Remarks on uniqueness and strong solutions to
deterministic and stochastic differential equations
F. Flandoli
Dipartimento di Matematica Applicata “U. Dini”
Università di Pisa, Italy
March 12, 2008
Abstract
Motivated by open problems of well posedness in fluid dynamics, two
topics related to strong solutions of SDEs are discussed. The first one on
stochastic flows for SDEs with non regular drift helps to solve a stochastic
transport equation where the corresponding deterministic equation is not
well posed. The second one is a concept of strong superposition solution
motivated by problems where uniqueness is not true or not known.
1
1.1
Introduction
Motivations from fluid dynamics
The long time project where the topics of this note take their origin is the
attempt to improve the theory of fluid dynamics by means of random perturbations in the governing fluid equations. One of the millennium prize problems
(see the presentation of Fefferman [11]) is concerned with the 3D Navier-Stokes
equations in a domain D ⊂ R3
∂u
+ u · ∇u + ∇p = ν4u + f
∂t
divu = 0, u|t=0 = u0
(1)
which are a rather good model for incompressible, viscous, constant-density
fluids; here u : D ×[0, ∞) → R3 is the velocity field of the fluid, p : D ×[0, ∞) →
R is the pressure field, both unknown, f : D ×[0, ∞) → R3 is the body force and
ν > 0 is the kinematic viscosity. Without details, let us just recall that, chosen
suitable boundary conditions, if u0 is a divergence free square integrable field
and f satisfies natural integrability conditions, there exists at least one weak
Leray-Hopf solutions (u, p), a notion which includes the property
Z
Z TZ
2
2
sup
|u (x, t)| dx +
k∇u (x, t)k dxdt < ∞
t∈[0,T ]
D
0
1
D
and the energy in equality
Z
Z tZ
1
2
2
|u (x, t)| dx + ν
k∇uk dxds
2 D
0
D
Z
Z tZ
1
2
≤
|u (x, 0)| dx +
u · f dxds.
2 D
0
D
Additional properties are known at least for simple domains, like
Z TZ
3/2
|p (x, t)| dxdt < ∞
0
D
Z
sup
k∇u (x, t)k dx < ∞
t∈[0,T ]
D
but they do not improve sufficiently the theory. The existence or absence of
singularities is a version of the above mentioned millennium open problem,
another version being the uniqueness of the weak Leray-Hopf solutions.
Since stochastic ordinary differential equations with nondegenerate diffusion
term have unique solutions under very general assumptions on the drift coefficient, where the corresponding deterministic equations may have multiple solutions (see for instance [30], [22]), it is natural to investigate stochastic versions
of the previous equations to see whether the theory is more complete. There are
at least two interesting kind of stochastic perturbations that we could introduce,
that we call (bilinear) multiplicative noise and additive noise, respectively. The
equations take the form
du + (u · ∇u + ∇p − ν4u − f ) dt
X
X
=
bk · ∇u ◦ dW k +
σj ej dβ j
j
k
where bk : D × [0, ∞) → R3 are given, {ej } is a complete orthonormal
© system
ª
of the L2 -space where the equation is settled, σj are real numbers, W k , β j
are independent one-dimensional Brownian motions on some probability space
(Ω, F, P ). Existence of weak Leray-Hopf solutions of the martingale problem
for these equations have been proved under various generality, see for instance
[15] and [26]. Uniqueness of weak solutions remains an open problem as well as
the existence or absence of singularities, but a few progresses have been made,
that we are going to summarize.
A non degenerate additive noise, a priori, looks more promising, having in
mind the finite dimensional theory. However, direct tools like Girsanov transformation looks absolutely far from applicable, so one has to try more difficult
approaches. Concerning singularities, thanks to a generalization of CaffarelliKohn-Nirenberg theory to the stochastic case, it was possible to prove that
time-stationary solutions have no singularities with probability one, at every
given time t, see Flandoli and Romito [19]; unfortunately, trajectories could
become singular sometimes so uniqueness remains open. Later, see [7], this P a.s. regularity at given times t has been understood also by other methods, but
2
regularity of trajectories was not improved. One of the main progress has been
the direct solution of the Kolmogorov equation by Da Prato and Debussche [7].
However the function spaces where it can be solve at present are not sufficient
to apply Itô formula or other arguments and prove uniqueness. In spite of this,
in the works [7], [8], [20] it has been proved under various generalities that the
stochastic Navier-Stokes equations have Markov selections of weak solutions,
and that every such Markov selection is strong Feller if the noise is sufficiently
non degenerate. This property of existence of selections which depend continuously on initial conditions is completely unknown in the deterministic case,
where it looks a problem as difficult as the millennium one. About additive
noise, let us mention another interesting feature: the energy inequality, under
expectation E over the space (Ω, F, P ), takes the form
·Z
¸
·Z t Z
¸
1
2
2
E
|u (x, t)| dx + νE
k∇uk dxds
2
D
0
D
·Z
¸
X
1
2
≤ E
|u (x, 0)| dx + t
σj2 .
2
D
j
Compared to the deterministic
energy inequality, we see that white noise injects
P
a given amount of energy j σj2 per unit of time, opposite to deterministic forces
RtR
f which act through the “work” term 0 D u · f dxds, still depending on the
unknown u, so its amplitude is not clear. This simplicity of the energy balance
is at the basis of some investigations
on turbulence, see [23], [16].
P
b
·
∇u ◦ dW k (◦ is Stratonovich operation)
The multiplicative noise
k
k
looks a priori more difficult than additive noise, degenerate in a sense. However
Mikulevicius and Rozovskii [26] insisted on it and we now believe it could be
of
P great interest kfor the question of singularities and uniqueness. The term
k bk · ∇u ◦ dW , under the assumption divbk = 0, produces a rotation in
the L2 space where u lives. There is no energy input due to this term. Its
effect is to re-distribute energy between modes. As such, it could be that it
depletes the rather well-organized dynamical formation of singularities, which is
a phenomenon (if it exists) of too rapid cascade of energy from larger to smaller
scales. Perhaps the random re-distribution poses obstacles to this procedure.
We cannot answer this question at present on the Navier-Stokes equation but
we may give a positive answer in the case of a very simplified related model, the
transport equation.
1.2
Content of the note
Given the previous motivations, in this note we discuss two topics related to
strong solutions of stochastic ordinary differential equations. The first one is the
existence of the stochastic flow of diffeomorphisms for the stochastic differential
equation in Rd :
dXt = b (Xt ) dt + σdW (t) ,
3
t ≥ 0,
X0 = x
(2)
when b : Rd → Rd is only Hölder continuous and bounded and σ is a non-zero
number. Strong solutions to such equation are known for even less regular b,
the best result (after previous investigations of Zvonkin [33], Veretennikov [31],
Portenko [27], among others) being proved by Krylov and Röckner [22], who
require only that b is of class Lp for p > d. However, strong solutions are useful
when they allow us to treat pathwise the dependence on certain parameters,
like the differentiability with respect to them (otherwise weak solution usually
suffice). For this reason we find interesting to have a differentiable stochastic
flow and not only strong solutions for every initial condition. For this result
we need b Hölder continuous, at present. The stochastic flow becomes a central
ingredient to solve uniquely (in a strong sense) a stochastic transport equation
(a linear stochastic partial differential equation), where the corresponding deterministic transport equation is open. This is a small step in the direction of
the problems described in the previous section.
The second topic we address in this note is the concept of superposition
solution, which is motivated by the fact that certain equations may have non
uniqueness and the physically interesting objects to investigate are not single
solutions but proper superpositions of them. We describe ideas around this
concept from the deterministic literature, essentially known, culminating in a
definition of strong superposition solution of a stochastic equation, that apparently is presented here for the first time. One of the reasons to investigate these
solutions is that they are strong, even if generalized, and they “always” exist,
contrary to classical strong solutions. We limit ourselves to finite dimensional
equations but clearly one of our motivations is do develop some mathematics
applicable to fluid dynamic problems when well posedness is not know or not
true.
2
Stochastic transport equation
The following deterministic PDE in Rd is called transport equation:
∂u (x, t)
+ b (x) · ∇u (x, t) = 0, (x, t) ∈ Rd × [0, ∞)
∂t
u (x, 0) = u0 (x) , x ∈ Rd .
If b : Rd → Rd is Lipschitz continuous, This equation can be ¡solved
¢ in several
p
d
ways. One of the result is that, given p¡ ≥ 1, for every
u
∈
L
R
there exists
0
¡ ¢¢
a unique distributional solution u ∈ C [0, ∞); Lp Rd , represented in the form
¡
¢
u (x, t) := u0 ϕ−1
t (x)
where ϕt is the flow of diffeomorphisms generated by the ordinary differential
equation
dXtx
= b (Xtx ) , t ≥ 0, X0x = x
(3)
dt
4
(ϕt x = Xtx ). In the sequel, for simplicity of exposition, let us always assume
divb = 0
which is also the most interesting case compared to the fluid dynamic equations
described in the introduction. Di Perna ¡and¢P.L. Lions [9] extended in a suitable
sense this result to the case b ∈ W 1,1 Rd and Ambrosio [1] to fields b is of
bounded variations. They replaced the pointwise solution of equation (3) with
the concept of Lagrangian flow, and they used a direct analysis of the transport
equation, and of the associated continuity equation, to construct uniquely the
flow. Part of these results have been also extended by Figalli [12] to a stochastic
version of equation (3) and the related parabolic equations.
In all these results some regularity assumption on ∇b is imposed, at least
L1 or a signed finite measure. When we only assume that b is Hölder continuous, equation (3) may present several pathologies (non-uniqueness, coalescence)
with consequences on the transport equation. One could speculate that similar
phenomena could appear for the equations of fluid dynamics, also because theories of Kolmogorov and of Onsager would predict that turbulent fluids are only
Hölder continuous with exponent 13 . Here comes the contribution of random
perturbations: the following stochastic version of the transport equation, with
(bilinear) multiplicative noise
dt u (t, x) + (b (x) · ∇u (t, x)) dt + σ∇u (t, x) ◦ dW (t) = 0
(4)
u (0, x) = u0 (x)
¡
¢
is well posed under the assumption b ∈ Cbα Rd for some α > 0 and σ 6= 0;
W (t) is a Brownian motion in Rd on a probability space (Ω, F, P ). Denote by
(Ft ) the filtration associated to the Brownian motion. For this equation one can
prove the following result. Let p ≥ 1 be given.
¡ ¢
Theorem 1 Given u0 ∈ Lp (Rd ), b ∈ Cb0,α Rd for some α > 0 and σ 6= 0,
strong unique solutions exist for the SDE (2) and they generate a stochastic
flow of diffeomorphisms ϕt (ω). Moreover, there exists a unique strong (in the
probabilistic sense) distributional solution in Lp of the Cauchy problem (4),
namely a measurable function (u(t,Rx, ω), t ≥ 0, x ∈ Rd , ω ∈ Ω) such that
p
(i) for P -a.e. ω ∈ Ω, supt∈[0,T ] Rd |u(t, x, ω)| dx < ∞
R
(ii) for every test function θ ∈ C0∞ (Rd ), the process t 7→ Rd u(t, x)θ(x)dx is
a continuous (Ft )-semimartingale
(iii) for every test function φ ∈ C0∞ (Rd ), we have:
Z
Z t Z
Z
ds
u(s, x)b(x) · ∇φ(x)dx +
u0 (x)φ(x)dx
u(t, x)φ(x)dx =
Rd
Rd
0
+
d Z t µZ
X
i=1
0
Rd
Rd
¶
u(s, x)Di φ(x)dx ◦ dWi (s) .
This unique solution is given by
u(t, x, ω) := u0 (ϕ−1
t (ω) x).
5
(5)
Full details of the proof and additional facts are given in [17]. The idea of
the proof, however, is easy to describe. Under the assumption b ∈ Cb0,α , for
every µ > 0 the (vector valued) elliptic PDE
σ2
4G + b · ∇G = µG − b
2
(6)
has a unique solution G ∈ Cb0,2+α . The new variable
Yt = φ (Xt ) := Xt + G (Xt )
satisfies
σ2
4G (Xt ) dt
2
= dXt − b (Xt ) dt + µG (Xt ) dt + σ∇G (Xt ) dWt
= σdWt + µG (Xt ) dt + σ∇G (Xt ) dWt .
dYt = dXt + (b · ∇G) (Xt ) dt + σ∇G (Xt ) dWt +
Since
lim k∇GkLip = 0
µ→∞
We may choose µ so large that φ is a Cb2+ε diffeomorphism, so Y satisfies the
traditional SDE
¡
¢
£
¡
¢¤
dYt = µG φ−1 (Yt ) dt + σ 1 + ∇G φ−1 (Yt ) dWt
where the coefficients are at least of class Cb1+α . This implies existence and
0
uniqueness of a strong solution Y and existence of a C 1+α stochastic flow of
diffeomorphisms, for every α0 < α, see [Kunita]. The proof that u(t, x, ω) :=
u0 (ϕ−1
t (ω) x) is the unique distributional solution is lengthy but not difficult,
see [17].
Remark 2 The transport equation is a very simplified linearized version of fluid
dynamic equations. Not only it is linear, but also we assume b given and deterministic (we assumed it only space dependent but extensions to time dependent
b is only a technical problem). In view of nonlinear terms like u · ∇u appearing
in usual fluid dynamic equations, there is also the difficulty that b = u is a
random field, beside the fact it is not a priori given. The generalization of the
method of the previous section to random field b is not a technical matter, since
the are even easy counterexamples. Therefore the result described above can be
only considered as a very preliminary step.
3
3.1
Superposition solution of deterministic ODEs
Definitions
The following presentation is based on [9], [1], [2], [12], [18].
6
Consider the Cauchy problem in Rd
dXt
= b (Xt )
dt
X0 = x
t≥0
where b : Rd → Rd is continuous and satisfies
some
³
´ growth condition for non
2
explosion of solutions, like hb (x) , xi ≤ C |x| + 1 , x ∈ Rd , for some constant
C > 0. We call this Cauchy problem (CPx ). On the contrary we call (ODE)
t
the differential equation dX
dt = b (Xt ),
¡ t ≥ 0.
¢
Denote by W the metric space C [0, ∞); Rd with the topology of uniform
convengence on compact intervals, and by BW its Borel σ-field.
Given x ∈ Rd , denote by C (x) the set of all continuous functions (Xt )t≥0
which satisfy the Cauchy problem (CPx ), namely
Z
Xt = x +
t
b (Xs ) ds
t ≥ 0.
0
Lemma 3 C (x) is a non empty compact set (hence C (x) ∈ B W ) and it is
connected. For every w ∈ W, the function x 7→ dist (w, C (x)) is measurable.
The claim C (x) 6= ∅ is Peano theorem, and closedness follows by a similar
argument using Ascoli-Arzelà theorem. A proof that it is connected has been
given by [29]. Measurability of x 7→ dist (w, C (x)) requires details from [5].
p
Example 4 Take d = 1, b (x) = 2sign (x) |x|: the set C (0) is not a singleton.
It contains the two extremal curves Xt± = ±t2 , the zero curve Xt0 = 0, and
infinitely many others easily computable.
Common tools to deal with non-uniqueness are the various concepts of multivalued or generalized flows, see for instance [4], [28]; this is useful to develop
some dynamical systems theory.
However one can deal with non-uniqueness also by means of probabilistic
tools, which, a priori, let us say philosophically, are more suitable if we want to
give different weights to different solutions. Let us introduce some notations.
Given a Polish space X, with Borel σ-field B X , denote by Pr (X) the set of all
Borel probability measures on X. The set Pr (X) is a metric space with the
weak (narrow) topology, the topology of convergence on test functions of class
Cb (X) (the space of continuous bounded functions on X).
Following the notations and definitions of [6], given a probability space
(Ω, F, P ) and a Polish space X, we introduce the set PrΩ,P (X) (simply PrΩ (X)
if P is clear) of all measurable mappings ω 7→ µω from Ω to Pr (X) (we call
ω 7→ µω “random measure”). To be more precise, we identify two random
measures that coincide P -a.s. and denote by PrΩ,P (X) the set of equivalence
classes. It is a Polish space itself with the following convergence, called narrow convergence of random measures: a sequence (µω
n ) ⊂ PrΩ,P (X) narrowly
7
converges to (µω ) ∈ PrΩ,P (X) if
Z
Z
µω
(f
)
Z
(ω)
P
(dω)
→
µω (f ) Z (ω) P (dω)
n
Ω
Ω
for every f ∈ Cb (X) and every integrable Z on (Ω, F, P ). See [6] for other
characterizations and several results on this topology.
Finally, given
(µω ) ∈
¡
¢
·
X
PrΩ,P (X), we may define the measure P ⊗ µ on Ω × X, F × B
as
Z
(P ⊗ µ· ) (A × B) =
µω (B) P (dω) , A ∈ F, B ∈ B X .
A
The random measure (µω ) can be reconstructed from P ⊗ µ· : it is P -a.s. equal
(up to an obvious identification) to the regular conditional probability distribution of P ⊗ µ· with respect to the σ-field F × X.
Definition 5 We call superposition solution of Cauchy problem (CPx ) any
probability measure µ ∈ Pr (W) such that
µ (C (x)) = 1.
Example 6 Going back to example 4,
µ∗ =
1
1
δ + + δX −
2 X
2
is an example of superposition solution of (CP0 ).
Definition 7 More generally, We call superposition solution of the (ODE) any
µ ∈ Pr (W) of the form
Z
µ=
µx µ0 (dx)
Rd
¡ ¢
where µ0 ∈ Pr Rd and (µx ) ∈ PrRd ,µ0 (W) with the property
¡
¢
µ0 x ∈ Rd : µx (C (x)) = 1 = 1.
We may also call such µ superposition solution of the (ODE) with initial distribution µ0 .
Superposition solutions of the Cauchy problem (CPx ) correspond to the case
µ0 = δx .
Remark
8 ¢Following [1], a superposition solution¡ of¢ the (ODE) is any µ ∈
¡
Pr Rd × W of the form µ = µ0 ⊗ µ· with µ0 ∈ Pr Rd and (µx ) ∈ PrRd ,µ0 (X)
with the property
¡
¢
µ0 x ∈ Rd : µx (C (x)) = 1 = 1.
8
¡
¢
The two formulations are equivalent. If µ ∈ Pr Rd × W is a solution in the
sense of this remark, µ = µ0 ⊗ µ· , then the measure µ ∈ Pr (W) defined as
Z
µ := E µ0 [µ· ] =
µx µ0 (dx) = π2 (µ0 ⊗ µ· )
Rd
R
is a solution in the sense of the definition above. Viceversa, if µ = Rd µx µ0 (dx) ∈
Pr (W) is a solution in the sense of the definition above, then µ := µ0 ⊗ µ· is
a solution in the sense of this remark. Thus the equivalence is obvious. What
could appear less obvious a priori is that a solution
µ in the sense of the sense
R
of the definition has a unique decomposition as Rd µx µ0 (dx), or that it carries
the same degree of information as µ = µ0 ⊗ µ· , since an expectation is used to
produce µ from µ. There is no reduction of information: Given such µ one can
reconstruct µ0 and µ· : given µ ∈ Pr (W), µ0 is uniquely given byπ0 µ, while µ·
is the regular conditional probability distribution of µ with respect to the σ-field
associated to π0 . To summarize, the two formulations are equivalent from any
viewpoint.
¡
¢
Remark 9 If µ ∈ Pr Rd × W is a superposition solution in the sense of the
previous remark, then
¡
¢
µ (x, w) ∈ Rd × W : w (0) = x = 1.
3.2
Relations with other notions
It is sometimes useful (for instance to perform computations) to know that
superposition solutions may be re-interpreted as stochastic processes. We have
the following rather obvious representation result.
Lemma 10 If µ is a superposition solution
¡ of (ODE)
¢ with initial distribution
µ0 , then the canonical process (ξt )t≥0 on W, B W , µ , defined as ξt (w) = w (t),
t ≥ 0, w ∈ W, having law µ, satisfies
Z t
ξt (w) = ξ0 (w) +
b (ξs (w)) ds t ≥ 0
0
for µ-a.e. w ∈ W and ξ0 has law µ0 .
9
¢
¡
R·
Proof. The set w ∈ W : ξ· (w) − ξ0 (w) − 0 b (ξs (w)) ds ≡ 0 belongs to
B W since it is defined by a continuous function. Then we simply have
µ
¶
Z ·
µ w ∈ W : ξ· (w) − ξ0 (w) −
b (ξs (w)) ds ≡ 0
0
µ
¶
Z ·
b (w (s)) ds ≡ 0
= µ w ∈ W : w (·) − w (0) −
0
¡
¢
= µ w ∈ W : w ∈ C (x0 ) for some x0 ∈ Rd
Z
¡
¢
µx w ∈ W : w ∈ C (x0 ) for some x0 ∈ Rd µ0 (dx)
=
d
ZR
µx (w ∈ W : w ∈ C (x)) µ0 (dx)
≥
Rd
= 1.
The other claims are obvious. The proof is complete.
Remark 11 Similarly, if µ is a superposition solution of (ODE)³ in the sense of
´
d
remark 8, with initial distribution µ0 , then the process (ξt )t≥0 on Rd × W, B R ⊗ B W , µ0 ⊗ µ·
defined as ξt (x, w) = w (t), t ≥ 0, x ∈ Rd , w ∈ W, has law
Z
E µ0 [µ· ] =
µx µ0 (dx) = π2 (µ0 ⊗ µ· )
Rd
and satisfies
Z
t
ξt (x, w) = x +
b (ξs (x, w)) ds
t≥0
0
for (µ0 ⊗ µ· )-a.e. (x, w) ∈ Rd × W.
We can now state some easy equivalences of notions. Recall (see [30]) that
µ ∈ Pr (W) is called a solution of the martingale
problem
¡
¢ for (ODE) if, denoted
as above by (ξt )t≥0 the canonical process on W, BW , µ defined as ξt (w) = w (t),
t ≥ 0, w ∈ W, then the stochastic process
Z t
f (ξt ) − f (ξ0 ) −
(b · ∇f ) (ξs ) ds t ≥ 0
0
¡ ¢
is a martingale (with respect to the filtration of (ξt )t≥0 ), for every f ∈ C0∞ Rd
(the space of all smooth compact support f : Rd → R). We speak of the
martingale problem for (CPx ) when π0 µ = δx .
Proposition 12 The following conditions are equivalent:
i) µ is a superposition solution of (ODE)
ii) µ is a solution¡ of ¢the martingale problem for (ODE)
iii) (µt )t≥0 ⊂ Pr Rd is narrowly measurable in t and satisfies
Z t
µt (f ) = µ0 (f ) +
µs (b · ∇f ) ds t ≥ 0
0
10
(7)
¡ ¢
for all f ∈ C0∞ Rd .
Remark 13 By equivalence with (iii) we mean: if µ satisfies (i) or (ii), then
µt = πt µ satisfies (iii); viceversa, if (µt )t≥0 satisfies (iii), then there is µ satisfying (i) (or (ii)), such that µt = πt µ. In particular, in (iii) we do not assume
that the family (µt )t≥0 comes from some µ ∈ Pr (W). In a sense, the coherence
of this family (to be marginal of a distribution in path space) is embodied into the
continuity equation. That (iii) implies (i) is a result of [1], called “superposition
principle”.
Proof. The proof that (i) implies (ii) is an easy exercise, having lemma 10.
Property (ii) implies (iii): since the stochastic process
Z
t
f (ξt ) − f (ξ0 ) −
(b · ∇f ) (ξs ) ds
t≥0
0
is a µ-martingale, then its µ-expectation is zero. The only difficult part is that
(iii) implies (i). We do not repeat the non-trivial proof, see [2]. We simply
mention that it makes use of a regularized (ODE) corresponding to a mollification of (µt )t≥0 , problem where the existence of a measure µε ∈ Pr (W) is rather
classical, and constructs a measure µ ∈ Pr (W) behind (µt )t≥0 as a weak limit
of (µε ).
Remark 14 Formally equation (7) may be written as the partial differential
equation
∂µt
+ div (bµt ) = 0
∂t
that we may call continuity equation associated to the (ODE). It is also a
particular case of the Fokker-Planck equation. Equation (7) is the weak or distributional formulation of the continuity equation.
3.3
Existence, uniqueness, flow properties
Since C (x) 6= ∅ for every x ∈ Rd , existence of superposition solutions for (CPx )
is trivial, and existence of superposition solutions for (ODE) with a given initial
distribution µ0 may be proved by a measurable selection theorem, that we do
not discuss since it is a particular case of the Markov selection result described
below.
About uniqueness for (CPx ), given x ∈ Rd the following conditions are
equivalent:
i) there is only one superposition solution of (CPx )
ii) C (x) is a singleton
iii) there is X ∈ W such that µ = δX for every superposition solution µ of
(CPx ).
The proof is very easy. For uniqueness concerning superposition solutions of
(ODE) with a given initial distribution µ0 , see the end of this section.
11
Concerning some kind of flow property for superposition solutions, that may
be useful to start investigations in the vein of dynamical systems (see a notion of Markov attractor in [18]), a relevant condition is Chapman-Kolmogorov
equation for marginals, or Markov property. We call selection of superposition solutions a family {µx }x∈Rd of superposition solutions indexed by x ∈ Rd
such that the mapping x 7→ µx is measurable. This may look like an element
of PrRd (W) but there are two differences: we require µx to be defined for all
x ∈ Rd , and no measure on Rd is involved. A selection of superposition solutions
{µx }x∈Rd will be called a Markov selection if
Z
x
µt+s = µys µxt (dy) t, s ≥ 0, x ∈ Rd
where as usual we set µxt = πt µx .
In the set-up of martingale solutions, there is a theorem of Krylov [21], see
also Stroock-Varadhan [30], which says that there exist at least one Markov
selection associated to (??). Since martingale and superposition solutions coincide, the theorem holds true in the set-up of superposition solutions. Thus one
has:
Proposition 15 There exist at least one Markov selection for the (ODE).
Stroock-Varadhan [30] prove also that uniqueness of Markov selections is
equivalent to the property that C (x) is a singleton for every x ∈ Rd , namely to
the well posedness for (ODE). Thus Markov selection are not a way to select a
distinguished unique object. Other criteria are necessary.
Can we read uniqueness for (CPx ) from the continuity equation (7)? It looks
difficult because we typically expect singular measures µt (the continuity equation, as a Fokker-Planck equation, is certainly degenerate). More promising is
to consider the continuity equation (7) with the purpose of uniqueness of superposition solutions to (ODE) with initial distribution µ0 absolutely continuous
0
with respect to Lebesgue measure, with density denoted in the sequel by ∂µ
∂x .
First, we have:
¡ ¢
Proposition 16 Given µ0 ∈ Pr Rd , the following conditions are equivalent:
i) there is only one superposition solution of (ODE) with initial distribution
µ0
ii) C (x) is a singleton for µ0 -a.e. x ∈ Rd
iii) there is a measurable mapping x 7→ X x from Rd to W such that µ0 (µx = δX x ) =
1 for every superposition solution µ of (ODE)
iv) there is only one solution (µt )t≥0 ⊂ Pr (W), narrowly measurable in t,
of equation (7) with initial condition µ0 .
Proof. (ii) implies (iii). There is a Borel set B ⊂ W of full µ0 -measure such
x
that C (x) = {X x } for all x ∈ B. Set
/ B. The mapping x 7→ X x
R X = 0 for x ∈
is measurable by lemma 3. If µ = Rd µx µ0 (dx) is a superposition solution of
(ODE), from µx (C (x)) = 1 for µ0 -a.e. x ∈ Rd we deduce µx = δX x for µ0 -a.e.
x ∈ Rd . (iii) is proved.
12
(iii) implies (i), because two superposition solutions of (ODE) have the same
(µx ) (up to µ0 -null sets).
(i) implies (ii). Let x 7→ X x be a measurable selection from the family
(C (x))x∈Rd . Set D (x) = C (x) {X x }. The set N of all x such that D (x) 6= ∅
is a Borel set. Let x 7→ Y x be a measurable selection from the family (D (x))x∈N ,
and extend it to zero outside N . The measures
Z
µX =
δX x µ0 (dx)
Rd
Z
µY =
δY x µ0 (dx)
Rd
are superposition solution of (ODE) with initial distribution µ0 , hence they
coincide, namely
δX x = δY x for µ0 -a.e. x ∈ Rd .
Since Y x 6= X x on N , we have µ0 (N ) = 0. (ii) is proved.
Finally, (i) and (iv) are equivalent, by proposition 12. The proof is complete.
In [9], [1], [2], using the concept of renormalized solutions, the following
statement is proved; it is similar to the statement on the transport equation
mentioned in the previous section (this is a further link between the subjects of
this note). To avoid technicalities, we restrict to the important case divb = 0.
¡ ¢
Theorem 17 If b ∈ W 1,1 Rd or even if it is just of bounded variation, then
¡ d¢
∞
0
for every initial distribution µ0 with ∂µ
R the continuity equation (7)
∂x ∈ L
¡ d¢
∞
t
has only one solution (µt )t≥0 ⊂ Pr (W) with the property ∂µ
R for
∂x ∈ L
every t ≥ 0.
We cannot deduce that C (x) is a singleton for a.e. x ∈ Rd¡, since
the
¢
∞
d
t
uniqueness for (7) is proved only in the class where ∂µ
∈
L
R
holds.
∂x
However it is possible to deduce the uniqueness of a special kind of flow, called
Lagrangian flow, see [9], [1], [2].
3.4
Zero-noise limit
From a physical viewpoint, solutions of the Cauchy problem (CPx ) that are
stable under random perturbations are more natural than others. By stability
under random perturbation we mean here the following concept. Consider the
stochastic differential equation (the same as (2))
dXtε = b(Xtε )dt + εdWt ,
X (0) = x0
where (Wt )t≥0 is a d-dimensional standard Brownian motion and ε > 0. If
¡ ¢
b ∈ Lp Rd with some reasonable growth condition, there is uniqueness in law,
¡ ¢
see [27]. We have also seen above in theorem 1 that if b ∈ Cb0,α Rd for some
α > 0 then this equation even defines a stochastic flow. Call Pxε0 the unique law,
13
on B W . The limit as ε → 0 of Pxε0 , in the weak topology, is a complex subject.
In general, at least if b satisfies the assumptions at the
of section 3.1
© beginning
ª
we can only prove that for every x0 ∈ Rd , the family Pxε0 ε>0 is tight and each
limit point of it is a superposition solution of the Cauchy problem (CPx ). But
there could be subsequences with different limits, at least in principle.
Fortunately there are at least a few examples of non-well posed (ODE) where
it is possible to prove that Pxε0 converges to a unique superposition solution, see
[3]. A very interesting example is the Cauchy problem
p
dXtε = 2sign(Xtε ) |Xtε |dt + εdWt , X0ε = 0
whose law P0ε weakly converges to the nontrivial superposition solution µ∗ of
example 6. This example is basic for the motivation of this theory: for certain
non-well posed systems the physically natural objects are nontrivial superposition solutions. Thus superposition solutions are the correct objects to deal
with.
Related to the zero-noise limit, let us remark that other a priori natural
limits are less suitable for selection. Given any superposition solution µ of
(CPx ), it is possible to construct a sequence of smooth fields bn (x, t) such the
corresponding solutions tend to µ, see [2]. Using only autonomous fields bn (x)
one cannot approximate all superposition solution, but at least several ones, see
[2] and [10], where exactly the example treated above is illustrated. Thus the
zero-noise limit looks more strict in identifying physically interesting limits.
3.5
Conclusions
To summarize the theory in the deterministic outlined above, we could say that
we have a notion of superposition solution, based on probabilistic tools, which is
mathematically quite satisfactory because of a series of natural equivalences with
other formulation (martingale solutions, solutions of the continuity equation),
a representation result in terms of processes useful to perform computations,
the possibility to build-up Markov flows; and it looks physically relevant due to
some example of zero-noise limit. If we insist on non-well posed systems and
want to develop mathematics for them, we think that the main open problem is
to devise good selection criteria, namely to be able to identify special families of
superposition solutions with distinguished properties. Markov selections are at
present only a minor step in this direction, useful mainly for dynamical systems
argument (not only: see for instance [20]); indeed, if the system is not well
posed, then there are several Markov selections and it is not clear which one
is physically more relevant. Zero-noise limit would be one of the most natural
approaches but it is at present limited to very few examples. A breakthrough
would be, in our opinion, to devise variational principles for selection purposes.
There is some similarity between this topic and invariant measures for chaotic
dynamical systems, a theory where variational principles exist for zero-noise
measures, under the assumptions of the theory of Bowen-Ruelle-Sinai.
14
4
Strong superposition solutions of SDEs
The aim of this section is to investigate the analog of the concepts described
above in the case of stochastic equations. Results for equations driven by rough
paths are given in [13]. We discuss here Itô type equations.
Consider the SDE
X
dXt = b (Xt ) dt +
σk (Xt ) dWtk
(8)
k
X0 = x ∈ Rd
with bounded continuous coefficients b, σk : Rd → Rd . Boundedness is assumed
only
¡ k ¢for simplicity, we need a growth condition to avoid explosion. The processes
Wt t≥0 , k = 1, 2, ... are independent Brownian motions on a probability space
(Ω, F, P ).
We shall denote the previous stochastic Cauchy problem by (SCPx ), while
the stochastic equation itself by (SDE).
4.1
Pathwise superposition solutions, additive noise
Consider the simple case when σk are given vectors in Rd (constant fields), that
we call additive noise case. We assume
X
2
kσk k < ∞
k
P
which guarantees that the R -valued stochastic process t 7→ k σk Wtk is a.s.
continuous. The stochastic equation has a pathwise meaning:
Z t
X
Xt (ω) = x +
b (Xs (ω)) ds +
σk Wtk (ω) .
d
0
k
d
For P -a.e. ω ∈ Ω, and for a given x ∈ R , we denote by C (x, ω) the set of all
continuous functions (γt )t≥0 which satisfy
Z t
X
γt = x +
b (γs ) ds +
σk Wtk (ω) t ≥ 0.
0
k
Proposition 18 For P -a.e. ω ∈ Ω and every x ∈ Rd , C (x, ω) is a non empty
compact connected set. For every w ∈ W, the function (x, ω) 7→ dist (w, C (x, ω))
is measurable.
To prove this statement it is sufficient to prove the result for the equation
Z t
eb (xs , s) ds t ≥ 0
xt = x +
0
where eb (x, t) = b (x + g (t)), g ∈ W, that is similar to the deterministic case
described above, and then apply a simple random transformation.
15
Definition 19 We call pathwise superposition solution of the Cauchy problem
(SCPx ) any random probability measure (µω ) ∈ PrΩ,P (W) such that
µω (C (x, ω)) = 1
for P -a.e. ω ∈ Ω
This strategy of definition can be applied to stochastic equations with general
(not additive) noise any time we can interpret pathwise the equation, perhaps
after a random transformation.
4.2
Strong superposition solutions, general case
Having in mind proposition 12, it is also natural to give the following general
definition of random superposition solution based on a suitable continuity equation, in place of the pathwise sets C (x, ω) which are restricted to particular
classes of SDEs.
Notice that, if (Xt ) is a solution of equation (8), for every smooth compact
support function f : Rd → R we have
Z t
XZ t
(σk · ∇f ) (Xs ) dWsk
f (Xt ) = f (x) +
(Lf ) (Xs ) ds +
0
where
Lf =
0
k
¡
¢
1X
trace σk σk∗ D2 f + b · ∇f.
2
k
Setting
µω
t
:= δXt (ω) , we have
Z
µt (f ) = µ0 (f ) +
t
µs (Lf ) ds +
0
XZ
k
0
t
µs (σk · ∇f ) dWsk .
(9)
This is the stochastic continuity equation (or stochastic Fokker-Planck equation)
associated to equation (8). Formally we could write
X
div (σk µt ) dWtk .
dµt = L∗ µt dt −
k
Definition 20 We call strong superposition solution of the Cauchy problem
ω
(SCPx ) any random probability measure (µω ) ∈ PrΩ,P (W) such that µω
t := πt µ
is adapted to the filtration of the Brownian motion and the stochastic continuity
equation (9) is satisfied, with µ0 = δx .
If (Ft ) denotes the filtration associated to the Brownian motion, adaptedness
¡ d¢
ω
of (µω
t ) to (Ft ) means that (t, ω) 7→ µt (ϕ) is (Ft )-adapted for every ϕ ∈ Cb R .
Remark 21 Taking expectation in (9) we readily have that the measure ρ ∈
Pr (W) defined as
ρ = E [µ· ]
16
has marginals ρt = πt ρ satisfying the classical Fokker-Planck equation
Z t
ρt (f ) = ρ0 (f ) +
ρs (Lf ) ds
0
¡
∞
for every f ∈ C0
4.3
¢
Rd , namely
∂ρt
∂t
= L∗ ρt in distributional form.
Existence of strong pathwise superposition solutions
To construct superposition solutions we use the following very simple principle.
Lemma 22 Let {X n } be a sequence of random variables on a probability space
n
(Ω, F, P ) with values in a Polish space W and let ρn denote the
© law of X ª on
W
B . If {ρn } is tight, then the sequence of random measures ω 7→ δX n (ω) is
tight in PrΩ,P (W).
©
ª
Moreover, if (µω ) ∈ PrΩ,P (W) is a limit point of ω 7→ δX n (ω) , then E P [µ· ]
is a limit point of ρn ; and viceversa,
© if ρ is a limit
ª point of ρn , then there exists
a limit point (µω ) ∈ PrΩ,P (W) of ω 7→ δX n (ω) such that E P [µ· ] = ρ.
©
ª
n (ω)
Proof.
Tightness
of
ω
→
7
δ
in PrΩ,P (W) corresponds to tightness
X
£
¤
P
of E δX n (·) in W, see [6]. Given a compact set K, we have
£
¤
E P δX n (·) (K) = P (X n ∈ K) = ρn (K)
and thus it is sufficient to use the assumption ©
of tightness of
ª {νn }.
n (ω) . It is the limit of
Let (µω ) be a limit
point
in
Pr
(W)
of
ω
→
7
δ
Ω,P
X
©
ª
some subsequence ω 7→ δX nk (ω) , namely
· Z
¸
· Z
¸
EP Z
f (w) δX nk (·) (dw) → E P Z
f (w) µω (dw)
W
W
for every f ∈ Cb (W) and Z ∈ L1 (Ω, P ). This implies
·Z
¸
P
nk
P
ω
E [f (X (·))] → E
f (w) µ (dw)
W
which means that ρn©k weakly converges
to E P [µ· ]. Thus, to every limit point
ª
ω
(µ ) ∈ PrΩ,P (W) of ω 7→ δX n (ω) it is associated a limit point E [µ· ] ∈ Pr (W)
of ρn .
(W). The sequence
© Viceversa, ªassume ρnk weakly converges to some ρ ∈ Pr
ω 7→ δX nk (ω) is precompact, and all its limit points (µω ) ∈ PrΩ,P (W) have
the property E [µ· ] = ρ. The proof is complete.
Theorem 23 In the additive noise of section 4.1, there exists a strong pathwise
superposition solution (µω ) ∈ PrΩ,P (W) of (SCPx ), with E [µ· ] being a solution
of the martingale problem.
17
Proof. We give the main arguments of the proof, full details being reported
in [18]. For every positive integer n, consider the integral equation
Xtn = x +
Z (t− n1 )∨0
0
b (Xsn ) ds +
X
k
σk W(kt− 1 )∨0 .
n
It has a unique continuous
¡
¢ adapted solution defined explicitly by iteration. For
every p > 1 and α ∈ 0, 12 , it is classical to prove the following estimates
"
#
E
sup |Xtn |
p
p
E [kXtn kW α,p ] ≤ C.
≤ C,
t∈[0,T ]
Here we have denoted by k.kW α,p the following semi-norm:
Z
kf kW α,p =
T
0
Z
T
p
|f (t) − f (s)|
|t − s|
0
1+αp
dsdt.
Hölder topologies can be used in place of these Sobolev ones. The consequence
of these estimates is that the laws ρn the law of X·n on W are tight. Classical arguments would imply the existence of martingale solutions, but let us
continue
in another
direction. By lemma 22, the sequence of random measures
©
ª
ω 7→ δX·n (ω) is tight in PrΩ,P (W). By [6], it is relatively compact in the narω
row topology
¡ d ¢ of PrΩ,P (W). Let (µ ) ∈ PrΩ,P (W) be any limit point. For every
∞
f ∈ C0 R we have
Z t
(Lf ) (Xsn ) 10≤s≤(t− 1 )∨0 ds
f (Xtn ) = f (x) +
n
0
XZ t
(σk · ∇f ) (Xsn ) 10≤s≤(t− 1 )∨0 dWsk
+
k
n
0
and thus we have
µnt (f ) = µn0 (f ) +
+
Z (t− n1 )∨0
0
X Z (t− n1 )∨0
k
0
µns (Lf ) ds
µns (σk · ∇f ) dWsk
where µnt = δXtn . The convergence of δX n to µ· in the narrow topology of
ω
PrΩ,P (W) allows us to pass to the limit and prove that
¢ ) satisfies the stochas¡ (µ
∞
d
tic continuity equation (9). Indeed, given f ∈ C0 R , the first three terms
converge weak star in L∞ (Ω, P ); for the stochastic term we need the following
fact that we do not prove here: if µn narrowly converges to µ in PrΩ,P (W) then
¡ ¢
Rt
for every f ∈ C0∞ Rd the stochastic integral 0 µns (f ) dWsk converge weak star
in L∞ (Ω, P ).
18
Since Xtn are strong solutions, adaptedness for (µω ) is again a consequence
of the narrow convergence. Finally, define the functional Fnω on W as
Fnω (γ)
°
°
Z (t− n1 )∨0
°
°
X
°
°
k
b (γs ) ds −
σk W(t− 1 )∨0 (ω)° ∧ 1
= sup °γt − x −
°
n
t∈[0,T ] °
0
k
and similarly for F ω (same expression but as n → ∞). These are elements of
the space of random continuous test functions where narrow convergence takes
place, see [6] definition 3.9, hence we have
Z
Z
F ω (γ) µnω (dγ) P (dω) →
F ω (γ) µω (dγ) P (dω) .
Ω
Ω
Moreover, since on a P -full measure set of ω’s we have that for µnω -a.e. γ
Fnω (γ) = 0
(10)
F ω (γ)
°Z
°
° t
´°
X ³
°
°
k
k
= sup °
b (γs ) ds +
σk Wt (ω) − W(t− 1 )∨0 (ω) ° ∧ 1
°
n
t∈[0,T ] ° (t− 1 )∨0
k
n
with some work we can prove that
Z
|F ω (γ) − Fnω (γ)| µnω (dγ) P (dω) → 0.
Ω
This implies, again by property (10), that
Z
F ω (γ) µω (dγ) P (dω) = 0
Ω
hence there is a set N ∈ F , P (N ) = 0, such that for all ω ∈
/N
µω (F ω ) = 0
which means µω (C (x, ω)) = 1. The proof is complete.
Remark 24 Clearly the above proof can be adapted to prove the existence of
strong superposition solutions in the case of state-dependent diffusion coefficients, a case however in which we cannot speak of pathwise solutions in general.
4.4
Conclusions
The topics treated above (essentially only existence) for stochastic superposition
solutions are just a beginning of investigation. A large number of questions
have to be answered, like the equivalence between different definitions, relation
19
with pathwise uniqueness and with the construction of Yamada-Watanabe [32],
relation with the concept of strong statistical solutions of Le Jan and Raymond
[24], existence of selections with flow properties, direct analysis of the stochastic
continuity equation, zero-noise limit and others. Some answers are given in
[18]. The results of [24] on a related concept seem to indicate that in some
cases a martingale solution could have only one associated strong superposition
solution. If this is true in sufficient generality, the question of selection of the
physically most interesting superposition solutions could reduce to the problem
of selection of the physically most interesting martingale solutions. The role
of superposition solutions could be, in such a case, to indicate whether the
emerged physical martingale solutions contain an intrinsic degree of splitting
(non-uniqueness) or not, as the very interesting examples of [24] highlight.
References
[1] L. Ambrosio, Transport equation and Cauchy problem for BV vector fields,
Invent. Math. 158 (2004), no. 2, 227–260.
[2] L. Ambrosio, G. Crippa, Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields,
UMI-Lecture Notes, to appear, 2006.
[3] R. Bafico, P. Baldi, Small random perturbation of Peano phenomena,
Stochastics, vol 6 (1982), 279-292.
[4] J.M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci. 7 (1997), no. 5,
475-502.
[5] C. Castaing, M. Valadier, Convex analysis and measurable multifunctions,
Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York,
1977.
[6] H. Crauel, Random probability measures on Polish spaces, Stochastics
Monographs, 11, Taylor & Francis, London, 2002.
[7] G. Da Prato, A. Debussche, Ergodicity for the 3D stochastic Navier-Stokes
equations, J. Math. Pures Appl. (9) 82 (2003), no. 8, 877–947.
[8] A. Debussche, C. Odasso, Markov solutions for the 3D stochastic NavierStokes equations with state dependent noise, J. Evol. Equ. 6 (2006), no. 2,
305–324.
[9] R. J. DiPerna, P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989), no. 3, 511–547.
[10] Weinan E, E. Vanden-Eijnden, A note on generalized flows, Phys. D 183
(2003), no. 3-4, 159–174.
20
[11] C. L. Fefferman, Existence and smoothness of the Navier-Stokes equations,
the millennium prize problems, Clay Math. Inst., Cambridge 2006, 57-67.
[12] A. Figalli, Existence and uniqueness of martingale solutions for SDEs with
rough or degenerate coefficients, 2007, to appear on J. Funct. Anal.
[13] F. Flandoli, M. Caruana, Lagrangian flows and continuity equation for
stochastic equations driven by rough paths, in preparation.
[14] F. Flandoli, strong superposition solutions to SDEs, in preparation.
[15] F. Flandoli, D. Gatarek, Martingale and stationary solutions for stochastic
Navier-Stokes equations, Probab. Theory Related Fields 102 (1995), no. 3,
367–391.
[16] F. Flandoli, M. Gubinelli, M. Hairer, M. Romito, Remarks on scaling laws
in turbulent fluids, to appear on Comm. Math. Phys.
[17] F. Flandoli, M. Gubinelli, E. Priola, Well posedness for a class of ODEs
and linear PDEs by stochastic perturbations, preprint, 2008.
[18] F. Flandoli, J. A. Langa, Markov attractors: a probabilistic approach to
multivalued flows, preprint 2007.
[19] F. Flandoli, M. Romito, Partial regularity for the stochastic Navier-Stokes
equations, Trans. Amer. Math. Soc. 354 (2002), no. 6, 2207–2241.
[20] F. Flandoli, M. Romito, Markov selections for the 3D stochastic NavierStokes equations, Probab. Theory Related Fields 140 (2008), no. 3-4, 407–
458.
[21] N. V. Krylov, The selection of a Markov process from a Markov system of
processes, (Russian) Izv. Akad. Nauk SSSR, Ser. Mat. 37 (1973), 691-708.
[22] N. V. Krylov, M. Röckner, Strong solutions of stochastic equations with
singular time dependent drift, Probab. Theory Relat. Fields 131 (2005),
154-196.
[23] A. Kupiainen, Statistical theories of turbulence, in Advances in Mathematical Sciences and Applications, Gakkotosho, Tokyo, 2003.
[24] Y. Le Jan, O. Raymond, Integration of Brownian vector fields, The Annals
of Probab. 30 (2002), no. 2, 826-873.
[25] A. J. Majda, A. L. Bertozzi, Vorticity and incompressible flow, Cambridge
Texts in Applied Mathematics, 27, Cambridge University Press, Cambridge, 2002.
[26] R. Mikulevicius, B. L. Rozovskii, Global Lˆ2-solutions of stochastic NavierStokes equations, The Annals of Probab. 33 (2005), no. 1, 137–176.
21
[27] N. I. Portenko, On the theory of diffusion processes, Probability theory
and mathematical statistics (Kiev, 1991), 259–267, World Sci. Publ., River
Edge, NJ, 1992.
[28] J. Simsen, C. B. Gentile, On attractors for multivalued semigroups defined
by generalized flows, Set Valued Anal., in press.
[29] G. Stampacchia, Le trasformazioni funzionali che presentano il fenomeno
di Peano, (Italian) Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat.
(8) 7, (1949), 80–84.
[30] D. W. Stroock, S. R. S. Varadhan, Multidimensional Diffusion Processes,
Springer-Verlag, Berlin-New York, 1979.
[31] A. Ju. Veretennikov, Strong solutions and explicit formulas for solutions of
stochastic integral equations. (Russian) Mat. Sb. (N.S.) 111(153) (1980),
no. 3, 434–452.
[32] T. Yamada, S. Watanabe, On the uniqueness of solutions of stochastic
differential equations, J. Math. Kyoto Univ. 11 (1971), 155-167.
[33] A. K. Zvonkin, A transformation of the phase space of a diffusion process
that will remove the drift. (Russian) Mat. Sb. (N.S.) 93(135) (1974), 129–
149, 152.
22