Mar 12, 2008 - well posed. The second one is a concept of strong superposition solution ... Uniqueness of weak solutions remains an open problem as well as.
Remarks on uniqueness and strong solutions to deterministic and stochastic differential equations F. Flandoli Dipartimento di Matematica Applicata “U. Dini” Universit`a 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 Itˆo 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 H¨older 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 R¨ockner [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 H¨older 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 H¨older 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 H¨older 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-Arzel`a 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 Itˆo 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.
H¨older 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.
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