Multidimensional stochastic differential equations with distributional drift

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by  Franco Flandoli

MULTIDIMENSIONAL STOCHASTIC DIFFERENTIAL
EQUATIONS WITH DISTRIBUTIONAL DRIFT
FRANCO FLANDOLI1, ELENA ISSOGLIO2, ANDFRANCESCO RUSSO3
Abstract. Thispaperinvestigatesatime-dependentmultidimensional
stochastic differential equation with drift being a distribution in a suit-
5
ableclassofSobolevspaceswithnegativederivationorder. Thisisdone
1
through a careful analysis of the corresponding Kolmogorov equation
0
whose coefficient is a distribution.
2

l
u
J Key words and phrases: Stochastic differential equations; distribu-

9 tional drift; Kolmogorov equation.
2

AMS-classification: 60H10; 35K10; 60H30; 35B65.
]
R
P 1. Introduction
.
h Let us consider a distribution valued function b : [0,T] → S′(Rd), where
t
a S′(Rd)is thespaceof tempereddistributions. An ordinarydifferential equa-
m
tion of the type
[
(1) dX = b(t,X )dt, X = x ,
t t 0 0
2
v x ∈ Rd, t ∈ [0,T], does not make sense, except if we consider it in a
0
0
very general context of generalized functions. Even if b is function valued,
1
without a minimum regularity in space, problem (1), is generally not well-
0
6 posed. A motivation for studying (1) is for instance to consider b as a
1. quenched realization of some (not necessarily Gaussian) random field. In
0 the annealed form, (1) is a singular passive tracer type equation.
4
Let us consider now equation (1) with a noise perturbation, which is
1
expected to have a regularizing effect, i.e.,
:
v
i (2) dX = b(t,X )dt+dW , X = x ,
X t t t 0 0
r for t ∈ [0,T], where W is a standard d-dimensional Brownian motion. For-
a
mally speaking, the Kolmogorov equation associated with the stochastic
differential equation (2) is
∂ u= b·∇u+ 1∆u on [0,T]×Rd,
(3) t 2
u(T,·) = f on Rd,
(cid:26)
1DipartimentoMatematica, Largo Bruno Pontecorvo 5, C.A.P. 56127, Pisa,
Italia [email protected]
2Department of Mathematics, University of Leeds, Leeds, LS2 9JT, UK
[email protected]
3Unit´e de Math´ematiques appliqu´ees, ENSTA ParisTech, Universit´e
Paris-Saclay, 828, boulevard des Mar´echaux, F-91120 Palaiseau, France
[email protected]
Date: July 30, 2015.
1
2 MULTIDIMENSIONAL SDES WITH DISTRIBUTIONAL DRIFT
for suitable final conditions f. Equation (3) was studied in the one-dimen-
sional setting for instance by [23] for any time independent b which is the
derivative in the distributional sense of a continuous function and in the
multidimensional setting by [13], for a class of b of gradient type belonging
to a given Sobolev space with negative derivation order. The equation in
[13] involves the pointwise product of distributions which in the literature is
defined by means of paraproducts.
The point of view of the present paper is to keep the same interpretation
of the product as in [13] and to exploit the solution of a PDE of the same
nature as (3) in order to give sense and study solutions of (2). A solution
X of (2) is often identified as a diffusion with distributional drift.
Of course the sense of equation (2) has to be made precise. The type of
solution we consider will be called virtual solution, see Definition 25. That
solution will fulfill in particular the property to be the limit in law, when
n → ∞, of solutions to classical stochastic differential equations
(4) dXn = dW +b (t,Xn)dt, t ∈ [0,T],
t t n t
whereb = b⋆φ and(φ )is asequenceof mollifiers converging to theDirac
n n n
measure.
Diffusions in the generalized sense were studied by several authors begin-
ning with, at least in our knowledge [20]; later on, many authors considered
specialcases ofstochasticdifferentialequations withgeneralized coefficients,
it is difficult to quote them all: in particular, we refer to thecase whenb is a
measure, [4, 7, 18, 22]. [4] has even considered the case when b is a not nec-
essarily locally finite signed measure and the process is a possibly exploding
semimartingale. In all these cases solutions were semimartingales. In fact,
[8] considered special cases of non-semimartingales solving stochastic differ-
ential equations with generalized drift; those cases include examples coming
from Bessel processes.
The case of time independent SDEs in dimension one of the type
(5) dX = σ(X )dW +b(X )dt, t ∈ [0,T],
t t t t
where σ is a strictly positive continuous function and b is the derivative
of a real continuous function was solved and analyzed carefully in [10] and
[11], which treated well-posedness of the martingale problem, Itˆo’s formula
under weak conditions, semimartingale characterization and Lyons-Zheng
decomposition. Theonlysupplementaryassumptionwastheexistenceofthe
functionΣ(x)= 2 x b dy aslimitofappropriateregularizations. Alsoin[1]
0 σ2
theauthorswereinterestedin(2)andtheyprovidedawell-stated framework
R
when σ and b are γ-H¨older continuous, γ > 1. In [23] the authors have also
2
shown that in some cases strong solutions (namely solutions adapted to the
completed Brownian filtration) exist and pathwise uniqueness holds.
As far as the multidimensional case is concerned, it seems that the first
paper was [2]. Here the authors have focused on (2) in the case of a time
independent drift b which is a measure of Kato class.
Comingback totheone-dimensionalcase, themainideaof [11]was theso
called Zvonkin transform which allows to transform the candidate solution
process X into a solution of a stochastic differential equation with continu-
ous non-degenerate coefficients without drift. Recently [16] has considered
MULTIDIMENSIONAL SDES WITH DISTRIBUTIONAL DRIFT 3
other types of transforms to study similar equations. Indeed the transfor-
mation introduced by Zvonkin in [27], when the drift is a function, is also
stated in the multidimensional case. In a series of papers the first named
author and coauthors (see for instance [9]), have efficiently made use of a
(multidimensional) Zvonkin type transform for the study of an SDE with
measurable not necessarily bounded drift, which however is still a function.
Zvonkin transform consisted there to transform a solution X of (2) (which
makes sense being a classical SDE) through a solution ϕ :[0,T]×Rd → Rd
of a PDE which is close to the associated Kolmogorov equation (3) with
some suitable final condition. The resulting process Y with Y = ϕ(t,X )
t t
for t ∈ [0,T] is a solution of an SDE for which onecan show strongexistence
and pathwise uniqueness.
Here we have imported that method for the study of our time-dependent
multidimensional SDE with distributional drift.
Thepaperisorganized as follows. InSection 2weadaptthetechniques of
[13], based on pointwise products for investigating existence and uniqueness
for a well chosen PDE of the same type as (3), see (6). In Section 3 we
introduce the notion of virtual solution of (2). The construction will be
based on the transformation X = ψ(t,Y ) for t ∈ [0,T], where Y is the
t t
solution of (34) and ϕ(t,x) = x+u(t,x),(t,x) ∈ [0,T]×Rd, with u being
the solution of (6). Section 3.3 shows that the virtual solution is indeed the
limit of classical solutions of regularized stochastic differential equations.
2. The Kolmogorov PDE
2.1. Setting and preliminaries. Let b bea vector field on [0,T]×Rd,d ≥
1, which is a distribution in space and weakly bounded in time, that is
b ∈ L∞([0,T];S′(Rd;Rd)). Let λ > 0. We consider the parabolic PDE in
[0,T]×Rd
∂ u+Lbu−(λ+1)u = −b on [0,T]×Rd,
(6) t
u(T) = 0 on Rd,
(cid:26)
where Lbu = 1∆u+ b · ∇u has to be interpreted componentwise, that is
2
(Lbu) = 1∆u +b·∇u for i = 1,...,d. A continuous function u : [0,T]×
i 2 i i
Rd → Rd will also be considered without any comment as u : [0,T] →
C(Rd;Rd). In particular we will write u(t,x) = u(t)(x) for all (t,x) ∈
[0,T]×Rd.
Remark 1. All the results we are going to prove remain valid for the equa-
tion
∂ u+Lb1u−(λ+1)u= −b on [0,T]×Rd,
t 2
u(T)= 0 on Rd,
(cid:26)
where b ,b both satisfy the same assumptions as b. We restrict the discus-
1 2
sion to the case b = b = b to avoid notational confusion in the subsequent
1 2
sections.
Clearly we have to specify the meaning of the product b ·∇u as b is a
i
distribution. In particular, we are going to make use in an essential way
the notion of paraproduct, see [21]. We recall below a few elements of this
theory;inparticular, whenwesay that thepointwise productexists in S′ we
4 MULTIDIMENSIONAL SDES WITH DISTRIBUTIONAL DRIFT
meanthatthelimit(7)existsinS′. For shortnesswedenotebyS′ andS the
spaces S′(Rd;Rd) and S(Rd;Rd) respectively. Similarly for the Lp-spaces,
1 ≤ p ≤ ∞. We denote by h·,·i the dual pairing between an element of S′
and an element of S.
We now recall a definition of a pointwise product between a function and
a distribution (see e. g. [21]) and some useful properties.
Suppose we are given f ∈ S′(Rd). Choose a function ψ ∈ S(Rd) such that
0 ≤ ψ(x) ≤ 1 for every x ∈ Rd, ψ(x) = 1 if |x| ≤ 1 and ψ(x) = 0 if |x| ≥ 3.
2
Then consider the following approximation Sjf of f for each j ∈ N
∨
ξ
Sjf(x):= ψ fˆ (x),
2j
(cid:18) (cid:18) (cid:19) (cid:19)
that is in fact the convolution of f against the smoothing rescaled function
ψ associated with ψ. This approximation is used to define the product fg
j
of two distributions f,g ∈ S′ as follows:
(7) fg := lim SjfSjg,
j→∞
if thelimit exists inS′(Rd). Theconvergence in thecase weareinterested in
is part of the assertion below (see [12] appendix C.4, [21] Theorem 4.4.3/1).
′
Definition 2. Let b,u :[0,T] → S be such that
′
(i) the pointwise product b(t)·∇u(t) exists in S for a.e. t ∈[0,T],
(ii) there are r ∈ R, q ≥ 1 such that b,u,b·∇u∈L1 [0,T];Hr .
q
′
We say that u is a mild solution of equation (6) in S i(cid:0)f, for ever(cid:1)y ψ ∈ S
and t ∈[0,T], we have
T
(8) hu(t),ψi = hb(r)·∇u(r),P (r−t)ψidr
Zt
T
+ hb(r)−λu(r),P (r−t)ψidr.
Zt
Here (P (t)) denotes the heat semigroup on S generated by 1∆−I,
t≥0 2
defined for each ψ ∈ S as
(P (t)ψ)(x) = p (x−y)ψ(y)dy,
t
Rd
Z
where p (x) is the heat kernel p (x) = e−t 1 exp −|x|2d and |·| is the
t t (2tπ)d/2 2t d
usual Euclidean norm in Rd. The semigroup (P(t)) (cid:16) exten(cid:17)ds to S′, where
t≥0
it is defined as
(PS′(t)h)(ψ) = hh, pt(·−y)ψ(y)dyi,
Rd
Z
for every h ∈ S′, ψ ∈ S.
Thefractional Sobolev spaces Hr arethe socalled Bessel potential spaces
q
and will be defined in the sequel.
Remark 3. If b,u,b·∇u a priori belong to spaces L1 [0,T];Hri for dif-
qi
ferent r ∈ R, q ≥ 1, i = 1,2,3, then (see e.g. (21)) there exist common
i i
(cid:0) (cid:1)
r ∈ R, q ≥ 1 such that b,u,b·∇u∈ L1 [0,T];Hr .
q
(cid:0) (cid:1)
MULTIDIMENSIONAL SDES WITH DISTRIBUTIONAL DRIFT 5
The semigroup (PS′(t))t≥0 maps any Lp Rd into itself, for any given
p ∈ (1,∞); the restriction (P (t)) to Lp Rd is a bounded analytic
p t≥0 (cid:0) (cid:1)
semigroup, with generator −A , where A = I − 1∆, see [6, Thm. 1.4.1,
p p (cid:0) (cid:1)2
1.4.2]. The fractional powers of A of order s ∈ R are then well-defined,
p
see [19]. The fractional Sobolev spaces Hs(Rd) of order s ∈ R are then
p
Hs(Rd) := A−s/2(Lp(Rd)) for all s ∈ R and they are Banach spaces when
p p
s/2 s/2
endowed with the norm k·k = kA (·)k . The domain of A is then
Hs p Lp p
p
the Sobolev space of order s, that is D(As/2) = Hs(Rd), for all s ∈ R. Fur-
p p
thermore, the negative powers A−s/2 act as isomorphism from Hγ(Rd) onto
p p
Hγ+s(Rd) for γ ∈ R.
p
We have defined so far function spaces and operators in the case of scalar
valued functions. The extension to vector valued functions must be under-
stood componentwise. For instance, the space Hs Rd,Rd is the set of all
p
vector fields u : Rd → Rd such that ui ∈ Hs Rd for each component ui of
p (cid:0) (cid:1)
u; the vector field P (t)u : Rd → Rd has components P (t)ui, and so on.
p (cid:0) (cid:1) p
Since we use vector fields more often than scalar functions, we shorten some
of the notations: we shall write Hs for Hs Rd,Rd . Finally, we denote by
p p
−β −β −β
H the space H ∩H with the usual norm.
p,q p q (cid:0) (cid:1)
Lemma 4. Let 1 < p,q < ∞ and 0 < β < δ and assume that q > p∨ d.
δ
Then for every f ∈ Hδ(Rd) and g ∈ H−β(Rd) we have fg ∈ H−β(Rd) and
p q p
there exists a positive constant c such that
(9) kfgkHp−β(Rd) ≤ ckfkHpδ(Rd)·kgkHq−β(Rd).
For the following the reader can also consult [25, Section 2.7.1]. Let us
considerthespacesC0,0(Rd;Rd)andC1,0(Rd;Rd)definedastheclosureof S
with respectto thenormkfkC0,0 = kfkL∞ andkfkC1,0 = kfkL∞+k∇fkL∞,
respectively. For 0< α < 1 we will consider the Banach spaces
C0,α = {f ∈C0,0(Rd;Rd) :kfk <∞},
C0,α
C1,α = {f ∈C1,0(Rd;Rd) :kfk <∞},
C1,α
endowed with the norms
|f(x)−f(y)|
kfkC0,α := kfkL∞ + sup |x−y|α
x6=y∈Rd
|∇f(x)−∇f(y)|
kfkC1,α := kfkL∞ +k∇fkL∞ + sup |x−y|α ,
x6=y∈Rd
respectively.
From now on, we are going to make the following standing assumption
on the drift b and on the possible choice of parameters:
Assumption 5. Let β ∈ 0, 1 , q ∈ d , d and set q˜:= d . The drift
2 1−β β 1−β
b will always be of the type (cid:16) (cid:17)
(cid:0) (cid:1)
b ∈ L∞ [0,T];H−β .
q˜,q
(cid:16) (cid:17)
6 MULTIDIMENSIONAL SDES WITH DISTRIBUTIONAL DRIFT
δ
1−β
β
β 1 1−β =: 1 1
d q d q˜ p
Figure 1. The set K(β,q).
Remark 6. The fact that b ∈ L∞ [0,T];H−β implies, for each p ∈ [q˜,q],
q˜,q
that b ∈ L∞ [0,T];H−β . (cid:16) (cid:17)
p
(cid:16) (cid:17)
Moreover we consider the set
d
(10) K(β,q) := κ= (δ,p) : β < δ < 1−β, < p < q ,
δ
(cid:26) (cid:27)
which is drawn in Figure 1. Note that K(β,q) is nonempty since β < 1 and
2
d < q < d.
1−β β
Remark 7. As a consequence of Lemma 4, for 0 < β < δ and q > p∨ d
δ
and if b ∈ L∞([0,T];H−β) and u ∈ C0([0,T];H1+δ), then for all t ∈ [0,T]
q p
−β
we have b(t)·∇u(t) ∈H and
p
kb(t)·∇u(t)kHp−β ≤ ckbk∞,Hq−βku(t)kHpδ,
having used the continuity of ∇ from H1+δ to Hδ. Moreover any choice
p p
(δ,p) ∈ K(β,q) satisfies the hypothesis in Lemma 4.
Definition 8. Let (δ,p) ∈ K(β,q). We say that u ∈ C [0,T];H1+δ is a
p
mild solution of equation (6) in H1+δ if
p (cid:0) (cid:1)
T T
(11) u(t)= P (r−t)b(r)·∇u(r)dr+ P (r−t)(b(r)−λu(r))dr,
p p
Zt Zt
for every t ∈ [0,T].
Remark 9. Notice that b·∇u∈ L∞ [0,T];H−β byRemark 7. ByRemark
p
6, b ∈ L∞ [0,T];H−β . Moreover(cid:16)λu ∈ L∞ [(cid:17)0,T];H−β by the embed-
p p
ding H1+δ(cid:16)⊂ H−β. Th(cid:17)erefore the integrals in(cid:16)Definition 8(cid:17)are meaningful
p p
−β
in H .
p
Note that setting v(t,x) := u(T −t,x), the PDE (6) can be equivalently
rewritten as
∂ v = Lbv−(λ+1)v +b on [0,T]×Rd,
(12) t
v(0) = 0 on Rd.
(cid:26)
MULTIDIMENSIONAL SDES WITH DISTRIBUTIONAL DRIFT 7
Thenotionofmildsolutionsofequation(12)inS′andinH1+δ areanalogous
p
to Definition 2andDefinition 8, respectively. Inparticular the mildsolution
in H1+δ verifies
p
t t
(13) v(t) = P (t−r)(b(r)·∇v(r))dr+ P (t−r)(b(r)−λv(r))dr.
p p
Z0 Z0
Clearly the regularity properties of u and v are the same.
ForaBanach spaceX wedenotetheusualnorminL∞([0,T];X) bykfk
∞,X
for f ∈ L∞([0,T];X). Moreover, on the Banach space C([0,T];X) with
norm kfk := sup kf(t)k for f ∈ C([0,T];X), we introduce a
∞,X 0≤t≤T X
(ρ)
family of equivalent norms {k·k , ρ≥ 1} as follows:
∞,X
(14) kfk(ρ) := sup e−ρtkf(t)k .
∞,X X
0≤t≤T
Nextwestate amappingpropertyoftheheatsemigroup P (t)onLp(Rd):
p
itmapsdistributionsoffractionalorder−β intofunctionsoffractionalorder
1 + δ and the price one has to pay is a singularity in time. The proof is
analogous to the one in [13, Prop. 3.2] and is based on the analyticity of the
semigroup.
Lemma 10. Let 0 < β < δ, δ+β < 1 and w ∈ H−β(Rd). Then P (t)w ∈
p p
H1+δ(Rd) for any t > 0 and moreover there exists a positive constant c such
p
that
(15) kPp(t)wkHp1+δ(Rd) ≤ ckwkHp−β(Rd)t−1+δ2+β.
Proposition 11. Let f ∈ L∞ [0,T];H−β and g : [0,T] → H−β for β ∈ R
p p
defined as (cid:16) (cid:17)
t
g(t)= P (t−s)f(s)ds.
p
Z0
Then g ∈ Cγ [0,T];H2−2ǫ−β for every ǫ > 0 and γ ∈ (0,ǫ).
p
(cid:16) (cid:17)
γ
Proof. First observe that for f ∈ D(A ) there exists C > 0 such that
p γ
(16) kP (t)f −fk ≤ C tγkfk ,
p Lp γ Hp2γ
for all t ∈ [0,T] (see [19, Thm 6.13, (d)]).
Let 0 ≤ r < t ≤ T. We have
t r
g(t)−g(r) = P (t−s)f(s)ds− P (r−s)f(s)ds
p p
Z0 Z0
t r
= P (t−s)f(s)ds+ (P (t−s)−P (r−s))f (s)ds
p p p
Zr Z0
t
= P (t−s)f(s)ds
p
Zr
r
+ AγP (r−s) A−γP (t−r)f(s)−A−γf(s) ds,
p p p p p
Z0
(cid:0) (cid:1)
8 MULTIDIMENSIONAL SDES WITH DISTRIBUTIONAL DRIFT
so that
kg(t)−g(r)kHp2−2ǫ−β
t
≤Zr kPp(t−s)f(s)kHp2−2ǫ−βds
r
+Z0 kAγpPp(r−s) A−pγPp(t−r)f(s)−A−pγf(s) kHp2−2ǫ−βds
t (cid:0) (cid:1)
≤ kA1−ǫ−β/2P (t−s)f(s)k ds
p p Lp
Zr
r
+ kA1−ǫ−β/2+γP (r−s) A−γP (t−r)f(s)−A−γf(s) k ds
p p p p p Lp
Z0
= : (S1)+(S2). (cid:0) (cid:1)
Let us consider (S1) first. We have
t
(S1) ≤ kA1−ǫP (t−s)k kA−β/2f(s)k ds
p p Lp→Lp Lp
Zr
t
≤Zr Cǫ(t−s)−1+ǫkf(s)kHp−βds
≤Cǫ(t−r)ǫkfk∞,Hp−β,
having used [19, Thm 6.13, (c)]. Using again the same result, the term (S2),
together with (16), gives (with the constant C changing from line to line)
r
(S2) = A1−ǫ+γP (r−s) P (t−r)A−γ−β/2f(s)−A−γ−β/2f(s) ds
p p p p p
Z0 (cid:13)r (cid:16) (cid:17)(cid:13)Lp
(cid:13) (cid:13)
≤ C (cid:13)(r−s)−1+ǫ−γ P (t−r)A−γ−β/2f(s)−A−γ−β/2f(s) (cid:13)ds
p p p
Z0r (cid:13) (cid:13)Lp
(cid:13) (cid:13)
≤ C (r−s)−1+ǫ−γ((cid:13)t−r)γkAp−γ−β/2f(s)kHp2γds (cid:13)
Z0
r
≤ C(t−r)γZ0 (r−s)−1+ǫ−γkf(s)kHp−βds
r
≤ C(t−r)γZ0 (r−s)−1+ǫ−γkfk∞,Hp−βds
≤ C(t−r)γrǫ−γkfk∞,Hp−β.
Therefore we have g ∈ Cγ [0,T];H2−2ǫ−β for each 0 < γ < ǫ and the
p
proof is complete. (cid:16) (cid:17) (cid:3)
The following lemma gives integral bounds which will be used later. The
proof makes use of the Gamma and the Beta functions together with some
basic integral estimates. We recall the definition of the Gamma function:
∞
Γ(a) = e−tta−1dt,
Z0
and the integral converges for any a ∈ C such that Re(a) > 0.
MULTIDIMENSIONAL SDES WITH DISTRIBUTIONAL DRIFT 9
Lemma 12. If 0 ≤ s < t ≤ T < ∞ and 0 ≤ θ < 1 then for any ρ ≥ 1 it
holds
t
(17) e−ρrr−θdr ≤ Γ(1−θ)ρθ−1.
Zs
Moreover if γ > 0 is such that θ+γ < 1 then for any ρ ≥ 1 there exists a
positive constant C such that
t
(18) e−ρ(t−r)(t−r)−θr−γdr ≤ Cρθ−1+γ.
Z0
Lemma 13. Let 1 < p,q < ∞ and 0 < β < δ with q > p ∨ d and let
δ
β+δ < 1. Then for b ∈ L∞([0,T];H−β) and v ∈ C([0,T];H1+δ) we have
p,q p
(i) ·P (·−r)b(r)dr ∈ C([0,T];H1+δ);
0 p p
(ii) ·P (·−r)(b(r)·∇v(r))dr ∈ C([0,T];H1+δ) with
R0 p p
R · (ρ)
(ρ)
P (·−r)(b(r)·∇v(r))dr ≤ c(ρ)kvk ;
(cid:13)Z0 p (cid:13)∞,Hp1+δ ∞,Hp1+δ
(cid:13) (cid:13)
(iii) λ (cid:13)·P (·−r)v(r)dr ∈ C([0,T];H(cid:13)1+δ) with
(cid:13)0 p (cid:13)p
R · (ρ)
(ρ)
λ P (·−r)v(r)dr ≤ c(ρ)kvk ,
(cid:13) Z0 p (cid:13)∞,Hp1+δ ∞,Hp1+δ
(cid:13) (cid:13)
where the cons(cid:13)tant c(ρ) is independ(cid:13)ent of v and tends to zero as ρ tends to
(cid:13) (cid:13)
infinity.
Observe that (δ,p) ∈ K(β,q) satisfies the hypothesis in Lemma 13.
Proof. (i) Lemma 10 implies that P (t)b(t) ∈ H1+δ for each t ∈ [0,T].
p p
Choosing ǫ = 1−β−δ, Proposition 11 implies item (i).
2
(ii)Similarlytopart(i),thefirstpartfollowsbyProposition11. Moreover
t
sup e−ρt P (t−r)(b(r)·∇v(r))dr
p
0≤t≤T (cid:13)Z0 (cid:13)Hp1+δ
(cid:13) (cid:13)
≤ c0≤sut≤pT Z(cid:13)(cid:13)0te−ρt(t−r)−1+δ2+βkv(r)kHp1+(cid:13)(cid:13)δkb(r)kHq−βdr
t
≤ ckbk∞,Hq−β 0≤sut≤pT Z0 e−ρrkv(r)kHp1+δe−ρ(t−r)(t−r)−1+δ2+βdr
(ρ) δ+β−1
≤ ckvk∞,Hp1+δkbk∞,Hq−βρ 2 < ∞.
· (ρ) (ρ)
Thus P (·−r)(b(r)·∇v(r))dr ≤ c(ρ)kvk .
0 p ∞,Hp1+δ ∞,Hp1+δ
(iii) Si(cid:13)mRilarly to parts (i) and (ii) t(cid:13)he continuity property follows by Propo-
(cid:13) (cid:13)
sition 11. Then
t t
sup e−ρt P (t−r)v(r)dr ≤ c sup e−ρtkv(r)k dr
0≤t≤T (cid:13)Z0 p (cid:13)Hp1+δ 0≤t≤T Z0 Hp1+δ
(cid:13) (cid:13)
(cid:13) (cid:13) ≤ ckvk(ρ) ρ−1 < ∞. (cid:3)
(cid:13) (cid:13) ∞,Hp1+δ
10 MULTIDIMENSIONAL SDES WITH DISTRIBUTIONAL DRIFT
2.2. Existence. Let us now introduce the integral operator I (v) as the
t
right hand side of (13), that is, given any v ∈ C([0,T];H1+δ), we define for
p
all t ∈ [0,T]
t t
(19) I (v) := P (t−r)(b(r)·∇v(r))dr+ P (t−r)(b(r)−λv(r))dr.
t p p
Z0 Z0
ByLemma13,theintegraloperatoriswell-definedanditisalinearoperator
on C([0,T];H1+δ).
p
Let us remark that Definition 8 is in fact meaningful under the assump-
tions of Lemma 13, which are more general than the ones of Definition 8
(see Remark 15).
Theorem 14. Let 1 < p,q < ∞ and 0 < β < δ with q > p ∨ d and let
δ
β+δ < 1. Then for b ∈ L∞([0,T];H−β) there exists a unique mild solution
p,q
v to the PDE (13) in H1+δ. Moreover for any 0< γ < 1−δ−β the solution
p
v is in Cγ([0,T];H1+δ).
p
Proof. By Lemma 13 the integral operator is a contraction for some ρ large
enough, thus by the Banach fixed point theorem there exists a unique mild
solution v ∈ C([0,T];H1+δ) to the PDE (13). For this solution we obtain
p
H¨older continuity in time of order γ for each 0 < γ < 1−δ−β. In fact each
term on the right-hand side of (19) is γ-H¨older continuous by Proposition
11 as b,b·∇v,v ∈ L∞([0,T];H−β). (cid:3)
p
Remark 15. By Theorem 14 and by the definition of K(β,q), for each
(δ,p) ∈ K(β,q) there exists a unique mild solution in H1+δ. However notice
p
that the assumptions of Theorem 14 are slightly more general than those of
Assumption 5 and of the set K(β,q). Indeed, the following conditions are
not required for the existence of the solution to the PDE (Lemma 13 and
Theorem 14):
• the condition d < p appearing in the definition of the region K(β,q)
δ
is only needed in order to embed the fractional Sobolev space H1+δ
p
into C1,α (Theorem 16).
• the condition q < d appearing in Assumption 5 is only needed in
β
Theorem 19 in order to show uniqueness for the solution u, indepen-
dently of the choice of (δ,p) ∈ K(β,q).
The following embedding theorem describes how to compare fractional
Sobolev spaces with different orders and provides a generalisation of the
Morrey inequality to fractional Sobolev spaces. For the proof we refer to
[25, Thm. 2.8.1, Remark 2].
Theorem 16. Fractional Morrey inequality. Let 0 < δ < 1 and d/δ <
p < ∞. If f ∈ H1+δ(Rd) then there exists a unique version of f (which
p
we denote again by f) such that f is differentiable. Moreover f ∈ C1,α(Rd)
with α = δ−d/p and
(20) kfk ≤ ckfk , k∇fk ≤ ck∇fk ,
C1,α Hp1+δ C0,α Hpδ