Stochastic transport equation with singular drift
aa r X i v : . [ m a t h . P R ] F e b STOCHASTIC TRANSPORT EQUATION WITH SINGULAR DRIFT
DAMIR KINZEBULATOV, YULIY A. SEM¨ENOV, AND RENMING SONG
Abstract.
We prove existence, uniqueness and Sobolev regularity of weak solution of the Cauchyproblem of the stochastic transport equation with drift in a large class of singular vector fields con-taining, in particular, the L d class, the weak L d class, as well as some vector fields that are not evenin L ε loc for any ε > Introduction
Throughout this paper we assume d ≥
3. Let B t be a Brownian motion in R d defined on aprobability space (Ω , F , P ) with respect to a complete and right-continuous filtration F t . Let ◦ denote the Stratonovich multiplication. Set L p ≡ L p ( R d ) ≡ L p ( R d , dx ), L p loc ≡ L p loc ( R d ) , W ,p ≡ W ,p ( R d ) , W ,p loc ≡ W ,p loc ( R d ) , C ∞ c ≡ C ∞ c ( R d ). We denote by k · k p → q the operator norm k · k L p → L q .The subject of this paper is the problem of existence, uniqueness and Sobolev regularity of weaksolution to the Cauchy problem for the stochastic transport equation (STE) du + b · ∇ udt + σ ∇ u ◦ dB t = 0 on (0 , ∞ ) × R d ,u | t =0 = f, (1)where u ( t, x ) is a scalar random field, σ = 0, f is in L p or W ,p , and b : R d → R d is in the class of form-bounded vector fields (see definition below), a large class of singular vector fields containing, inparticular, vector fields b with | b | ∈ L d , or with | b | in the weak L d class, as well as some vector fields b with | b | 6∈ L ε loc for any ε > ∂ t u + b · ∇ u = 0(corresponding to σ = 0 in (1)) is in general not well posed already for a bounded but discontinuous b .Moreover, in that case, even if the initial function f is regular, one can not hope that the correspondingsolution u will be regular immediately after t = 0. This, however, changes if one adds the noise term σ ∇ u ◦ dB t , σ >
0. For the stochastic STE (1), a unique weak solution exists and is regular forsome discontinuous b . This effect of regularization and well-posedness by noise, demonstrated by theSTE, attracted considerable interest in the past few years, as a part of the more general programof establishing well-posedness by noise for SPDEs whose deterministic counterparts arising in fluiddynamics are not well-posed, see [BFGM, GM] for detailed discussions and further references.In [BFGM], the authors establish existence, uniqueness and Sobolev W ,p -regularity (up to theinitial time t = 0, with p large) for weak solutions of (1) with time-dependent drift b satisfying | b ( · , · ) | ∈ L q (cid:0) [0 , ∞ ) , L r + L ∞ (cid:1) , dr + 2 q Mathematics Subject Classification. (actually, [BFGM] allows b = b + b with b satisfying the condition above and b being continu-ously differentiable with at most linear growth at infinity; their uniqueness result imposes additionalassumptions on div b ). They apply this result to study the SDE X t = x − Z ts b ( r, X r ) dr + σ ( B t − B s ) , (2)constructing, in particular, a unique, W ,p -regular stochastic Lagrangian flow that solves (2) fora.e. x ∈ R d . The STE can be viewed as the equation behind both the SDE (via path-wise interpretationof the STE and the SDE, see [BFGM]) and the parabolic equation ( ∂ t − σ ∆ + b · ∇ ) v = 0 (arisingfrom (1) upon taking expectation, i.e. v = E [ u ], see, if needed, (8) below).In this paper, we show that the regularity and well-posedness for (1) hold for a much larger classof drifts b , at least in the time-independent case b = b ( x ) (see, however, Remark 2 below concerningtime-dependent b ). Definition . A Borel vector field b : R d → R d is said to be form-bounded with relative bound δ > b ∈ F δ , if | b | ∈ L and there exists a constant λ = λ δ ≥ k| b | ( λ − ∆) − k → ≤ √ δ. It is easily seen that the condition b ∈ F δ can be stated equivalently as a quadratic form inequality k bϕ k ≤ δ k∇ ϕ k + c δ k ϕ k , ϕ ∈ W , , for a constant c δ (= λδ ). Let us also note that b ∈ F δ , b ∈ F δ ⇒ b + b ∈ F δ , √ δ := p δ + p δ . Examples.
1. Any vector field b ∈ L d ( R d , R d ) + L ∞ ( R d , R d )is in F δ for δ > ε > b = f + h with k f k d < ε , h ∈ L ∞ ( R d , R d ). It follows from H¨older’s inequality and the Sobolev embedding theoremthat for any g ∈ L , k| b | ( λ − ∆) − g k ≤ k f k d k ( λ − ∆) − g k dd − + k h k ∞ λ − k g k ≤ c k f k d k g k + k h k ∞ λ − k g k ≤ ( c + 1) ε k g k for λ = ε − k h k − ∞ .
2. The class F δ also contains vector fields having critical-order singularities, such as b ( x ) = ±√ δ d − | x | − x (by Hardy’s inequality ( d − k| x | − ϕ k ≤ k∇ ϕ k , ϕ ∈ W , ).3. More generally, the class F δ contains vector fields b with | b | in L d,w (the weak L d space). Recallthat a Borel function h : R d → R is in L d,w if k h k d,w := sup s> s |{ x ∈ R d : | h ( x ) | > s }| /d < ∞ . TOCHASTIC TRANSPORT EQUATION 3
By the Strichartz inequality with sharp constant [KPS, Prop. 2.5, 2.6, Cor. 2.9], if | b | in L d,w , then b ∈ F δ with p δ = k| b | ( λ − ∆) − k → ≤ k b k d,w Ω − d d k| x | − ( λ − ∆) − k → ≤ k b k d,w Ω − d d d − , where Ω d = π d Γ( d + 1) is the volume of the unit ball in R d .We also note that if h ∈ L ( R ), T : R d → R is a linear map, then the vector field b ( x ) = h ( T x ) e ,where e ∈ R d , is in F δ with appropriate δ , but | b | may not be in L d,w loc .4. More generally, the class F δ contains vector fields in the Campanato-Morrey class and the Chang-Wilson-Wolff class, with δ depending on the respective norms of the vector field in these classes, see[CWW].5. We note that there exists b ∈ F δ such that | b | 6∈ L ε loc ( R d , R d ) for any ε >
0, e.g., consider | b ( x ) | = C B (0 , α ) − B (0 , − α ) (cid:12)(cid:12) | x | − (cid:12)(cid:12) − ( − ln (cid:12)(cid:12) | x | − (cid:12)(cid:12) ) β , β > , < α < . We emphasize that the condition b ∈ F δ is not a refinement of | b | ∈ L d + L ∞ in the sense that F δ is not situated between L d + L ∞ and L p + L ∞ , p < d . In contrast to the elementary sub-classes of F δ listed above, the class F δ is defined in terms of the operators that, essentially, constitute the equationin (1).The key result of this paper is the Sobolev regularity of solutions u to the Cauchy problem for theSTE (1): sup t ∈ [0 ,T ] (cid:13)(cid:13) E |∇ u | q (cid:13)(cid:13) ≤ C k∇ f k q q , q = 1 , , . . . , (3)provided that b is in F δ with δ smaller than a certain explicit constant, see Theorem 2. This is astochastic (parabolic) counterpart of the Sobolev regularity estimates for solutions of the correspondingdeterministic elliptic equation established in [KS]. More precisely, in [KS] the authors consider theoperator − ∆ + b · ∇ , b ∈ F δ with 0 < δ < ∧ (cid:0) d − (cid:1) , d ≥ v to the elliptic equation ( µ − ∆ + b · ∇ ) v = f in L q for 2 ∨ ( d − ≤ q < √ δ : k∇ v k qdd − ≤ K k f k q , (4)with K depending only on d , q , the relative bound δ and c δ . The estimate (4) is needed in [KS] torun a Moser-type iteration procedure that yields the Feller semigroup corresponding to − ∆ + b · ∇ .It was established in [KiS2] that, given b ∈ F δ with δ < ∧ (cid:0) d − (cid:1) , this Feller semigroup determines,for every starting point x ∈ R d , a weak solution to the SDE X t = x − Z t b ( X r ) dr + √ B t (5)(see also [KiS] where the authors consider drifts in a larger class).The approach to studying SDEs via regularity theory of the STE, developed in [BFGM], can becombined with Theorem 2 to obtain strong existence and uniqueness for (2) with b ∈ F δ (cf. Remark1 below), albeit potentially excluding a measure zero set of starting points x ∈ R d . For results on DAMIR KINZEBULATOV, YULIY A. SEM¨ENOV, AND RENMING SONG strong existence and uniqueness for any x ∈ R d , with b satisfying (in the time-independent case) | b | ∈ L p + L ∞ with p > d or p = d , see [Kr1, Kr2, KrR].We conclude this introduction with a few remarks concerning the criticality of the singularities ofform-bounded drifts.1. In [BFGM, Sect. 7], the authors show that the SDE (5) with drift b ( x ) = β | x | − x and startingpoint x = 0 does not have a weak solution if β > d −
2. In view of Example 2 above, this drift b belongs to F δ with √ δ = β d − , so by the result of [KiS2] cited above, the weak solution to (5) with x = 0 exists as long as β > β < if d = 3, β < d ≥ d ≥ β < d − using [KiS3, Corollary 4.10]). Thus, the weak well-posedness of (5) is sensitive tochanges in the value of the constant multiple β of b (equivalently, changes in the value of the relativebound δ ). In this sense, the singularities of b ∈ F δ are critical.Let us note that the diffusion process with drift b ( x ) = c | x | − x , c ∈ R , was studied earlier in [W].2. Let b ∈ F δ . There is a quantitative dependence between the value of the relative bound δ andthe regularity properties of solutions to the corresponding equations (PDEs or STEs). Indeed, theadmissible values of q in (4), as well as in (3), depend on the value of δ . This dependence is lost if oneconsiders b with | b | ∈ L d + L ∞ since any such b has arbitrarily small relative bound, cf. Example 1.3. Concerning the difference between classes F δ and its subclass L d + L ∞ , let us also note thefollowing: if v is a weak solution of the elliptic equation ( λ − ∆ + b · ∇ ) v = f , λ > f ∈ C ∞ c with | b | ∈ L d + L ∞ and v ∈ W ,r for r large (e.g. by (4)), then, by H¨older’s inequality,∆ v ∈ L rdd + r loc . However, for b ∈ F δ , one can only say that (cf. Example 5 above)∆ v ∈ L dd +2 loc (one can in fact show that v ∈ W , ). That is, in case b ∈ F δ , there are no W ,p estimates on solution v for p large.See [KiS3] for detailed discussions of remarks 2 and 3 above. Notations.
Denote h f, g i = h f g i := Z R d f gdx (all functions considered below are assumed to be real-valued).Set ρ ( x ) ≡ ρ κ,θ ( x ) := (1 + κ | x | ) − θ , κ > , θ > d , x ∈ R d . It is easily seen that |∇ ρ ( x ) | ≤ θ √ κρ ( x ) , x ∈ R d . (6)Below we will be applying (6) to ρ with κ chosen sufficiently small.For any p >
1, we use p ′ to denote its conjugate p/ ( p − L pρ ≡ L p ( R d , ρdx ). Denote by k · k p,ρ the norm in L pρ , and by h· , ·i ρ the inner product in L ρ .Set W , ρ := { g ∈ W , | k g k W , ρ := k g k ,ρ + k∇ g k ,ρ < ∞} . TOCHASTIC TRANSPORT EQUATION 5
Define constants β q := 1 + 4 qd, q = 1 , , . . . Put J T := [0 , T ]. 2. Main results
Below we consider the Cauchy problem for the STE du + µ udt + b · ∇ udt + σ ∇ u ◦ dB t = 0 on (0 , ∞ ) × R d ,u | t =0 = f ∈ L p , p ≥ , (CP)where µ ≥
0. Since solutions of the Cauchy problems (1) and (CP) will differ by a multiple e − µt , itsuffices to prove the well-posedness of (CP).Let us first make a few preliminary remarks.1. We can rewrite the equation in (CP), using the identity relating Stratonovich and Itˆo integrals Z t ∇ u ◦ dB s = Z t ∇ udB s − d X k =1 [ ∂ x k u, B k ] t , B t = ( B kt ) dk =1 , (7)as du + µudt + b · ∇ udt + σ ∇ udB t − σ u = 0 . (8)2. If b ∈ C ∞ c ( R d , R d ) and f ∈ C ∞ c , then (see [Ku, Theorem 6.1.9]) there exists a unique adaptedstrong solution of (CP) u ( t ) − f + µ Z t uds + Z t b · ∇ uds + σ Z t ∇ u ◦ dB s = 0 a.s. , t ∈ J T , given by e − µt u ( t ) = f (Ψ − t ) , t > , (9)where Ψ t : R d × Ω → R d is the stochastic flow for the SDE X t = x − Z t b ( X r ) dr + σB t , (10)i.e. there exists Ω ⊂ Ω, P (Ω ) = 1, such that, for all ω ∈ Ω , Ψ t ( · , ω )Ψ s ( · , ω ) = Ψ t + s ( · , ω ), Ψ ( x, ω ) = x , and1) for every x ∈ R d , the process t Ψ t ( x, ω ) is a strong solution of (10),2) Ψ t ( x, ω ) is continuous in ( t, x ), Ψ t ( · , ω ) : R d → R d are homeomorphisms, and Ψ t ( · , ω ), Ψ − t ( · , ω ) ∈ C ∞ ( R d , R d ).We first state our basic existence result. Recall that b ∈ F δ if k bϕ k ≤ δ k∇ ϕ k + c δ k ϕ k , ϕ ∈ W , , for some constant c δ ≥ DAMIR KINZEBULATOV, YULIY A. SEM¨ENOV, AND RENMING SONG
Theorem 1.
Assume that d ≥ , b ∈ F δ with √ δ < σ β . Let T > , p ≥ . Provided that κ is chosensufficiently small, there are constants µ (cid:0) δ, c δ , p (cid:1) ≥ , C = C ( δ, c δ , p ) > and C = C ( δ, c δ , p, T ) > such that for any µ ≥ µ (cid:0) δ, c δ , p (cid:1) , for every f ∈ L p there exists a function u ∈ L ∞ ( J T , L (Ω , L ρ )) for which the following are true. ( i ) sup t ∈ J T k E u ( t ) k p ≤ k f k p , Z J T k∇ v p k ds ≤ C k f k p p , (11) E (cid:10) ρ (cid:12)(cid:12) ∇ Z J T uds (cid:12)(cid:12) (cid:11) ≤ C k f k p , (12) where v := E u and v p := v | v | p − , so, in particular, for a.e. ω ∈ Ω , ∇ R T u ( s, · , ω ) ds ∈ L ( R d , R d ) and hence b · ∇ Z J T u ( s, · , ω ) ds ∈ L , and, for every test function ϕ ∈ C ∞ c , we have a.s. for all t ∈ J T , h u ( t ) , ϕ i − h f, ϕ i + µ h Z t uds, ϕ i + (cid:10) b · ∇ Z t uds, ϕ (cid:11) − σ (cid:10)Z t udB s , ∇ ϕ (cid:11) + σ (cid:10) ∇ Z t uds, ∇ ϕ (cid:11) = 0 . (13)( ii ) For any sequence of smooth vector fields b m ∈ C ∞ c ( R d , R d ) , m = 1 , , . . . , that are uniformlyform-bounded in the sense that b m ∈ F δ with c δ independent of m , and are such that b m → b in L ( R d , R d ) as m → ∞ , we have for initial functions f ∈ C ∞ c , u m ( t ) → u ( t ) in L (Ω , L ρ ) uniformly in t ∈ J T , where u m is the unique strong solution to (CP) ( with b = b m ) . An example of such smooth approximating vector fields { b m } is given in the next section.The next theorem establishes the Sobolev regularity of u up to the initial time t = 0. Theorem 2.
Assume that d ≥ , b ∈ F δ with √ δ < σ β and f ∈ W , . Let κ be sufficiently smalland µ ( δ, c δ , be the constant in Theorem 1 with p = 2 . For µ ≥ µ ( δ, c δ , , let u be the processconstructed in Theorem 1. There exists µ ( δ, c δ ) ≥ µ ( δ, c δ , such that for µ ≥ µ ( δ, c δ ) , the followingare true. (a) E u , E |∇ u | ∈ L ∞ ( J T , L ) , so u ∈ L ∞ ( J T , L (Ω , W , ρ )) ; (b) for any test function ϕ ∈ C ∞ c , the process t
7→ h u ( t ) , ϕ i is ( F t ) -progressively measurable and hasa continuous ( F t ) -semi-martingale modification that satisfies a.s. for every t ∈ J T , h u ( t ) , ϕ i − h f, ϕ i + µ Z t h u, ϕ i ds + Z t (cid:10) b · ∇ u, ϕ (cid:11) ds − σ Z t h u, ∇ ϕ i dB s + σ Z t (cid:10) u, ∆ ϕ (cid:11) ds = 0 . (14) Moreover, if √ δ < σ β q for some q = 1 , , . . . , then there exists constants µ ( δ, c δ , q ) ≥ µ ( δ, c δ , q ) (with µ ( δ, c δ , equal to the µ ( δ, c δ ) above) and C = C ( δ, c δ , q ) > such that when µ ≥ µ ( δ, c δ , q ) TOCHASTIC TRANSPORT EQUATION 7 and f ∈ W , q , we have sup ≤ α ≤ (cid:13)(cid:13) E |∇ u | q (cid:13)(cid:13) L − α ( J T ,L dd − α ) ≤ C k∇ f k q q . (15) In particular, there exists C > such that sup t ∈ J T E h ρ |∇ u | q i ≤ C k∇ f k q q . (16) If q > d , then for a.e. ω ∈ Ω , t ∈ J T , the function x u ( t, x, ω ) is H¨older continuous, possibly aftermodification on a set of measure zero in R d (in general, depending on ω ). Theorem 3.
Assume that d ≥ , b ∈ F δ with √ δ < σ β and f ∈ W , . Provided κ is sufficientlysmall, there exists µ = µ ( δ, c δ ) ≥ such that for µ ≥ µ ( δ, c δ ) , (CP) has a unique solution in theclass of functions satisfying (a) , (b) of Theorem 2. A function satisfying (a), (b) of Theorem 2 will be called a weak solution of (CP). This definitionof weak solution is close to [BFGM, Definition 2.13]. It should be noted however that the authors in[BFGM] prove their uniqueness result, in the time-dependent case, in a larger class of weak solutions(not requiring any differentiability, see [BFGM, Definition 3.3]) but under additional assumptions on b . Specialized to the time-dependent case, they assume that b satisfiesdiv b ∈ L d + L ∞ (17)in addition to | b | ∈ L d + L ∞ . The latter is needed to establish (15) for solutions of the adjoint equationto the STE, i.e. the stochastic continuity equation (which allows to prove an even stronger result: theuniqueness of weak solution to the corresponding random transport equation), see [BFGM, Sect. 3].We expect that an analogue of (17) for b ∈ F δ can be found with some additional effort. However,we will not address this matter in this paper. Of course, in the case b ∈ F δ , div b = 0, one has (15) forsolutions to the stochastic continuity equation, so one can prove the uniqueness for (CP) by repeatingthe argument in [BFGM, Sect. 3].The proof of the uniqueness result in Theorem 3 (see Section 6) adopts the method of [BFGM,Sect. 3]. Remark 1 (On applications to SDEs) . Armed with Theorems 1 and 2, one can repeat the argumentin [BFGM, Sect. 4] to prove the following result. Assuming that b ∈ F δ with δ sufficiently small, thereexists a stochastic Lagrangian flow for SDE (10), i.e. a measurable map Φ : J T × R d × Ω → R d suchthat, for a.e. x ∈ R d , the process t Φ t ( x, ω ) is a strong solution of the SDE (10):Φ t ( x, ω ) = x − Z t b ( s, Φ r ( x, ω )) dr + σB t ( ω ) , a.s. , t ∈ J T , (18)and Φ t ( x, · ) is F t -progressively measurable. If also √ δ < σ β q , q = 1 , , . . . , then Φ t ( · , ω ) ∈ W , q loc ( t ∈ J T ) for a.e. ω ∈ Ω. Moreover, Φ t is unique, i.e. any two such stochastic flows coincide a.s. forevery t > x . Remark 2 (STE with time-dependent b ) . The proof of the key result of this paper (Proposition 2below, i.e. a priori Sobolev regularity of solutions of the STE) carries over, without change, to thetime-dependent form-bounded vector fields:
DAMIR KINZEBULATOV, YULIY A. SEM¨ENOV, AND RENMING SONG
Definition . A vector field b ∈ L (cid:0) [0 , ∞ ) × R d , R d (cid:1) is said to be form-bounded with relative bound δ >
0, written as b ∈ e F δ , if | b | ∈ L ([0 , ∞ ) × R d ) and Z ∞ k b ( t, · ) φ ( t, · ) k dt δ Z ∞ k∇ φ ( t, · ) k dt + Z ∞ g ( t ) k φ ( t, · ) k dt for some g = g δ ∈ L [0 , ∞ ), for all φ ∈ C ∞ c ([0 , ∞ ) × R d ).The class e F δ contains vector fields | b ( · , · ) | ∈ L q (cid:0) [0 , ∞ ) , L r + L ∞ (cid:1) , dr + 2 q , with δ that can be chosen arbitrarily small (using H¨older’s inequality and the Sobolev embeddingtheorem). Another example is | b ( t, x ) | c | x − x | − + c | t − t | − (cid:0) log( e + | t − t | − ) (cid:1) − − ε , ε > , ( t, x ) ∈ [0 , ∞ ) × R d , which belongs to the class e F δ with δ = c (2 / ( d − (using Hardy’s inequality).We plan to address the regularity theory of the STE with b ∈ e F δ elsewhere.3. A priori estimates
Assume b ∈ F δ . In the remainder of this paper, we fix some b m ∈ C ∞ c ( R d , R d ) such that b m → b in L ( R d , R d ) as m → ∞ and for every m = 1 , , . . . k b m ϕ k ≤ δ k∇ ϕ k + c δ k ϕ k , ϕ ∈ W , with c δ independent of m (see example of such b m below). Let f ∈ C ∞ c . Let u m be the unique strongsolution to u m ( t ) − f + µ Z t u m ds + Z t b m · ∇ uds + σ Z t ∇ u m ◦ dB s = 0 a.s. , t ∈ J T = [0 , T ] . (19)Then, by [Ku, Section 6.1], for any p, r ≥ α = ( α , . . . , α d ) of non-negativeintegers, E ( | D α u m | p ) ∈ L ∞ ( J T × R d )and Z R d (1 + | x | r ) (cid:0) E | u m | p + E |∇ u m | p (cid:1) dx ∈ L ∞ ( J T ) . Remark 3 (Example of { b m } ) . Denote by m the indicator of {| x | ≤ m, | b ( x ) | ≤ m } , and by η m ∈ C ∞ c a [0 , η m = 1 on B (0 , m ). Consider b m := η m e ǫ m ∆ ( m b ) , ( ∗ )where ǫ m ↓ { γ m } ↓ { ǫ m } ↓ b m so that b m ∈ F δ m with δ m = ( √ δ + √ γ m ) ↓ δ and c δ m ≤ c δ starting from some m on . TOCHASTIC TRANSPORT EQUATION 9
Since b ∈ F δ , there exists λ ≥ k| b | ( λ − ∆) − k → ≤ √ δ . Then c δ = λδ . We claim that,we can select { ǫ m } ↓ k| b m | ( λ − ∆) − k → ≤ p δ m . ( ∗∗ )Once this claim is proven, we will have c δ m = λδ m ≤ c δ starting from some m on, which implies therequired. Now we prove the claim. We have b m = m b + ( b m − m b ) , where, clearly, k| m b | ( λ − ∆) − k → ≤ √ δ for every m , while b m − m b ∈ L d . It follows from H¨older’sinequality and the Sobolev embedding theorem that for any g ∈ L , k| b m − m b | ( λ − ∆) − g k ≤ k b m − m b k d k ( λ − ∆) − g k dd − ≤ c k b m − m b k d k g k . It is easily seen that, for every m , the norm k b m − m b k d can be made smaller than c − γ m by selecting { ǫ m } ↓ k ( b m − m b )( λ − ∆) − k → ≤ γ m . Now ( ∗∗ ) follows.Finally, to have b m form-bounded with the original relative bound δ , it suffices to multiply b m in( ∗ ) by δδ m . (Although, to carry out the proofs of Theorems 1-3, the last step is not necessary since allour assumptions on δ are strict inequalities.)We prove the next proposition under more general assumptions on δ and p than in Theorem 1. Proposition 1.
Let b ∈ F δ with √ δ < σ . Let T > , p ∈ ( p c , ∞ ) , p c := (cid:0) − √ δσ (cid:1) − . Let f ∈ C ∞ c , let b m and u m be as above. There exist constants µ ( δ, c δ , p ) ≥ , C = C ( δ, c δ , p ) > and C = C ( δ, c δ , p, T ) > independent of m such that for any µ ≥ µ (cid:0) δ, c δ , p (cid:1) and m = 1 , , . . . , thefollowing are true: ( i ) sup t ∈ J T k E u m ( t ) k p ≤ k f k p , Z J T k∇ v p k ds ≤ C k f k p p , ( E ) where v := E u and v p := v | v | p − ; ( ii ) if √ δ < σ , then E (cid:10) ρ (cid:18) ∇ Z J T u m ( s ) ds (cid:19) (cid:11) ≤ C k f k p . ( E ) Proposition 2.
Let b ∈ F δ and f ∈ C ∞ c , let b m and u m be as above. For every q ≥ , thereexists constants µ ( δ, c δ , q ) ≥ and C = C ( δ, c δ , q ) > independent of m such that if √ δ < σ β q and µ ≥ µ ( δ, c δ , q ) , then sup ≤ α ≤ (cid:13)(cid:13) E |∇ u m | q (cid:13)(cid:13) L − α (cid:0) [0 ,T ] ,L dd − α (cid:1) ≤ C k∇ f k q q . ( E ) Proof of Proposition 1.
For brevity, we write u for u m in this proof. The identity (7) allows us torewrite (19) as u ( t, · ) − f + µ Z t uds + Z t b m · ∇ uds + σ Z t ∇ udB s − σ Z t ∆ uds = 0 a.s., t ∈ J T . (20) Below we will be appealing to (20).We first prove ( E ). Applying Itˆo’s formula to u , we obtain, in view of (20), u ( t ) − f = − µ Z t u ds − Z t b m · ∇ u ds − σ Z t ∇ u dB s + σ Z t ∆ u ds. Since t R t ∇ u dB s is a martingale, v = E u satisfies ∂ t v = − µv − b m · ∇ v + σ v, v (0) = f . We multiply the last equation by v | v | p − and integrate by parts (recall that v p = v | v | p − ),1 p ∂ t h| v p | i + 2 µ h| v p | i + 4 pp ′ σ h|∇ v p | i − p h b m · ∇ v p , v p i ≤ , so applying the quadratic inequality we have (for ε > ∂ t h| v | p i + 2 pµ h| v | p i + 2 σ p ′ h|∇ v p | i − (cid:18) ε h|∇ v p | i + 14 ε h b m v p i (cid:19) ≤ . Finally, by our assumption on b m , ∂ t h| v | p i + 2 pµ h| v | p i + 2 σ p ′ h|∇ v p | i − (cid:18) ε h|∇ v p | i + δ ε h|∇ v p | i + c δ ε h| v | p i (cid:19) ≤ . Taking ε = √ δ in the last inequality and integrating with respect to t , we obtain for t > h| v ( t ) | p i + 2 (cid:18) σ p ′ − √ δ (cid:19) Z t h|∇ v p | i ds + (cid:20) pµ − c δ √ δ (cid:21) Z t h| v | p i ds ≤ k f k pp , where σ p ′ − √ δ > p > p c . Taking µ ≥ c δ √ δp , we arrive at ( E ).Now we deal with ( E ). Let µ ≥ c δ √ δp as above. By ( E ),sup t ∈ J T (cid:10) ρ E u ( t ) (cid:11) ≤ k ρ k p ′ sup t ∈ J T k E u ( t ) k p ≤ c k f k p , (21)since θ > d in the definition of ρ .We multiply (20) by ρ R t uds , integrate, and take expectation, to get E (cid:10) ρ Z t uds, u ( t ) (cid:11) = E (cid:10) ρ Z t uds, f (cid:11) − E (cid:10) ρ Z t uds, b m · ∇ Z t uds (cid:11) (22) − σ E (cid:10) ρ Z t uds, Z t ∇ udB s (cid:11) + σ E (cid:10) ρ Z t uds, Z t ∆ uds (cid:11) + µ E (cid:10) ρ Z t uds, Z t uds (cid:11) =: I + I + I + I + I . Denote the left-hand side of (22) by I . Set U := Z t uds. By H¨older’s inequality and (21), E (cid:10) ρU (cid:11) ≤ t (cid:10) ρ Z t E u ds (cid:11) ≤ t c k f k p . (23) TOCHASTIC TRANSPORT EQUATION 11
Integrating by parts in I and using the quadratic inequality, we have2 σ I = − E (cid:10) ρ |∇ U | (cid:11) − E (cid:10) U ∇ ρ, ∇ U (cid:11) ≤ − E (cid:10) ρ |∇ U | (cid:11) + αE (cid:10) |∇ ρ | U (cid:11) + 14 α E (cid:10) |∇ ρ ||∇ U | (cid:11) ( α > ≤ − (cid:18) − θ √ κ α (cid:19) E (cid:10) ρ |∇ U | (cid:11) + θ √ καT c k f k p . Substituting the last estimate into (22), we obtain σ (cid:18) − θ √ κ α (cid:19) E (cid:10) ρ |∇ U | (cid:11) ≤ σ θ √ καT c k f k p + | I | + | I | + | I | + | I | + | I | . (24)We now estimate | I i | , i = 0 , , , ,
5. By (21) and (23), | I | ≤ (cid:0) E (cid:10) ρU (cid:11)(cid:1) (cid:0) E (cid:10) ρu ( t ) (cid:11)(cid:1) ≤ c k f k p . Similarly, | I | ≤ c k f k p , | I | ≤ µc k f k p . Next, applying the quadratic inequality, we get | I | ≤ ν E (cid:10) ρb m U (cid:11) + 14 ν E (cid:10) ρ |∇ U | (cid:11) ( ν > (cid:0) in the first term, we apply b m ∈ F δ with ϕ := √ ρU ) ≤ ν (cid:0) δ E h|∇ ( √ ρU ) | i + c δ E h ρU i (cid:1) + 14 ν E (cid:10) ρ |∇ U | (cid:11) (in the first term, we use ( a + c ) ≤ (1 + ǫ ) a + (1 + 1 ǫ ) c , ǫ > ≤ νδ (1 + ǫ ) E (cid:10) |√ ρ ∇ U | (cid:11) + νδ (cid:0) ǫ (cid:1) E (cid:10) | U ∇√ ρ | (cid:11) + νc δ E h ρU i + 14 ν E (cid:10) ρ |∇ U | (cid:11) (in the second term, we apply (6) and then use (23); also, we apply (23) in the last term) ≤ (cid:18) νδ (1 + ǫ ) + 14 ν (cid:19) (cid:10) ρ |∇ U | (cid:11) + T c k f k p , c = c ( ν, δ, c δ , θ, κ, ǫ ) . In the current setting, we have R t ∇ udB s = ∇ R t udB s (see, for instance, [HN]). Thus, integrating byparts, we obtain I = σ E (cid:10) ρ ∇ U, Z t udB s (cid:11) + σ E (cid:10) U ∇ ρ, Z t udB s (cid:11) , so | I | ≤ σ (cid:0) E (cid:10) ρ |∇ U | (cid:11)(cid:1) (cid:0) E (cid:10) ρ (cid:18)Z t udB s (cid:19) (cid:11)(cid:1) + σ (cid:0) E (cid:10) |∇ ρ | U (cid:11)(cid:1) (cid:0) E (cid:10) |∇ ρ | (cid:0)Z t udB s (cid:1) (cid:11)(cid:1) (we use (6) and apply the Itˆo isometry) ≤ σ (cid:16) E (cid:10) ρ |∇ U | (cid:11)(cid:17) (cid:0) E (cid:10) ρ Z t u ds (cid:11)(cid:1) + θ √ κσ (cid:0) E (cid:10) ρU (cid:11)(cid:1) (cid:18) E (cid:10) ρ Z t u ds (cid:11)(cid:19) (we apply the quadratic inequality in the first term and then use (23)) ≤ σγ E (cid:10) ρ |∇ U | (cid:11) + σT c γ k f k p + θ √ κσT c k f k p ( γ > . Substituting the above estimates on | I | , | I | , | I | , | I | and | I | in (24), we obtain (cid:18) σ − νδ (1 + ǫ ) − ν − σγ − σ θ √ κ α (cid:19) E (cid:10) ρ |∇ U | (cid:11) ≤ c k f k p for an appropriate constant c = c ( α, γ, ν, δ, θ, κ, ǫ, c δ , µ ) < ∞ . Take ν = (2 √ δ ) − . Since √ δ < σ byassumption, we can select γ , ǫ sufficiently small and α sufficiently large so that σ − (cid:0) νδ + 14 ν (cid:1) − νδε − σγ − σ θ √ κ α > , and thus ( E ) follows with constant C = c (cid:0) σ − νδ (1 + ǫ ) − ν − σγ − σ θ √ κ α (cid:1) − . (cid:3) Remark 4.
In Proposition 1, the interval ( p c , ∞ ) of admissible values of p decreases to the empty setas √ δ ↑ σ . In fact, one can show that if b ∈ F δ , √ δ < σ and b m ∈ C ∞ c are as above, then the limit s - L p - lim m e − t Λ m (loc. uniformly in t ≥ , p > p c , where Λ m = − σ ∆ + b m · ∇ , D (Λ m ) = W ,p , exists and determines a L ∞ contraction, quasi contrac-tion holomorphic semigroup in L p , say, e − t Λ , see [KiS3, Theorems 4.2, 4.3]. The operator Λ is anappropriate operator realization of the formal operator − σ ∆ + b · ∇ in L p . One can compare thisresult with the example in [BFGM, Sect. 7], where the authors show that the SDE X t = − Z t b ( X s ) ds + σB t , b ( x ) = √ δ d − | x | − x ∈ F δ , corresponding to operator − σ ∆ + b · ∇ , does not have a weak solution if √ δ > σ . Proof of Proposition 2.
For any multiindex I with entries in { , . . . , d } , i.e., an element of { , . . . , d } ×· · · × { , . . . , d } , say, p times, we write | I | = p . For any such multiindex I and l ∈ { , . . . , d } , wedenote by I − l the multiindex obtained from I by dropping an index of value l . Let I − l + k bethe multiindex I with an index of value l dropped and replaced with an index of value k . It does notmatter from which component the value l is dropped. TOCHASTIC TRANSPORT EQUATION 13
For brevity, we write u for u m in this proof. Set w r := ∂ x r u, ≤ r ≤ d, where u is the strong solution of (19), and w I := Y r ∈ I ∂ x r u. Step 1.
We apply Itˆo’s formula in Stratonovich form to w I , obtaining w I ( t ) − Y r ∈ I ∂ x r f = X r ∈ I Z t w I − r ( s ) ◦ dw r ( s ) . Next, differentiating (20) in x r and then substituting the resulting expression for dw r into the previousformula, we obtain w I ( t ) − Y r ∈ I ∂ x r f = − µ Z t w I ds − X r ∈ I Z t w I − r (cid:0) b m · ∇ w r + ∂ x r b · ∇ u (cid:1) ds − σ X r ∈ I Z t w I − r ∇ w r ◦ dB s . Let b km , k = 1 , . . . , d , be the components of the vector field b m . We have w I ( t ) − Y r ∈ I ∂ x r f = − µ Z t w I ds − X r ∈ I Z t w I − r (cid:0) b m · ∇ w r + ∂ x r b m · ∇ u (cid:1) ds − σ Z t ∇ w I ◦ dB s (we use Z t ∇ w I ◦ dB s = Z t ∇ w I dB s − d X k =1 [ ∂ x k w I , B k ] t )= − µ Z t w I ds − X r ∈ I Z t w I − r (cid:0) b m · ∇ w r + ∂ x r b m · ∇ u (cid:1) ds − σ Z t ∇ w I dB s + σ Z t ∆ w I ds = − µ Z t w I ds − Z t b m · ∇ w I ds − X r ∈ I d X k =1 Z t ∂ x r b km w I − r + k ds − σ Z t ∇ w I dB s + σ Z t ∆ w I ds. Put v I := E [ w I ] . Since t R t ∇ w I dB s is a martingale, v I satisfies v I ( t ) − Y r ∈ I ∂ x r f = − µ Z t v I ds − Z t b m · ∇ v I ds − X r ∈ I d X k =1 Z t ∂ x r b km v I − r + k ds + σ Z t ∆ v I ds, i.e., ∂ t v I = − µv I + σ v I − b m · ∇ v I − X r ∈ I d X k =1 ∂ x r b km v I − r + k , v I (0) = Y r ∈ I ∂ x r f. (25) Step 2.
We multiply the equation in (25) by v I , and integrate:12 ∂ t (cid:10) v I (cid:11) + µ h v I i + σ (cid:10) ( ∇ v I ) (cid:11) = − (cid:10) v I , b m · ∇ v I (cid:11) − (cid:10) v I , X r ∈ I d X k =1 ∂ x r b km v I − r + k (cid:11) . Then, for every t ∈ J T ,12 (cid:10) v I ( t ) (cid:11) − (cid:10) v I (0) (cid:11) + µ Z t v I ds + σ Z t (cid:10) ( ∇ v I ) (cid:11) ds (26)= − Z t (cid:10) v I , b m · ∇ v I (cid:11) ds − Z t (cid:10) v I , X r ∈ I d X k =1 ∂ x r b km v I − r + k (cid:11) ds =: − S I − S I . We estimate | S I | and | S I | as follows: | S I | ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z t (cid:10) v I , b m · ∇ v I (cid:11) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ γ Z t (cid:10) ( ∇ v I ) (cid:11) ds + 14 γ Z t (cid:10) v I b m (cid:11) ds (we use b m ∈ F δ )) ≤ (cid:18) γ + δ γ (cid:19) Z t (cid:10) ( ∇ v I ) (cid:11) ds + c δ γ Z t h v I i . (27)Next, integrating by parts, and applying the quadratic inequality, we have | S I | = (cid:12)(cid:12)(cid:12)(cid:12) − Z t X r ∈ I d X k =1 h ( v I − r + k ∂ x r v I + v I ∂ x r v I − r + k ) b km i (cid:12)(cid:12)(cid:12)(cid:12) ds ≤ α Z t X r ∈ I d X k =1 (cid:10) ( ∂ x r v I ) + ( ∂ x r v I − r + k ) (cid:11) ds + 14 α Z t X r ∈ I d X k =1 (cid:10) v I − r + k ( b km ) + v I ( b km ) (cid:11) ds. Let q = 1 , , . . . . Summing over all I with | I | = 2 q and noticing that every multiindex of length 2 q is counted 4 qd times, we obtain X I | S I | ≤ αqd X I Z t (cid:10) |∇ v I | (cid:11) ds + qdα X I Z t (cid:10) v I b m (cid:11) ds (use b m ∈ F δ in the second term) ≤ αqd X I Z t (cid:10) |∇ v I | (cid:11) ds + qdδα X I Z t (cid:10) |∇ v I | (cid:11) ds + qdc δ α X I Z t (cid:10) v I (cid:11) ds. Also, by (27), we have X I | S I | ≤ (cid:18) γ + δ γ (cid:19) X I Z t (cid:10) |∇ v I | (cid:11) ds + c δ γ X I Z t h v I i . Now, armed with the last two estimates, we sum both sides of (26) over all I with | I | = 2 q to obtain12 X I (cid:10) v I ( t ) (cid:11) + µ Z t v I ds + κ Z t X I (cid:10) |∇ v I | (cid:11) ds ≤ X I (cid:10) v I (0) (cid:11) + (cid:20) qdc δ α + c δ γ (cid:21) X I Z t h v I i , where κ := σ − γ − δ γ − αqd − qdδα . TOCHASTIC TRANSPORT EQUATION 15
The maximum κ ∗ := max α,γ> κ = σ − √ δ − qd √ δ is attained at α = √ δ , γ = √ δ . For this choice of α and γ , we have κ ∗ = σ − β q √ δ . Since β q √ δ < σ by assumption, we have κ ∗ > X I (cid:10) v I ( t ) (cid:11) + (cid:0) µ − ˆ c (cid:1) Z t v I ds + κ ∗ Z t X I (cid:10) |∇ v I | (cid:11) ds ≤ X I (cid:10) v I (0) (cid:11) , where ˆ c := qdc δ √ δ + c δ √ δ . Thus, choosing µ ≥ ˆ c , we obtain12 sup τ ∈ [0 ,t ] X I (cid:10) v I ( τ ) (cid:11) + κ ∗ Z t X I (cid:10) |∇ v I | (cid:11) ds ≤ X I (cid:10) v I (0) (cid:11) . Step 3 . Recalling that v I = E (cid:2)Q r ∈ I ∂ x r u (cid:3) , v I (0) = Q r ∈ I ∂ x r f , we obtain from the previous estimate:sup t ∈ J T X ≤ k ≤ d (cid:10) ( E ( ∂ x k u ) q ) (cid:11) ≤ c (cid:10) |∇ f | q (cid:11) , (28) X ≤ k ≤ d Z t (cid:10) |∇ E ( ∂ x k u ) q | (cid:11) ds ≤ c (cid:10) |∇ f | q (cid:11) , (29)for appropriate positive constants c , c . By the Sobolev embedding theorem, Z t (cid:10) ( ∇ E |∇ u | q ) (cid:11) ds ≥ c Z t (cid:10) ( E |∇ u | q ) dd − (cid:11) d − d ds, so (29) yields k E |∇ u | q k L ( J T ,L dd − ) ≤ c k∇ f k q q , for appropriate constant c > k E |∇ u | q k L ∞ ( J T ,L ) ≤ c k∇ f k q q , weobtain ( E ). (cid:3) Proof of Theorem 1
Recall that k · k p,ρ denotes the norm in L p ( R d , ρdx ), and h· , ·i ρ the inner product in L ( R d , ρdx ).We assume throughout this section that b ∈ F δ and b m , m = 1 , , . . . are as in the beginning of theprevious section. Lemma 1.
Let b ∈ F δ , and let b m be as above. Then the following are true: ( i ) k b √ ρ k < ∞ . ( ii ) k b √ ρ B c (0 ,R +1) k ↓ as R → ∞ . ( iii ) h ρ | b − b m | i → as m → ∞ .Proof. ( i ) Using b ∈ F δ , and applying (6) and h ρ i < ∞ , we have k b √ ρ k ≤ δ k∇√ ρ k + c δ h ρ i < ∞ . ( ii ) For any R ≥
1, let η R be a [0 , η R ( x ) = 1 if | x | > R + 1; η R ( x ) = 0 if | x | ≤ R ; and sup R ≥ k∇ η R k ∞ ≤ C . Then k b √ ρη R k ≤ δ k∇ [ √ ρη R ] k + c δ h ρη R i . We have ∇ [ √ ρη R ] = √ ρ ( ∇ ρ ) η R + √ ρ ∇ η R =: S + S . Using (6), we have k S k ≤ C h ρη R i → R → ∞ . Next, we use sup R ≥ k∇ η R k ∞ ≤ C to get k S k ≤ C (1 + κR ) − θ h B (0 ,R +1) − B (0 ,R ) i = c d C (1 + κR ) − θ R d → R → ∞ since θ > d . This completes the proof of ( ii ).( iii ) This is a consequence of ( ii ) and b m → b in L ( R d ).The proof of Lemma 1 is complete. (cid:3) Lemma 2.
Let β √ δ < σ , f ∈ C ∞ c and u m be the strong solution to (19) . Provided that κ > inthe definition of ρ is chosen sufficiently small, there exists µ (cid:0) δ, c δ (cid:1) ≥ such that for any µ ≥ µ (cid:0) δ, c δ (cid:1) , lim n,m →∞ sup t ∈ J T k E | u n ( t ) − u m ( t ) | k ,ρ = 0 . Proof.
Set g ≡ g n,m := u n − u m , n, m = 1 , , . . . , then g ( t ) + µ Z t gds + Z t b m · ∇ gds + Z t ( b n − b m ) · ∇ u m ds + σ Z t ∇ gdB s − σ Z t ∆ gds = 0 . Applying Itˆo’s formula, we obtain g ( t ) = − µ Z t g ds − Z t b m · ∇ g ds − Z t g ( b n − b m ) · ∇ u m ds − σ Z t ∇ g dB s + σ Z t ∆ g ds, so denoting h := E [ g ] we arrive at ∂ t h + 2 µh − σ h + b m · ∇ h + 2( b n − b m ) · E [ g ∇ u m ] = 0 , h (0) = 0 . Multiplying this equation by ρh and integrating by parts, we obtain12 k h ( t ) k ,ρ + 2 µ Z t k h k ,ρ ds + σ Z t k∇ h k ,ρ ds + σ Z t h ( ∇ ρ ) h, ∇ h i (30)+ Z t h b m · ∇ h, h i ρ ds + 2 Z t h h ( b n − b m ) · E [ g ∇ u m ] i ρ ds = 0 . Since our assumption on δ is a strict inequality, using (6) and selecting κ sufficiently small, we canand will ignore in what follows the terms containing ∇ ρ .Applying the quadratic inequality and using b m ∈ F δ , we obtain (cf. the proof of ( E )) σ Z t k∇ h k ,ρ ds + Z t h b m · ∇ h, h i ρ ds ≥ (cid:18) σ − √ δ (cid:19) Z t k∇ h k ,ρ ds − c δ √ δ Z t k h k ,ρ ds, where σ − √ δ > δ . TOCHASTIC TRANSPORT EQUATION 17
We obtain from (30):12 sup τ ∈ [0 ,t ] k h ( τ ) k ,ρ + (cid:18) σ − √ δ (cid:19) Z t k∇ h ( s ) k ,ρ ds + (cid:20) µ − c δ √ δ (cid:21) Z t k h k ,ρ ds ≤ Z t h h | b n − b m | · E [ | g ∇ u m | ] i ρ ds. Select µ ≥ c δ √ δ . Then the previous estimate yields12 sup τ ∈ [0 ,t ] k h ( τ ) k ,ρ ≤ Z t h h | b n − b m | · E [ | g ∇ u m | ] i ρ ds, so it remains to show that Z t h h | b n − b m | · E [ | g ∇ u m | ] i ρ ds → n, m → ∞ . We estimate h h | b n − b m | · E [ | g ∇ u m | ] i ρ ≤ h| b n − b m | h ( E [ g ]) ( E [ |∇ u m | ]) i ρ ≡ h| b n − b m | h ( E [ |∇ u m | ]) i ρ ≤ h| b n − b m | i ρ h h E [ |∇ u m | ] i ρ ≤ h| b n − b m | i ρ h h E [ |∇ u m | ] i ≤ h| b n − b m | i ρ h h i h ( E [ |∇ u m | ]) i (we apply Proposition 1, and (28) with q = 1) ≤ c h| b n − b m | i ρ k f k k∇ f k (we apply Lemma 1( iii )) → n, m → ∞ . The proof of Lemma 2 is complete. (cid:3)
Lemma 2 allows to prove that { u m } is a Cauchy sequence in L ∞ ( J T , L (Ω , L ρ )). Lemma 3.
Let β √ δ < σ , f ∈ C ∞ c and u m be the strong solution to (19) . Provided that κ > inthe definition of ρ is chosen sufficiently small, it holds that u m converges in L (Ω , L ρ ) to a process u ,uniformly in t ∈ J T .Proof. Let κ be small enough and µ greater than or equal to the µ ( δ, c δ ). Let µ ≥ µ ( δ, c δ ). Then byLemma 2, sup t ∈ J T E k ( u n ( t ) − u m ( t )) k ,ρ ≤ h ρ i sup t ∈ J T k E | u n ( t ) − u m ( t ) | k ,ρ → m , n → ∞ . Thus, we can define u ( t ) := s - L (Ω , L ρ )- lim m u m ( t ) uniformly in t ∈ J T . The proof is complete (cid:3)
We are in position to give the proof of Theorem 1.
Proof of Theorem 1.
It suffices to carry out the proof for f ∈ C ∞ c , and then use a density argument.It follows from the assumption √ δ < σ β that p ≥ p c , ∞ ), p c = (cid:0) − √ δσ (cid:1) − .(Indeed, p c < √ δ < σ . In particular, p c < √ δ < σ β since β > µ ( δ, c δ , p )be the constant from Proposition 1. Assume that µ ≥ µ ( δ, c δ , p ). Then the conclusions of Proposition1 are valid.We prove ( i ) first. We do this in two steps. Step 1.
Selecting κ sufficiently small so that Lemma 3 applies, we obtain that u m converges in L (Ω , L ρ ) to a process u , uniformly in t ∈ J T . Thus u ∈ L ∞ ( J T , L ( R d , L (Ω)), and we have for all t ∈ J T , u m → u in L ∞ ( J T , L (Ω , L ρ )) , (31)which yields Z t u m ds → Z t uds in L (Ω , L ρ ); (32)the latter, ( E ) and a standard weak compactness argument yield ∇ Z t u m ds → ∇ Z t uds weakly in L (Ω , L ρ ( R d , R d )) . (33) Step 2.
Given a test function ϕ ∈ C ∞ c , we multiply (19) by ρϕ , integrate and write (we take µ = 0to shorten calculations) h u m ( t ) − u ( t ) , ρϕ i + h u ( t ) , ρϕ i − h f, ρϕ i = − (cid:10) ( b m − b ) · ∇ Z t u m ds, ρϕ (cid:11) − (cid:10) b · ∇ Z t u m ds, ρϕ (cid:11) + σ (cid:10)Z t ( u m − u ) dB s , ∇ ρϕ (cid:11) + σ (cid:10)Z t udB s , ∇ ρϕ (cid:11) (34) − σ (cid:10) ∇ Z t ( u m − u ) ds, ∇ ρϕ (cid:11) − σ (cid:10) ∇ Z t uds, ∇ ρϕ (cid:11) . Let us now note the following. In view of (31) and (33), h u m ( t ) − u ( t ) , ρϕ i ≡ h u m ( t ) − u ( t ) , ϕ i ρ → L (Ω). Similarly, using (33) and (6), (cid:10) ∇ Z t ( u m − u ) ds, ∇ ρϕ (cid:11) → L (Ω) , (a)and, since ϕ | b | ∈ L ρ (using that ϕ has compact support), (cid:10) b · ∇ Z t u m ds, ρϕ (cid:11) → (cid:10) b · ∇ Z t uds, ρϕ (cid:11) weakly in L (Ω) . (b)By ( E ), k∇ R t u m ds k L (Ω ,L ρ ) ≤ c with c < ∞ independent of m , and ϕ | b m − b n | → L ρ (in fact,in L ). Thus (cid:10) ( b m − b ) · ∇ Z t u m ds, ρϕ (cid:11) → L (Ω) . (c)Finally, let us show that (cid:10)Z t ( u m − u ) dB s , ∇ ρϕ (cid:11) → L (Ω) . (d) TOCHASTIC TRANSPORT EQUATION 19
Indeed, using Itˆo’s isometry, we have using (6) E (cid:12)(cid:12)(cid:12)(cid:12)(cid:10)Z t ( u m − u ) dB s , ∇ ρϕ (cid:11)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c E h (cid:12)(cid:12) Z t ( u m − u ) dB s (cid:12)(cid:12) i ρ h| ϕ | i ρ = c h E Z t ( u m − u ) ds i ρ → . The convergence (d) follows.Thus, using (a)-(d), we can pass to the L (Ω)-weak limit in (34) as m → ∞ , obtaining that u satisfies (13) (with test functions ϕρ which, clearly, exhaust C ∞ c ).The estimates in (11), (12) now follow from Proposition 1.The last assertion ( ii ) is Lemma 3 proved above.The proof of Theorem 1 is complete. (cid:3) Proof of Theorem 2
Proof of Theorem 2.
Part (a) follows from Theorem 1( i ). The last assertion, (15), follows from Propo-sition 2 and Lemma 3. So we only need to prove part (b).Since the weak- L ( J T × Ω) limit of any sequence of ( F t )-progressively measurable processes on J T remains ( F t )-progressively measurable and t
7→ h u m ( t ) , ϕ i is ( F t )-progressively measurable for every m , in view of (32), the process t
7→ h u ( t ) , ϕ i is ( F t )-progressively measurable as well. The proof of(14) follows closely the proof of (13) above except that now, instead of ( E ), we appeal to the Sobolevregularity estimate (16) with q = 1.The existence of a continuous ( F t )-semi-martingale modification of t
7→ h u ( t ) , ϕ i is a consequenceof the identity (14).The proof of Theorem 2 is complete. (cid:3) Proof of Theorem 3 (weak uniqueness)
The fact that (CP) has at least one weak solution was proved in Theorem 2. We now prove itsuniqueness. We adopt the argument of [BFGM, Sect. 3]. We will need the following definitions andresults. Let us fix a version of the Brownian motion B t having continuous trajectories B t ( ω ) for every ω ∈ Ω . Lemma 4.
Let b ∈ F δ with √ δ < σ β and f ∈ W , . Let u = u ( t, x, ω ) be a weak solution to (CP) .Then for a.e. ω ∈ Ω , ˜ u ω ( t, x ) := u ( t, x + σB t ( ω ) , ω ) is a weak solution to the Cauchy problem ∂ t ˜ u ω + µ ˜ u ω + ˜ b ω · ∇ ˜ u ω = 0 , ˜ u ω | t =0 = f, where ˜ b ω ( t, x ) := b ( x + σB t ( ω )) , (35) that is, the following are true:
1) ˜ u ω ∈ L ∞ ( J T , W , ρ ) ; for every ψ ∈ C ( J T , C ∞ c ) , the function t
7→ h ˜ u ω ( t ) , ψ ( t ) i has a continuous representative, i.e. acontinuous function which coincides with t
7→ h ˜ u ω ( t ) , ψ ( t ) i for a.e. t ∈ J T ; for every ψ ∈ C ( J T , C ∞ c ) , this continuous representative of t
7→ h ˜ u ω ( t ) , ψ ( t ) i satisfies for every t ∈ J T , h ˜ u ω ( t ) , ψ ( t ) i = h f, ψ (0) i + µ Z t h ˜ u ω ( s ) , ψ ( s ) i ds + Z t h ˜ u ω ( s ) , ∂ s ψ ( s ) i ds − Z t h∇ ˜ u ω ( s ) , ˜ b ω ( s ) ψ ( s ) i ds. The proof of Lemma 4 follows closely the proof of [BFGM, Prop. 3.4] (taking into account thedefinition of the weak solution to (CP)) and we omit the details.Consider the terminal value problem dv m + µv m dt + ∇ · ( b m v m ) dt + σ ∇ v m ◦ dB t = 0 , t ∈ [0 , t ∗ ] , v m | t = t ∗ = v ∈ C ∞ c , (36)where b m ∈ C ∞ c ( R d , R d ) ( m = 1 , , . . . ) (since b m are bounded and smooth, we have strong existenceand uniqueness for this equation).The following is an analogue of [BFGM, Cor. 3.8]. Lemma 5. ˜ v ωm ( t, x ) := v m ( t, x + σB t ( ω )) satisfies, for a.e. ω ∈ Ω , ˜ v ωm ∈ C ([0 , t ∗ ] , C ∞ c ) and ∂ t ˜ v ωm + µ ˜ v ωm + ∇ · ( b ωm ˜ v ωm ) = 0 , ˜ v ωm ( t ∗ , x ) = v ( x + σB t ∗ ( ω )) . We will also need
Lemma 6.
Let √ δ < σ . There exist a constant µ ( c δ ) ≥ and a sufficiently small κ > (in thedefinition of ρ ) such that sup t ∈ J T k ρ − E [ v m ( t )] k ≤ k ρ − v k , µ ≥ µ ( c δ ) , m = 1 , , . . . where v m is the strong solution to (36) .Proof. Without loss of generality, we will carry out the proof for the forward equation, and will dropthe subscript m from b m . Set w := E [ v ]. Arguing as in the proof of Proposition 1, we obtain that w satisfies ∂ t w + 2 µw − σ w − ∇ · ( bw ) + b · ∇ w = 0 , w (0) = v . (37)We first carry out the proof for ρ ≡
1. Multiplying the previous equation by w and integrating, weobtain 12 ∂ t h| w | i + 2 µ h| w | i + σ h|∇ w | i + 3 h∇ w, bw i = 0 . Applying the quadratic inequality and the form-boundedness condition b ∈ F δ , we get that, for any γ >
0, 12 ∂ t h| w | i + (2 µ − γc δ ) h| w | i + (cid:20) σ − γδ + 14 γ ) (cid:21) h|∇ w | i ≤ , and so, selecting µ ( c δ ) := γc δ and µ ≥ µ ( c δ ), we obtain12 h| w ( t ) | i + (cid:20) σ − γδ + 14 γ ) (cid:21) Z t h|∇ w | i ds ≤ h| v | i . Upon maximizing the coefficient in the square brackets in γ (thus, selecting γ = √ δ ), we obtain thatthe coefficient is positive since √ δ < σ . In particular, it follows that sup t ∈ J T k E [ v m ( t )] k ≤ k v k . TOCHASTIC TRANSPORT EQUATION 21
In presence of ρ − , we argue as above but get new terms containing ∇ ρ − , which we bound appealingto the estimate |∇ ρ − | = (cid:12)(cid:12)(cid:12)(cid:12) ∇ ρρ (cid:12)(cid:12)(cid:12)(cid:12) ≤ θ √ κρ − (by (6)) , with κ selected sufficiently small. (Note that to justify k ρ − E [ v m ( t )] k < ∞ we can appeal to quali-tative Gaussian upper bound on the heat kernel of (37).) (cid:3) Let us note that the assumption of the theorem β √ δ < σ implies √ δ < σ .We are now in position to complete the proof of Theorem 3. Proof of Theorem 3.
Let µ and κ be as in Lemma 6. In view of the linearity of the stochastic transportequation, it suffices to show that a weak solution u to (CP) with initial condition u (0) = 0 must beidentically zero for all t ∈ J T . In view of Lemma 4, it suffices to show that ˜ u ω corresponding to u isidentically zero a.s.Let v ∈ C ∞ c . It follows from Lemma 5 that, for a.e. ω ∈ Ω, ˜ v ω ( s ) ∈ C ( J T , C ∞ c ). Thus by Lemma4, for a.e. ω ∈ Ω with ψ ( s ) := ˜ v ω ( s ), for all 0 < t ∗ ≤ T , h ˜ u ω ( t ∗ ) , v ( · + σB t ∗ ( ω )) i ( • )= µ Z t ∗ h ˜ u ω ( s ) , ˜ v ωm ( s ) i d + Z t ∗ h ˜ u ω ( s ) , ∂ s ˜ v ωm ( s ) i ds − Z t ∗ h∇ ˜ u ω ( s ) , ˜ b ω ( s )˜ v ωm ( s ) i ds = Z t ∗ h∇ ˜ u ω , (˜ b ωm ( s ) − ˜ b ω ( s ))˜ v ωm i ds =: I. Step 1. Let us first show that E (cid:12)(cid:12)(cid:12)(cid:12)Z t ∗ h∇ u, ( b − b m ) v m n i ds (cid:12)(cid:12)(cid:12)(cid:12) → m ↑ ∞ . ( •• )We have E (cid:12)(cid:12)(cid:12)(cid:12)Z t ∗ h∇ u, ( b − b m ) v m i ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z t ∗ (cid:10) | b − b m | E (cid:2) |∇ u | (cid:3) E (cid:2) | v m | (cid:3) (cid:11) ds ≤ (cid:18)Z t ∗ (cid:10) ρ | b − b m | (cid:11) ds (cid:19) (cid:18)Z t ∗ (cid:10) ( E (cid:2) |∇ u | (cid:3) ) (cid:11) ds (cid:19) (cid:18)Z t ∗ (cid:10) ρ − ( E (cid:2) | v m | (cid:3) ) (cid:11) ds (cid:19) . The first integral converges to 0 as m ↑ ∞ by Lemma 1( iii ), the second integral is finite by thedefinition of weak solution before Theorem 3, and the third integral is bounded from above uniformlyin m by √ t ∗ k ρ − v k < ∞ , see Lemma 6. Thus, ( •• ) follows.Step 2. By Step 1, there exists a subset Ω t ∗ ,v ⊂ Ω of probability 1 and a sequence m k ↑ ∞ suchthat for every ω ∈ Ω t ∗ ,v , Z t ∗ h∇ u, ( b − b m k ) v m k i ds → m k ↑ ∞ . Making the change of variable x x + σB t ( ω ) and using the fact that c − t ∗ ,w ρ ( · ) ≤ ρ ( · + σB t ( ω )) ≤ c t ∗ ,w ρ ( · ) for some constant c t ∗ ,w > ω ∈ Ω t ∗ ,v , I → m k ↑ ∞ . Fix a countable dense subset D of C ∞ c ( R d ) and define˜Ω := \ t ∗ ∈ [0 ,T ] ∩ Q ,v ∈ D Ω t ∗ ,v , a full measure set in Ω. Applying the diagonal argument (and so passing to a subsequence of { ε k } ),we obtain by ( • ) and Step 2 that for every ω ∈ ˜Ω, ˜ u ω ( t ) = 0 for all t ∈ [0 , T ] ∩ Q . Since t
7→ h ˜ u ω ( t ) , ϕ i , ϕ ∈ C ∞ c ( R d ) is continuous, we obtain that ˜ u ω ( t ) = 0 for all t ∈ [0 , T ] for all ω ∈ ˜Ω, as needed.The proof of Theorem 3 is complete. (cid:3) References [BFGM] L. Beck, F. Flandoli, M. Gubinelli, M. Maurelli, Stochastic ODEs and stochastic linear PDEs with critical drift:regularity, duality and uniqueness.
Electron. J. Probab. , (2019), Paper No. 136, 72 pp.[CWW] S.Y.A. Chang, J.M. Wilson, T.H. Wolff, Some weighted norm inequalities concerning the Schr¨odinger operator, Comment. Math. Helvetici , (1985), p. 217-246.[GM] B. Gess, M. Maurelli, Well-posedness by noise for scalar conservation laws, Comm. Partial Differential Equations (2018), no. 12, p. 1702-1736.[HN] J. E. Hutton, P. I. Nelson, Interchanging the order of differentiation and stochastic integration. Stoc. Proc. Appl. , (1984), p. 371–377.[KiS] D. Kinzebulatov, Yu.A. Sem¨enov, Brownian motion with general drift, Stoc. Proc. Appl. , (2020), p. 2737-2750[KiS2] D. Kinzebulatov, Yu.A. Sem¨enov, Feller generators and stochastic differential equations with singular (form-bounded) drift, Osaka J. Math. , to appear.[KiS3] D. Kinzebulatov, Yu. A. Sem¨enov, On the theory of the Kolmogorov operator in the spaces L p and C ∞ . Ann. Sc. Norm. Sup. Pisa (5) , (2020), p. 1573-1647.[KPS] V. F. Kovalenko, M. A. Perelmuter, Yu. A. Semenov. Schr¨odinger operators with L l/ w ( R l )-potentials. J. Math.Phys. (1981), p. 1033-1044.[KS] V. F. Kovalenko, Yu. A. Semenov, C -semigroups in L p ( R d ) and C ∞ ( R d ) spaces generated by differential expression∆ + b · ∇ . (Russian) Teor. Veroyatnost. i Primenen. , (1990), 449-458; translation in Theory Probab. Appl. , (1990), p. 443-453.[Kr1] N. V. Krylov, On diffusion processes with drift in L d , Preprint , arXiv:2001.04950.[Kr2] N. V. Krylov, On time inhomogeneous stochastic Itˆo equations with drift in L d +1 , Preprint , arXiv:2005.08831.[KrR] N. V. Krylov, M. R¨ockner. Strong solutions of stochastic equations with singular time dependent drift.
Probab.Theory Related Fields , (2005), p. 154-196.[Ku] H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge Studies in Advanced Mathematics,vol. 24, Cambridge University Press, Cambridge, 1990.[W] R. J. Williams, Brownian motion with polar drift,
Trans. Amer. Math. Soc. , (1985), p. 225-246. D´epartement de math´ematiques et de statistique, Universit´e Laval, Qu´ebec, QC, G1V 0A6, Canada
Email address : [email protected] University of Toronto, Department of Mathematics, Toronto, ON, M5S 2E4, Canada
Email address : [email protected] Department of Mathematics, University of Illinois, Urbana, IL 61801, USA
Email address ::