STOCHASTIC FLOWS OF DIFFEOMORPHISMS FOR ONE-DIMENSIONAL SDE WITH DISCONTINUOUS DRIFT

The existence of a stochastic ﬂow of class C 1, α , for α < 12 , for a 1-dimensional SDE will be proved under mild conditions on the regularity of the drift. The diffusion coefﬁcient is assumed constant for simplicity, while the drift is an autonomous BV function with distributional derivative bounded from above or from below. To reach this result the continuity of the local time with respect to the initial datum will also be proved.


Introduction
The problem of the existence and smoothness of the stochastic flow under conditions of low regularity of the coefficients has been much studied. Apart from the intrinsic interest of the problem, there is an interest due to the range of possible applications of these results to PDE theory. For example, in [3] the uniqueness of the stochastic linear transport equation with Hölder continuous drift was proved, through new results about stochastic flows of class C 1,α . Because of the greater regularity of stochastic flows compared to deterministic flows, there are cases in which a PDE admits infinitely many solutions in the deterministic case, but it becomes well posed if it is perturbed by a stochastic noise. In addition in [2] it is proved that in some cases, through a zero-noise limit, it is possible to find a criterion to select one particular solution. We consider an equation of the form A complete proof of existence of the stochastic flows of class C 1,α is known only when b is Hölder continuous and bounded, see [3]. In the 1-dimensional case, an important example that deals with discontinuous b has been studied in [5]. Moreover there are preliminary results in [6]. The class of bounded variation (BV) fields b emerges from these works as a natural candidate for the flow property, although only a few partial properties have been proved. Moreover, BV fields are the most general class considered also in the deterministic literature, see [1]: in any dimension, when b ∈ BV and the negative part of the distributional divergence of b is bounded, a generalized notion of flow exists and is unique. The aim of this work is to give a complete proof of existence of the stochastic flows of class C 1,α , for α < 1 2 , in dimension one, when b ∈ BV and the positive or the negative part of the distributional derivative of b is bounded. This result, although restricted to the 1-dimensional case, is stronger than the deterministic one both because we accept a bound on b from any side, and because we construct a flow of class C 1,α , not only a generalized flow. The partial results of the paper [6] suggest the problem whether b ∈ BV is sufficient. We cannot reach this result without a one-side control on b . The fact that a similar assumption is imposed in [1] is maybe an indication that it is not possible to avoid it.

Flow of homeomorphisms and known results
All results contained in this paper will be proved under the following hypothesis: We will first suppose b 1 ∈ W 1,∞ . Under this hypothesis we will prove that the local time of the stochastic differential equations (SDE) solutions is Hölder continuous with respect to the initial data. Thanks to this result we will prove the existence of the stochastic flow of class C 1,α , for α < 1 2 . At the end of the paper, using standard facts on the backward equations, we will show that the results proved hold if we replace the hypothesis b 1 ∈ W 1,∞ with the hypothesis b 2 ∈ W 1,∞ . A result about one-dimension stochastic flows, under the hypothesis b ∈ BV, was given in [6]. There it was first proved the non-coalescence property through an elegant proof different from the one proposed in these notes. Then, the flow continuity was proved, and this property, together with the continuity of the flow of the backward equation, implies the homeomorphic property of the flow. However the proof of the continuity of the flow appears to be incomplete. Indeed, in order to apply Kolmogorov's lemma, the following inequality is shown: where τ is a stopping time depending on x and y. This doesn't appear sufficient to apply Kolmogorov's lemma. Note that, as a standard consequence of the pathwise uniqueness we have that ∀h > 0, a.s., From this inequality, using Gronwall's lemma we obtain: This fact, together with the proof of the non-coalescence property, contained in [6], is sufficient to prove the existence of a stochastic flow of homeomorphisms. Therefore the proof of the homeomorphic property contained in [6] can be corrected easily under our hypothesis 1 and 2.
However we are interested in stronger results about smoothness of the flow. In particular we are interested in the smoothness of the inverse flow, which is a basic ingredient, for instance, in the analysis of stochastic transport equations. While the homeomorphic property implies only the continuity of the inverse flow, we will prove that the inverse flow is of class C 1,α for α < 1 2 . Notation Throughout the paper we will assume given a stochastic basis with a 1-dimensional Brownian motion (Ω, ( t ) t≥0 , , P, W t ). Moreover, for each 0 ≤ s < t we denote by s,t the completed σalgebra generated by W u − W r for s ≤ r ≤ u ≤ t. We will use the following notation: We will denote by X x t the unique solution of the stochastic differential equation (1).

Local-time continuity with respect to the initial data
Definition 3.1. Let X x t be the unique solution of equation (1), and let a ∈ . We will denote by L a t (X x ) its local time at a, i.e. the continuous and increasing process such that Further details about local time can be found in [8].

Remark 1.
Recall the following inequality which is used, for example, to prove the continuity with respect to (a, t) of the local time: Let X t = X 0 + A t + M t be a continuous semimartingale, where M t is a continuous local martingale, vanishing in 0 and A t is a continuous process with bounded variation, vanishing in 0. Suppose that sup t≤T |M t | ∨ sup t≤T |A t | ≤ K. Then it holds: There exists a modification of L a t (X x ) which is jointly continuous in (a, t, x), and it is Hölder continuous in (a, x), of order α, for α < 1 2 . Proof. Define an increasing sequence of stopping times as follows: We have T n ↑ ∞ a.s. Denote by X T n the unique stopped solution of equation (1). It is sufficient to prove the theorem for L a t∧T n (X x ). Note that ∀t ≥ 0 |X t∧T n − x| ≤ 2n, and X T n satisfies the hypothesis of remark 1. By definition of L a t (X x ) it follows: Thanks to the inequality |X x t − X y t | ≤ e K t |x − y|, which holds a.s. the first term on the right hand admits a modification jointly continuous in (a, t, x), and lipschitz continuous in (a, x). In particular it is Hölder continuous in (a, x), of order α, for α < 1 2 . The second term is obviously continuous in (a, t, x) and Hölder continuous in (a, x), of order α, for α < 1 2 . We now prove that the third one admits a modification jointly continuous in (a, t, x), and Hölder continuous in (a, x), of order α, for α < 1 2 . We will apply Kolmogorov's lemma to the space C([0, ∞)), endowed with the sup norm. It will be useful to apply remark 1. Let (a, x) ∈ 2 and (b, y) ∈ 2 . We will consider only the case b > a and y > x. In the other cases the following estimates are similar. It holds: We need to estimate the last term to apply Kolmogorov's lemma: Using Itô formula and the boundness of f we will obtain the L p -boundness of 1 Thus we have: Therefore, thanks to Kolmogorov's lemma we have proved that admits a modification jointly continuous in (a, t, x), and Hölder continuous in (a, x), of order α, for α < 1 2 . To complete the proof we have to show that t∧T n 0 sgn(X x s − a)dW s admits a modification jointly continuous in (a, t, x), and Hölder continuous in (a, x), of order α, for α < 1 2 . We have: This inequalities and Kolmogorov's lemma prove that t∧T n 0 sgn(X x s − a)dW s admits a modification jointly continuous in (a, t, x), and Hölder continuous in (a, x), of order α, for α < 1 2 . The proof is complete.
From now on we will consider only the continuous version of L a t (X x ).
is continuous.
Proof. Note that ∀s ≤ t, and ∀u ∈ it holds |X u

Existence of the stochastic flow of diffeomorphisms
We will now prove the non-coalescence property of the solutions of equation (1). This result has been already proved in [6] under more general hypothesis. However, for the sake of completeness we will give a proof based on the continuity of L a t (X x ). The following lemma, which appears in [8], and in [7] with a complete proof, will be useful. Proof. Fix x ∈ , h > 0 and T ≥ 0. From corollary 3.3, it follows that the process s → sup t∈[0,T ] sup a∈ L a t+s (X x )− L a t (X x ), is continuous and vanishing in 0. Therefore ∀ ∈ (0, 1) a.s. exists s ,x (ω) > 0 such that This inequality implies: In the same way we obtain: In particular a.s. N ,T,x (ω) := T s ,x (ω) < ∞, and thus a.s. we have Remark 2. Using corollary 3.3, with the same argument of the preceding proof, it is possible to prove that given an interval . This fact will be used in the next theorem, which is crucial to prove the existence of a flow of class C 1,α . Theorem 4.3. Let x, y, t ∈ such that x < y, and t ≥ 0. Then, a.s. Proof.
Step 1. Fix x < y and t ≥ 0. Fix > 0, and let s ,x, y (ω) be defined as in remark 2. Moreover define N ,t,x, y (ω) := t s ,x, y (ω) . Define , ∀ i ∈ , t i (ω) = (i × s ,x, y (ω)) ∧ t. Obviously we have, a.s., that i ≥ N ,t,x, y (ω) implies t i (ω) = t. Define g : Ω × + → R + as: Note that, with the same reasoning used in corollary 3.3 and remark 2, it is possible to show that g is a.s. continuous, increasing and vanishing in 0. Let z, w ∈ [x, y] such that z < w. Then it holds: Observe that in the last summation a.s. only a finite number of terms are different from 0. Define: Note that the following estimate holds: Finally define the random set: The following properties are immediate: ρ A i, ,z,w ≥ 0, ρ A i, ,z,w (a)da = 1, and has support contained in [− 1 1− (w − z)e K t , 0]. We will use the following notation: . Similarly we define: ρ B i, ,z,w satisfies properties similar to those of ρ A i, ,z,w . In particular its support is contained in Step 2. Thanks to estimates (3) and (4) we have: Using the decomposition b * = 2b 1 − b, and the relation, which holds for α ≤ β, , we obtain: Step 3. From the occupation time formula we have: Step 4. It holds: From this equality it follows: 2. a.s. φ s,t is a diffeomorphism ∀0 ≤ s ≤ t ≤ T and φ s,t , φ −1 s,t , Dφ s,t , and Dφ −1 s,t are continuous in (s, t, x), and of class C α in x, for α < 1 2 . 3. a.s. φ s,t (x) = φ u,t (φ s,u (x)) ∀0 ≤ s ≤ u ≤ t ≤ T and x ∈ , and φ s,s (x) = x.
Moreover an explicit expression for Dφ s,t (x) is given by: Proof.
Step 1. We first give the proof under the assumption b 1 ∈ W 1,∞ . It will take the first three steps. Let D be the set of dyadic numbers, and let D T := D ∩ [0, T ]. Define ∀x ∈ D and ∀t ∈ D T , X 0,x t (ω) = X x t (ω). Note the two following facts: 1. Being D a countable set, there exists a negligible set A 0 such that, ∀ω ∈ A 0 and ∀x ∈ D X x · is a continuous solution of equation (1). In particular since it holds ∀ω ∈ A 0 and x ∈ D, the family of processes {X x . } x∈D is uniformly equicontinuous.
2. Thanks to the countability of the set D × D × D T , there exists a negligible set A 1 ⊃ A 0 such that ∀ω ∈ A 1 ∀x, y ∈ D such that x < y, and ∀t ∈ D T it holds These facts imply that, ∀ω ∈ A 1 is well-defined the continuous extension of X 0,· · (ω) on × [0, T ]. Denote by φ 0,t (x)(ω) this extension. Note that, ∀ω ∈ A 1 , the family {φ 0,· (x)(ω)} x∈ is uniformly equicontinuous; more precisely, In particular we have that a.s. |φ 0,t (x) − x| is bounded in x. This fact, together with continuity in x, implies that, ∀ω ∈ A 1 , φ 0,t (·) is surjective. Moreover it's immediate to verify that, ∀ω ∈ A 1 , and ∀x, y ∈ , such that x < y, it holds: This fact, together with the continuity of L a t (X x ) with respect to (a, t, x) implies that ∀ω ∈ A 1 , ∀0 ≤ t ≤ T, φ 0,t (x)(ω) is differentiable in x, and its derivative is exp L a t (X x )Db(da) . So ∀t ∈ [0, T ] a.s. φ 0,t is a surjective and differentiable function whose derivative is strictly positive everywhere, and therefore is a diffeomorphism. Moreover exp L a t (X x )Db(da) is of class C α with respect to x for α < 1 2 .
Step 2. We now prove that ∀x ∈ a.s. φ 0,t (x) = X x t ∀0 ≤ t ≤ T. Fix x ∈ . Because of the a.s. continuity of both φ 0,t (x) and X x t with respect to t, and because of the countability of D T , it is Remark 3. A classic approach to the problem of the existence of a C 1,α stochastic flow is the use of Itô formula to remove the drift (see [9] and later works on this approach, or a variant in [3]).
Nevertheless, under the assumptions of this paper, there are some difficulties to use this approach. Indeed, suppose it is possible to prove the existence of a solution of the following parabolic equation and that the solution is sufficiently smooth, so that we can apply Itô formula: Suppose moreover that u(t, ·) is a diffeomorphism. Through the change of variable Y t = u(t, X t ) the original problem has been reduced to the problem of the existence of the flow of the following stochastic equation where σ(t, y) = ∂ x u(t, u −1 t ( y)). To solve this problem using well-known theorems, we should prove σ ∈ C 1,α for some α > 0. But σ ∈ C 1,α implies ∂ x x u(t, ·) ∈ C α . So the discontinuity of the term b(x)∂ x u(t, x) would be balanced by the term ∂ t u(t, x). However, examples such that ∂ t u(t, x) is continuous, and b(x) is discontinuous can be shown. Consider, for example, b(x) = sgn(x). It is possible to prove that u(t, x) = E[X t,x T ] and ∂ t u(t, x) = P(X t,x T < 0) − P(X t,x T > 0). This term, as shown in [4], is continuous in x. Thus we have obtained ∂ x x u(t, ·) ∈ C α . So, even if it is possible to prove the existence and smoothness of the solution of equation (14), it is not possible to prove that σ ∈ C 1,α , and it is not possible to prove the existence of the stochastic flow through well-known theorems.