MULTIDIMENSIONAL SDES WITH SINGULAR DRIFT AND UNIVERSAL CONSTRUCTION OF THE POLYMER MEASURE WITH WHITE NOISE POTENTIAL

Transcription

1 Submitted to the Annal of Probability arxiv: arxiv: MULTIDIMENSIONAL SDES WITH SINGULAR DRIFT AND UNIVERSAL CONSTRUCTION OF THE POLYMER MEASURE WITH WHITE NOISE POTENTIAL BY GIUSEPPE CANNIZZARO, AND KHALIL CHOUK, Univerity of Warwick and Techniche Univerität Berlin AbtractWe tudy exitence and uniquene of olution for tochatic differential equation with ditributional drift by giving a meaning to the Stroock-Varadhan martingale problem aociated to uch equation. The approach we exploit i the one of paracontrolled ditribution introduced in [17]. A a reult we make ene of the three dimenional polymer meaure with white noie potential. 1. Introduction. The aim of the preent paper i to give a meaning to Stochatic Differential Equation SDE) of the form 1.1) dx t = V t, X t )dt + db t, X = x where B i a d-dimenional Brownian motion, x a point in and V i a function of time taking value in the pace of ditribution S, ). Of coure, a it i written, 1.1) doe not make any ene unle we impoe certain retriction concerning the regularity or integrability or both) of the drift V. The cae of V being a mooth enough vector-field ha been deeply invetigated and i nowaday well-undertood. Upon auming V L p loc, + ) Rd ) for p > d +, it i till poible to obtain local pathwie exitence and uniquene a hown in [6]. When V i an effective ditribution, the majority of reult deal with the time-homogenou ituation i.e. V i taken to be independent of time), ee for example [3, 1, 11], and exitence and uniquene can be determined either in the weak or trong ene, depending on the interplay between it regularity and integrability. When V C[, T ], S, )) with a non-trivial dependence on time, the picture become even more blurred, ince it i already unclear how to define a convenient notion of olution. Neverthele ome advance have been recently made in [9], where the author invetigate the cae of a time dependent ditributional drift taking value in a cla of Sobolev pace with negative derivation order on. Thi work wa carried out while G.C. wa a PhD tudent at Techniche Univerität Berlin MSC 1 ubject claification: Primary 6K35, 6K35; econdary 6K35 Keyword and phrae: Singular SDE, Singular SPDE, Paracontrolled Calculu, Polymer Meaure 1

2 Our attempt i to generalize the work of F. Delarue and R. Diel. In [7], they contruct olution to SDE with V t, ) = x Y t, ) and Y a 1/3 + ε)-hölder function in pace on ome interval I R, by formulating a Stroock-Varadhan martingale problem for 1.1). What we aim at i to go beyond the one dimenional cae and conider a ditributional drift on for d 1. More preciely we tudy the cae of V C[, T ], C β, )) for β <, where C β, ) i the Beov- Hölder pace of ditribution on ee.1) for the exact definition). In the ame pirit a [7], we prove well-poedne for the martingale problem correponding to the generator G V of the diffuion 1.1), which i given by 1.) G V = t V. In general, one would want to ay that a probability meaure P on Ω = C[, T ], ), endowed with the uual Borel σ-algebra BC[, T ], )), olve the martingale problem related to G V tarting at x, if the canonical proce X, X t ω) = ωt), atifie 1. PX = x) = 1,. for any T T and ϕ D, where D i a et of function on [, T ], the proce 1.3) { ϕt, X t ) } G V ϕ), X )d i a quare integrable martingale with repect to P. t T The problem here lie in the fact that if we chooe D imply a the pace of mooth function and V C[, T ], C β, )), with β <, then G V ϕ i not a function anymore but a ditribution with the ame regularity a V ) and, once again, it i not clear what meaning to attribute to G V ϕ), X ). The point here i that we need to determine a uitable domain D for which G V ϕ i a continuou function of time, bounded in pace. In other word, we need to olve the following Partial Differential Equation PDE), that we will refer to a the generator equation, 1.4) G V ϕ = f, ϕt, ) = ϕ T for f C[, T ], L )) and a ufficiently large cla of terminal condition ϕ T. Once thi i done, we can replace the aertion 1.3) with the requirement that the proce 1.5) { ϕt, X t ) f, X )d } t

3 i a quare integrable martingale for every f C[, T ], L )) and ϕ the olution of 1.4). However, PDE of the type 1.4), auming β 3, ), cannot be claically handled ince the preumed olution i not expected to be mooth enough to allow to define the ill-poed term V ϕ. To bypa it, F. Delarue and R. Diel in [7] adopt the technique exploited by M.Hairer in [19] and, more preciely, they make ue of Lyon Rough Path theory to interpret the ill-defined product a a rough integral. Depite the poibility of overcoming the well-poedne iue, rough path theory ha the dramatic diadvantage of being crucially attached to the one parameter etting o that there i imply no hope to go beyond the one-dimenional cae with thoe technique. Thi i preciely the point in which the paracontrolled ditribution approach, developed in [17] or alternatively the Theory of Regularity Structure []), come into play. In thi context though, the poibility of olving equation that are not claically well-poed come at a price. More pecifically, in cae β 3, 1 ], we are not allowed to take any V C[, T ], C β, )) but only thoe that can be enhanced to a rough ditribution, V ee Definition 3.6). In other word, we need to be able to build in ome way, tarting from V, an additional object atifying uitable regularity requirement but depending only on V itelf. We refrain from detailing the contruction here and we limit ourelve to looely tate the reult. THEOREM 1.1. Let β 3, ), γ, β+) and V C[, T ], C β, )). If β 3, 1 ], aume further that V can be enhanced to a rough ditribution V. Then, there exit a non-trivial Banach pace, D C[, T ], C γ )), uch that for any ϕ T C γ ) and f C[, T ], L )), 1.4) admit a unique olution in D. Moreover, the map aigning to ϕ T, f and V the olution to the generator equation i jointly locally Lipchitz continuou. If we now formulate the Stroock-Varadhan martingale problem for the SDE 1.1), by requiring point 1. tated before and 1.5) to be a quare integrable martingale for every f C[, T ], L )), with ϕ D and D the Banach pace determined in the previou theorem, then we can indeed prove it well-poedne. THEOREM 1.. Let β 3, ) and V C[, T ], C β, )). If β 3, 1 ], aume further that V can be enhanced to a rough ditribution V. Then, there exit a unique probability meaure P which olve the martingale problem with generator G V tarting at x a decribed above), for every x. The natural quetion at thi point i if and when it i poible to build, given V C[, T ], C β, )), it enhancement V. The example are variou for 3

4 4 d = 1, the one decribed in [7, Section 5] would do) but probably one of the mot intereting cae i the one that allow to contruct the and 3 dimenional polymer meaure with white noie potential. The Polymer meaure with white noie potential i a ingular meaure on the pace of continuou function that i formally given by T ) 1.6) Q T dω) = Z 1 exp ξω )d W T dω) where W i the Wiener meaure on C[, T ], ), d =, 3, ξ a patial white noie on the d-dimenional toru T d independent of W, and Z i an infinite renormalization contant. A it i written, the expreion in 1.6) i of coure enele ince we are exponentiating the integral in time of a white noie, which i a ditribution, over a Brownian path and dividing then by an infinite contant, all operation that require to be given a meaning to. Even if eemingly unrelated, we will ee that, if it were well-poed, under the polymer meaure the canonical proce, X t ω) = ω t ha the ame law a the olution to the SDE given by 1.7) dx t = ht t, X t )dt + db t where B i a brownian motion with repect to W and h, the olution to the KPZ-type equation 1.8) t h = 1 h + 1 h + ξ, h, ) = in which ξ i the ame pace white noie a the one appearing in 1.6). Summarizing, if we are able to decribe the law of 1.7) then we can alo give a quenched decription of the infiniteimal dynamic of the polymer itelf, in other word, make ene of it. It i not difficult to gue, from the KPZ-type equation above, that h ha regularity lightly le than in dimenion and lightly le than 1 in dimenion 3 thu, in principle, falling into the cope of Theorem 1.. But of coure to be able to apply it, we will need to prove well-poedne of 1.8), which i non-trivial given the ingularity of the noie, and, for thi, we will exploit once more the paracontrolled ditribution approach. Once local exitence and uniquene for the previou Stochatic Partial Differential Equation SPDE) i etablihed and one ha hown that, in d = 3, V t, ) = def ht t, ) can be enhanced to a rough ditribution, we obtain the following reult.

5 THEOREM 1.3. Let ξ ε be a mollified verion of the noie and Q ε T the polymer meaure defined in 1.6) with ξ ε replacing ξ. Then, there exit a meaure Q T and T = T ξ) >, independent of the choice of the mollifier, uch that for all T < T, Q ε T = Q T. The lat part of our work will conit in determining ome of the propertie of the Polymer Meaure built in the previou theorem. At firt notice that, the contruction above i local in the ene that we can prove that the meaure formally given in 1.6) exit only up to a poibly finite exploion time T, depending, in principle, on the feature of the noie. We want to how that uch an exploion doe not occur. Our proof relie on the trict poitivity of the olution to the Parabolic Anderon Equation PAM), formally given by t u = 1 u + uξ with initial condition identically equal to 1 and we provide a novel proof of thi in Section 7. valid for both d = and 3. At lat, looking at the way in which the Polymer meaure 1.6) i written, it might eem that Q T i abolutely continuou with repect to the Wiener one. Thi i definitely not the cae. In principle, ince Q T i the meaure decribing the law of the olution to 1.7), looking at the SDE one guee correctly) that the drift cannot be of Cameron-Martin type. The actual proof doe not make ue of the previou heuritic but intead focue on the renormalization propertie of 1.8) o that in the end we have the following tatement. THEOREM 1.4. In the aumption of Theorem 1.3, let T and Q T be a tated above. Then, in both dimenion d = and 3, T can be choen to be + and the meaure Q T i ingular with repect to the Wiener one. A a lat remark, we point out that the contruction of the Polymer meaure we carried out before i rather univeral in the ene that it doe not rely on the pecific feature of the noie. Indeed, given that we are able to prove well-poedne of an equation of the type 1.8) driven by a generic noie ξ then the ame argument apply. The ame hold true for the proof of the ingularity. For the continuou directed random polymer, i.e. the one formally given by the expreion 1.6), but with a pace-time white noie in patial dimenion 1, an analogou reult wa obtained in [1]. Our proof follow a completely different approach, which in turn can be traightforwardly adapted to recover their reult. 5

6 6 PLAN OF THE PAPER. In Section, we introduce Beov pace and the main element of paracontrolled calculu that will be needed in the ret of the paper. Section 3 i dedicated to the generator equation. We prove that it admit a unique olution and that the flow i a locally Lipchitz map. A anticipated in the introduction, thi i then crucial for Section 4, in which we define the Martingale problem aociated to the SDE 1.1) and prove it well-poedne. The lat three ection are devoted to the Polymer meaure: it contruction Section 5), the KPZ-type equation thank to which it i poible Section 6) and it propertie Section 7). ACKNOWLEDGEMENTS. We want to thank Profeor P.K.Friz and M.Gubinelli for the numerou dicuion and fruitful advice. While thi work wa developed GC wa funded by the RTG 1845 and Berlin Mathematical School. KC i upported by European Reearch Council under the European Union Seventh Framework Programme FP7/7-13) / ERC grant agreement nr NOTATION. We collect here ome notation we will ue throughout the paper. In the preent work we will alway conider function/ditribution on, d 1 arbitrary but fixed, with value in R n, o, in order to lighten the notation, if B, R n ) i a pace of function/ditribution from to R n, then we will denote by B R n = def B, R n ), and B = def B, R) Let δ, η R, T > and T [, T ). Let D, D ) be a Banach pace and ζ, ζ : [T T, T ] B be two function. We will ay that ζ C δ η, T,T D and ζ C η, T,T D if ζ C δ η, T,T D < and ζ Cη, T,T D < repectively, where def ζ C δ η, T D = up T t) η ft) f) D,T <t T T,T ] t δ, ζ Cη, T,T D def = up T t) η ft) D. t T T,T ] In cae the norm on ζ doe not depend on η, i.e. η =, or T = T, we will imply remove the correponding ubcript. We will ay that a b if there exit a contant C > uch that a Cb.. Beov Space and Paracontrolled Calculu. In thi firt paragraph we want to introduce the definition of the function pace we will be uing throughout the ret of the work and recall the main ingredient of the paracontrolled calculu 1. Let χ, ϱ D be non-negative radial function uch that 1 For a thorough introduction on Beov Space ee [], or [17] for the main definition and propertie we will ue from now on.

7 1. The upport of χ i contained in a ball and the upport of ϱ i contained in an annulu;. χξ) + j ϱ j ξ) = 1 for all ξ ; 3. uppχ) uppϱ j.)) = for i 1 and uppϱ i.)) uppϱ j.)) = when i j > 1. χ, ϱ) atifying the above propertie are aid to form a dyadic partition of unity. For the exitence of a dyadic partition of unity ee [, Propoition.1]. Let now F denote the Fourier tranform and χ, ϱ) be a dyadic partition of unity. Then, the Littlewood-Paley block are defined a 1 u = F 1 χf u), j u = F 1 ϱ j )F u) for j where ϱ j ) = def ϱ j ) and, for α R, p, q [1, + ], the Beov pace Bp,qR α d, R n ) i.1) Bp,qR α d, R n ) = {u S, R n qb ); u αp,q = jqα j u }. q L p,r n ) < + j 1 We will often deal with the pecial cae p = q =, o we et C α, R n ) = def B, R α d, R n ) and denote by u α = u B α, it norm. Such a notation i alo jutified by the fact that, for non-integer α >, C α, R n ) coincide with the uual pace of α-hölder continuou function. In order to manipulate tochatic term and exploit propertie of the element in Wiener chao, we will bound their norm in Beov pace with finite p = q and then get back to the pace C α. To do o, the following Beov embedding will prove to be fundamental. PROPOSITION.1. Let 1 p 1 p + and 1 q 1 q +. For all R the pace Bp 1,q 1 i continuouly embedded in B d 1 1 ) p 1 p p,q, in particular we have u α d u B α p,p. p.1. Operation with Beov-Hölder ditribution. Let f, g be two ditribution in S ). Upon uing the Littlewood-Paley decompoition of f and g, we can formally write their product a fg = f g + f g + f g where the firt and the lat ummand at the right hand ide are called paraproduct while the econd reonant term, and they are repectively defined by f g = g f = i f j g and f g = i f j g. j 1 i<j 1 j 1 i j 1 With thee notation at hand, we can tate the following propoition. 7

8 8 PROPOSITION. Bony Etimate, []). g C β, Let α, β R. Let f C α and if α, then f g C β and f g β f L g β if α <, then f g C α+β and f g α+β f α g β if α + β >, then f g C α+β and f g α+β f α g β Summarizing, the previou propoition tell u that the product of general f C α and g C β i well-defined if and only if α + β > and in thi cae fg C δ, where δ = min{α, β, α + β} ee [17, Lemma.1], for the proof in thi pecific context). One of the key reult of the paracontrolled analyi carried out in [17], i a commutation relation between the operator and, that we here recall ee [17, Lemma.4]). PROPOSITION.3 Commutator Lemma). Let α, β, γ R be uch that α, 1), α + β + γ > and β + γ <. Then, for f, g and h mooth, the operator allow for the bound Rf, g, h) = f g) h fg h) Rf, g, h) α+β+γ f α g β h γ hence, it can be uniquely extended to a bounded trilinear operator on C α C β C γ. In the following Propoition, which ummarize [6, Lemma.5] and [17, Lemma A.8], we decribe the action of the heat kernel on Beov-Hölder function and it relation with the paraproduct. PROPOSITION.4 Schauder Etimate). Let P t = e 1 t be the heat flow, θ and α R. Let f C α and < t then we have P t f α+θ t θ f α and P t Id)f α θ t θ f α. If α [, 1], the latter bound become P t Id ) f L t α f α Moreover if α < 1 and β R, the following commutator etimate hold.) P t f g) f P t g α+β+θ t θ f α g β for all g C β.

9 For notational convenience, let u define If)t) = def t P t f)d, where the operator P t wa introduced in Propoition.4. Since we will be working with function exploding at a certain rate a t goe to and we will need to undertand what happen when we convolve them with the heat kernel, we collect in the following corollary ome imple reult. COROLLARY.5. Let t [, T ], α, β R, γ, δ [, 1), γ, γ] and ε, 1]. Let f C η,t C α. Then, 1. if α β > 1 and ϑ def = α β γ + δ + 1 >, we have t δ If)t) β T ϑ up [,T ] γ f) α. if α ε > 1, γ < δ, α ε γ + δ + 1 > and < t, we have 9 δ If)t) If)) L t ε T ϑ up [,T ] γ f) α where ϑ = δ γ if δ > γ and ϑ = δ γ otherwie. PROOF. The proof i a rather traightforward application of Propoition.4, o we omit it for the ake of conciene ee Corollary.5 in [5]). REMARK.6. The reader hould keep in mind that the convolution with P t allow to gain θ regularity in pace at the price of an exploion a time goe to of order θ/. One ha then to adjut the choice of the parameter o that they fit the aumption of the previou Corollary and thi will be done implicitly throughout the paper not to heavy the preentation. 3. Solving the Generator equation. The aim of thi ection i to how exitence and uniquene of olution for the generator equation connected to the SDE 1.1), i.e. the PDE 3.1) t u + 1 u + V u = f, ut, ) = ut ) where T > i arbitrary but fixed, u T i the terminal condition and f C T C β, for β 3, ]. Let t, x) [, T ) R and, for a function ψ, let J T ψ) be defined by J T ψ)t) = T def t R P d r t ψr)dr, where P t = e 1 t i the uual heat flow. Uing the previou notation, the mild formulation of our generator equation read 3.) ut) = P T t u T + J T f + u V ) t).

10 1 Now, ince V C T C β, Schauder etimate Propoition.4) ugget that the olution u to the previou equation cannot have patial regularity better than β +. According to Propoition., the product between u and V i well-poed if and only if the um of the regularitie of the factor i trictly poitive, which, in the preent cae, read β β = β + 1 >, i.e. β > 1. Therefore, for β 1, ), we can directly apply Bony and Schauder etimate and contruct the olution to the equation directly. Even if pace, notation and tool might look different, thi cae can be eaily hown to correpond to the one treated in [9] ee Remark 3.3). On the other hand, to overcome the 1 barrier, another method ha to be exploited and paracontrolled ditribution mut be introduced The Young cae: β 1, ). In order to contruct the olution of the generator equation we will ue a fixed point argument, i.e. we will introduce a uitable map and prove it i a contraction on a uitable pace, hence admitting a unique fixed point according to Banach Fixed Point theorem. To do o, let u fix a terminal time T >, α 1 β, β + ), a terminal condition u T C β+ and f C T C β. Given a function u in C T,T C α, for T [, T ), we define the map Γ T u) a 3.3) Γ T u)t) = def P T t u T + J T f + u V ) t) where J T i the operator defined above and we omitted the dependence on pace. Notice that Γ T u)t) α P T t u T α + J T f + u V ) t) α u T β+ + u T β+ + β α T +1 β α f CT C β + T +1 u V CT C β β α T +1 f CT C β + V ) C T C β u C T,T C α where the econd inequality i a imple application of Corollary.5 and the lat follow by Bony etimate Propoition.. Therefore, etting γ = β α + 1 we have Γ T u) C T,T C α ut C β+ + T γ ) f CT C β + V C T C β u C T,T C α. The next propoition ummarize what we have obtained o far and how how to build a local in time olution to 3.) for V C[, T ], C β, )), β 1, ). PROPOSITION 3.1. Let T >, β 1, ) and α 1 β, β + ). For u T, f, V ) C α C T C β C T C β, let Γ T be the map on C T,T C α defined by 3.3). Then there exit γ > uch that the following bound hold true 3.4) Γ T u) C T,T C α ut C β+ + T γ ) f CT C β + V C T C β u C T,T C α

11 11 and 3.5) Γ T u) Γ T v) C T,T C α T γ V CT C β u v C T,T C α Hence, there exit T [, T ) depending only on V CT C β ), and a unique function u C[T T, T ], C α ) that olve the generator equation 3.). PROOF. The bound 3.4) i proved above and an analogou argument how that 3.5) hold true a well. Therefore there exit T, T ) ufficiently cloe to T and depending only on V CT C β uch that the map Γ T i a trict contraction of C[T T, T ], C α ) in itelf and, by Banach fixed point theorem, it admit a unique fixed point. We have now all the element in place to tate and prove the following theorem. THEOREM 3.. Let β 1, ), α 1 β, β + ) and T >. For any u T, f, V ) C α C T C β C T C β, there exit a unique olution u C T C α to the generator equation 3.1), where the product u V i defined according to Propoition.. Moreover, the olution u atifie u C ε T C ρ ut α + f CT C β + u C T C α V C T C β for every ρ and ε uch that ρ + ε α. At lat, the flow of the generator equation, i.e. the map aigning to every triplet u T, f, V ) C α C T C β C T C β the olution u to 3.1), i a locally Lipchitz continuou map. PROOF. Thank to Propoition 3.1, we already know that there exit T [, T ) and a unique function u C[T T, T ], C α ) that olve the generator equation 3.). Now, ince the equation i linear and conequently the T determined above depend only on V and not on u T, we can extend our olution to the whole interval [, T ], iterating the contruction we jut carried out, o that the reulting u i defined on the whole interval [, T ]. The time regularity of the olution can be eaily obtained by an interpolation argument. Finally, taking V, Ṽ C T C β, f, f C T C β, u T, ũ T C T C β+ and denoting by u V rep. uṽ ) the olution of the equation G V u = f rep. G Ṽ u = f) with terminal condition u T rep. ũ T ), it i eay to how that, if then max{ u T, ũ T, f, f, V, Ṽ } R u V uṽ CT C α R u T ũ T + f f + V Ṽ which prove that the flow i indeed a locally Lipchitz map for more detail, ee for example the proof of an analogou reult in [15]).

12 1 REMARK 3.3. A we pointed out before, the analyi performed in thi ection correpond to the cae treated in [9] with the only difference that we preferred to work with Hölder pace of negative regularity intead of Sobolev pace. There i no doubt that we could have ued the latter pace a well ince Bony and Schauder etimate Propoition. and.4) hold alo for thee pace ee [, Chapter ]). 3.. The rough cae: β 3, 1 ]. The analyi of the rough cae i more ubtle and require a better undertanding of the tructure of the olution to the generator equation. Let u aume for the moment that V i a mooth function. Thank to Bony decompoition of the product we can write 3.) a 3.6) ut) = J T f + u V ) + u t) where u t) = def P T t u T + J T u V + u V ) What we ee at thi point i that when V i a ditribution in C[, T ], C β, )) the only ill-defined term of the equation 3.6) i the reonant term contained in u. Neverthele, Propoition.) ugget that, if it were well-poed, u t) C θ 1 for θ < β +. A we announced before, we need ome inight regarding the expected tructure of the olution. Indeed, even if it i not poible to make ene of the ill-poed product for all ditribution belonging to pace whoe regularitie do not um up to a trictly poitive quantity, maybe it i poible to identify a uitable ubpace for which it i. To recognize uch a ubpace we begin with the following lemma, which allow to commute the heat kernel J T and the paraproduct. LEMMA 3.4. Let T >, θ [1, β + ), ρ > θ 1 and h C T C β. For R T d [, T ), let g C T,T C θ be uch that g C ρ T,T L. Then the following inequality hold J T g h) g J T h) C T,T C T κ g θ 1 C T,T C + g ) θ C ρ T,T L h CT C β with κ = def min { 1 θ β, ρ θ 1 } >. PROOF. By direct computation, for t [T T, T ], we can expre the right hand ide of the inequality a the um of two term I 1 and I, repectively given by I 1 t) = I t) = T t T t Pr t gr) hr)) gr) P r t hr) ) dr, gr) gt)) P r t hr) dr.

13 13 Uing the commutation reult in.) we directly get I 1 t) θ 1 T t r t) θ β gr) θ hr) β dr T 1 θ β)/ g C T,T C θ h C T C β For I we apply Schauder etimate and obtain I t) θ 1 T t T and thi end the proof. t gr) gt) L r t) θ+1)/ dr h CT C θ ) r t) ρ θ+1)/ dr g C ρ T,T L h C T C β,rd T 1 ρ+ θ+1 ) g C ρ T,T L h C T C β The previou Lemma ugget that, at leat at a formal level, the olution u of our equation admit the following expanion 3.7) u = J T f) + u J T V ) + u where u = u + J T u V ) u J T V ) hould be more regular than u itelf. On the one hand, equation 3.7) convey the algebraic tructure we expect the olution of 3.) to have and on the other, it tell u that u, in term of regularity exhibit the ame behaviour a J T V ). Thi i exactly the core idea of the paracontrolled approach developed in [17] and it will allow u to conveniently define the ill-poed term. We are now ready to introduce the pace of paracontrolled ditribution aociated to the equation 3.1). DEFINITION 3.5. Let T >, 4 θ 1 3 < α < θ < β+ and ρ >. For f C T C β and T [, T ), we define the pace of paracontrolled ditribution D α,θ,ρ a the et of couple of ditribution u, u ) C T,T C θ C T,T C α 1 uch that u t) = def ut) u t) J T V )t) J T f)t) C α 1 for all T T t T. We equip D α,θ,ρ u, u ) D α,θ,ρ with the norm def = u C T,T C + u θ C ρ T,T L + u C T,T C α 1 + u Cα 1, T,T C α 1 and we introduce the metric d D α,θ,ρ, defined for all u, u ), v, v ) D α,θ,ρ by d D α,θ,ρ u, u ), v, v ) ) = u, u ) v, v ) D α,θ,ρ

14 14 Endowed with the metric d D α,θ,ρ, the pace D α,θ,ρ, d ) D α,θ,ρ i a complete metric pace. The advantage of the paracontrolled formulation i that the problem of wellpoedne for the product can be tranferred from the function u, that we have to determine and i therefore unknown, to V, or better J T V ), which on the other hand i given. To ee how thi work, take u, u ) D α,θ,ρ j = 1,..., d, we get 3.8) j u = J T j f) + d u, i J T j V i ) + U, j, U, j = j u + i=1 o that the reonant term, for V mooth, can be written a j u V j = J T j f) V j +. Differentiating u, for d j u, i J T V i ) i=1 d u, i J T j V i ) ) V j + U, j V j i=1 By Bony paraproduct etimate we immediately deduce that U, j i α )-regular in pace and, ince α > 4 3, we conclude that the lat ummand i well-defined even when V t) C β. In order to make ene of the econd ummand we need to R exploit the commutator d in Propoition.3 which give d u, i J T j V i ) ) V j = i=1 d u, i J T j V i ) V i) + i=1 d Ru, i, J T j V i ), V j ) where the lat ummand of the previou can be extended in a continuou way to V C T C β ince 3α 4 >. The only term which are till ill-poed are J T j V i ) V i for i, j = 1,..., d. Notice though that they do not depend on u anymore but only on V, o if we can build them in ome way, we are done and we can make ene of the product. Thi i the reaon why we introduce the notion of rough ditribution. DEFINITION 3.6 Rough Ditribution). Let β 3, ] 1, γ < β + and T >. Set H γ = C T C γ C T C γ 3. We define the pace of rough ditribution R a d X γ { = def def cl H γ Kη) = η, J T j η i ) η j ) ) i,j=1,...,d, η CT C } where cl H γ { } denote the cloure of the et in bracket with repect to the topology of H γ and, for a function ψ : R, J T ψ) i the olution of the equation i=1 t + 1 ) J T ψ) = ψ, J T ψ)t, ) =.

15 We denote by V = V 1, V ) a generic element of X γ and whenever V 1 = V we ay that V i a lift or enhancement) of V. REMARK 3.7. The reader familiar with rough path theory can appreciate the imilarity of the pace introduced above with the pace of rough path aociated to a given path. Let u point out that, a in the above-mentioned ituation, there i in general no canonical choice for the extra-term J T j η i ) η j when η ha pace regularity γ. However there are everal imple cae, a the one in [7, Section 5] for d = 1, in which thi contruction can be uccefully carried out. To witne, let u conider the time-independent one dimenional ituation with V = x Y and Y C β+1. Then, for x R T J T xy )t, x) = t P r t x y)dry y)dy = P T t Y x) R t where the firt equality follow by the fact that P t i the fundamental olution to the heat equation. Then, P T t Y C β+3 and P T t Y x Y i well-poed if and only if β > 3 which i well-beyond the 3 barrier. In force of the previou definition and thank to the computation above, u V can be decompoed a d d u V = J T j f) V j + u, i J T j V i ) V j ) j=1 + i,j=1 d Ru, i, J T j V i ), V j ) + i,j=1 d U, j V j which ugget that the left hand ide of the previou hould be a continuou functional of u, u ) D α,θ,ρ and V X γ. Thi i exactly what the next propoition prove. PROPOSITION 3.8. Let T > and 4 3 < α < θ < γ < β +. Let V = V 1, V ) X γ be an enhancement of V, f be either a function in C T L or coincide with one of the component of V 1, i.e. f {V1 i, i = 1,..., d}, and, for T [, T ), u, u ) D α,θ,ρ. Define u V by 3.9) u V = def d H j f, V) + j=1 + d i,j=1 u, i V i,j j=1 d d Ru, i, J T j V1), i V j 1 ) + U,j V j 1, i,j=1 j=1 15

16 16 where, in cae f = V j 1, Hj f, V) = def V j,j, while if f C T L, H j f, V) = def J T j f) V j 1, and U i given by the expreion in 3.8). Then u V i welldefined and the following etimate hold u V ) Cα 1, T,T C γ 3 1 F f CT L V 1 CT C γ R ) d V X γ + u, u ) ) D α,θ,ρ where F = def {f C T L }. At lat, under the previou aumption, the product u V, defined according to Bony decompoition and equation 3.9), i welldefined. PROOF. In order to prove the bound in the tatement, one ha to conider each of the ummand in 3.9) eparately. For the firt three, it i an immediate conequence of the aumption V X γ, Bony etimate Propoition.) and the commutator lemma Propoition.3) repectively. For the lat, notice that, by the definition of U given in 3.8), the fact that u, u ) D α,θ,ρ and, again, Bony paraproduct etimate, we have U, j α T t) α 1 u, u ) D α,θ,ρ + T γ α u, u ) D α,θ,ρ V CT C β which immediately give, for α > 1 up T t) α 1 U, j t) α u, u ) γ 1 ) D α,θ,ρ 1 + T V CT t [T T C β,t ] Hence, by applying Bony etimate for we get the expected bound on the fourth ummand a well. The lat part of the tatement i once more a conequence of Propoition.. At thi point we have all we need in order to et up our fixed point argument. Indeed, let V = V 1, V ) X θ be an enhancement of V, i.e. V = V 1, and et M T to be the map from D α,θ,ρ to C T C α given by 3.1) M T u, u ) = J T f) + J T u V ) + Ψ T t for u, u ) D α,θ,ρ, α < θ and ΨT t according to Propoition 3.8. Set = P T t u T, where the term u V i defined 3.11) M T :D α,θ,ρ C[T T, T ], C α )) C[T T, T ], C α 1, )) u, u ) Mu, u ), u)

17 We can now prove that thi map i a contraction in the pace D α,θ,ρ it admit a unique fixed point. 17 and therefore PROPOSITION 3.9. Let < T < 1, 4 θ 1 3 < α < θ < γ < β +, ρ, γ 1 ). Let u T C γ, V C T C β, V = V 1, V ) X γ be an enhancement of V and f be either a function in C T L or coincide with one of the component of V 1, i.e. f {V1 i, i = 1,..., d}. Then, for T [, T ), there exit κ >, depending only on α, θ, ρ and γ, uch that the map M T defined by 3.11) atifie the following etimate 3.1) M T u, u ) D α,θ,ρ 1 F f CT L V C T C β + u T γ V X γ ) 1 + T κ u, u ) D α,θ,ρ ) where F = def {f C T L } and 3.13) M T u, u ) M T v, v ) D α,θ,ρ ) ) 1 + V X γ 1 + T κ u, u ) v, v ) ) D α,θ,ρ and i therefore a trict contraction in D α,θρ for T T mall enough. PROOF. Let u, u ) D α,θ,ρ. In order to prove that M T u, u ) = M T u, u ), u) D α,θ,ρ it uffice to etimate the term M T u, u ) = Ψ T t + J T f + u V ), M T u, u ) = def u and M T u, u ) = def M T u, u ) J T f) u J T V ) in uitable norm. More preciely we have to control the following quantity M T u, u ) D α,θ,ρ def = M T u, u ) C T,T C θ + M T u, u ) C ρ T,T L + M T u, u ) C T,T C α 1 + M T u, u ) Cα 1, T,T C α 1 Let u begin with firt. According to the definition of M T u, u ) we have to etimate the C T,T C θ -norm of 3.14) Ψ T, J T f), J T u V ), J T u V ), J T u V ). Since the heat-flow P t i a bounded linear operator from C θ to itelf we get immediately that up Ψ T t θ u T θ u T γ t T

18 18 By Corollary.5 we have J T f)t) θ { T γ θ f CT L if f C T L T γ θ f CT C γ if f {V k ; k = 1,..., d} Let u focu on J T u V ) and J T u V ). Applying once more Corollary.5 and Bony etimate we obtain J T u V )t) θ J T u V )t) θ T γ θ T γ 1 u C T,T C θ 1 u C T,T C θ 1 V CT C γ V CT C γ γ θ T u, u ) D α,θ,ρ V X γ γ 1 T u, u ) D α,θ,ρ V X γ We will now treat the reonant term J T u V ). By the firt part of Corollary.5 and Propoition 3.8 we directly ee that it C θ -norm i bounded by T γ α θ up γ α θ T t [T T,T ] T t) α 1 u V t) γ 3 1F f CT L V 1 CT C γ + ) V X γ + u, u ) ) ) D α,θ,ρ where F = def {f C T L } and thi complete the tudy of the firt term. Conider now M T u, u ) C ρ. In thi cae we have to bound the derivative T L Rd of the term in 3.14) in the C ρ T,T L -norm. Thank to Propoition.4 we ee that Ψ T t Ψ T = P t 1)P u T t ρ u T ρ t ρ u T γ The econd part of Corollary.5 guarantee that, for < t T, J T f)t) J T f)) t ρ Analogouly, by Bony etimate we get J T u V ) ) t) J T u V ) ) ) t ρ and J T u V ) ) t) J T u V ) ) ) t ρ { γ 1 T ρ f CT L if f C T L T γ 1 ρ f CT C γ if f {V k ; k = 1,..., d} T γ 1 ρ u C T,T C θ 1 T θ+γ ρ u C T,T C θ 1 which imply the correct bound. At lat, by Corollary.5 we have V CT C γ V CT C γ J T u V ) ) t) J T u V ) ) ) t ρ γ ρ α T up t [T T,T ] T t) α 1 u V t) γ 3

19 which in turn can be bounded via Propoition 3.8. Concerning, the o called Gubinelli derivative, by definition, M T u, u ) = u and, by aumption, u C T,T C θ. Hence 19 M T u, u ) C T,T C α 1 u C T,T C α u C T,T C θ u, u ) D α,θ,ρ However thi i not yet the needed bound due to the miing factor T to ome poitive power. Let u oberve that ut) α 1 ut) ut ) α 1 + u T α Now it uffice to notice that we can etimate the firt ummand in two different way iθ 1) u C T,T C i ut) ut )) θ T t) ρ u C ρ T,T L where i i the i-th Littlewood-Paley block. Then interpolating thi two bound we get ut) ut ) α 1 T ρ1 ε) u ε C T,T C θ u 1 ε T ρ1 ε) u, u ) C ρ T,T L D α,θ,ρ with ε = def α 1 θ 1, 1). Therefore M T u, u ) C T,T C α 1 = u C T,T C α 1 T ρ1 ε) u, u ) D α,θ,ρ + u T θ and we can now move to the term involving the remainder M T u, u ). By definition, M T u, u ) i given by M T u, u ) = M T u, u ) J T f) u J T V ) = Ψ T + J T u V ) u J T V ) ) + J T u V ) + J T u V ) Now, by Schauder etimate we directly have T t) α 1 Ψ T t) α 1 T t) γ α u T γ which give the needed bound for the term Ψ T. Lemma 3.4 and the fact that α < θ imply T t) α 1 J T u V ) u J T V ) CT C α 1 T κ T t) α 1 u C T,T C θ + u C ρ T,T L ) V CT C γ

20 For J T u V ) ) we exploit once more Corollary.5 and Bony etimate, o that T t) α 1 J T u V )t) α 1 T θ+γ α 1 u C T,T C θ V C T C γ At thi point it remain only to bound the norm of the term J T u V ). Again by Corollary.5 and Propoition 3.8 we have T t) α 1 J T u V )t) α 1 T γ α up T t) α 1 u V t) γ 3 t [,T ] T γ α 1 F f CT L V 1 CT C γ V X γ ) 1 + u, u ) D α,θ,ρ where F = def {f C T L }. Now, putting all the previou etimate together we conclude the validity of the bound 3.1). Notice that the map u, u ) M T u, u ) Ψ T J T f) i linear and therefore 3.13) can be obtained by the previou computation, imply replacing u with u v and u with u v. At lat, thank to 3.13), we ee that there exit T = T V X γ ) > mall enough uch that the map M T i a trict contraction from D α,θ,ρ T,T,V into itelf. A in the Young cae, the previou propoition repreent the crucial technical tool through which we can tate and prove the following theorem. THEOREM 3.1. Let β 3, 1 ], 4 3 < θ < γ < β + and T >. Let S c be the operator aigning to every triplet u T, f, η) C γ C T C C T C the olution u C T C θ to equation 3.1). Then, there exit a locally Lipchitz continuou map S r : C γ C T L {V k, k = 1,..., d} ) X γ C T C θ that extend S c in the following ene S c u T, f, η)t) = S r u T, f, Kη))t), for all t T and u T, f, η) C γ C T C C T C. Moreover, for any ρ < θ 1, S r take value in C θ T L and S r C ρ T L. PROOF. A in the proof of Theorem 3. and thank to Propoition 3.9, we can apply Banach fixed point theorem and get the exitence of a unique olution u, u) D α,θ,ρ T,T,V to 3.1). Moreover, for T > fixed, T i independent on the terminal condition u T, hence we can iterate our fixed point procedure on [T T, T T ], [T 3T, T T ],... and extend our olution to the whole interval [, T ]. Since the olution u i obtained through a fixed point procedure on the pace of paracontrolled ditribution, it i well-known that it give rie to a continuou flow u T, f, V) S r u T, f, V) ee [17] for more detail). ) )

21 Let V be a mooth function and V = V, J T j V i ) V j ) j it enhancement. The algebraic expanion given by the equation 3.9) implie that the term u V, defined in Propoition 3.8, coincide with the uual product and, therefore, the olution u contructed via the fixed point argument outlined above correpond to the claical one by uniquene. Therefore the relation S c u T, f, V ) = S r u T, f, V, J T j V i ) V j )) i jutified, where we recall that S c i the flow of the equation 1 G V u = h ut, ) = u T and thi complete the proof of Theorem The Martingale Problem. In the previou ection, we olved the generator equation and Theorem 3. and 3.1 repreent the formal verion of what wa looely tated in Theorem 1.1. A it wa mentioned in the introduction, thi wa the firt tep we had to undertake in order to be able to formulate and prove wellpoedne for the SDE, formally given by 4.1) dx t = V t, X t )dt + db t, X = x where B i a d-dimenional Brownian motion, x a point in and V i a function of time taking value in C β, for β 3, ). Before proceeding, let u introduce a imple convention that collect under one name the rough and the Young regime. DEFINITION 4.1. Let β 3, ). We ay that V C T C β i a ground drift if either β 1, ) or β 3, 1 ] and that V can be lifted to an element V X γ, for ome γ < β +. We are now ready to formulate a uitable Stroock-Varadhan martingale problem for 4.1), namely DEFINITION 4.. Let T > and V C T C β be a ground drift according to Definition 4.1. Let Ω = C[, T ], ) and F = BC[, T ], )), the uual Borel σ-algebra on it. We ay that a probability meaure P on Ω, F), endowed with the canonical filtration F t ) t T, olve the martingale problem with generator G V tarting at x, if the canonical proce X t ω) = ωt) atifie the two following propertie 1. PX = x) = 1

22 . For every τ T, f C T L and every u τ C β+ the proce { } ut, X t ) f, X )d t [,τ] i a quare integrable martingale under P, where u i the olution of the generator equation 3.1) contructed in Theorem 3. and 3.1. The next theorem guarantee that the Stroock-Varadhan Martingale Problem formulated in the previou definition i indeed well-poed ee alo Theorem 1.). THEOREM 4.3. Let T > and V C T C β be a ground drift according to Definition 4.1. Then there exit a unique probability meaure P on Ω, F, F t ) t T ) which olve the martingale problem with generator G V tarting at x, for every x. Moreover, the canonical proce X t ω) = ωt) under P i trong Markov. PROOF. We will focu on the cae β 3, 1 ], the cae β > 1 being analogou. From now on we will take ρ, θ, γ) R 3 a in Theorem 3.1, V C T C β uch that there exit V n a mooth regularization of V for which, a n, KV n ) converge to V in H γ, where the operator K i defined according to Definition 3.6. Exitence: Let X n be the unique trong olution of the SDE 4.) dx n t = V n t, X n t )dt + db t, X = x. For i = 1,..., d, let u n = u n,1,..., u n,d ) be uch that for every i, u n,i i the unique olution of the equation G V n u n,i = V n,i, u T x) =. Take < < t < T and apply Itô formula to the proce {u n t, X n t )} t, o that u n t, X n t ) u n, X n ) = V n r, X n r )dr + = X n t X n B t B ) + u n r, X n r )db r u n r, X n r )db r where the lat equality i a direct conequence of the fact that X n olve 4.) by contruction. In order to prove tightne for the equence X n ) n we want to apply Kolmogorov criterion, therefore we need to bound the p-th moment of the increment of X n, uniformly in n. For p 1, by tandard propertie of the Brownian motion B and Burkholder-Davi-Gundy inequality, we obtain 4.3) E [ X n t X n p ] E [ u n t, X n t ) u n, X n ) p ] [ ) p/ ] + t p/ + E u n r, Xr n ) dr

23 3 Notice that the lat term of the previou can be bounded by u n r, X n r ) dr u n r,.) dr t ) u n C T C θ where we recall that θ > 1 and hence C θ 1 i continuouly embedded in L ). Adding and ubtracting u n, X t ), the firt ummand in 4.3) become u n t, X t ) u n, X ) u n t) u n ) + u n ) X n t X n Now, for the firt term we can exploit the regularity in time of our olution, while for the econd u n ) = u n T ) u n ) T ρ u n C ρ L, ince R we choe u n a the olution to the generator equation with zero terminal condition. d Since u n converge to the olution u contructed in the Theorem 3.1 in the topology of D α,θ,ρ T,V, each of the norm of un i bounded by the analogou of u and 4.3) become E[ X n t X n p ] t p θ u C θ T L + T pρ u p C ρ T L E[ X n t X n p ] + t p/ 1 + u p C T C θ ) At thi point, the bound 3.1) in Propoition 3.9 guarantee that it i poible to chooe T > uch that T 1 + V H γ ) 1. Pulling the econd ummand of the right hand ide to the left hand ide, we obtain E[ X n t X n p ] t p θ u C θ T L + t p/ 1 + u p C T C θ ) for all < < t < T, uniformly in n the right hand ide doe not depend on n anymore). Denote by X n,1 t) = X n T + t). Since T doe not depend on the initial condition x and the olution u i defined on the whole interval [, T ], we can repeat the previou argument o that E[ X n t+t Xn +T p ] = E[ X n,1 t X n,1 p ] t p/ for all, t T, uniformly in n. Now, when T t T we have that E[ X n t X n p ] p E[ X n t X n T p ]+E[ X n T Xn p ] T t p/ + T p/ t p/ Iterating the procedure over [T, 3T ], [3T, 4T ],..., we finally get up E[ Xt n X n p ] t p/ n for all, t T. At thi point, we can apply Kolmogorov criterion which implie tightne of the equence X n ) n in C[, T ], ).

24 4 It remain to how that every limiting proce olve our martingale problem. To thi purpoe, let X n ) n be a converging ubequence, τ T, f, u τ ) C T L C γ and u n be the olution to the generator equation G V n u n = f with terminal condition u τ. Applying Itô formula to u n t, Xt n ) we obtain Let Z n t u n t, X n t ) u n, x) f, X n )d = denote the left hand ide of the previou. Then u n, X n )db E Z n t T u n CT L T u C T L which implie that Z n t, t T ) i a bounded equence of quare integrable martingale. Now, ince for every n, Z n i a martingale, we have that 4.4) E[Z n t Z n )F X n r, r )] = hold for any continuou functional F : C[, ], ) R. At thi point, to complete the proof, we only need to pa to the limit in the previou equality. Let u oberve that, thank to the fact that X n converge in ditribution to X and u n, u n ) converge uniformly to u, u), alo Z n converge in ditribution to Z t = ut, X t ) u, x) f, X )d. Analogouly, Zt n Z n )F Xr n, r ) converge in ditribution to Z t Z )F X r, r ) and, ince Zt n, t T ) i a equence with uniformly bounded econd moment, which in particular implie that Zt n Z n )F Xr n, r )) n i a uniformly integrable family, we can interchange limit and expectation in the identity 4.4) by dominated convergence theorem and Skorohod repreentation theorem), o that at lat we get which prove the claim. E[Z t Z )F X r, r )] = Uniquene and trong Markov property: Let P 1 and P be two olution of the martingale problem tarting at x. Let f C[, T ], L )) and u be the olution of the generator equation G V u = f with zero terminal { condition. Since under both P 1 and P the canonical proce X i uch that ut, X t ) } t f, X )d i a martingale, we have u, x) = E Pi [ut, X T ) for i = 1,. Therefore, E P1 [ T T ] [ T ] f, X )d = E Pi f, X )d ] [ T ] f, X )d = E P f, X )d t [,T ]

25 Since the previou hold for every f C[, T ], L )), we conclude that the proce X ha the ame marginal under P 1 and P. By a traightforward adaptation of [8, Theorem 4.] the main difference lying on the fact that our generator i time-dependent, but that doe not affect the proof in any ene), we deduce that it ha the ame finite dimenional ditribution and it i Markov with repect to both probability meaure, which in turn guarantee uniquene. For the trong Markov property we need intead [8, Theorem 4.6 and 4.] Contruction of the Polymer Meaure. In thi ection we will contruct the o called polymer meaure in dimenion d =, 3 and how how to exploit the technique developed o far to prove Theorem 1.3. More concretely, our purpoe i to make ene of 5.1) Q T dω) = Z 1 exp T ) ξω )d W T dω) where W i the Wiener meaure on C[, T ], ), d =, 3, ξ a patial white noie on the d-dimenional toru T d independent of W, and Z i an infinite renormalization contant. Let u recall that the periodic pace Gauian white noie i a centered Gauian random field which formally atifie 5.) E[ξx)ξy)] = δx y) for any two point x, y T d, where, again, T d i the d-dimenional toru and d = or 3. A the covariance function in 5.) ugget, the white noie i too ingular for 5.1) to make ene. In order to have an expreion that we can manipulate, we conider a mollified verion of the noie, defined by 5.3) ξ ε = k Z d mεk)ˆξk)e k where {ˆξk)} k Z d i a family of tandard normal random variable with covariance E[ˆξk 1 )ˆξk )] = 1 {k1 = k }, e k i the Fourier bai L T d ) and m a mooth radial function with compact upport uch that m) = 1. Now, given ξ ε, let Q ε be the meaure defined by T Q ε T dω) = Zε 1 exp ) [ T ξ ε ω )d Wdω), Zε = E W exp and h ε : R + T d R be the local in time olution to the equation 5.4) t h ε = 1 hε + 1 hε + ξ ε c ε h, x) = )] ξ ε ω )d

26 6 where c ε i a contant that will be characterized in Theorem 6.1. For ξ ε mooth, h ε i known to exit and be regular, therefore the proce M ε t ω ) = h ε T, ω )dω M ε t ω) = h ε T, ω ) d where M i the quadratic variation of M, i clearly a quare integrable martingale. Giranov theorem then implie that, under the meaure defined by Q ε T dω) = exp M εt 1 ) M ε T Wdω), the canonical proce ha the ame law a the olution X ε to the SDE dx ε t = V ε t, X ε t )dt + db t, X = x when one chooe V ε t, x) to be h ε T t, x). But now, applying Itô formula to h ε T t, X ε t ) and recalling that h ε olve 5.4) we conlude that Q ε T dω) = Q ε T dω). At thi point we can take advantage of Theorem 4.3, whoe applicability i enured by the next propoition, which guarantee the exitence of a unique limiting meaure for the equence Q ε T ) ε and conequently for the equence Q ε T ) ε. PROPOSITION 5.1. Let h ε be the local in time olution to 5.4) for d =, 3 and V ε t, x) = h ε T t, x). Then, there exit T > uch that for all T T, V ε t, x) i a ground drift according to definition 4.1, i.e. we have 1. for d = the proce V ε converge almot urely in C[, T ], C β T )) for all β < to ome element V.. for d = 3 and all β < 1/ the proce KV ε ) converge almot urely in H β+ T 3 ) to ome element V X β+. Moreover in both cae the limit i independent of the choice of the mollifier m. REMARK 5.. Notice that we are applying Theorem 4.3 to ditribution defined on the toru and not on the full pace. Thi i completely harmle ince the pace C γ T d ) can be een a the pace of periodic ditribution lying in C γ ee alo [17, Appendix A] for a dicuion on thi apect). Let u tre the fact that the proof of Propoition 5.1 boil down to a well-poedne reult for the equation 5.5) t h = 1 h + 1 h + ξ, h, x) =.

27 In the one dimenional cae with ξ a pace-time white noie, the previou i nothing but the celebrated Kardar-Parii-Zhang equation [4], which wa uccefully tudied by M.Hairer in [19] and ubequently by M. Gubinelli and N.Perkowki in [16]. The regularity iue one encounter when dealing with the three dimenional verion are morally the ame thee author had to face and the technique we will exploit are omewhat imilar to their epecially to [16]). For the ake of completene, we will prove Propoition 5.1 pointing out the difficultie one ha to overcome and illutrating the main tep one need to undertake in order to olve 5.5), till keeping it a concie a poible and referring the intereted reader to the quoted paper. 6. A KPZ-type equation driven by a purely patial white noie. The aim of thi ection i to prove well-poedne of the KPZ-type equation, introduced in 5.5) to make ene of the polymer meaure with white-noie potential. We will focu on the three-dimenional cae, ince in dimenion the reult follow by analogou, but impler, argument. Let u conider the cae of non-zero initial condition, h, and write 5.5) in it mild formulation 6.1) ht) = P t h + I h )t) + Iξ) t def where P t = e 1 t i the heat flow, for a function f on, T ] T 3, If)t) = def P t f)d and ξ i the uual pace white noie on T 3, i.e. a centered Gauian random field whoe covariance function i formally given a in 5.). The problem with the previou equation lie in the fact that, ince a a random ditribution, ξ C θ T d ) for θ < d which in d = mean θ < 1 while in d = 3, θ < 3 ) tantard Schauder etimate ugget that the patial regularity of h cannot be better than θ + and therefore the non-linearity in 6.1), for both d = and 3, i not well-defined. Now, let u point out that the term determining the regularity of h i Iξ), o maybe, upon ubtracting it to the potential olution, what def remain i more regular. In other word, one define h 1 = h Iξ), derive the equation it hould olve and, a before, guee it regularity. For example, etting X = Iξ), h 1 hould atify h 1 t) = P t h + I X )t) + I X h 1 + h 1 )t) and it regularity hould be a the one of I X ). If it were well-poed, thi lat term would be θ + 4-Hölder in pace which i trictly greater than θ + o that indeed h 1 i more regular than h. While in dimenion thi i enough given that X and I X ) can be contructed and belong to the correct Beov-Hölder pace, all the other term atify Bony condition), it i till not ufficient in d = 3 and o, we proceed further in the expanion. 7

28 8 The problem i that after ubtracting a finite number of term, there will be no more gain in regularity and omething ele i needed in order to define the ill-poed product and conequently olve the equation. Thi i exactly the point in which the paracontrolled approach, a we will ee in what follow, enter the game. Now that we have given a heuritic idea of what i going on, let u be more formal. We begin by defining the object that will appear in our expanion. Let η be a mooth function and et 6.) X t η) = def Iη)t), X t η) = def I X )t), X t η) = def I X X)t), X t η) = def I X X)t), X t η) = def I X )t). A announced before, in cae η i the pace white noie, the previou tochatic procee are not analitically well-defined and we will have to exploit tochatic calculu tool in order to make ene of them and prove that they atify certain regularity requirement. Now, let h be the olution of 6.1) driven by η and v be given by v = def h Xη) X η) X η). Plugging thi expreion back into 6.1), we ee that v olve ) 6.3) vt) = P t h + 4X t η) + I v Xη) t) + R v t) where R v i defined a 6.4) R v t) = def X t η) + I X η) X η) + v ) + X η) + v ) ) t) At thi point, we will plit the analyi of the equation in two ditinct module. On one ide, with purely analytical argument, we will identify a uitable ubpace of the pace of ditribution, depending on the procee defined above, for which it i poible to make ene of the ill-poed operation in 6.3) and formulate a fixed point map that i continuou in thee data. On the other, we will exploit probabilitic technique to contruct uch procee tarting with a white noie ξ and prove they have the expected regularity, through a regularization procedure. NOTATION. From now on, all the function and ditribution we will conider will live on the d-dimenional toru. Since no confuion can occur, we will indicate the function pace with the ame notation introduced in Section, but the domain will not be but T d.