Path-by-path uniqueness of multidimensional SDE's on the plane with nondecreasing coefficients

In this paper we study path-by-path uniqueness for multidimensional stochastic differential equations driven by the Brownian sheet. We assume that the drift coefficient is unbounded, verifies a spatial linear growth condition and is componentwise nondeacreasing. Our approach consists of showing the result for bounded and componentwise nondecreasing drift using both a local time-space representation and a law of iterated logarithm for Brownian sheets. The desired result follows using a Gronwall type lemma on the plane. As a by product, we obtain the existence of a unique strong solution of multidimensional SDEs driven by the Brownian sheet when the drift is non-decreasing and satisfies a spatial linear growth condition.


Introduction
In this work, we consider the following system of stochastic integral equations on the plane with additive noise: Borel measurable satisfying some conditions that will be specified later and W = (W s,t , (s, t) ∈ R 2 + ) is a d-dimensional Brownian sheet given on a filtered probability space (Ω, F , (F s,t , (s, t) ∈ R 2 + ), P) with ∂W = 0, where ∂W stands for the restriction of W to the boundary ∂D = {0} × R + ∪ R + × {0} of D = R 2 + . We endow D with the partial order " " (respectively "≺") defined by (s, t) (s ′ , t ′ ) when s ≤ s ′ and t ≤ t ′ , respectively (s, t) ≺ (s ′ , t ′ ) when s < s ′ and t < t ′ . Observe that (1.1) is a particular case of the more general non-Markovian type equation X s,t − X s,0 − X 0,t + X 0,0 = t 0 s 0 b(ξ, ζ, X ξ,ζ )dξ dζ + t 0 s 0 a(ξ, ζ, X ξ,ζ ) dW ξ,ζ for (s, t) ∈ R 2 + , (1.2) where a : R 2 + × R d → R d × R d is a Borel measurable matrix function. Note that (1.2) appears as an integral equation when one rewrites the following quasilinear stochastic hyperbolic differential equation ∂ 2 X s,t ∂s∂t = b(s, t, X s,t ) + a(s, t, X s,t ) ∂ 2 W s,t ∂s∂t , (1.3) where the notation " ∂ 2 Ws,t ∂s∂t " designates a white noise on D. As pointed out by Farré and Nualart [18] (see also [31]), a formal π 4 rotation transforms (1.3) into a nonlinear stochastic wave equation. This idea, thanks to Walsh [37], has been used by Carmona and Nualart [10] to provide existence and uniqueness results for the 1-dimension stochastic wave equation with some initial conditions Y (0, ·) and ∂Y ∂t (0, ·), where t varies in R + , x varies in R andẆ denotes a white noise in time as well as in space. Reformulation of (1.4) using a π 4 rotation allows use of the rectangular increments of both t and x (see e.g. [31,Section 1]).
The problem (1.1) can also be interpreted as a noisy analog of the so-called Darboux problem given by where σ and τ are absolutely continuous on [0, T ]. Using Caratheodory's theory of differential equations, Deimling [15] proved an existence theorem for the system (1.5)-(1.6) when b is Borel measurable in the first two variables and bounded and continuous in the last three variables.
Existence and uniqueness of solutions to stochastic differential equations (SDEs) driven by a Brownian sheet has been widely studied. In the time homogeneous case, Cairoli [8] proved that (1.2) has a unique strong solution when the coefficients are Lipschitz continuous. This result was generalised to the time dependent coefficients by Yeh [39] under an additional growth condition. Weak existence of solutions to (1.2) was derived in [40] assuming that the coefficients are continuous, satisfy a growth condition and the initial value has moment of order six. In all of the above mentioned works, the coefficients are at least continuous. Nualart and Tindel [26], show that (1.2) has a unique strong solution when the drift is nondecreasing and bounded. Their results were extended to SDEs driven by two parameter martingales in [27] (see also [10; 18] for further extensions). In [17], the authors generalised the above results to SDEs driven by a fractional Brownian sheet.
In this work we are concerned with a different uniqueness question. In particular, we look at the notion of path-by-path uniqueness introduced by Davie [13] (see also Flandoli [19]). Let V, resp. ∂V be the space of continuous R d -valued functions on D, resp. ∂D. The following definition can be seen as a counterpart of [19,Definition 1.5] in the case of two parameter processes. Definition 1.1. We say that the path-by-path uniqueness of solutions to (1.1) holds when there exists a full P-measure set Ω 0 ⊂ Ω such that for all ω ∈ Ω 0 the following statement is true: there exists at most one function y ∈ V which satisfies T 0 T 0 |b(ξ, ζ, y ξ,ζ )|dξdζ < ∞, ∂y = x, for some x ∈ ∂V and T > 0 and y s,t = x + One of the motivations for studying path-by-path uniqueness comes from the regularisation by noise of random ODEs. For instance, let v be a continuous function and let us consider the following one parameter equation in R d We know that there exists a unique solution to the above equation when b is Lipschitz in x, uniformly in t, with uniform linear growth. Observe that uniqueness also holds when b is only locally Lipschitz. Under some weak conditions on b the corresponding equation without v might be ill-posed or uniqueness could not be valid. For example when b is merely bounded and measurable one may ask if there is a notion of uniqueness if v has some specific features. In other words, can we find a path v that regularises the equation? The result obtained in [13] shows that when b ∈ L ∞ , the Brownian path regularises the drift b in the sense of Definition 1.1. In addition, as shown in [4,Section 1.8.5], path-by-path uniqueness is much stronger than pathwise uniqueness. Indeed Shaposhnikov and Wresch [34,Section 4] exhibit SDEs such that strong solutions exist, pathwise uniqueness holds and path-by-path uniqueness fails to hold. This is another motivation for studying path-by-path uniqueness even when pathwise uniqueness holds.
In the case of Brownian motion, when the drift is bounded and measurable, and the diffusion is reduced to the identity, the path-by-path uniqueness of equation (1.2) was proved by Davie in [13]. This result was extended in several directions. For non-constant diffusion, Davie in [14] proved path-by-path uniqueness of solution to (1.2), interpreting the equation in the rough path sense. In [11], the authors showed that path-by-path uniqueness holds if the Brownian motion is replaced by a d-dimensional fractional Brownian motion of Hurst parameter H ∈ (0, 1). It is also assumed that the drift b can be merely a distribution as long as H is sufficiently small. In [29], Priola considered equations driven by a Lévy process assuming that the drift is Hölder continuous (see [30] for the non-constant diffusion coefficient case).
Path-by-path uniqueness is closely related to the regularisation by noise problem for ordinary (or partial) differential equations (ODEs or PDEs) which has recently drawn a lot of attention. Beck, Flandoli, Gubinelli and Maurelli [4] proved a Sobolev regularity of solutions to the linear stochastic transport and continuity equations with drift in critical L p spaces. Such a result does not hold for the corresponding deterministic equations. Butkovsky and Mytnik [6] analysed the regularisation by noise phenomenon for a non-Lipschitz stochastic heat equation and proved path-by-path uniqueness for any initial condition in a certain class of a set of probability one. Amine, Mansouri and Proske [2] investigated path-by-path uniqueness for transport equations driven by the fractional Brownian motion of Hurst index H < 1/2 with bounded vector-fields. In [11; 20] the authors solved the regularisation by noise problem from the point of view of additive perturbations. In particular, Galeati and Gubinelli [11] considered generic perturbations without any specific probabilistic setting. Amine, Baños and Proske [1] constructed a new Gaussian noise of fractional nature and proved that it has a strong regularising effect on a large class of ODEs. More recently, Harang and Perkowski [21] studied the regularisation by noise problem for ODEs with vector fields given by Schwartz distributions and proved that if one perturbs such an equation by adding an infinitely regularising path, then it has a unique solution. Kremp and Perkowski [23] looked at multidimensional SDEs with distributional drift driven by symmetric α-stable Lévy processes for α ∈ (1,2]. In all of the above mentioned works, the driving noise considered are one parameter processes. In what follows, we make use of the Girsanov theorem to show that the path-by-path uniqueness in our setting is equivalent to the uniqueness of a random ODE on the plane. For any x ∈ ∂V and any ω such that the path (s, t) −→ W s,t is continuous, we denote by S(x, ω) the set of functions in V that solve (1.7). Under linear growth and monotonicity conditions on b, we prove that S(x, ω) has at most one element. By a vector translation argument, it suffices to show that S(0, ω) has no more than one element.
As in [13, Section 1], we show the path-by-path uniqueness on D 1 = [0, 1] 2 . Precisely, we consider the integral equation where the drift is of spatial linear growth. There is no loss of generality in reducing the problem to D 1 since we can repeat the argument on any square [m, m + 1] × [ℓ, ℓ + 1], (m, ℓ) ∈ N 2 , m > 0. We first suppose that b is bounded and monotone. Let is well-defined for P-a.e. y ∈ V 1 0 . Moreover, if y ∈ V 1 0 is chosen random, with law d P = LdP, then, by Girsanov theorem (see for example [8; 12; 22]), the path W defined by has law P. This means that y is a solution to (1.8) with W defined by (1.9). Path-by-path uniqueness of solutions to (1.8) holds if and only if for P-a.e. y ∈ V 1 0 , z = y is the only solution to with W given by (1.9), which is equivalent to saying that for P-a.e. y ∈ V 1 0 , the only solution to is u = 0 (see e.g. [13,Section 1]). Since P is absolutely continuous with respect to P, it is enough to show that, if W is an R d -valued Brownian sheet, then, with probability one, there is no nontrivial solution u ∈ V 1 0 of This is the statement of Theorem 3.4 which is extended to unbounded monotone drifts in Theorem 3.2. Our proof of Theorem 3.4 relies on some estimates for an averaging operator along the sheet (see Lemma 3.6). This result plays a key role in the proof of a Gronwall type lemma (see Lemma 3.9) which enables us to prove path-by-path uniqueness of solutions to (1.1). The Yamada-Watanabe principle for one dimensional SDEs driven by Brownian sheets was derived in [28] (see also [41]). More precisely, the authors show that combining weak existence and pathwise uniqueness yields existence of a unique strong solution in the two parameter setting. This result can be extended to the multidimensional case (see e.g. [36,Remark 2]). When b is of linear growth, we can show (see Lemma A.1) that the SDE (1.2) has a weak solution. The latter together with path by path uniqueness (and thus pathwise uniqueness) implies the existence of a unique strong solution to the SDE (1.2) and therefore generalises some results in [17; 26] to the multidimensional case. To the best of our knowledge, such a result has not been derived in the multidimensional case.
The remainder of the paper is structured as follows. In Section 2, we recall some basic definitions and concepts. The main results are stated and proved in Section 3. In section 4, we prove some preliminary results whereas Section 5 is devoted to the proof of a number of auxiliary results.

Basic definitions and concepts
In this section we recall some basic definitions and concepts for SDEs on the plane. We start with the definitions of filtered probability space and d-dimensional Brownian sheet that can be found in [28; 39].
Definition 2.1. We call a filtered probability space any probability space (Ω, F , P) with a family (F s,t , (s, t) ∈ D) of sub-σ-algebras of F such that (1) F 0,0 contains all null sets in (Ω, F , P), is a right-continuous system in the sense that Definition 2.2. We call a one-dimensional (F s,t )-Brownian sheet on a filtered probability space (Ω, F , (F s,t , (s, t) ∈ D), P) any real valued two-parameter stochastic process W (0) = (W s,t , (s, t) ∈ D) satisfying the following conditions: s,t is F s,t -measurable for every (s, t) ∈ D.
In the following, we discuss the notions of weak and strong solutions to the SDE (1.2) (see for example [39,Section 2]). We start with the definition of a weak solution.
-adapted, has continuous sample paths and, P-a.s., Remark 2.4. When the drift b satisfies a linear growth condition, weak existence holds for (1.8) (see Theorem A.1).
We now turn to the notion of strong solution. Let B(V) (respectively B(∂V)) be the σ-algebra of Borel sets in the space V (respectively ∂V) of all continuous R d -valued functions on D (respectively ∂D) with respect to the metric topology of uniform convergence on compact subsets of D. The subsequent definitions are borrowed from [28]. Definition 2.6. Let T(∂V × V) be the class of transformations F of ∂V × V into V which satisfies the condition that for every probability measure λ on (∂V, B(∂V)), there exists a transformation F λ of ∂V × V into V such that Definition 2.7. Let (X, W ) be a weak solution to the SDE (1.2) on a filtered probability space (Ω, F , {F s,t , (s, t) ∈ D}, P) and let λ be the probability distribution of ∂X. We call (X, W ) a strong solution to (1.2) if there exists a transformation F λ of ∂V × V into V satisfying Conditions 1 and 2 of Definition 2.6 such that Here is a well known concept of uniqueness associated to strong solutions of (1.2) provided such solutions exist.
Definition 2.8. We say that the SDE (1.2) has a unique strong solution if there exists F ∈ T(∂V × V) such that, is a weak solution of (1.2) on a filtered probability space (Ω, F , (F s,t , (s, t) ∈ D), P) and the probability distribution of ∂X is denoted by λ, then X = F λ [∂X, W ] P-a.s. on Ω.
There are two classical notions of uniqueness associated to weak solutions (see e.g. [28, Definitions 1.2 and 1.7]). Definition 2.9. We say that the solution to the SDE (1.2) is unique in the sense of probability distribution if whenever (X, W ) and (X ′ , W ′ ) are two solutions of (1.2) on two possibly different filtered probability spaces and ∂X = x = ∂X ′ for some x ∈ ∂V, then X and X ′ have the same probability distribution on (V, B(V)).
Definition 2.10. We say that the pathwise uniqueness of solutions to the SDE (1.2) holds if whenever (X, W ) and (X ′ , W ) with the same W are two solutions to (1.2) on the same probability space and ∂X = ∂X ′ , then X = X ′ for P-a.e. ω ∈ Ω.

Main results
In this section, we present the main results of this paper. We assume the following conditions on the drift: Borel measurable function satisfying the spatial linear growth condition, that is, there exists a constant M such that More precisely for x, y ∈ R d , we have: Recall that using the Girsanov theorem, path-by-path uniqueness holds if there exists Ω 1 ⊂ Ω with P(Ω 1 ) = 1 such that for any ω ∈ Ω 1 , there is no nontrivial solution u ∈ C([0, 1] 2 , R d ) to the following system of integral equations Let us also consider the set Q = [−1, 1] d and its dyadic decomposition. Recall that x ∈ Q is called a dyadic number if it is a rational with denominator a power of 2. The next theorem is equivalent to Theorem 3.4.
. Let b be as in Theorem 3.4. Then there exists Ω 1 ⊂ Ω with P(Ω 1 ) = 1 such that for any ω ∈ Ω 1 , u = 0 is the unique solution in V 0 to the system of integral equations (3.1).
The proof of Theorem 3.5 is carried out in two main steps. In the first step, we use a two-parameter Wiener process to regularise (3.1) on dyadic intervals. In the second step we show a Gronwall type lemma (see Lemma 3.9). The regularisation is as follows: For any positive integer n, we divide [0, 1] into 2 n intervals I n,k =]k2 −n , (k+1)2 −n ] and define ̺ nkk ′ by The next three lemmas whose proofs are given in Section 5 provide an estimate for ̺ nkk ′ (x, y) and ̺ nkk ′ (0, x) for every dyadic numbers x, y ∈ Q. Lemmas 3.6 and 3.7 are counterparts of Lemmas 3.1 and 3.2 in [13] for the Brownian sheet. The proof of Lemma 3.6 uses the local time-space integration formula for the Brownian sheet as given in [5]. Lemma 3.8 follows from Lemma 3.7 using the fact that the set of dyadic numbers is dense in R.
for all dyadic numbers x, y ∈ Q and all choices of integers n, k, is a positive random constant that does not depend on x, y, n, k and k ′ . Lemma 3.7. Suppose b is as in Lemma 3.6. Then there exists a subset Ω 2 of Ω with P(Ω 2 ) = 1 such that for all ω ∈ Ω 2 , for any choice of n, k, k ′ , and any choice of a dyadic number x ∈ Q where C 2 (ω) is a positive random constant that does not depend on x, n, k and k ′ .
Observe that the proofs of the above two results do not require the monotonic argument on the drift b.
Lemma 3.8. Suppose b is as in Theorem 3.4. Let Ω 2 be a subset of Ω such that, for any ω ∈ Ω 2 , (3.2) holds for every n, k, k ′ , and every dyadic number x ∈ Q. Then for any ω ∈ Ω 2 , any n, k, k ′ , and any x ∈ Q, where C 2 (ω) is a positive random constant that does not depend on x, n, k and k ′ .
The subsequent result is a Gronwall type lemma and constitutes the main result in the second step of the proof of Theorem 3.5. Its proof is found in Section 5.  exists Ω 1 ⊂ Ω with P(Ω 1 ) = 1 and a positive random constant C 1 such that for any ω ∈ Ω 1 , any sufficiently large positive integer n, any (k, k ′ ) ∈ {1, 2, . . . , 2 n } 2 , any β(n) ∈ 2 −4 3n/4 , 2 −4 2n/3 , and any solution u to the system of integral equations we have We are now ready to prove Theorem 3.5.

Preliminary results
In order to prove the auxiliary lemmas provided in the previous section, we need some preliminary results that have been obtained by applying a local time-space integration formula for Brownian sheets (see [5] for related results). Let us first recall the notion of local time in the plane of the Brownian sheet. Let (W       s,t , 0 ≤ t ≤ 1). Then the following holds Next, we consider the local time process in the plane L := (L x s,t ; x ∈ R, s ≥ 0, t ≥ 0) as defined in [37, Section 2] (see also [38,Section 6,Page 157 Then it holds Let us now consider the norm · defined by Consider the set H of measurable functions f on [0, 1] 2 × R such that f < ∞. Endowed with · , the space H is a Banach space. In the following, we define a stochastic integral over the space with respect to the local time for the elements of H. This extends the definition in [16]. We say that f ∆ : [0, 1] 2 × R → R is an elementary function if there exist two sequences of real numbers (  and that the limit does not depend on the choice of the sequence (f n ) n∈N . This limit is called integral of f with respect to L. Similar results were obtained in [16].
Let f : [0, 1] 2 × R d → R be a continuous function such that for any (s, t) ∈ [0, 1] 2 , f (s, t, ·) is differentiable and for any i ∈ {1, · · · , d}, the partial derivative ∂ xi f is continuous. We also know from [5, Proposition 3.1] that for a d-dimensional Brownian sheet W s,t := (W (1) s,t , · · · , W (d) s,t ); s ≥ 0, t ≥ 0 defined on a filtered probability space and for any (s, t) ∈ [0, 1] 2 and any i ∈ {1, · · · , d}, we have where W (i) := ( W Then there exist positive constants α and C (independent of ∇ y b, a, a ′ , ε and ε ′ ) such that Here ∇ y b denotes the gradient of b with respect to the third variable, |·| is the usual norm on R d and the R d -valued for all i ∈ {1, . . . , d}.
Proof. The proof of (4.5) is based on the local time-space integration formula (4.4) and the Barlow-Yor inequality. Fix (a, a ′ , ε, ε ′ ) ∈ [0, 1] 4 . Since x −→ e αεε ′ dx 2 is a convex function, we deduce from the Jensen inequality that In order to obtain (4.5) it suffices to prove that for every i ∈ {1, 2, . . . , d}, there exist positive constants α = α i and C = C i such that s,t , 0 ≤ s, t ≤ 1) denotes a standard Brownian sheet independent of the process (Y Using once more the convexity of the function x −→ e 3αx 2 for any α > 0, we obtain Hence to get the desired estimate, we need to prove that for every k ∈ {1, 2, 3}, there exist positive constants α k and C k such that E exp(α k I 2 k ) ≤ C k . For every s ∈]0, 1], s −1/2 Y s,v , 0 ≤ v ≤ 1 is a standard Brownian motion with respect to the filtration F 1,· := (F 1,t , t ∈ [0, 1]). Therefore the process is an Itô integral with respect to F 1,· and thus a square-integrable F 1,· -martingale. In addition, for any constant α ∈ R + , the following expansion formula holds Moreover, by the Jensen inequality and the Barlow- The above expression if finite for α such that 8αc 2 1 e < 1, i.e. α < 1/8c 2 1 e (by ratio test). Hence, there exists positive constants α 1 and C 1 such that

Similarly for
there exists positive constants α 2 and C 2 such that It remains to estimate the term I 3 . By the Jensen inequality, we have Note that for every (s, t) ∈]0, 1] is a standard normal random variable. Therefore (4.6) yields The proof of (4.5) is completed by taking α = min( 1 64 , α 2 , α 3 ). For every 0 ≤ a < h ≤ 1, 0 ≤ a ′ < h ′ ≤ 1 and for (x, y) ∈ R d define the function ̺ by: As a consequence of Proposition 4.3, we have: (1) For every (x, y) ∈ R 2d , x = y and every (ε, ε ′ ) ∈ [0, 1] 2 , we have (2) For any (x, y) ∈ R 2 and any η > 0, we have Proof. We start by showing (4.7). Note that it is enough to show this when b is compactly supported and differentiable. Indeed, if b is not differentiable, then, since the set of compactly supported and differentiable functions is dense in L ∞ ([0, 1] 2 ×R d ), there exists a sequence (b n , n ∈ N) of compactly supported and differentiable functions which converges a.e. to b on [0, 1] 2 × R d and the desired result will follow from the Vitali's convergence theorem. Using the mean value theorem and the Cauchy-Schwartz inequality, we have Using the Minkowski inequality, the Jensen inequality and Proposition 4.3 applied to the function (s, t, z) −→ b(s, t, z + y + ξ(x − y)), we obtain This ends the proof of (4.7).

Proof of the auxiliary results
In this section we prove the auxiliary results stated in Section 3. Consider Q = [−1, 1] d and its dyadic decomposition.

Regularization by noise.
Proof of Lemma 3.6. Let Q denote the set of couples (x, y) of dyadic numbers in Q. For every integer m ≥ 0, we define Then Q m has no more than 2 2d(m+2) elements and we have Q = m∈N Q m . Consider the set E δ,n defined by Define also E δ by E δ := ∞ n=0 E δ,n . Then, we deduce from (4.8) that where C and α are the deterministic constants in Proposition 4.3. Moreover, (E δ , δ ∈ Q + ) is a nonincreasing family and for δ ≥ δ 0 := 2(d + 1)α −1 , we have Let us now define Ω 1 by Then P(Ω 1 ) = 1 and for every ω ∈ Ω 1 , there exists δ ω > 0 such that for all choices of n, k, k ′ , m, and all choices of couples (x, y) in Q m . Now, fix ω ∈ Ω 1 , choose two dyadic numbers x, y in Q and define m by For r ≥ m, define x r = (2 −r ⌊2 r x 1 ⌋, . . . , 2 −r ⌊2 r x d ⌋) and y r = (2 −r ⌊2 r y 1 ⌋, . . . , 2 −r ⌊2 r y d ⌋), where ⌊·⌋ is the integer part function. Observe that (x m , y m ) ∈ Q m , (x r , x r+1 ) ∈ Q r+1 and (y r , y r+1 ) ∈ Q r+1 . Since for Moreover since ̺ nkk ′ (x m , x m ) = 0 and ̺ nkk ′ (y m , y m ) = 0, we obtain for every integer q ≥ m + 1. In addition for some integer q ≥ m + 1, we have x r = x and y r = y for all r ≥ q, therefore It follows from (5.4) and (5.2) that Using the following facts: Choosing C 1 (ω) = 48dδ ω yields the desired result.
Appendix A. Appendix In this section we provide a weak existense result for SDEs driven by Brownian sheet under the linear growth condition Then (1.8) has a weak solution.