕ is β − conv ex if the map ϕ + β 2 | · | 2 is conv ex, and w e denote C conv w := C w ∩ C conv , with C conv :=  ϕ ∈ C 0 ( R d ) : ϕ is β − conv ex  = −C conc . 2.4 Dual p oten tial maps and reduced dual problem Due to the particular choice (2.2) of the cost function, the HJB equation (2.7) turns out to b e amenable to in teresting manipulation leading to more explicit solution structur e. W e shall use extensively the Moreau transform, also called inf-conv olution: T + β [ ϕ ]( x ) := inf y ∈ R d ϕ ( y ) + β 2 | x − y | 2 , x ∈ R d . Notice that ψ := T + β [ ϕ ] is β − conca ve as T + β [ ϕ ] − β 2 | · | 2 is an inﬁmum of aﬃne functions. W e also directly verify that the op erator: T − β [ ψ ] := − T + β [ − ψ ] satisﬁes T − β ◦ T + β [ ϕ ] = ϕ for all β − con vex map ϕ . (2.8) In particular ϕ = T − β [ ψ ] is called for this reason a dual p otential map . Our main c haracterization result of the SBB problem reduces the dual problem V ( µ 0 , µ T ) to the following optimization problem o ver the set of dual p oten tial maps: V red ( µ 0 , µ T ) := sup ϕ ∈C conv w J ( ϕ ) , with J ( ϕ ) := µ T  T + β [ ϕ ]  − µ 0  T + β [ u ϕ T ]  (2.9) e u ϕ s ( y ) := N s ∗ e ϕ ( y ) , ( s, y ) ∈ Q T , where N s := (2 π s ) − d 2 e − | . | 2 2 s is the heat k ernel in R d . The justiﬁcation of this expression for the dual problem is clariﬁed by the subsequen t Lemma 5.1 (3-b) below, which connects T + β [ u ϕ T ] to the HJB equation corresp onding to the control problem (2.6). Remark 2.3. The restriction of the dual functions to be β − con vex in (2.9) can be relaxed. T o see this, observe that ϕ ≥ ˜ ϕ := T − β ◦ T + β [ ϕ ] ∈ C conv w and T + β [ ˜ ϕ ] = T + β [ ϕ ] by (2.8). This implies that J ( ˜ ϕ ) ≥ J ( ϕ ) whic h means that the β − conv exit y restriction on the dual p oten tial do es not aﬀect the dual maximizati on problem. 7 2.5 A relaxed dual form ulation Unfortunately , the form ulation (2.5) of the dual problem V does not guarantee existence of an optimal p oten tial map. F or this reason, we need to in tro duce an appropriate relax- ation so as to allo w for existence while preserving the same v alue. First, from the previous considerations, w e already kno w that it is suﬃcien t to restrict the dual maps to those β − concav e maps. As β − conca ve maps are b ounded from ab o ve b y a quadratic function, it remains to specify the asymptotic b eha vior of ψ − at inﬁnit y . Instead, w e introduce the following relaxed dual space by enforcing an appropriate integrabilit y condition: ¯ C conc := n ψ is ﬁnite-v alued and β − concav e : µ 0  T + β h u T − β [ ψ ] T i < ∞ o , and we introduce our relaxed dual form ulation as ¯ V ( µ 0 , µ T ) := sup ψ ∈ ¯ C conc µ T ( ψ ) − µ 0 ( V ψ 0 ) , (2.10) with the v alue function of the standard sto c hastic con trol problem V ψ 0 deﬁned in (2.6). 3 Main results The main ob jectiv e of this pap er is to establish strong dualit y b etw een the problems SBB of (2.3) and V of (2.5) whic h requires to b e relaxed to (2.10) to ensure existence, and actually can b e expressed under the reduced dual form V red of (2.9). Theorem 3.1. L et µ 0 , µ T ∈ P 2 ( R d ) . Then (i) SBB = V = ¯ V = V red ∈ (0 , ∞ ) at ( µ 0 , µ T ) ; (ii) Under the additional c ondition β T > 1 : (a) The r e duc e d dual pr oblem V red has a solution ˆ ϕ ∈ C conv w , which induc es a solution ˆ ψ := T − β [ ˆ ϕ ] ∈ ¯ C conc of the dual pr oblem ¯ V ; (b) The primal pr oblem SBB has a solution ˆ P ∈ P ( µ 0 , µ T ) ; (c) the map Y t := id − 1 β ∇ T + β [ u ˆ ϕ T − t ] is a wel l-deﬁne d one-to-one map on Q T with natur al extension to ¯ Q T , and Y := Y ( X ) is a ˆ Q − Br ownian motion with d ˆ Q d ˆ P    F t := e − u ˆ ϕ T − t ( Y t ) . (d) Denoting ν t = N t ∗ ν 0 , the dual optimizer ˆ ϕ is char acterize d by the SBB system d Y 0 # µ 0 d ν 0 = e u ˆ ϕ T and d Y T # µ T d ν T = e ˆ ϕ . W e rep ort the pro of of the duality (i) in Section 4, and the remaining part (ii) in Section 6. T o b etter understand the last characteri zation of our solution, recall that the Shr ¨ odinger bridge corresponds to β = ∞ regime where Y t = id for all t ∈ [0 , T ], while 8 the Bass martingale corresponds to the β = 0 regime where ˆ ϕ = 0. W e observe here that the characterization (ii-c) in the last statement is comp osed by a com bination of the Schr ¨ odinger bridge and the Bass martingale: • Similar to the structure of the Shr ¨ odinger bridge, the ﬂo w of probability measures ( ν t ) t ≤ T satisﬁes the forw ard F okker-Planc k equation, while the map ( t, y ) 7→ h t ( y ) := e u ˆ ϕ T − t ( y ) is a solution of the backw ard heat equation; • The map Y is our substitute of the Brenier map in the Bass martingale comp onen t of our c haracterization, and its inv erse X is expressed as X t = id + 1 β ∇ log h t . W e represen t the SBB system in the standard graphical form adopted in the Shr ¨ odinger bridge literature as: µ T − − − − → m T := Y T # µ T − − − − → dm T dν T = e ˆ ϕ − − − − → ν T x   x      Bass Shr ¨ odinger          µ 0 − − − − → m 0 := Y 0 # µ 0 − − − − → dm 0 dν 0 = e u ˆ ϕ T − − − − → ν 0 4 Pro of of the ﬁrst dualit y results In order to pro ve the ﬁrst duality result SBB = V in Theorem 3.1 (i), w e focus on the dep endence of the v alue function SBB on its second argument. W e th us ﬁx µ 0 ∈ P 2 ( R d ) and we analyse in this section the map F : P 2 ( R d ) − → R deﬁned b y: F ( m ) := SBB( µ 0 , m ) , m ∈ P 2 ( R d ) . Lemma 4.1. The map F is c onvex and c ontinuous on P 2 ( R d ) . Pr o of. 1. W e ﬁrst sho w that F is con vex on P 2 ( R d ). Let m 0 , m 1 ∈ P 2 ( R d ) and λ ∈ (0 , 1). Fix η > 0 and choose P i ∈ P 0 ( µ 0 , m i ) with E P i  R T 0 c ( ν P i t )d t  ≤ F 0 ( m i ) + η , i = 0 , 1. Consider the enlarged space ˜ Ω := Ω × { 0 , 1 } with canonical pro cess ( ˜ X , U ) and canonical ﬁltration ˜ F = F ˜ X ∨ F U , F ˜ X = {F ˜ X t = σ ( e X s , s ≤ t ) } t ≤ T and F U = {F U t = σ ( U ) } t ≤ T . Let ˜ P be the probabilit y measure under whic h U is a Bernoulli( λ ) v ariable indep enden t of ˜ X 0 , e X has conditional la w P i giv en { U = i } , i = 0 , 1, and thus its c haracteristics ˜ ν are deﬁned b y ˜ ν i := ( ˜ α i , ˜ σ i ) conditionally on { U = i } . By the Mark ovian pro jection argumen t of Gy ¨ ongi [7], we see that P := ˜ P ◦ e X − 1 ∈ P ( µ 0 , m λ ), m λ := (1 − λ ) m 0 + λm 1 , with c haracteristics ν P t = ( α P t , σ P t ) deﬁned by ( α P t , σ P t ( σ P t ) ⊺ ) = E e P [( e α t , e σ t ( e σ t ) ⊺ ) |F ˜ X t ], and by the con vexit y of the cost function c in ( a, bb ⊺ ), we see that F ( m λ ) ≤ E P Z T 0 c ( ν P t ) dt ≤ λ E P 0 Z T 0 c ( ν P 0 t ) d t + (1 − λ ) E P 1 Z T 0 c ( ν P 1 t ) d t ≤ λF ( m 1 ) + (1 − λ ) F ( m 0 ) + 2 η . 9 The required con vexit y of F now follows from the arbitrariness of η ∈ [0 , 1]. 2. W e next sho w the contin uit y of F . W e shall pro ve in Step 3 below that F ( m ) ≤ (1 + C 0 κ ) F ( m ′ ) + d (1 + β ) κ, whenever κ := W 2 ( m, m ′ ) ≤ T 2 . (4.1) Let m n ∈ P 2 ( R d ) with m n → m in σ ( M 2 , C w ). Then m n ⇀ m and, since | . | 2 ∈ C w , R | . | 2 d m n → R | . | 2 d m , which implies W 2 ( m n , m ) → 0. Then, as κ n := W 2 ( m n , m ) ≤ T 2 for large n , it follo ws from (4.1) that: F ( m ) ≤ lim inf n →∞ (1 + C 0 κ n ) F ( m n ) + d (1 + β ) κ n = lim inf n →∞ F ( m n ) . Applying (4.1) again with m and m n exc hanged, we obtain the rev erse inequalit y . Hence lim n →∞ F ( m n ) = F ( m ) , whic h pro ves the contin uit y of F . 3. It remains to prov e (4.1). Fix η > 0 and pic k P ∈ P 0 ( µ 0 , m ′ ) such that E P Z T 0 c ( ν P t )d t ≤ F ( m ′ ) + η . (4.2) 3-a. Time change on [0 , T − κ ] . Set θ := T T − κ and deﬁne on the same space e X u := X θu for u ∈ [0 , T − κ ]. Let e P b e the law of e X on C ([0 , T − κ ]; R d ). Then e X 0 ∼ µ 0 and e X T − κ = X T ∼ m ′ . A direct c hange-of-v ariables computation in the martingale problem sho ws that e X has c haracteristics e α u = θ α P θu and e σ u = √ θ σ θu , and therefore: E e P Z T − κ 0 c ( e α t , e σ t ) d t = 1 2 θ E P Z T 0  | α P t | 2 + β θ 2 | √ θ ( σ t − I ) + ( √ θ − 1) I d | 2  d t. Note that | √ θ b + ( √ θ − 1) I d | 2 = θ | b | 2 + 2 √ θ ( √ θ − 1) b · I + ( √ θ − 1) 2 | I d | 2 , and b y Y oung’s inequalit y 2 √ θ ( √ θ − 1) b · I ≤ εθ ( √ θ − 1) 2 | b | 2 + 1 ε | I d | 2 = ( θ 2 − θ ) | b | + 1 ε | I d | 2 for ε := θ − 1 ( √ θ − 1) 2 = √ θ +1 √ θ − 1 . Then | √ θ b + ( √ θ − 1) I d | 2 ≤ θ 2 | b | + ( √ θ − 1) 2 + √ θ − 1 √ θ +1 ) | I d | 2 = θ 2 | b | + θ ( √ θ − 1) √ θ +1 d , and w e deduce from (4.2) that E e P Z T − κ 0 c ( e α t , e σ t ) d t ≤ θ E P Z T 0 c ( α P t , σ P t ) d t + T β d ( √ θ − 1) 2( √ θ + 1) ≤ θ ( F ( m ′ ) + η ) + T β d ( √ θ − 1) 2( √ θ + 1) . 3-b. T erminal c orr e ction on [ T − κ, T ] . Let π b e an optimal W 2 -coupling betw een m ′ and m , disin tegrated as π ( dy | x ) m ′ ( dx ). Enlarge the space and sample Y ∼ π ( · | e X T − κ ). Then Law( Y ) = m and E | Y − e X T − κ | 2 = W 2 ( m ′ , m ) 2 = κ 2 . Deﬁne b X t := 1 { t ∈ [0 ,T − κ ] } e X t + 1 { t ∈ [ T − κ,T ] } n e X T − κ + t − ( T − κ ) κ  Y − e X T − κ  o . 10 On [ T − κ, T ], b X has drift b α t = ( Y − e X T − κ ) /κ and diﬀusion b σ t ≡ 0, hence E Z T T − κ c ( b α t , b σ t ) dt = W 2 ( m ′ , m ) 2 2 κ + β d κ = κ 2 + β d κ. (4.4) Let b P be the la w of b X on the canonical space. By a Marko vian pro jection argumen t as in Step 1 of the current pro of, we see that b P ∈ P 0 ( µ 0 , m ) and F ( m ) ≤ E b P Z T 0 L ( α b P t , Γ b P t ) dt ≤ E Z T 0 L ( b α t , b Γ t ) dt ≤ θ ( F ( m ′ ) + η ) + T β d ( √ θ − 1) 2( √ θ + 1) + κ 2 + β d κ, where the last inequalit y follo ws from (4.3) and (4.4). Substitutin g θ := T T − κ induces the required estimate (4.1) b y the arbitrariness of η > 0. ⊔ ⊓ W e are no w ready for the ﬁrst dualit y result in Theorem 3.1 (i). Prop osition 4.2. F or µ 0 , µ T ∈ P 2 ( R d ) , we have SBB( µ 0 , µ T ) = V ( µ 0 , µ T ) . Pr o of. 1. Deﬁne ¯ F : M 2 → R ∪ { + ∞} b y ¯ F ( m ) :=  F ( m ) , if m ∈ P 2 ( R d ) , + ∞ , if m ∈ M 2 \ P 2 ( R d ) . Since F is conv ex on P 2 ( R d ), the map ¯ F is conv ex on M 2 . Moreov er, ¯ F is σ ( M 2 , C w ) − lo wer semicon tinuous on M 2 . Then, ¯ F ( µ T ) = ¯ F ∗∗ ( µ T ) b y the F enc hel-Moreau theorem, where: ¯ F ∗∗ ( m ) = sup ψ ∈C w m ( ψ ) − ¯ F ∗ ( ψ ) and ¯ F ∗ ( ψ ) = sup m ∈M 2 m ( ψ ) − ¯ F ( m ) . Since ¯ F = + ∞ on M 2 \ P 2 ( R d ), we hav e ¯ F ∗ ( ψ ) = sup m ∈P 2 m ( ψ ) − F ( m ). Hence F ( µ T ) = ¯ F ( µ T ) = ¯ F ∗∗ ( µ T ) . 2. W e no w complete the required dual represen tation by identifying the map ¯ F ∗∗ to the v alue V ( µ 0 , µ T ) introduced in (2.5). First, ¯ F ∗ ( ψ ) = sup m ∈M 2 m ( ψ ) − ¯ F ( m ) = sup m ∈P 2 m ( ψ ) − F ( m ) = sup m ∈P 2 m ( ψ ) − inf P ∈P ( µ 0 ,m ) J 0 ( P ) = sup P ∈P ( µ 0 ) − J ψ ( P ) , and ¯ F ∗∗ ( µ T ) = sup ψ ∈C w µ T ( ψ ) − ¯ F ∗ ( ψ ) = sup ψ ∈C w µ T ( ψ ) − sup P ∈P ( µ 0 ) − J ψ ( P ) . 11 Since ¯ F ( µ T ) = F ( µ T ), w e now prov e that ¯ F ∗∗ ( µ T ) = V ( µ 0 , µ T ) as deﬁned in (2.5). This is similar to Lemma 3.5 in [12] whic h was established under their co ercivit y condition on the cost function. 2-a. W e ﬁrst sho w that it suﬃces to pro v e the required result for potential maps ψ with | ψ + | ∞ < ∞ . Indeed, for ψ ∈ C w , deﬁne ψ k := ψ ∧ k , and notice that as ψ k ↑ ψ , c ≥ 0, and ψ − ( X T ) ∈ L 1 ( P ), it follo ws from the monotone con vergence theorem that J ψ k ( P ) ↑ J ψ ( P ). By the standard approximate optimizer, we also obtain the monotone con vergence inf P ∈P ( µ 0 ) J ψ k ( P ) ↑ inf P ∈P ( µ 0 ) J ψ ( P ). 2-b. The inequality inf P ∈P 0 ( µ 0 ) J ψ ( P ) ≥ − µ 0 ( V ψ 0 ) w as already established right b efore in tro ducing the dual proble V ( µ 0 , µ T ) in (2.5). W e next prov e the reverse inequality b y means of a measurable selection argumen t. Since ψ + is b ounded and c ≥ 0, w e hav e J ψ ( P ) ≤ | ψ + | ∞ for all P ∈ P ( µ 0 ), hence V ψ 0 ( x ) < ∞ for all x . F or ﬁxed η > 0, it follows from the measurable selecti on result, see e.g. [12, App endix, Thm. A.1], th at there exists a µ 0 -measurable map x 7→ P x,η ∈ P 0 ,x suc h that J ψ ( P x,η ) ≤ − V ψ 0 ( x ) + η for µ 0 -a.e. x. Deﬁne P η := R R d P x,η ( · ) µ 0 ( dx ) on Ω. As P η [ · | X 0 = x ] = P x,η , µ 0 -a.s., we ha ve P η ∈ P ( µ 0 ), and it follows from F ubini’s theorem that J ψ ( P η ) = Z R d J ψ ( P x,η ) µ 0 ( dx ) ≤ − Z R d V ψ 0 ( x ) µ 0 ( dx ) + η . This sho ws that P η is an η − optimizer, and we deduce the required result b y the arbi- trariness of η > 0. ⊔ ⊓ In preparati on for the pro of of the reduced dual formulation in the next subsection, w e no w sho w that w e ma y restrict the dual maximization to upper bounded β –concav e p oten tials ψ : C conc w, ↑ := C conc ∩ C w, ↑ with C w, ↑ := { ψ ∈ C w : | ψ + | ∞ < ∞} . Lemma 4.3. We have V ( µ 0 , µ T ) = V ( µ 0 , µ T ) = V ( µ 0 , µ T ) := sup ψ ∈C c onc w, ↑ { µ T ( ψ ) − µ 0 ( V ψ 0 ) } . Pr o of. (a) W e ﬁrst restrict the dual maximization to C w, ↑ . F or arbitrary ψ ∈ C w , w e ha ve ψ n := ψ ∧ n ∈ C w, ↑ , n ≥ 1. As ψ n ↑ ψ and ψ n ≥ ψ 1 ∈ L 1 ( µ T ), it follo ws by monotone con vergence that µ T ( ψ n ) ↑ µ T ( ψ ) . On the other hand, monotonicit y in the terminal pay oﬀ giv es V ψ n 0 ↑ V ψ 0 p oin t wise. As A := | ψ w | ∞ < ∞ , and the Wiener measure Q 0 ,x ∈ P 0 ( δ x ) started from δ x has charac- teristics ν Q 0 ,x = (0 , I d ) inducing the cost c ( ν Q 0 ,x ) = 0, we hav e V ψ n 0 ( x ) ≥ E Q 0 ,x [ ψ n ( X T )] ≥ − A (1 + E Q 0 ,x | X T | 2 ) = − A (1 + | x | 2 + dT ) , for all n ≥ 1 . Then V ψ n 0 ≥ V ψ 1 0 ∈ L 1 ( µ 0 ), and w e get by monotone conv ergence µ 0 ( V ψ n 0 ) ↑ µ 0 ( V ψ 0 ) . Hence µ T ( ψ n ) − µ 0 ( V ψ n 0 ) − → µ T ( ψ ) − µ 0 ( V ψ 0 ), and w e deduce from the arbitrariness of ψ ∈ C w that V ( µ 0 , µ T ) = sup C w, ↑ { µ T ( ψ ) − µ 0 ( V ψ 0 ) } . 12 (b) W e next show that we may restrict further the dual maximization to β –conca v e p oten tials. Fix now ψ ∈ C w, ↑ and deﬁne its β –conca ve en velope ˆ ψ := T + β ◦ T − β [ ψ ] ∈ C conc w , so that ψ ≤ ˆ ψ ≤ | ψ + | ∞ . By monotonicity in the terminal pay oﬀ, V ψ t ≤ V ˆ ψ t for all t < T . On the other hand, notice that V ψ is l.s.c. as the suprem um of con tinuous maps, and it follows from the easy part of dynamic programming principle (DPP) that V ψ is a l.s.c. viscosit y supersolution of the HJB equation (2.7) on [0 , T ) × R d . Since the Hamiltonian H is inﬁnite outside its domain dom( H ) := { ( p, A ) : A < β I d } , w e deduce that V ψ t is β –concav e for every t < T . As V ψ t ≥ ψ (tak e ν ≡ 0), the minimalit y of ˆ ψ yields V ψ t ≥ ˆ ψ for all t < T . Applying the easy part of the DPP at time b et ween times t and T ′ < T pro vides V ψ t ( x ) ≥ E P h ˆ ψ ( X T ′ ) − Z T ′ t c ( ν P r ) dr i − → T ′ ↗ T E P h ˆ ψ ( X T ) − Z T t c ( ν P r ) dr i for all P ∈ P t ( δ x ) , b y dominated and monotone conv ergence (note ˆ ψ is b ounded ab ov e and has quadratic gro wth). By arbitrariness of P , this shows that V ψ t ≥ V ˆ ψ t , and thus V ψ t = V ˆ ψ t on [0 , T ) × R d . In particular V ψ 0 = V ˆ ψ 0 , and µ T ( ψ ) − µ 0 ( V ψ 0 ) ≤ µ T ( ˆ ψ ) − µ 0 ( V ˆ ψ 0 ) , inducing the required result. (c) Fix ψ ∈ C conc w and ( t, x ) ∈ [0 , T ) × R d . Fix any y ∈ R d and consider on [ t, T ] the deterministic admissible characteristics α s ≡ y − x T − t , Γ s ≡ 0. Then X T = y a.s. and R T t c ( α s , Γ s ) ds = | y − x | 2 2( T − t ) + β d 2 ( T − t ), since | I d | 2 = tr( I d ) = d . Hence V ψ t ( x ) ≥ ψ ( y ) − | y − x | 2 2( T − t ) − β d 2 ( T − t ) ≥ − C t,ψ (1 + | x | 2 ) . (4.5) F or n ≥ 1 set ψ n := ψ ∧ n . Then ψ n ↑ ψ and ψ n ( y ) ≥ ψ 1 ( y ), so (4.5) yields the uniform-in- n b ound V ψ n t ( x ) ≥ ψ 1 ( y ) − | y − x | 2 2( T − t ) − β d 2 ( T − t ) ≥ − C t (1 + | x | 2 ) , n ≥ 1 , with C t < ∞ dep ending on t, β , d, y , ψ 1 ( y ) but not on n . (d) Fix ψ ∈ C conc w and set ψ n := ψ ∧ n . (d1) µ T ( ψ n ) ↑ µ T ( ψ ) in the extende d sense. Since ψ n = ψ ∧ n , w e hav e ( ψ n ) − = ψ − and ( ψ n ) + ↑ ψ + . Because ψ ( x ) ≤ C (1 + | x | 2 ) and µ T ∈ P 2 ( R d ), w e hav e ψ + ∈ L 1 ( µ T ). If R R d ψ − dµ T = ∞ , then R R d ψ n dµ T = R R d ( ψ n ) + dµ T − R R d ψ − dµ T = −∞ for all n, so µ T ( ψ n ) ↑ µ T ( ψ ) = −∞ . If instead R R d ψ − dµ T < ∞ , then ψ − ∈ L 1 ( µ T ), and w e obtain µ T ( ψ n ) ↑ µ T ( ψ ) by monotone con v ergence. 13 (d2) We ak duality on the enlar ge d class. Fix P ∈ P 0 ( µ 0 , µ T ). If E P R T 0 L ( α P s , Γ P s ) ds = + ∞ , then there is nothing to prov e. W e may th us assume that E P R T 0 L ( α P s , Γ P s ) ds < ∞ . Let { P x } x ∈ R d b e a regular conditional la w of P given X 0 . Then P x ∈ P 0 ( δ x ) for µ 0 –a.e. x , and b y deﬁnition V ψ n 0 ( x ) ≥ E P x h ψ n ( X T ) − R T 0 L ( α P s , Γ P s ) ds i . Int egrating with resp ect to µ 0 and using disintegration, µ T ( ψ n ) − µ 0 ( V ψ n 0 ) ≤ E P R T 0 L ( α P s , Γ P s ) ds. T aking the inﬁm um o ver P ∈ P 0 ( µ 0 , µ T ) yields µ T ( ψ n ) − µ 0 ( V ψ n 0 ) ≤ V ( µ 0 , µ T ) , n ≥ 1 . (4.6) (d3) Pass n → ∞ . Since ψ n ↑ ψ , monotonicit y in the terminal pay oﬀ giv es V ψ n 0 ↑ V ψ 0 p oin t wise, hence V ψ n 0 ≤ V ψ 0 . Moreo ver, by Step (c) with t = 0, there exists C < ∞ suc h that V ψ n 0 ( x ) ≥ − C (1 + | x | 2 ) n ≥ 1 , so that µ 0 ( V ψ n 0 ) and µ 0 ( V ψ 0 ) are w ell-deﬁned in ( −∞ , + ∞ ] and µ 0 ( V ψ n 0 ) ≤ µ 0 ( V ψ 0 ). Thus from (4.6), µ T ( ψ n ) − µ 0 ( V ψ 0 ) ≤ µ T ( ψ n ) − µ 0 ( V ψ n 0 ) ≤ V ( µ 0 , µ T ) . Letting n → ∞ and using µ T ( ψ n ) ↑ µ T ( ψ ) from (d1), we obtain µ T ( ψ ) − µ 0 ( V ψ 0 ) ≤ V ( µ 0 , µ T ) , ψ ∈ C conc . (4.7) Since C conc w, ↑ ⊂ C conc , part (b) yields V ( µ 0 , µ T ) = sup ψ ∈C conc w, ↑ { µ T ( ψ ) − µ 0 ( V ψ 0 ) } ≤ sup ψ ∈C conc { µ T ( ψ ) − µ 0 ( V ψ 0 ) } . Com bining with (4.7), w e conclude V ( µ 0 , µ T ) = sup ψ ∈C conc n µ T ( ψ ) − µ 0 ( V ψ 0 ) o . ⊔ ⊓ 5 Reduced dual v eriﬁcation W e now complemen t the ﬁrst duality result SBB = V , as established in the last section, b y justifying equality with the reduced dual v alue V red as claimed in Theorem 3.1 (i). W e start with the analysis of the regularity of the maps u ϕ . Lemma 5.1. Fix T , β > 0 and ψ ∈ C c onc . Then φ := T − β [ ψ ] , and: (1) h φ t := N T − t ∗ e φ > 0 for al l t ≤ T , and h φ ∈ C 1 , 3 ( Q T ) satisﬁes ∂ t h φ + 1 2 ∆ h φ = 0 on Q T . (2) ˜ u φ := log h φ ∈ C 1 , 3 ( Q T ) , with D 2 ˜ u φ t + κ ( t ) I ⪰ 0 , κ ( t ) := β 1+ β ( T − t ) , t < T , and ∂ t ˜ u φ + 1 2  ∆ ˜ u φ + |∇ ˜ u φ | 2  = 0 on Q T . 14 (3) v φ := T + β [ ˜ u φ ] ∈ C 1 , 2 ( Q T ) , and the minimum in T + β [ ˜ u φ t ]( x ) is uniquely attaine d at Y t ( x ) , for al l ( t, x ) ∈ Q T . Mor e over, (3-a) ∇ v φ = β (id − Y ) , ∂ t v φ = ∂ t ˜ u φ ( Y ) , and D 2 v φ − β I = − β 2  β I + D 2 ˜ u φ ( Y )  − 1 . In p articular, − 1 T − t I ⪯ D 2 v φ t ≺ β I , for al l t < T . (3-b) v φ is a solution of the HJB e quation (2.7) . (3-c) If, in addition, ψ ∈ C c onc w, ↑ , then v φ t → ψ lo c al ly uniformly as t ↑ T , and v φ has quadr atic gr owth uniformly in t , i.e. ther e exists a c onstant C < ∞ such that | v φ t ( x ) | ≤ C (1 + | x | 2 ) , ( t, x ) ∈ Q T . Pr o of. By deﬁnition of C conc , we ha ve µ 0 ( v ψ 0 ) < ∞ . Then v ψ 0 ( x ∗ ) < ∞ for some x ∗ ∈ R d , and b y deﬁnition of T + β , we deduce that ˜ u φ 0 ( y 0 ) < ∞ for some y 0 ∈ R d . Equiv alen tly , ( N T ∗ e φ )( y 0 ) < ∞ . W e con tinue the proof in several steps. 1. W e ﬁrst sho w that h φ and ˜ u φ are ﬁnite on Q T . Fix 0 < t < T and x ∈ R d . Then N T − t ( x − z ) N T ( y 0 − z ) =  T T − t  d 2 e q x ( z ) with q x ( z ) := − | x − z | 2 2( T − t ) + | y 0 − z | 2 2 T . Notice that q x ( z ) = −  1 2( T − t ) − 1 2 T  | z | 2 +  x T − t − y 0 T  · z − | x | 2 2( T − t ) + | y 0 | 2 2 T is a conca ve quadratic polynomial in z since t < T . Hence it is b ounded ab o ve on R d , so there exists C t,x < ∞ suc h that N T − t ( x − z ) ≤ C t,x N T ( y 0 − z ) , z ∈ R d . Multiplying b y e φ ( z ) and in tegrating, w e obtain 0 < N T − t ∗ e φ ( x ) ≤ C t,x ( N T ∗ e φ )( y 0 ) < ∞ for all ( t, x ) ∈ Q T . (5.1) 2. Fix a compact strip K = [ t 0 , τ ] × B R ⊂ (0 , T ) × R d , 0 < t 0 < τ < T . Cho ose s ∗ ∈ ( T − t 0 , T ). By (5.1), ( N s ∗ ∗ e φ )( y 0 ) < ∞ . F or every m ulti-index α and every m ∈ { 0 , 1 } with | α | + 2 m ≤ 3, w e hav e ∂ m s D α x N s ( x − z ) = P m,α,s ( x − z ) N s ( x − z ), where P m,α,s is a p olynomial of degree at most 3, and its co eﬃcien ts are b ounded for s ∈ [ T − τ , T − t 0 ]. Since T − t ∈ [ T − τ , T − t 0 ] on K , the same Gaussian comparison as in the previous Step 1 giv es sup ( t,x ) ∈ K   ∂ m s D α x N T − t ( x − z )   ≤ C K,m,α N s ∗ ( y 0 − z ). Because R R d N s ∗ ( y 0 − z ) e φ ( z ) dz < ∞ , diﬀerentiat ion under the integral sign is justiﬁed on K . Therefore h φ ∈ C 1 , 3 ( Q T ), ∂ t h φ + 1 2 ∆ h φ = 0. Since h φ > 0, also ˜ u φ = log h φ ∈ C 1 , 3 ( Q T ), ∂ t ˜ u φ + 1 2  ∆ ˜ u φ + |∇ ˜ u φ | 2  = 0. 3. Fix t ∈ (0 , T ) and set s := T − t, κ ( t ) := β 1+ β s . Let G ( y ) := φ ( y ) + β 2 | y | 2 . Since φ is β –conv ex, G is conv ex. W e compute ˜ u φ t ( x ) = log R R d (2 π s ) − d 2 e φ ( y ) − | x − y | 2 2 s d y . A direct completion of the square giv es − β 2 | y | 2 − | x − y | 2 2 s + κ ( t ) 2 | x | 2 = − 1+ β s 2 s   y − x 1+ β s   2 . Therefore ˜ u φ t ( x ) + κ ( t ) 2 | x | 2 = − d 2 log(2 π s ) + log Z R d e G ( y ) − 1+ β s 2 s | y − x 1+ β s | 2 d y . = − d 2 log(2 π s ) + log Z R d e G ( z + x 1+ β s ) − 1+ β s 2 s | z | 2 d z , 15 b y the c hange of v ariables z = y − x 1+ β s . F or eac h ﬁxed z , the map x 7− → G  z + x 1+ β s  − 1+ β s 2 s | z | 2 is con vex. By H ¨ older’s inequalit y , the logarit hm of the in tegral of its exponential is conv ex. Hence x 7− → ˜ u φ ( t, x ) + κ ( t ) 2 | x | 2 is conv ex, and therefore D 2 ˜ u φ t + κ ( t ) I ⪰ 0 . (5.2) 4. F or ( t, x, y ) ∈ (0 , T ) × R d × R d , deﬁne F ( t, x, y ) := ˜ u φ t ( y ) + β 2 | x − y | 2 . By (5.2), D 2 y F ( t, x, y ) = D 2 ˜ u φ t ( y ) + β I ⪰ ( β − κ ( t )) I . Since β − κ ( t ) = β 2 ( T − t ) 1+ β ( T − t ) > 0, the map y 7→ F ( t, x, y ) is strongly conv ex for each ﬁxed ( t, x ). T o see it attains its minimum, ﬁx ( t, x ). By T aylor ’s form ula at y = 0, F ( t, x, y ) ≥ F ( t, x, 0) + ∇ y F ( t, x, 0) · y + β − κ ( t ) 2 | y | 2 . Hence F ( t, x, y ) ≥ β − κ ( t ) 2 | y | 2 − C t,x | y | − C t,x , which tends to + ∞ as | y | → ∞ . Th us F ( t, x, · ) is co erciv e, so it has a unique minimizer Y t ( x ), and v φ t ( x ) := F  t, x, Y t ( x )  . W e next prov e that Y is contin uous on Q T . Let ( t n , x n ) → ( t, x ) and set y n := Y ( t n , x n ). F or all large n , w e ha ve t n ∈ [ t/ 2 , ( t + T ) / 2]. On this compact time in terv al, c ∗ := inf r ∈ [ t/ 2 , ( t + T ) / 2] ( β − κ ( r )) > 0. Using strong conv exit y at y = 0, F ( t n , x n , y ) ≥ F ( t n , x n , 0) + ∇ y F ( t n , x n , 0) · y + c ∗ 2 | y | 2 . As ( t n , x n ) sta ys in a compact set, b oth F ( t n , x n , 0) and ∇ y F ( t n , x n , 0) are b ounded uniformly in n , so F ( t n , x n , y ) ≥ c ∗ 2 | y | 2 − C | y | − C . Because y n minimizes F ( t n , x n , · ), F ( t n , x n , y n ) ≤ F ( t n , x n , 0), and the right-hand side is uniformly b ounded. Hence ( y n ) is b ounded. Passing to a conv ergent subsequence if needed, y n → y ∗ , and con tinuit y of F gives F ( t, x, y ∗ ) = lim n F ( t n , x n , y n ) ≤ lim n F ( t n , x n , y ) = F ( t, x, y ), for all y ∈ R d . Th us y ∗ minimizes F ( t, x, · ), and b y uniqueness y ∗ = Y t ( x ). Hence Y is con tinuous. F or ﬁxed t , the minimizer satisﬁes the ﬁrst-order condition ∇ y F ( t, x, Y ( t, x )) = G t ( x, Y t ( x )) = 0 , with G t ( x, y ) := ∇ ˜ u φ t ( y ) − β ( x − y ) . (5.3) Notice that G t is C 1 with D y G t ( x, y ) = D 2 ˜ u φ t ( y ) + β I d , whic h is inv ertible b y (5.2). Then Y t is C 1 b y the implicit functions theorem. Diﬀeren tiating (5.3) in x , w e obtain ∇ Y t ( x ) = β  D 2 ˜ u φ t ( Y t ( x )) + β I d  − 1 . (5.4) W e no w derive the env elop e identities. Since v φ t ( x ) = ˜ u φ t ( Y t ( x )) + β 2 | x − Y t ( x ) | 2 , the c hain rule and (5.3)-(5.4) pro vide ∇ v φ t ( x ) = β ( x − Y ( t, x )) = ∇ ˜ u φ t ( Y t ( x )) (5.5) D 2 v φ t ( x ) = β I d − β 2  D 2 ˜ u φ t ( Y ( t, x )) + β I d  − 1 , (5.6) and D 2 v φ inherits the con tinui ty of Y . 16 It remains to compute ∂ t v φ . W e do not diﬀeren tiate Y in time. Fix ( t, x ) and let h > 0. Since Y t ( x ) is admissible in the minimization deﬁning v φ t + h ( x ), v φ t + h ( x ) ≤ F  t + h, x, Y t ( x )  , hence v φ t + h ( x ) − v φ t ( x ) h ≤ F ( t + h, x, Y t ( x )) − F ( t, x, Y t ( x )) h = ˜ u φ t + h ( Y t ( x )) − ˜ u φ t ( Y t ( x )) h , and then lim sup h ↓ 0 v φ t + h ( x ) − v φ t ( x ) h ≤ ∂ t ˜ u φ t ( Y t ( x )). F or the reverse inequality , since Y t + h ( x ) is admissible in the minimization deﬁning v φ t ( x ), v φ t ( x ) ≤ F  t, x, Y t + h ( x )  , so v φ t + h ( x ) − v φ t ( x ) h ≥ F ( t + h, x, Y t + h ( x )) − F ( t, x, Y t + h ( x )) h = ˜ u φ t + h ( Y t + h ( x )) − ˜ u φ t ( Y t + h ( x )) h . Because Y is con tin uous and ∂ t ˜ u φ is con tin uous, Y t + h ( x ) → Y t ( x ) and ∂ t ˜ u φ t ( Y t + h ( x )) → ∂ t ˜ u φ t ( Y t ( x )). Therefore lim inf h ↓ 0 v φ t + h ( x ) − v φ t ( x ) h ≥ ∂ t ˜ u φ t ( Y t ( x )). Hence ∂ t v φ t ( x ) = ∂ t ˜ u φ t ( Y t ( x )) . (5.7) Since Y and ∂ t ˜ u φ are contin uous, ∂ t v φ is contin uous. Th us v φ ∈ C 1 , 2 ( Q T ). 5. Let λ b e an eigen v alue of D 2 ˜ u φ t ( Y t ( x )), and let γ b e the corresp onding eigen v alue of D 2 v φ t ( x ). By (5.6), γ = β − β 2 β + λ = β λ β + λ . Since (5.2) gives λ ≥ − κ ( t ), and λ 7→ β λ/ ( β + λ ) is increasing on ( − β , ∞ ), − β κ ( t ) β − κ ( t ) ≤ γ < β . Using β κ ( t ) β − κ ( t ) = 1 T − t , w e obtain − 1 T − t I ⪯ D 2 v φ ( t, · ) ≺ β I d . Recall that the Hamiltonian for A < β I d , H ( p, A ) = 1 2 | p | 2 + β 2 (tr( I − 1 β A ) − 1 − d ). With p = ∇ v φ t ( x ) and A = D 2 v φ t ( x ), and using (5.6),  I − 1 β D 2 v φ t ( x )  − 1 = I + 1 β D 2 ˜ u φ t ( Y t ( x )) . Therefore H ( ∇ v φ , D 2 v φ )( t, x ) = 1 2 |∇ ˜ u φ t ( Y t ( x )) | 2 + 1 2 ∆ ˜ u φ t ( Y t ( x )). Using (5.5), (5.7), and the Cole–Hopf equation for ˜ u φ , we obtain ∂ t v φ + H ( ∇ v φ , D 2 v φ ) = 0 on Q T . 6. Assume no w that ψ ∈ C conc w, ↑ . First notice that − C w ≤ ψ ≤ C for some constant C implies that C ≥ φ ( y ) ≥ − C + sup x − C | x | 2 − β 2 | x − y | 2 ≥ C ′ w ( y ), for some constant C ′ . As φ is bounded from abov e, we deduce that h φ t → e φ lo cally uniformly as t ↑ T . The lo cal uniform conv ergence ˜ u φ ( t, · ) → φ is inherited from that of h φ t to wards e φ . By the stabilit y of the Moreau en velop under lo cal uniform conv ergence, we see that v φ t = T + β [ ˜ u φ t ] → T + β [ φ ] = ψ lo cally uniformly . W e next justify the claimed growth for v φ . First, v φ t ≤ ˜ u φ t ≤ sup φ . Moreov er, by the β − con vexit y of φ , for eac h y there exists ξ ∈ ∂ ( φ + β 2 | · | 2 )( y ) such that φ ( z ) ≥ φ ( y ) + ( ξ − β y ) · ( z − y ) − β 2 | z − y | 2 , for all z ∈ R d . 17 Using this inequalit y with s = T − t and z = y + W s , W s ∼ N (0 , sI ), we get h φ t ( y ) = E  e φ ( y + W s )  ≥ e φ ( y ) E  e ( ξ − β y ) · W s − β 2 | W s | 2  = e φ ( y ) (1 + β s ) − d/ 2 e s 2(1+ β s ) | ξ − β y | 2 ≥ e φ ( y ) (1 + β T ) − d/ 2 . Then ˜ u φ t ( y ) = log h φ t ( y ) ≥ φ ( y ) − d 2 log(1 + β T ) =: φ ( y ) − C T , and v φ t ( x ) = inf y n ˜ u φ t ( y ) + β 2 | x − y | 2 o ≥ inf y n φ ( y ) + β 2 | x − y | 2 o − C T = ψ ( x ) − C T . Since ψ ∈ C conc w, ↑ , this pro ves the claimed quadratic growth of v φ . ⊔ ⊓ W e no w ha ve all ingredients to iden tify the maps V ψ and v φ b y means of a v eriﬁcation argumen t. Lemma 5.2. L et ψ ∈ C conc w, ↑ so that φ := T − β [ ψ ] ∈ C conv w, ↑ . Then V ψ = v φ on Q T . Pr o of. In the rest of this pro of, we denote t m := T − 1 m , m ∈ N . 1. Let P ∈ P t,x b e suc h that E P R T t c ( ν P s ) ds < ∞ , and recall from Lemma 2.2 that E P [sup r ∈ [ t,T ] | X r | 2 ] < ∞ . By the C 1 , 2 regularit y of v := v φ and its quadratic gro wth established in Lemma 5.1 (3), and since the stochastic integral in Itˆ o’s form ula is a true martingale, we obtain E P [ v t m ( X t m )] − v t ( x ) = E P h Z t m t  ∂ s v s + α P s ·∇ v s + 1 2 σ P s ( σ P s ) ⊺ : D 2 v s  ( X s )d s i ≤ E P h Z t m t c ( ν P s )d s i , where the last inequality follo ws from the fact that v is a sup ersolution of the HJB equation (2.7). Since X t m → X T P -a.s., v t m → ψ lo cally uniformly , as m ↗ ∞ , and | v t m ( X t m ) | ≤ C (1 + sup r ∈ [ t,T ] | X r | 2 ) ∈ L 1 , we see that v t m ( X t m ) → ψ ( X T ) in L 1 ( P ) b y dominated con vergence. Then, as the cost function c ≥ 0, w e may no w deduce b y monotone conv ergence that v t ( x ) ≥ E P h ψ ( X T ) − Z T t c ( α P s , σ P s ) ds i , and by arbitrariness of P ∈ P t,x , we get v t ( x ) ≥ V ψ t ( x ). 2. W e next pro ve the rev erse inequalit y v t ( x ) ≤ V ψ t ( x ). By Lemma 5.1 (3), the minimizer Y t ( x ) in v t ( x ) = T + β [ ˜ u t ]( x ) is unique. F or an arbitrary y ∈ R d , let Y be Q − Brownian motion on [ t, t m ] with Y t = y , and notice that the upper b oundedness of φ implies that Z s := h φ ( s,Y s ) h φ ( t,y ) is a p ositiv e b ounded martingale inducing an equiv alen t probability measure Q m on F T b y d Q m := Z t m d Q . By Girsano v’s theorem the pro cess W s := Y s − R s ∧ t m t ∇ log h φ ( r , Y r ) dr deﬁnes a Q m − Bro wnian motion on [0 , T ], and w e ma y then write the dynamics of Y as d Y s = ∇ log h φ ( s, Y s ) 1 s ∈ [ t,t m ] ds + dW s , Y t = y . 18 Denote X s := X s ( Y s ) 1 s ∈ [ t,t m ] + ( X t m + R s t m dW s ) 1 s ∈ ( t m ,T ] , where we recall that X s := id + 1 β ∇ ˜ u s . By Lemma 5.1 (3-a), it follows that ∇ v s ( X s ) = β ( X s − Y s ) = ∇ ˜ u s ( Y s ) = ∇ log h φ ( s, Y s ) , s ∈ [ t, t m ] . (5.8) Set b α s := ∇ v s , b σ s :=  I d − D 2 v s β  − 1 . Since Y s := id − 1 β ∇ v s is the inv erse of X s , diﬀeren tiating the identit y Y s ( X s ( y )) = y yields D X s ( y ) =  I d − D 2 v s ( X s ( y )) β  − 1 , and therefore D X s ( Y s ) = b σ s ( X s ) , s ∈ [ t, t m ]. By Lemma 5.1 (1)–(2) , w e ha ve ˜ u φ ∈ C 1 , 3 ( Q T ), and therefore ( s, y ) 7− → X s ( y ) := y + 1 β ∇ ˜ u φ s ( y ) is of class C 1 , 2 on Q T . Applying Itˆ o to X s = X ( s, Y s ) on [ t, t m ] using (5.8), we obtain dX s =  ∂ s X s + D X s ∇ ˜ u s + 1 2 ∆ X s  ( Y s ) ds + D X s ( Y s ) dW s =  ∇ ˜ u s + 1 β  ∇ ∂ s ˜ u s + D 2 ˜ u s ∇ ˜ u s + 1 2 ∇ ∆ ˜ u s  ( Y s ) ds + b σ s ( X s ) dW s . Since h φ solv es the backw ard heat equation, diﬀerentiating the PDE of ˜ u = log h φ giv es ∇ ∂ s ˜ u s + D 2 ˜ u s ∇ ˜ u s + 1 2 ∇ ∆ ˜ u s = 0. Hence, using also (5.8), w e get X t = x := X t ( y ) , dX s = b α s ( X s ) ds + b σ s ( X s ) dW s , s ∈ [ t, t m ] , and X s = X t m + W s − W t m for s ∈ [ t m , T ] . Let P t,x,m b e the law of X under Q m . Then P m t,x ∈ P t,x with characteristics ν P m t,x s = ( b α, b σ )( s, X s ) 1 s ∈ [ t,t m ] + (0 , I d ) 1 s ∈ [ t m ,T ] . As c ( ν P m t,x ) = 0 on [ t m , T ], it follows from the HJB equation (2.7) satisﬁed b y v that v t ( x ) = E P m t,x h v t m ( X t m ) − Z t m t c ( ν P m t,x s ) ds i = E P m t,x h ψ ( X T ) − Z T t c ( ν P m t,x s ) ds i ≤ V ψ t ( x )+ ε m , where ε m := E P m t,x [ v t m ( X t m ) − ψ ( X t m + δ m Z )] with δ m := √ T − t m and Z a stan- dard Gaussian r.v indep enden t of F t m . Deﬁne ψ m ( x ) := E [ ψ ( x + δ m Z )]. Then ε m = E P m t,x  v t m ( X t m ) − ψ m ( X t m )  . Since ( v t m , ψ m ) → ( ψ , ψ ) lo cally uniformly as m ↗ ∞ and b oth v t m and | ψ m | ha v e quadratic growth uniformly in m , we see that lim m →∞ ε m = 0, inducing the required inequalit y v t ( x ) ≤ V ψ t ( x ). ⊔ ⊓ Pro of of Theorem 3.1 (i) Observe that { φ = T − β [ ψ ] for some ψ ∈ C conc w, ↑ } = C conv w, ↑ . Then, com bining the ﬁrst duality result of Proposition 4.2 with Lemma 4.3 and the last v eriﬁcation argumen t in Lemma 5.2, w e see that SBB( µ 0 , µ T ) = V ( µ 0 , µ T ) = sup φ ∈C conv w, ↑ µ T ( T + β [ φ ]) − µ 0 ( v φ 0 ) . (5.9) It remains to pro ve that the v alue of the supremum on the righ t-hand side is not aﬀected b y enlarging the set of dual p oten tial maps from C conv w, ↑ to C conv w . T o see this, we int ro duce 19 for all φ ∈ C conv w , the sequence φ n := T − β [ ψ n ] with ψ n := ψ ∧ n and ψ := T + β [ φ ]. Then ψ n ∈ C conv w, ↑ so that T + β [ φ n ] = T + β ◦ T − β [ ψ n ] = ψ n , and the corresp onding φ n ∈ C conv w, ↑ satisﬁes φ n ≤ ϕ , implying that v φ n 0 ≤ v φ 0 . Therefore J ( φ n ) = µ T ( ψ n ) − µ 0 ( v φ n 0 ) ≥ µ T ( ψ n ) − µ 0 ( v φ 0 ) . This implies by monotone con vergence that lim inf n →∞ J ( φ n ) ≥ lim inf n →∞ µ T ( ψ n ) − µ 0 ( v φ 0 ) = µ T ( ψ ) − µ 0 ( v φ 0 ) = J ( φ ) . ⊔ ⊓ 6 Dual attainment Our ob jectiv e in this section is to prov e Theorem 3.1 (ii-a), namely the existence of p oten tial ˆ ψ attaining the dual v alue V ( µ 0 , µ T ), and a dual p otential ˆ ϕ attaining the reduced dual v alue V red ( µ 0 , µ T ). W e shall denote throughout: U ϕ η := u ϕ η T (0) = log E [ e ϕ ( W ηT ) ] , for some ﬁxed η ∈ (0 , 1) . Lemma 6.1. F or every ϕ ∈ C conv w, ↑ with ϕ (0) = 0 , we have: (i) Ther e is a c onstant ν η (dep ending on β , T , d, η ) such that − ν η ≤ U ϕ η ≤ − d 2 log( η ) + inf x ∈ R d u ϕ T ( x ) + | x | 2 2(1 − η ) T . (ii) Ther e exists a c onstant ν ′ η (dep ending on β , T , d, η ), such that for al l x ∈ R d : u ϕ s ( x ) ≥ − β | x | 2 2 + 1 { s> 0 }  β 2 | x | 2 4 s − ν ′ η + U ϕ η β 2 s  − 1 { s =0 } ( ν ′ η + U ϕ η ) | x | , s ∈ [0 , T ] , (6.2a) u ϕ s ( x ) ≤ U ϕ η + ν η − d 2 log  1 − s η T  + | x | 2 2( η T − s ) , s ∈ [0 , η T ) . (6.2b) Pr o of. (i) Using the Y oung inequality | x − y | 2 ≤ | x | 2 1 − η + | y | 2 η , we see that e u ϕ T ( x ) = Z e ϕ ( y ) − 1 2 T | x − y | 2 (2 π T ) d 2 dy ≥ η d 2 e − | x | 2 2(1 − η ) T Z e ϕ ( y ) − | y | 2 2 ηT (2 π η T ) d 2 dy = η d 2 e − | x | 2 2(1 − η ) T e U ϕ η , whic h pro vides the righ t-hand side inequalit y in (i). Next, by the β –con vexit y of ϕ , we obtain the Jensen inequalit y for all R > 0: ϕ ( x ) ≤ − 1 2 β | x | 2 + 1 | B R | Z B R ( x )  ϕ + 1 2 β | . | 2  ( y ) dy ≤ 1 2 β R 2 + 1 | B R | Z B R ( x ) ϕ ( y ) dy . 20 Applying Jensen’s inequalit y again to the exponential function, w e deduce that ϕ ( x ) ≤ 1 2 β R 2 + log  1 | B R | Z B R ( x ) e ϕ ( y ) dy  ≤ 1 2 β R 2 + log  1 | B R | e | x | 2 + R 2 2 ηT Z B R ( x ) e ϕ ( y ) − | y | 2 2 ηT dy  ≤ 1 2 β R 2 + log  (2 π η T ) d 2 | B R | e | x | 2 + R 2 2 ηT e U ϕ η  = U ϕ η + | x | 2 2 η T + ¯ ν η ( R ) , with ¯ ν η ( R ) := 1 2  β + 1 η T  R 2 + log (2 π η T ) d 2 | B R | . (6.3) As ϕ (0) = 0, this provides the left-hand side inequality in (i) with the ﬁnite constant ν η := inf R> 0 ¯ ν η ( R ) . (ii) Notice that (6.3) pro vides the required upp er bound in (6.2b) at s = 0. T o obtain the low er bound in (6.2a) at s = 0, note that the β –conv exity of ϕ implies that ϕ ( x ) + 1 2 β | x | 2 ≥ p · x for all p ∈ ∂ ϕ (0) , (6.4) as ϕ (0) = 0. Therefore | p | = p · p | p | ≤ β 2 + max B 1 ϕ ≤ β 2 + 1 2 η T + ν η + U ϕ η =: ν ′ η + U ϕ η , b y substituting R = 1 in (6.3). Then ϕ ( x ) + 1 2 β | x | 2 ≥ p · x ≥ − ( ν ′ η + U ϕ η ) | x | , whic h is exactly the case s = 0 in (6.2a), since u ϕ 0 = ϕ . By (6.4), the deﬁnition of u ϕ s , and a direct Gaussian calculation, e u ϕ s ( x ) = Z R d e ϕ ( y ) − | x − y | 2 2 s (2 π s ) d 2 dy ≥ Z R d e p · y − β 2 | y | 2 − | x − y | 2 2 s (2 π s ) d 2 dy = e | x s + p | 2 2( β + 1 s ) − | x | 2 2 s (1 + β s ) d 2 . Using the Y oung inequality p · x 1+ β s ≥ − λ 2 | x | 2 − 1 2 λ | p | 2 (1+ β s ) 2 , λ := β 2 s 2(1+ β s ) > 0 , this implies u ϕ s ( x ) ≥ − d 2 log(1 + β s ) − β 4 (2 − β s ) | x | 2 − | p | 2 (1 + β s ) β 2 s , whic h gives the case s > 0 in (6.2a) b y using the b ound | p | ≤ ν ′ η + U ϕ η from abov e. W e ﬁnally derive (6.2b) b y using the righ t-hand side of (6.3): u ϕ s ( x ) = log E  e ϕ ( x + W s )  ≤ U ϕ η + ν η + log E h e 1 2 ηT | x + W s | 2 i = U ϕ η + ν η − d 2 log  1 − s η T  + | x | 2 2( η T − s ) , s ∈ [0 , η T ) , 21 In the r est of this p ap er we assume µ 0 , µ T ∈ P 2 ( R d ) , δ := 1 − 1 β T > 0 and we ﬁx η ∈ ( δ, 1) . Pro of of Theorem 3.1 (ii-a) By (5.9), w e ma y consider a maximizing sequence ( ϕ n ) n ≥ 1 for Problem (2.9) satisfying: ϕ n ∈ C conv w, ↑ , ϕ n (0) = 0 , and J ( ϕ n ) := µ T  T + β [ ϕ n ]  − µ 0  T + β [ u ϕ n T ]  n →∞ − → V red . (6.5) As J ( ϕ + const) = J ( ϕ ), w e ma y set ϕ n (0) = 0 for all n ≥ 1. W e pro ceed in sev eral steps. 1. W e ﬁrst pro ve that the sequence  U ϕ n η  n ≥ 1 is b ounded . (6.6) First, U ϕ n η ≥ − ν η b y Lemma 6.1 (i). T o deriv e a uniform upp er b ound, we observ e that T + β [ ϕ n ] ≤ ϕ n (0) + β 2 | . | 2 = β 2 | . | 2 , so that Lemma 6.1 (i) yields u ϕ n T ( y ) ≥ U ϕ n η − | y | 2 2(1 − η ) T + d 2 log( η ) . Then T + β  u ϕ n T  ( x ) ≥ U ϕ n η + d 2 log( η ) + T + β h − | . | 2 2(1 − η ) T i ( x ) . Since η < δ , w e hav e 1 (1 − η ) T < β , and therefore T + β h − | . | 2 2(1 − η ) T i ( x ) = − β 2  β T (1 − η ) − 1  | x | 2 . It follows that, for all suﬃcien tly large n , V red − 1 ≤ J ( ϕ n ) = Z T + β [ ϕ n ]( x ) µ T ( dx ) − Z T + β  u ϕ n T  ( x ) µ 0 ( dx ) ≤ β 2 Z | x | 2 µ T ( dx ) − U ϕ n η − d 2 log( η ) + β 2  β T (1 − η ) − 1  Z | x | 2 µ 0 ( dx ) . Since µ 0 , µ T ha ve ﬁnite second momen ts and V red < ∞ , this pro ves (6.6). 2. W e next sho w that, after passing to a subsequence ( n k ) k , w e ma y ﬁnd a limit function: ϕ n k − → ˆ ϕ ∈ C conv w as k → ∞ , locally uniformly on R d . (6.7) Indeed, inequalities (6.2a)–(6.2b) of Lemma 6.1 (ii), ev aluated at s = 0, together with (6.6), sho w that the sequence ( ϕ n ) n is lo cally b ounded. As each ϕ n is β –con v ex and lo cally uniformly b ounded, it follows that ( ϕ n ) n is lo cally Lipsc hitz, uniformly in n ; see Ro c k afellar [11], Theorem 10.6, p. 88. The con vergence in (6.7) is then a direct consequence of the Arzel` a–Ascoli theorem. Notice that ˆ ϕ ∈ C conv w as it inherits the β − conv exit y of the maps ϕ n k , together with the quadratic b ounds in (6.2a)–(6.2b). 3. In this step, we provide the proof of existence of a dual p oten tial optimizer for the reduced dual problem V red . W e rely on the follo wing claim whic h will b e justiﬁed later in Steps 5 to 7 b elo w: T + β [ ϕ n k ] − → T + β [ ˆ ϕ ] as k → ∞ , locally uniformly on R d . (6.8) 22 (3a) Since ϕ n (0) = 0, we hav e T + β [ ϕ n ] ≤ β 2 | . | 2 ∈ L 1 ( µ T ) by Assumption 6. It then follo ws from (6.8) and F atou’s lemma that lim sup k →∞ µ T  T + β [ ϕ n k ]  ≤ µ T  T + β [ ˆ ϕ ]  . (3b) Using Lemma 6.1 (ii), (6.2a), with s = T , and the b oundedness of  U ϕ n η  n ≥ 1 pro ved in Step 1 that u ϕ n k T + β 2 | . | 2 ≥ − c 0 + β 2 4 T | . | 2 for some constan t c 0 . Then, T + β  u ϕ n k T  ( x ) = inf y n u ϕ n k T ( y ) + β 2 | x − y | 2 o ≥ − c 0 + β 2 | x | 2 + inf y n β 2 4 T | y | 2 − β x · y o = − c 0 +  β 2 − T  | x | 2 , for all x ∈ R d , where the last low er b ound is in L 1 ( µ 0 ) as µ 0 ∈ P 2 ( R d ). W e may then apply F atou’s lemma to get lim inf k →∞ µ 0  T + β  u ϕ n k T  ≥ µ 0  L  , L := lim inf k →∞ T + β  u ϕ n k T  . (6.9) After passing to a subsequence n k ′ = n x k ′ for ﬁxed x ∈ R d , we hav e L ( x ) = lim k ′ →∞ T + β  u ϕ n k ′ T  ( x ) ≥ lim sup k ′ →∞  β 2 | x | 2 − β x · y n k ′ − c 0 + β 2 4 T | y n k ′ | 2  , with y n k ′ a minimizer of T + β  u ϕ n k ′ T  ( x ). Then if L ( x ) < ∞ , this shows that the sequence ( y n k ′ ) k ′ is b ounded; after p ossibly passing to a further subsequence, we ma y therefore assume that y n k ′ → ˆ y ∈ R d . It follows that L ( x ) = lim k ′ →∞  u ϕ n k ′ T ( y n k ′ ) + β 2 | x − y n k ′ | 2  ≥ lim inf k ′ →∞ u ϕ n k ′ T ( y n k ′ ) + β 2 | x − ˆ y | 2 = log  lim inf k ′ →∞ E  e ϕ n k ′ ( y n k ′ + W T )   + β 2 | x − ˆ y | 2 ≥ u ˆ ϕ T ( ˆ y ) + β 2 | x − ˆ y | 2 ≥ inf y ∈ R d n u ˆ ϕ T ( y ) + β 2 | x − y | 2 o = T + β  u ˆ ϕ T  ( x ) , b y F atou’s lemma and the lo cal uniform con v ergence of ϕ n k ′ to ˆ ϕ , where u ˆ ϕ T ( y ) ∈ ( −∞ , ∞ ] is understo o d in the extended sense. The same inequalit y L ( x ) ≥ T + β  u ˆ ϕ T  ( x ) is trivially true if L ( x ) = ∞ . Plugging this into (6.9), we obtain lim inf k →∞ µ 0  T + β  u ϕ n k T  ≥ µ 0  T + β  u ˆ ϕ T  . (6.10) 23 (3c) Set A k := R T + β [ ϕ n k ] dµ T and B k := R T + β  u ϕ n k T  dµ 0 , k ≥ 1. By (3a), the sequence ( A k ) k is b ounded ab o ve, and by (6.5), we see that A k − B k = J ( ϕ n k ) − → V red . Hence ( B k ) k is b ounded abov e. T ogether with (6.10), this implies µ 0  T + β  u ˆ ϕ T  < ∞ , so that J ( ˆ ϕ ) is w ell deﬁned. Com bining the conclusion from (3a) with (6.10), we no w obtain V red = lim k →∞ J ( ϕ n k ) ≤ lim sup k →∞ Z T + β [ ϕ n k ] dµ T − lim inf k →∞ Z T + β  u ϕ n k T  dµ 0 ≤ Z T + β [ ˆ ϕ ] dµ T − Z T + β  u ˆ ϕ T  dµ 0 = J ( ˆ ϕ ) , whic h sho ws that V red = J ( ˆ ϕ ), as required. 4. W e no w prov e that the induced p oten tial ˆ ψ := T + β [ ˆ ϕ ] attains the supremum in the relaxed dual problem ¯ V . 4(a). Fix ( t, x ) ∈ [0 , T ) × R d and let P ∈ P t ( δ x ) satisfy E P R T t c ( ν P r ) dr < ∞ . Since P ◦ X − 1 t = δ x , we hav e X t = x , P -a.s. Hence, after subtracting at time t and deﬁning W s := W P s − W P t , α s := α P s , σ s := σ P s , s ∈ [ t, T ] , w e obtain X s = x + R s t α r dr + R s t σ r dW r , t ≤ s ≤ T , P -a.s., where W = ( W s ) s ∈ [ t,T ] is a d -dimensional P -Bro wnian motion on [ t, T ]. Fix y ∈ R d and deﬁne the auxiliary process Y y s := y + R s t α r dr + W s , s ∈ [ t, T ]. Then E P | Y y T | 2 ≤ 3 | y | 2 + 3( T − t ) E P Z T t | α r | 2 dr + 3 d ( T − t ) < ∞ . Since ˆ ϕ is ﬁnite and β –conv ex and ˆ ϕ (0) = 0, we hav e for all p ∈ ∂ ˆ ϕ (0): ˆ ϕ ( z ) ≥ p · z − β 2 | z | 2 ≥ −| p | | z | − β 2 | z | 2 , z ∈ R d . Since E P | Y y T | 2 < ∞ , this implies ˆ ϕ ( Y y T ) − ∈ L 1 ( P ). Let Ω t,T := C ([ t, T ]; R d ), let R t,y b e Wiener measure on Ω t,T started from y at time t , and let Q y := La w P ( Y y ) ∈ P (Ω t,T ). W e claim that H ( Q y | R t,y ) ≤ 1 2 E P Z T t | α r | 2 dr . (6.11) F or this, let n ≥ 1 and deﬁne τ n := inf n s ≥ t : R s t | α r | 2 dr ≥ n o ∧ T , α ( n ) r := α r 1 { r ≤ τ n } , and Y ( n ) ,y s := y + R s t α ( n ) r dr + W s , s ∈ [ t, T ]. Set Z n := exp  − R T t α ( n ) r · dW r − 1 2 R T t | α ( n ) r | 2 dr  . Since R T t | α ( n ) r | 2 dr ≤ n , No viko v’s criterion holds, so Z n is a true martingale and d e P n := 24 Z n d P deﬁnes a probabilit y measure. Under e P n , the process f W ( n ) := W + R . t α ( n ) r dr is a Bro wnian motion on [ t, T ]. Therefore, if Q n := La w P ( Y ( n ) ,y ), then La w e P n ( Y ( n ) ,y ) = R t,y . By contraction of relativ e en tropy under the measurable map ω 7→ Y ( n ) ,y ( ω ), H ( Q n | R t,y ) ≤ H ( P | e P n ) = E P  log d P d e P n  = E P [ − log Z n ] = 1 2 E P Z T t | α ( n ) r | 2 dr , b ecause E P [ R T t α ( n ) r · dW r ] = 0. Hence H ( Q n | R t,y ) ≤ 1 2 E P R T t | α ( n ) r | 2 dr . Also, sup s ∈ [ t,T ] | Y ( n ) ,y s − Y y s | ≤ R T τ n | α r | dr , th us E P h sup s ∈ [ t,T ] | Y ( n ) ,y s − Y y s | 2 i ≤ ( T − t ) E P R T τ n | α r | 2 dr − → 0. Th us Q n ⇒ Q y w eakly on Ω t,T . By low er semicontin uit y of relativ e en trop y , H ( Q y | R t,y ) ≤ lim inf n →∞ H ( Q n | R t,y ) ≤ 1 2 E P R T t | α r | 2 dr . F or m ≥ 1, deﬁne f m ( ω ) := ˆ ϕ ( ω T ) ∧ m, ω ∈ Ω t,T . Then f + m ≤ m and f − m = ˆ ϕ ( ω T ) − , so f m ∈ L 1 ( Q y ). Moreo ver, e f m ( ω ) ≤ e ˆ ϕ ( ω T ) , ω ∈ Ω t,T , hence E R t,y [ e f m ] ≤ E R t,y [ e ˆ ϕ ( ω T ) ] = ( N T − t ∗ e ˆ ϕ )( y ) = e u ˆ ϕ T − t ( y ) < ∞ . Applying the Donsk er–V aradhan v ariational formula to f m under R t,y , we see that u ˆ ϕ T − t ( y ) ≥ E Q y [ f m ] − H ( Q y | R t,y ) ↑ E Q y [ ˆ ϕ ( Y y T )] − H ( Q y | R t,y ) ≥ E Q y [ ˆ ϕ ( Y y T )] − 1 2 E P Z T t | α r | 2 d r , b y monotone con vergence and (6.11). Next, we hav e ˆ ψ ( X T ) ≤ ˆ ϕ ( Y y T ) + β 2 | X T − Y y T | 2 , P - a.s. As E P | X T − Y y T | 2 = | x − y | 2 + E P R T t | σ r − I d | 2 dr , we get E P h ˆ ψ ( X T ) − Z T t c ( ν r )d r i ≤ E P [ ˆ ϕ ( Y y T )] − 1 2 E P Z T t | α r | 2 d r + β 2  E P | X T − Y y T | 2 − E P Z T t | σ r − I d | 2 d r  ≤ u ˆ ϕ T − t ( y ) + β 2 | x − y | 2 . By the arbitrariness of y ∈ R d and P ∈ P t ( δ x ), this sho ws that V ˆ ψ t ( x ) ≤ ˆ v t ( x ) , (6.12) whic h pro ves the claim. 4(b). Since ˆ ϕ is ﬁnite and β –conv ex, the map ˆ ψ = T + β [ ˆ ϕ ] is ﬁnite and β –conca ve. Moreo ver, (6.12) implies V ˆ ψ 0 ≤ ˆ v 0 , and therefore µ 0 ( V ˆ ψ 0 ) ≤ µ 0 ( ˆ v 0 ) < ∞ . Hence ˆ ψ ∈ C conc and we may apply the dualit y result Lemma 4.3 together with (6.12) to obtain V ( µ 0 , µ T ) ≥ µ T ( ˆ ψ ) − µ 0 ( V ˆ ψ 0 ) ≥ µ T ( ˆ ψ ) − µ 0 ( ˆ v 0 ) = J ( ˆ ϕ ) = V red ( µ 0 , µ T ) , (6.13) b y Step 3. As V red ( µ 0 , µ T ) = V ( µ 0 , µ T ) b y Theorem 3.1 (i), w e get µ T ( ˆ ψ ) − µ 0 ( ˆ v 0 ) = J ( ˆ ϕ ) = V red ( µ 0 , µ T ) = V ( µ 0 , µ T ). Com bining this with (6.13), w e obtain µ T ( ˆ ψ ) − µ 0 ( V ˆ ψ 0 ) = V ( µ 0 , µ T ) , µ 0 ( V ˆ ψ 0 ) = µ 0 ( ˆ v 0 ) , 25 and then V ˆ ψ 0 = ˆ v 0 , µ 0 –a.s. by (6.12), pro ving that ˆ ψ attains the supremum in ¯ V ( µ 0 , µ T ). 5. In order to prov e the claimed con vergence of T + β [ ϕ n k ] in (6.8), ﬁx R > 0 and s ∈ (0 , η T ), where η ∈ (0 , δ ) is the parameter ﬁxed in Step 1 . Then   T + β [ ϕ n k ] − T + β [ ˆ ϕ ]   L ∞ ( B R ) ≤   T + β [ ϕ n k ] − T + β [ u ϕ n k s ]   L ∞ ( B R ) +   T + β [ u ϕ n k s ] − T + β [ u ˆ ϕ s ]   L ∞ ( B R ) +   T + β [ u ˆ ϕ s ] − T + β [ ˆ ϕ ]   L ∞ ( B R ) . W e shall pro ve in Step 5 below that, for ev ery ﬁxed s ∈ (0 , η T ), u ϕ n k s − → u ˆ ϕ s and T + β [ u ϕ n k s ] − → T + β [ u ˆ ϕ s ] , lo cally uniformly in R d . (6.14) Hence, for ev ery ﬁxed R > 0 and s ∈ (0 , η T ), lim sup k →∞   T + β [ ϕ n k ] − T + β [ ˆ ϕ ]   L ∞ ( B R ) ≤   T + β [ u ˆ ϕ s ] − T + β [ ˆ ϕ ]   L ∞ ( B R ) + θ R s , (6.15) where θ R s := sup n ≥ 1   T + β [ u ϕ n s ] − T + β [ ϕ n ]   L ∞ ( B R ) . In Step 6 we show that θ R s − → 0 as s ↘ 0 , and the same argument, applied to ˆ ϕ , yields   T + β [ u ˆ ϕ s ] − T + β [ ˆ ϕ ]   L ∞ ( B R ) − → 0 as s ↘ 0 . T ogether with (6.15), this pro ves (6.8). 6. In this step we justify (6.14). Fix s ∈ (0 , η T ) with η ∈ (0 , δ ) as in Step 1 , and set Φ := { ˆ ϕ, ϕ n , n ≥ 1 } . By Lemma 6.1 (ii), (6.2b), at time 0, together with the b oundedness of  U ϕ n η  n from Step 1 , there exists a constant C η < ∞ such that ϕ n ≤ C η + | . | 2 2 η T , n ≥ 1 , and therefore b ϕ ≤ C η + | . | 2 2 η T , (6.16) b y (6.7). In particular, as s < η T , we hav e u ϕ s ( x ) = log  N s ∗ e ϕ ( x )  ∈ R for all x ∈ R d , ϕ ∈ Φ. W e split the remaining argumen ts into three parts. (5a) F or ϕ ∈ Φ and R ≥ 1, deﬁne ζ ϕ R ( s, x ) := N s ∗ ( e ϕ 1 B c R )( x ) N s ∗ ( e ϕ 1 B R )( x ) . 26 Then there exist constants C s , a s > 0, dep ending only on ( β , T , d, η , s ), suc h that sup ϕ ∈ Φ ζ ϕ R ( s, x ) ≤ C s exp  − 1 4  1 s − 1 η T  R 2 + a s | x | 2  , x ∈ R d , R ≥ 1 . (6.17) Indeed, ϕ ( y ) ≤ C η + | y | 2 2 η T for all ϕ ∈ Φ by (6.16), and using | x − y | 2 2 s = | y | 2 2 η T + | x − y | 2 2 s − | y | 2 2 η T ≥ | y | 2 2 η T + 1 4  1 s − 1 η T  | y | 2 − a s | x | 2 , for a suitable constan t a s > 0, we obtain N s ∗ ( e ϕ 1 B c R )( x ) ≤ e C η Z B c R e | y | 2 2 ηT − | x − y | 2 2 s (2 π s ) d 2 dy ≤ C s e − 1 4  1 s − 1 ηT  R 2 + a s | x | 2 . (6.18) On the other hand, since Φ is lo cally bounded b elo w on B 1 , there exists m 1 < ∞ suc h that ϕ ( y ) ≥ − m 1 for all y ∈ B 1 , ϕ ∈ Φ . Hence, for R ≥ 1 and every ϕ ∈ Φ, N s ∗ ( e ϕ 1 B R )( x ) ≥ N s ∗ ( e ϕ 1 B 1 )( x ) ≥ (2 π s ) − d 2 e − m 1 Z B 1 e − | x − y | 2 2 s dy ≥ C ′ s e − a ′ s | x | 2 , for some constan ts C ′ s > 0 and a ′ s ≥ 0, indep enden t of ϕ and R . Combining this with (6.18) yields (6.17). (5b) Fix R > 0, and set γ k,R := ∥ ϕ n k − ˆ ϕ ∥ L ∞ ( B R ) − → 0 as k → ∞ b y (6.7). F or every ﬁxed x ∈ R d , the estimate in (5a) implies that sup ϕ ∈ Φ ζ ϕ R ( s, x ) − → 0 as R → ∞ . W e no w write N s ∗ e ϕ n k ( x ) ≥ e − γ k,R N s ∗ ( 1 B R e ˆ ϕ )( x ) = e − γ k,R 1 + ζ ˆ ϕ R ( s, x ) N s ∗ e ˆ ϕ ( x ) , N s ∗ e ϕ n k ( x ) ≤  1 + ζ ϕ n k R ( s, x )  N s ∗ ( 1 B R e ϕ n k )( x ) ≤  1 + ζ ϕ n k R ( s, x )  e γ k,R N s ∗ e ˆ ϕ ( x ) . Sending k → ∞ and then R → ∞ , we see that u ϕ n k s ( x ) − → u ˆ ϕ s ( x ), for all x ∈ R d . Moreo ver, the positiv e-time semicon v exity estimate from Section 5 impl ies that, for this ﬁxed s , the maps u ϕ n k s + κ ( s ) 2 | . | 2 are conv ex on R d for all k , with κ ( s ) := β 1+ β s . Since the limit is ﬁnite every where, Rock afellar [11], Theorem 10.8, p. 90, implies u ϕ n k s − → u ˆ ϕ s lo cally uniformly in R d . (5c) W e next pro ve the con vergence of the Moreau en v elop es. F or ev ery ϕ ∈ Φ, deﬁne G ϕ s ( y ) := u ϕ s ( y ) + β 2 | x − y | 2 . 27 As Φ is lo cally b ounded from b elo w on B 1 , the same argumen t as in (5a) giv es a quadratic low er bound u ϕ s ≥ − c s − a s | . | 2 , ϕ ∈ Φ, for some constan ts c s , a s > 0 dep ending only on ( β , T , d, η , s ). Hence, after p ossibly enlarging constants, u ϕ s ( y ) + β 2 | x − y | 2 ≥ − C s,r + c s,r | y | 2 , x ∈ B r , ϕ ∈ Φ , (6.19) for some constan ts C s,r > 0 and c s,r > 0. In particular, for ﬁxed ( s, r ), the righ t- hand side tends to + ∞ as | y | → ∞ , uniformly in x ∈ B r and ϕ ∈ Φ. On the other hand, b y (6.16) and s < η T , there exists C ′ s,r < ∞ such that u ϕ s (0) ≤ C ′ s,r for all ϕ ∈ Φ. Therefore, T + β [ u ϕ s ]( x ) ≤ u ϕ s (0) + β 2 | x | 2 ≤ C ′ s,r + β 2 r 2 , x ∈ B r , ϕ ∈ Φ . Com bining this with (6.19), w e deduce that every minimiser y of u ϕ s ( · ) + β 2 | x − ·| 2 for x ∈ B r and ϕ ∈ Φ satisﬁes | y | ≤ R s,r for some R s,r < ∞ independent of ϕ and x . Th us T + β [ u ϕ s ]( x ) = inf | y |≤ R s,r n u ϕ s ( y ) + β 2 | x − y | 2 o , x ∈ B r , ϕ ∈ Φ . The lo cal uniform con vergence of u ϕ n k s established in (5b) then implies T + β [ u ϕ n k s ] − → T + β [ u ˆ ϕ s ] , lo cally uniformly in R d , whic h pro ves (6.14). 7. It remains to prov e that, for the family Φ := { ϕ n , n ≥ 1 } , we ha ve θ R s := sup n ≥ 1   T + β [ u ϕ n s ] − T + β [ ϕ n ]   L ∞ ( B R ) − → 0 as s ↘ 0 , for all R > 0 . (6.20) Fix R > 0 and ρ ∈ (0 , 1). Since Φ is lo cally bounded and β –conv ex, it is locally equi- Lipsc hitz, so there exists a common Lipschitz constan t L = L R,ρ on B R + ρ . Denoting δ s ( ρ ) := P ( | W s | ≥ ρ ), we see that e u ϕ s ( x ) = N s ∗ e ϕ ( x ) ≥ N s ∗ ( e ϕ 1 B ρ ( x ) )( x ) ≥ e ϕ ( x ) − Lρ  1 − δ s ( ρ )  , ϕ ∈ Φ , x ∈ B R . (6.21) On the other hand, e u ϕ s ( x ) ≤ N s ∗ ( e ϕ 1 B ρ ( x ) )( x ) + N s ∗ ( e ϕ 1 B ρ ( x ) c )( x ) . The ﬁrst term is b ounded ab ov e b y N s ∗ ( e ϕ 1 B ρ ( x ) )( x ) ≤ e ϕ ( x )+ Lρ . F or the second term, b y the same Gaussian-tail argument as in Step 5 (a), using the ﬁxed parameter η from Step 1 , there exists a deterministic function g R,ρ ( s ), deﬁned for 0 < s < η T , suc h that g R,ρ ( s ) − → 0 as s ↘ 0 , and N s ∗ ( e ϕ 1 B ρ ( x ) c )( x ) ≤ g R,ρ ( s ) , ϕ ∈ Φ , x ∈ B R . 28 Hence e u ϕ s ( x ) ≤ e ϕ ( x )+ Lρ + g R,ρ ( s ) , ϕ ∈ Φ , x ∈ B R . T ogether with (6.21), this yields − Lρ + log  1 − δ s ( ρ )  ≤ u ϕ s ( x ) − ϕ ( x ) ≤ log  e ϕ ( x )+ Lρ + g R,ρ ( s )  − ϕ ( x ) ≤ Lρ + C R,ρ g R,ρ ( s ) , for some constant C R,ρ > 0, by the Lipschitz property of log on a b ounded inter v al aw ay from the origin, using the lo cal uniform b oundedness of Φ on B R . Since both δ s ( ρ ) → 0 and g R,ρ ( s ) → 0 as s ↘ 0, we obtain lim sup s ↘ 0 sup ϕ ∈ Φ ∥ u ϕ s − ϕ ∥ L ∞ ( B R ) ≤ Lρ − → ρ → 0 0 for all R > 0 . (6.22) W e next localize the minimizers in the deﬁnition of T + β [ u ϕ s ]. By the same low er-b ound argumen t as in Step 5 (c), for ev ery R > 0 there exists R ′ > 0 suc h that, for all suﬃcien tly small s ∈ (0 , η T ), all ϕ ∈ Φ, and all x ∈ B R , every minimizer in the deﬁnition of T + β [ u ϕ s ]( x ) b elongs to B R ′ . Denoting by y ϕ s ( x ) ∈ B R ′ suc h a minimizer, w e hav e T + β [ u ϕ s ]( x ) = u ϕ s  y ϕ s ( x )  + β 2   x − y ϕ s ( x )   2 ≤ ϕ  y ϕ s ( x )  + β 2   x − y ϕ s ( x )   2 + ∥ u ϕ s − ϕ ∥ L ∞ ( B R ′ ) ≤ T + β [ ϕ ]( x ) + ∥ u ϕ s − ϕ ∥ L ∞ ( B R ′ ) . On the other hand, b y Jensen’s inequalit y and the β –conv exit y of ϕ , w e ha ve u ϕ s ≥ ϕ − β d 2 s , and therefore T + β [ u ϕ s ]( x ) ≥ T + β [ ϕ ]( x ) − β d 2 s. Combining the last tw o estimates, we obtain   T + β [ u ϕ s ]( x ) − T + β [ ϕ ]( x )   ≤ ∥ u ϕ s − ϕ ∥ L ∞ ( B R ′ ) + β d 2 s, ϕ ∈ Φ , x ∈ B R , for all suﬃcien tly small s ∈ (0 , η T ). T aking the suprem um o ver ϕ ∈ Φ and x ∈ B R , and using (6.22) on B R ′ , we obtain (6.20). ⊔ ⊓ 7 Primal Attainme nt T o prov e existence of a primal optimizer b P for the problem SBB( µ 0 , µ T ) w e need the follo wing additional regularit y properties of the maps ˆ u, ˆ v . and the corresponding Y . Lemma 7.1. (i) The minimum value T + β [ ˆ u T ] is attaine d by a unique minimizer Y 0 which is Lipschitz-c ontinuous. (ii) The minimum value value ˆ ψ = T + β [ ˆ ϕ ] is attaine d by a non-empty c omp act set of minimizers Y T . (iii) F or al l t < T , we have | ˆ v w | L ∞ [0 ,t ] + | ∇ ˆ u √ w | L ∞ [ t,T ] < ∞ . 29 Pr o of. (i) By Step 3(c) in the pro of of Theorem 3.1 (ii-a) in Section 6, w e hav e ˆ v 0 = T + β [ ˆ u 0 ] ∈ L 1 ( µ 0 ) and in particular ˆ v 0 ( x ) < ∞ for µ 0 -a.e. x . Denote for such x b y F x ( y ) := ˆ u 0 ( y ) + β 2 | x − y | 2 , and notice that ˆ u 0 ( x ) + | x | 2 2 T = − d 2 log(2 π T ) + log R R d e ˆ ϕ ( y ) − | y | 2 2 T + x · y T d y is conv ex as a log-Laplace transform. Then D 2 ˆ u 0 ⪰ − 1 T I d and therefore D 2 F x ( y ) ⪰  β − 1 T  I d ≻ 0 , (7.1) as β T > 1. Th us F x attains a unique minimizer, and since ( x, y ) 7→ F x ( y ) is Borel, the measurable maximum theorem yields a Borel map Y 0 . F or x i ∈ R d and y i := Y 0 ( x i ), i = 1 , 2, the ﬁrst-order condition for the minimization writes x i = y i + β − 1 ∇ ˆ u 0 ( y i ). Denote δ x := x 1 − x 2 , δ y := y 1 − y 2 , then we obtain b y subtracting and taking the scalar product with δ y , w e obtain δx · δ y = | δ y | 2 + 1 β  ∇ ˆ u 0 ( y 1 ) − ∇ ˆ u 0 ( y 2 )  · δ y ≥ (1 − 1 β T ) | δ y | 2 , b y (7.1). By the Cauch y-Sch w arz inequality , this implies that | δ y | ≤ (1 − 1 β T ) − 1 | δ x | , which the required Lipsc hitz con tinuit y of Y . (ii) As ˆ ϕ is β -con vex, the map g 0 := ˆ ϕ + β 2 | . | 2 is proper, l.s.c. and conv ex. W e hav e ˆ ψ ( x ) = β 2 | x | 2 − g ∗ 0 ( β x ), By the ﬁrst dualit y result, the maximizing sequence may b e c hosen so that ψ n k := T + β [ ϕ n k ] ∈ C w , hence each ψ n k is ﬁnite on R d . T ogether with the lo cal uniform conv ergence ψ n k → ˆ ψ = T + β [ ˆ ϕ ], this implies that ˆ ψ is ﬁnite on all of R d . Hence g ∗ 0 ( p ) < ∞ for all p ∈ R d , and therefore g 0 is co erciv e, and the map y 7→ g 0 ( y ) − β x · y is co erciv e and l.s.c. for eac h ﬁxed x and, as such, attains its minim um with compact argmin set Y T ( x ). (iii) Fix no w τ < T and let us prov e for some constan t C τ that |∇ ˆ u s ( x ) | ≤ C τ (1 + | x | ) , s ∈ [ T − τ , T ] , x ∈ R d , (7.2) | ˆ v t ( x ) | ≤ C τ (1 + | x | 2 ) , 0 ≤ t ≤ τ , x ∈ R d , for all τ < T . (7.3) Deﬁne G s ( x ) := ˆ u s ( x ) + κ τ 2 | x | 2 , s ∈ [ T − τ , T ]. By (5.2), eac h G s is con vex. Moreov er, (5.1) implies G s ( x ) ≤ C τ (1 + | x | 2 ) , s ∈ [ T − τ , T ] , x ∈ R d . Since s 7→ ˆ u s (0) is contin uous on [ T − τ , T ], we also ha ve | G s (0) | ≤ C τ , s ∈ [ T − τ , T ]. Cho ose any p s ∈ ∂ G s (0). F or ev ery unit v ector e , p s · e ≤ G s ( e ) − G s (0) , − p s · e ≤ G s ( − e ) − G s (0). Hence | p s | ≤ C τ , s ∈ [ T − τ , T ]. By conv exit y , G s ( x ) ≥ G s (0) + p s · x ≥ − C τ (1 + | x | ). Now ﬁx x ∈ R d , s ∈ [ T − τ , T ], and a unit vector e , and set r := 1 + | x | . Since G s is con vex and diﬀerentiable, ∇ G s ( x ) · e ≤ G s ( x + re ) − G s ( x ) r , −∇ G s ( x ) · e ≤ G s ( x − re ) − G s ( x ) r . Using the quadratic upper b ound at x ± re and the aﬃne lo wer bound at x , we obtain |∇ G s ( x ) · e | ≤ C τ (1+ | x ± re | 2 )+ C τ (1+ | x | ) r . Since r = 1 + | x | and | x ± r e | ≤ | x | + r ≤ 1 +2 | x | , this yields |∇ G s ( x ) · e | ≤ C τ (1 + | x | ). T aking the suprem um ov er | e | = 1 yields |∇ G s ( x ) | ≤ C τ (1 + | x | ). Since ∇ ˆ u s ( x ) = ∇ G s ( x ) − κ τ x , we obtain (7.2). It remains to prov e (7.3). F or the upp er b ound, choose y = 0 in the Moreau env elop to see that ˆ v t ≤ ˆ u t (0) + β 2 | . | 2 ≥ C τ + β 2 | . | 2 , 0 ≤ t ≤ τ , where the b oundeded of ˆ u t (0) on [ T − τ , T ] w as deriv ed in the last paragraph. F or the low er b ound, Lemma 6.1(ii) yields that there exists C τ < ∞ suc h that u s ≥ − C τ − β 4 (2 − β s ) | . | 2 , s ∈ [ T − τ , T ]. Since s 7→ β 4 (2 − β s ) is decreasing and s ≥ T − τ , setting 30 λ τ := β 4  2 − β ( T − τ )  < β 2 , we obtain u s ≥ − C τ − λ τ | . | 2 , s ∈ [ T − τ , T ]. Consequen tly , w e hav e for t ∈ [0 , τ ] that v t ( x ) ≥ − C τ + inf y ∈ R d  β 2 | x − y | 2 − λ τ | y | 2  = − C τ − β λ τ β − 2 λ τ | x | 2 . ⊔ ⊓ Pro of of Theorem 3.1 (ii-b-c-d) 1. Deﬁne m 0 := Y 0 # µ 0 , ν 0 ( dy ) := e − ˆ u T ( y ) m 0 ( dy ) , ν T := ν 0 ∗ N T , m T ( dy ) := e ˆ ϕ ( y ) ν T ( dy ) . (7.4) Since µ 0 ∈ P 2 ( R d ) and Y 0 is Lipsc hitz b y Lemma 7.1 (i), it follows that m 0 = Y 0 # µ 0 ∈ P 2 ( R d ). Moreo ver, b y T onelli’s theorem and the deﬁnition of ˆ u T , m T ( R d ) = ν T ( e ˆ ϕ ) = ν 0 ( N T ∗ e ˆ ϕ ) = m 0 ( e ˆ u T e − ˆ u T ) = 1 . Hence m T ∈ P ( R d ). Since ν T ≪ Leb b y Gaussian conv olution, we also ha ve m T ≪ Leb. 1. Fix g ∈ C b ( R d ) and deﬁne ˆ ϕ ε := ˆ ϕ + εg , ˆ ψ ε := T + β [ ˆ ϕ ε ]. Set ˜ φ ε := T − β  T + β [ ˆ ϕ ε ]  , ¯ φ ε := ˜ φ ε − ˜ φ ε (0). Then ¯ φ ε ∈ C conv w and ¯ φ ε (0) = 0. Moreo ver J is in v ariant under addition of constan ts, so J ( ¯ φ ε ) = J ( ˜ φ ε ). By Remark 2.3, T + β [ ˜ φ ε ] = T + β [ ˆ ϕ ε ] , J ( ˜ φ ε ) ≥ J ( ˆ ϕ ε ). Since ˆ ϕ maximizes J o ver C conv w , we obtain, for all suﬃcien tly small | ε | , J ( ˆ ϕ ε ) ≤ J ( ¯ φ ε ) = J ( ˜ φ ε ) ≤ J ( ˆ ϕ ) . (7.5) T erminal derivatives. By Lemma 7.1 (ii), for eac h x ∈ R d the set Y T ( x ) is nonempt y and compact. Hence, b y Danskin’s theorem, ∂ + ˆ ψ ε   ε =0 ( x ) = min y ∈ Y T ( x ) g ( y ) , ∂ − ˆ ψ ε   ε =0 ( x ) = max y ∈ Y T ( x ) g ( y ). Moreov er, since g is b ounded, ˆ ϕ ( y ) − | ε || g | ∞ ≤ ˆ ϕ ε ( y ) ≤ ˆ ϕ ( y ) + | ε || g | ∞ , y ∈ R d , and therefore, b y monotonicity of T + β , ˆ ψ ( x ) − | ε || g | ∞ ≤ ˆ ψ ε ( x ) ≤ ˆ ψ ( x ) + | ε || g | ∞ , x ∈ R d . Th us    ˆ ψ ε ( x ) − ˆ ψ ( x ) ε    ≤ | g | ∞ , ε  = 0. Then dominated conv ergence yields ∂ ε µ T ( ˆ ψ ε )   ε =0 + = µ T  min Y g  and ∂ ε µ T ( ˆ ψ ε )   ε =0 − = µ T  max Y g  . Initial derivative. Deﬁne u ε,T := log( N T ∗ e ˆ ϕ + εg ) , v ε, 0 := T + β [ u ε,T ]. By Lemma 7.1 (i), the minimizer Y 0 ( x ) is unique for µ 0 -a.e. x , hence Danskin’s theorem giv es ∂ ε v ε, 0 ( x )   ε =0 = ∂ ε u ε,T ( Y 0 ( x ))   ε =0 µ 0 -a.e. x . Since g is b ounded, diﬀeren tiation under the Gaussian in- tegral is justiﬁed and yields G ( y ) := ∂ ε u ε,T ( y )   ε =0 = N T ∗ ( g e ˆ ϕ )( y ) h T ( y ) , | G ( y ) | ≤ | g | ∞ . Also, e −| ε || g | ∞ ( N T ∗ e ˆ ϕ )( y ) ≤ ( N T ∗ e ˆ ϕ + εg )( y ) ≤ e | ε || g | ∞ ( N T ∗ e ˆ ϕ )( y ), so | u ε,T ( y ) − u 0 ,T ( y ) | ≤ | ε || g | ∞ , y ∈ R d . By monotonicit y of T + β , this implies | v ε, 0 ( x ) − v 0 , 0 ( x ) | ≤ | ε || g | ∞ , x ∈ R d . Hence dominated con vergence applied to the diﬀerence quotien ts giv es ∂ ε µ 0 ( v ε, 0 )    ε =0 = µ 0  G ( Y 0 )  = m 0 ( G ) = ν 0  N T ∗ ( g e ˆ ϕ  = ν T  g e ˆ ϕ  = m T ( g ) , b y (7.4) and T onelli’s theorem. 31 Conclusion. Divide (7.5) b y ε and let ε → 0 ± . Using the deriv ativ e form ulas ab o v e, we obtain, for ev ery g ∈ C b ( R d ), µ T  min Y T g  ≤ m T ( g ) ≤ µ T  max Y T g  . (7.6) 2. Let Γ := { ( x, y ) : y ∈ Y T ( x ) } b e the graph of Y . Since m T ≪ Leb and ˆ ϕ is β - con vex, ˆ ϕ is diﬀerentiable m T -a.e. At ev ery diﬀerentiabilit y p oint y such that ( x, y ) ∈ Γ, then y minimizes z 7→ ˆ ϕ ( z ) + β 2 | x − z | 2 , so the ﬁrst-order optimalit y condition yields ∇ ˆ ϕ ( y ) + β ( y − x ) = 0. Thu s x = X T ( y ) := y + β − 1 ∇ ˆ ϕ ( y ). Therefore, for m T -a.e. y , the section Γ y := { x ∈ R d : ( x, y ) ∈ Γ } is con tained in the singleton { X T ( y ) } . Our ob jective in this step is to show that µ T = X T # m T . (7.7) Since ˆ ϕ is l.s.c. the graph Γ := { ( x, y ) : y ∈ Y T ( x ) } is closed and Γ = { c Γ = 0 } , where c Γ ( x, y ) := 1 ∧ dist(( x, y ) , Γ) is b ounded, con tinuous, and nonnegativ e. By Kantoro vich dualit y , using the standard notation f ⊕ h = f ( x ) + h ( y ), we hav e 0 ≤ inf π ∈ Π( µ T ,m T ) π ( c Γ ) = sup ( f ,h ) ∈ D µ T ( f ) + m T ( h ) , D := { ( f , h ) ∈ C b × C b : f ⊕ h ≤ c Γ  . F or ( f , h ) ∈ D , w e hav e max y ∈ Y T ( x ) h ( y ) ≤ − f ( x ) , x ∈ R d . Applying the right-h and side inequalit y in (7.6) with g = h , w e get m T ( h ) ≤ R max y ∈ Y T ( x ) h ( y ) µ T ( dx ) ≤ − µ T ( f ), hence ≤ 0. By arbitrariness of ( f , h ) ∈ D , se deduce that inf π ∈ Π( µ T ,m T ) R c Γ dπ = 0. Since the cost is b ounded and con tinuous, an optimal coupling π ⋆ ∈ Π( µ T , m T ) exists and satisﬁes π ⋆ (Γ) = 1, as the minimum v alue is 0. Disin tegrate π ⋆ with resp ect to its second marginal m T : π ⋆ ( dx, dy ) = π ⋆ y ( dx ) m T ( dy ). Since π ⋆ (Γ) = 1, for m T -a.e. y the measure π ⋆ y is supp orted on Γ y , hence on { X T ( y ) } . Th us π ⋆ y = δ X T ( y ) for m T -a.e. y . T aking ﬁrst marginals, we obtain (7.7). 3. Let R b e the law of a Brownian motion Y on [0 , T ] with initial law m 0 . Deﬁne Z T := e ˆ ϕ ( Y T ) − ˆ u 0 ( Y 0 ) and notice that E R [ Z T ] = E R  e − ˆ u 0 ( Y 0 ) E R [ e ˆ ϕ ( Y T ) | Y 0 ]  = 1. W e may then in tro duce the equiv alent probab ility measure d ˆ Q := Z T d R . F or an y b ounded Borel f , w e ha ve E ˆ Q [ f ( Y T )] = E R  f ( Y T ) e ˆ ϕ ( Y T ) − ˆ u T ( Y 0 )  = m T ( f ). Thus Y 0 ∼ m 0 and Y T ∼ m T under ˆ Q . The density pro cess is Z t := E R [ Z T |F t ] = e ˆ u t ( Y t ) − ˆ u 0 ( Y 0 ) , t ∈ [0 , T ). As ˆ u is a C 1 , 2 solution of the back ward heat equation, w e hav e by Itˆ o’s formula d Z t = Z t ∇ ˆ u t ( Y t ) · d W R t , t < T , and it follo ws from Girsano v’s theorem that Y t = Y 0 + Z t 0 ∇ ˆ u s ( Y s )d s + W t , 0 ≤ t ≤ T . Let X := Y + 1 β ∇ ˆ u ( Y ). Then X 0 ∼ µ 0 under ˆ Q and has con tinuous paths on [0 , T ) b y the C 1 , 2 ( Q ) regularit y of ˆ u . Moreov er, as Y 0 ∈ L 2 ( ˆ Q ) and ∇ ˆ u has linear growth b y Lemma 7.1 (iii), it follo ws from the BDG inequalit y and Gron wall’s lemma that : E ˆ Q h sup 0 ≤ s ≤ T | Y s | 2 i + E ˆ Q h sup 0 ≤ s ≤ T | X s | 2 i < ∞ . (7.8) 32 5. By Lemma 5.1(3-a) and Step 7 in the pro of of Theorem 3.1 (ii) in Section 6, the follo wing maximizers of the Hamiltonian H ( ∇ v t , D 2 v t ) are w ell deﬁned: ν t = ( a t , σ t ) , with a t = ∇ ˆ v t ( X t ) , σ t =  I d − β − 1 D 2 ˆ v t ( X t )  − 1 , t ∈ [0 , T ) , Fix n ≥ 1, τ < T , and deﬁne ρ n := τ ∧ inf { t ≤ τ : | X t | ≥ n } . Fix also 0 < ε < τ . Since ˆ v ∈ C 1 , 2 ( Q ), Itˆ o’s form ula on [ ε ∧ ρ n , ρ n ] gives E ˆ Q  ˆ v ρ n ( X ρ n ) − ˆ v ε ∧ ρ n ( X ε ∧ ρ n )  = E ˆ Q Z ρ n ε ∧ ρ n  ∂ t ˆ v + a t · ∇ ˆ v + 1 2 σ t σ ⊤ t : D 2 ˆ v  ( t, X t )d t = E ˆ Q Z ρ n ε ∧ ρ n c ( ν t ) dt. (7.9) Since β T > 1, choose δ > 0 so small that β − ( T − δ ) − 1 > 0. Then for t ∈ [0 , δ ] and x in a ﬁxed compact set K ⊂ R d , the function y 7→ ˆ u t ( y ) + β 2 | x − y | 2 is uniformly strongly con vex, hence has a unique minimizer Y t ( x ). Moreo ver, a T aylor expansion at y = 0 sho ws that these minimizers remain in a compact ball B R , uniformly for s ∈ [0 , δ ] and x ∈ K . Since ˆ u t → ˆ u 0 uniformly on B R as t ↘ 0, it follows that ˆ v t ( x ) = inf y  ˆ u t ( y ) + β 2 | x − y | 2  → inf y  ˆ u 0 ( y ) + β 2 | x − y | 2  = ˆ v 0 ( x ) , uniformly on K. Letting ε ↓ 0, lo cal uniform conv ergence of v t to v 0 near t = 0 and contin uit y of X give ˆ v ε ∧ ρ n ( X ε ∧ ρ n ) → ˆ v 0 ( X 0 ) ˆ Q -a.s. Moreo ver the conv ergence is dominated by an in tegrable random v ariable, since on { ρ n > 0 } the argument sta ys in the compact set [0 , T − 1 n ] × B n , while on { ρ n = 0 } the term is exactly ˆ v 0 ( X 0 ) ∈ L 1 ( ˆ Q ). W e then deduce from (7.9) that E ˆ Q  ˆ v ρ n ( X ρ n )  = E ˆ Q [ ˆ v 0 ( X 0 )] + E ˆ Q R ρ n 0 c ( ν t ) dt, and b y com bining the quadratic growth property of ˆ v in Lemma 7.1 with (7.8), we get by dominated and monotone conv ergence: E ˆ Q [ ˆ v τ ( X τ )] = E ˆ Q [ ˆ v 0 ( X 0 )] + E ˆ Q h Z τ 0 c ( ν t ) dt i = µ 0 ( v 0 ) + E ˆ Q h Z τ 0 c ( ν t ) dt i , τ < T . (7.10) In particular, E ˆ Q R τ 0 c ( ν t ) dt < ∞ for all τ < T . 6. Deﬁne F s ( y ) := ˆ u s ( y ) + β 2 | y | 2 , s > 0 , and F ( y ) := ˆ ϕ ( y ) + β 2 | y | 2 . Recall that F s → F lo cally uniformly on R d as s ↓ 0 and β X t = ∇ F T − t ( Y t ) , t < T . (7.11) Since Y has con tinuous paths and Y T ∼ m T , we hav e Y t → Y T a.s. as t ↑ T . Because m T ≪ Leb and F is ﬁnite conv ex, F is diﬀeren tiable at Y T for ˆ Q -a.e. sample point. Fix an arbitrary sequence t n ↑ T . Set f n := F T − t n , f := F , x n := Y t n , and x := Y T . Since f n → f lo cally uniformly , x n → x ˆ Q -a.s., eac h f n is diﬀerentiable, and f is diﬀeren tiable at x for ˆ Q -a.e. sample p oint, the standard stability result for gradien ts of con vex functions yields ∇ F T − t n ( Y t n ) → ∇ F ( Y T ), ˆ Q -a.s. Hence, by (7.11), X t n = 33 β − 1 ∇ F T − t n ( Y t n ) → β − 1 ∇ F ( Y T ) ˆ Q -a.s. By Step 4, for m T -a.e. y we hav e X T ( y ) = β − 1 ∇ F ( y ). Since Y T ∼ m T , it follo ws that X t n → X T ( Y T ) ˆ Q -a.s. Since the sequence t n ↑ T w as arbitrary , we conclude that X t → X T := X T ( Y T ) as t ↑ T , ˆ Q -a.s. Th us X admits a con tinuous extension to [0 , T ]. As Y T ∼ m T under ˆ Q and µ T = X T # m T b y Step 4, it follows that La w ˆ Q ( X T ) = µ T . Consequen tly ˆ P := La w ˆ Q ( X ) ∈ P ( µ 0 , µ T ) . 7. Fix τ ∈ (0 , T ) and let ( ˆ P ω ) ω b e a regular conditional probabilit y distribution of ˆ P giv en F τ . W e ﬁrst observe that for ˆ P -a.e. ω , the shifted canonical pro cess on [ τ , T ] under ˆ P ω starts from X τ ( ω ) and is a contin uous semimartingale with absolutely contin uous c haracteristics ( a r , σ r ) r ∈ [ τ ,T ) . Therefore, for ˆ P -a.e. ω , the Step 4 estimate on the shifted in terv al [ τ , T ) with horizon T − τ and initial p oin t X τ ( ω ) give E ˆ P ω h sup τ ≤ s

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