Rough linear PDE's with discontinuous coefficients - existence of solutions via regularization by fractional Brownian motion

We consider two related linear PDE's perturbed by a fractional Brownian motion. We allow the drift to be discontinuous, in which case the corresponding deterministic equation is ill-posed. However, the noise will be shown to have a regularizing effect on the equations in the sense that we can prove existence of solutions for almost all paths of the fractional Brownian motion.


Introduction
In this paper we study examples of the so called regularization by noise phenomenon for a class of linear equations perturbed by fractional Brownian motion. In short, this is the name given to the phenomenon that occurs when ill-posed deterministic equations becomes well-posed by adding stochastic terms.
More specifically, assume b ∈ L 1 (R d ; L ∞ ([0, T ]; R d )) ∩ L ∞ ([0, T ] × R d ; R d ) is a given function and let B H be a d-dimensional fractional Brownian motion (fBm). In this paper we will study two different but related linear stochastic PDE's where b acts as a drift term. The stochastic transport equation reads ∂ t u(t, x) + b(t, x) · ∇u(t, x) + c(t, x)u(t, x) + ∇u(t, x) ·Ḃ H t = 0, u(0, x) = u 0 (x) (1.1) where u 0 ∈ C 1 b (R), and we allow c to be a distribution in the sense that c is the distributional derivative of a bounded function. In the case that c = div b this is called the continuity equation which we also may define as the measure valued equation where µ 0 is a given measure. We see that u(t, x) is equal to the Radon-Nykodim derivative of µ t w.r.t. Lebesgue measure.
Both equations are related to the stochastic ordinary equation (1.3) in the sense that the push-forward µ t := (φ t ) µ 0 solves the continuity equation (1.2) and the composition u(t, x) := u 0 (φ −1 t (x)) exp{− t 0 c(r, φ r (φ −1 t (x))dr} solves the transport Since the trajectories of B H are P -a.s. Hölder continuous with exponent strictly smaller than H, and the solutions themselves cannot be expected to have higher regularity, also the integration theory by Young is out of reach for these equations.
As the title of the paper suggest, we shall interpret the integrals in the rough path setting, meaning we will use the iterated integrals of B H and the theory of controlled paths to give meaning to these integrals.
We will discuss the equations separately. For notational simplicity we write B for the fBm.
Rough PDE's with discontinuous coefficients

The stochastic transport equation
Integrating the linear transport equation in time gives u(t, x) + t 0 b(s, x) · ∇u(s, x)ds + t 0 c(s, x)u(s, x)ds + t 0 ∇u(s, x) · dB s = u 0 (x). (1.5) It is well known that the corresponding deterministic equation might develop discontinuities when b is irregular. Moreover, a weak formulation of the deterministic equation is not straightforward. Integrating against a test function η ∈ C ∞ c (R), we see that the term R b(t, x) · ∇u(t, x)η(x)dx does not allow for integration by parts unless there is some regularity on b. We will choose the noise in such a way that the solution is weakly differentiable, thus circumventing integration by parts. Notice, however, that we will still use a (spatially) weak formulation of the equation.

Related literature and main contributions
The linear transport equation has been studied extensively. When the noise term is removed, Di Perna and Lions [8], showed that when b ∈ L 1 ([0, T ]; W 1,1 loc (R d )) with linear growth and div b ∈ L 1 ([0, T ] × R d ), the equation is well posed. Notice that the regularity restrictions on b is needed in order to make a definition of a solution as indicated above.
These results were later generalized to the setting of bounded variation vector fields by Ambrosio in [1].
The stochastic version driven by Brownian motion with Stratonovich formulation, i.e. t 0 ∇u(s, x) • dB s , has also received some attention. We mention the results in [9] and [17], developed simultaneously and independently, using two somewhat different techniques.
An approach of using rough paths for regularization by noise was used in [4], building on [5]. The techniques of [4] and [5] are similar in spirit to the present paper in the sense that they both use calculation on the occupation measures. The main advantage of [4] and [5] is that they offer a more defined separation between the probabilistic considerations and the analysis of the involved ODE and PDE's, thus making the approach suitable for different types of driving noise. In the present paper one needs to carefully keep track of P -null sets because many of the estimates are only shown to be true under expectation. On the other hand it gives some flexibility since some of the expressions are semi explicit via the local time.
The paper [4] consider drifts for which div b ∈ L ∞ ([0, T ] × R d ), and allow for linear growth. When d = 1 this is restricts to (locally) Lipschitz drift, but when d > 1 this condition is much weaker than Lipschitz. Another difference from the current paper is that [4] considers H ∈ ( 1 3 , 1 2 ). For the technique in the current paper to work, we need to have H < 1 3 which makes the rough path theory a bit more involved. The main advantage of the technique of the present paper is that the solution can easily be seen to be smoother in space, so that there is no need for integration by parts on the drift term, which is the reason for restricting to bounded divergence on b in [4].
In addition, we include a part where d = 1 where the proof is much simpler. The proof is based on a local-time technique that was introduced in [18] to study the Stochastic Heat Equation.
The main contribution of the present paper is to introduce the notion of "local time solutions", see Definition 6.3. This notion is introduced to compensate for the lack of integration by parts for the drift term in (1.5), and in fact allows to define a notion of solution when the multiplicative term c in (1.5) is allowed to be a distribution. Section 6.3 presents examples where one can check that the local time solutions actually gives rise to solutions that fits into the framework of the rough path integration theory.

Notation
For Banach spaces V, W we denote L(V ; W ) the set of all continuous linear mappings from V to W . For simplicity we denote L(V ) := L(V ; R). If the spaces V and W are finite dimensional, and we can identify L(V ⊗ W ) with L(V ; L(W )). In particular, for a sufficiently smooth function f : V → W the k'th derivative is considered as a map . For an integer p the p-step truncated tensor algebra We recall the following Taylor formula for a function f : More specifically, we shall use the explicit formula We shall frequently use the space For simplicity the norm in the space L ∞ ([0, T ] × R d ; R d ) will be denoted · ∞ .

Elements of controlled rough paths
The theory of rough paths was first introduced by Terry Lyons in the late 90's, see [16]. The insight of this work is that even though solutions to ODE's driven by rough signals are typically not continuous as a function of the signals themselves, by adding extra information, namely the iterated integrals of the driving signals, one obtains a topology for which there is continuity of the solutions. The theory was further developed by Gubinelli, [12] and [13], who introduced the notion of controlled paths which defines spaces that are well suited for constructing solutions of the rough ODE's.
In the present paper we shall use controlled paths as one of our main tools. See [10] for an introduction. It should be noted that the full theory of rough paths is not necessary for the present paper since there will be cancellations of the "area" of the rough paths due to the structure of the noise term in (1.1) and (1.2). Still, this machinery is convenient to understand the equations as expansions in the driving noise.
Throughout this section we fix some γ ∈ (0, 1 2 ) and let p be the integer part of 1 γ .
A γ-rough path is a mapping that satisfies an algebraic (Chen's) relation and an analytic relation We denote by C γ the set of all rough paths equipped with the metric Given a function X ∈ C 1 ([0, T ]; R d ) we can consider its canonical lift to a rough path We denote by C γ g the closure of the canonical lift of C 1 ([0, T ]; R d ) in the rough path topology. 1 An element X ∈ C γ g will be referred to as a geometric rough path and it satisfies the identity Given a rough path X ∈ C γ , we shall say that a mapping We denote by D pγ X the set of all paths controlled by X, and we equip this linear space with the semi-norm 0 ) we get a norm which controls the · ∞ -norm of Y in the following way. We have Y 1 Sometimes written C 0,γ g in the literature, whereas C γ g is reserved for paths satisfying (2.4). While C 0,γ g is strictly included in C γ g one can use geodesic approximations and interpolation to show C γ g ⊂ C 0,γ g ⊂ C γ g for γ < γ, so that one can still approximate elements satisfying (2.4) at the expense of choosing a smaller γ.
If we consider two paths Y andỸ, controlled by X andX respectively, we introduce the "distance" Similar as above have the following estimate max n=1,...,p We define the total space equipped with its natural topology, i.e. the weakest topology such that is continuous.
If f is a scalar valued function with higher Hölder regularity, i.e. |f st | |t − s| β for some β ≥ pγ and a controlled path Y ∈ D pγ X we can define the product as a controlled path f Y.
is bilinear and continuous when β ≥ pγ.
Proof. To see that the mapping is well defined is sufficies to notice that satisfies the required time-regularity when β ≥ pγ. To see continuity of this map we can similarly write

Integration of controlled rough paths
Following [10] we denote by C α,β 2 ([0, T ]) the space of functions Ξ : We equip the space with the semi-norm Ξ α,β := Ξ α + δΞ β . The following result is sometimes referred to as the "sewing lemma": More specifically, where P denotes a partition of [s, t] and |P| its mesh. The limit can be taken along any sequence of partitions and is independent of this choice.
For a proof, see [10]. It is clear from (2.7) that C θ 2 ([0, T ]) ⊂ ker(I) for θ > 1. We are ready to define the integral of a controlled rough path. For X ∈ C γ and Y ∈ D pγ X let Chen's relation (2.1) gives X ut , so that From (2.2) and (2.5) each term can be bounded by C|t − s| (p+1)γ for an appropriate constant C. Consequently |δΞ sut | |t − s| (p+1)γ . Since (p + 1)γ > 1 we arrive at the following definition: Definition 2.3. Let X ∈ C γ and let Y ∈ D pγ X . We define the rough path integral of Y w.r.t. X as The rest of this section is devoted to obtaining a "local Lipschitz"-type estimate when we regard the above as a mapping Indeed, let X,X ∈ C γ and let Y andỸ be controlled by X andX respectively. Define Ξ as before andΞ EJP 25 (2020), paper 34. Lemma 2.5. Assume γ (0, X), Y X , |Y 0 | ≤ M for some constant M , and similarly for X andỸ. Then there exists a constant C M such that Ξ −Ξ γ,(p+1)γ ≤ C M (|Y 0 −Ỹ 0 | + Y;Ỹ X;X + γ (X,X)).
Proof. We begin by decomposing Using (2.6) we can find a constantC M such that Similarly,

Controlling solutions of ODE's
In this section we will show how to control solutions of ODE's perturbed by a rough When there is no chance of confusion we shall denote the solution of (2.9) by φ t for notational convenience. Notice that we shall later on be interested in φ t as a function of x, but for this section we leave it fixed. We have ). We shall lift the composition f (φ) to a controlled path in D pγ X .
Rough PDE's with discontinuous coefficients Lemma 2.6. Assume X is a geometric rough path. Then the mapping Proof. Begin by writing For a sufficiently smooth function g : In the above we have used that X satisfies (2.4) so that ∇ n g(φ s ) since ∇ n g only acts on symmetric tensors. Furthermore, the second term |φ st | m+1 |t − s| (m+1)γ , and the third term |t − s|.
, thus proving the lemma.

Remark 2.7.
We note that the symmetry of ∇ n g in the proof of Lemma 2.6 is the reason that the full generality of the theory of rough paths is not needed in this paper.
we may define f (φ r )dX r as the rough path integral of f (φ) w.r.t. X as in (2.8).

Stability w.r.t. the driving path
The purpose of this section is to prove local Lipschitz continuity of the mapping where φ is the solution to (2.9), f ∈ C p b (R d ; R d ) and f (φ) denotes the lift as described in the previous section. We begin with some trivial bounds, namely letX ∈ C γ and denote byφ the solution to (2.9) when we replace X byX, i.e.
by the induction hypothesis combined with (2.11).
The main result of this section is the following.
, which will prove the claim. Towards this goal, for a function g smooth enough, the remainder term of the Taylor expansion satisfies For the first term above we have |t − s| (m+1)γ ∇ m+1 g ∞ γ (X,X). For the second term we use, uniformly in r ∈ [0, 1] Together with the bound |φ This combined with (2.10) gives which ends the proof of the lemma.
Combining the above Lemma, Lemma 2.5 and Remark 2.4 we get Corollary 2.10. Let X ∈ C γ g . Then there exists a family of smooth paths X such that EJP 25 (2020), paper 34.

Stability w.r.t. the drift
Let us fix X ∈ C γ and we consider the ODE (2.9). Assume we have a sequence of functions b such that there exists a solution of for every > 0 to We will show stability in the sense of controlled rough paths when we assume that φ converges in an appropriate topology. This convergence will be shown to hold in Proposition 4.15 for our particular case. Lemma 2.11. Assume φ converges in C γ to the solution of (2.9). Then for any f ∈ where the above convergence is in C γ .
Proof. Note that the second claim follows from the first in connection with Remark 2.2.
To see the first claim, one has to show The proof follows the same lines as the proof of Lemma 2.9 with minor modifications, noting that X =X.

An Itô-Stratonovich formula
For the sake of being self-contained, we include a change-of-variable formula for our particular case. Let η ∈ C ∞ c (R d ) and assume φ · solves (2.9). If X is a smooth path usual calculus yields, We can generalize this to geometric rough paths.
Lemma 2.12. Suppose η ∈ C ∞ c (R d ) and X is a rough path above X. Then we have where the last term is the rough path integral.
where we used continuity of ∇η and dominated convergence in the last step. Note that the above reasoning does not use any regularity requirements on b.
Finally, we have by definition of the rough path integral.

Integrated ODE's
To emphasize that the solution of (2.9) depends on the initial value x, we denote its In later chapters we shall be interested in expressions on the form as a controlled path in order to define t 0 ν(f (φ r ))dX r in the rough path sense. Similar results as the previous chapters holds, summarized below.
The following holds.

The rough path integral
t 0 ν(f (φ r ))dX r is well defined. 2. Let X ∈ C γ g . Then there exists a family of smooth paths X such that Proof. Begin with the first assertion. Integrating (2.10) w.r.t. ν gives EJP 25 (2020), paper 34.
Since ν is finite and b is bounded we get for each k, n and q Using linearity, boundedness of b and dominated convergence the reader is invited to complete the remaining steps of the proof.

Fractional Brownian motion and Girsanov's theorem
In this section we introduce the fractional Brownian motion as well as the technical tools we shall need in the remainder of the paper. More specifically the representation in terms of a fractional integral operator allows us to formulate the appropriate version of Girsanov theorem. The notion of strong local non-determinism is then used to infer technical bounds that are useful for studying local time estimates later in the paper. Finally we mention how one can construct a rough path lift of the fractional Brownian motion.
Observe that B has stationary increments and its trajectories are Hölder continuous of index H − ε for all ε > 0.
Denote by E the set of step functions on [0, T ] and denote by H the Hilbert space defined as the closure of E with respect to the inner product The mapping 1 [0,t] → B t can be extended to an isometry between H and a Gaussian subspace of L 2 (Ω).
For a function f ∈ L 2 ([a, b]), we define the left fractional Riemann-Liouville integral and by D α a+ its inverse.
We define K H (t, s) as For a proof we refer to [19]. In particular, the moment-estimate is found in the proof of Theorem 3, [19].
In the absence of the independent increments one has for H = 1 2 , we shall use the following fact (see [   and so from Cramer's rule we get (Σ −1 ) j,j = (V ar(Z j |Z 1 , . . . , Z j , . . . , Z m )) −1 (3.4) From the above we can prove the following technical estimates on the fractional Brownian motion.
Rewrite the right hand side of (3.5) as where we have used [15,Theorem 1] in the first inequality. Then we get from (3.4) that where we have used (3.2) and |s j+1 − s j | ≤ 1 in the two last steps, respectively. Using The result follows.
As noted in Remark 2.7, it turns out that the structure of the noise will not see the full rough path lift of the fBm. Still we mention that the fractional Brownian motion can be lifted to a rough path, as was first done in [21]. We shall, however, refer to [20] for a different construction where the authors construct the iterated integrals using a Stratonovich-Volterra-type representation.
Then there exists a set Ω B with full measure such that Assume now that H is such that 1 H is not an integer. We can choose γ < H such that 1 γ = 1 H , and from the above theorem we have, P -a.s., B ∈ C γ g .
Let us remark that for H ∈ ( 1 4 , 1 2 ) there exists a lift of B to a rough path building the iterated integral from linear interpolation of B. For the method of the current paper to work we need smaller H, see Section 4. When H ∈ (0, 1 4 ) the dyadic interpolation fails to give a converging sequence of rough paths, see [6]. Nevertheless, the construction in [20] gives a geometric rough path so that we may approximate B by a sequence of lifted smooth paths, in the rough path topology.

Fractional Brownian motion SDE's
For this section we shall study a SDE driven by an additive fractional Brownian motion, i.e.  [2] as demonstrated in the next Proposition. For proofs the reader is referred to [2].
Denote by φ n (t, x) the solution to (4.1) when b is replaced by b n . Then for fixed (t, x) ∈ [0, T ] × R d the sequence is φ n (t, x) is relatively compact in the strong topology of L 2 (Ω). Once one has strong convergence, one can use a somewhat standard trick, see e.g. [14] or [19], to show that t 0 b n (r, φ n (r, x))dr → t 0 b(r, φ r (x))dr which gives that the limit solves (4.1).
Furthermore the following result shows how the fBm regularizes the flow of (4.1).
Using the two previous results together with weak compactness in L 2 (Ω; W k,p (U )) for an open and bounded U ⊂ R d we get the following result.  For every open and bounded U ⊂ R d the solution to (4.1) is k-times weakly differentiable in the sense that φ t ∈ L 2 (Ω; W k,p (U )) for every p > 1. Moreover, φ n (t) converges to φ t in the weak topology of L 2 (Ω; W k,p (U )).

The one-dimensional case
In this section we include a proof of Proposition 4.1 when d = 1 and H < 1 6 . From [19] it is already known that there exists a unique strong solution to this equation when b is of linear growth. From [19] it also becomes clear why the proof is simpler when d = 1 -one can use comparison of SDE's to generate the strong convergence as indicated in Section 4.1.3.
We shall restrict our attention to when b is bounded and integrable, but we are interested in how the solution depends on the initial value x. More specifically we will show the following.
ists a unique strong solution to (4.1). Moreover the mapping (x → φ t (x)) is weakly differentiable in the sense that for fixed t we have for all open and bounded U ⊂ R.
This theorem is proved in three steps. In the first step we establish an integration by parts formula for the fractional Brownian motion. In the second step we assume that b is smooth and has compact support. It is then well known that φ t (·) is smooth, and we use the integration by parts formula to bound φ t L 2 (Ω;W 1,p (U )) independently of b . In the third step we approximate a general b by smooth functions. We use comparison to generate strong convergence in L 2 (Ω) of the corresponding sequence of solutions. From step one and two we can bound the sequence in L 2 (Ω; W 1,p (U )) and argue via weak compactness to prove Theorem 4.4.

An integration by parts formula
The purpose of this section is to prove a integration by parts type formula involving a random variable inspired by local time calculus. More specifically, we have   We start by defining Λ b (t, z) as above, and prove that it is a well defined element of L p (Ω) for every p > 1. .
Proof. Since we assume m is an even integer, we may write where for notational convenience we have used B s0 = y and v m+1 = 0, ds = ds 1 . . . ds m , du = du 1 . . . du m and b ⊗m (s, y) := m j=1 b(s j , y). Using (3.5) the above is bounded by It remains to show that Λ b satisfies the integration by parts formula (4.2). Notice that one has to be careful interchanging the order of integration in (4.3). Indeed, if b = 1, one should think of R iue −iu(Bs−y) du = −∂ y δ Bs (y) where δ Bs (y) is the Donsker-Delta of B s , which is not a random variable in the usual sense (one could introduce the Donsker-Delta as a generalized random variable in the sense of White Noise theory, but we shall avoid this).
To circumvent this difficulty we define an approximating sequence It is immediate that for an appropriate constant, so that Λ b K (t, ·) is integrable if R t 0 |b(s, y)|dsdy < ∞. One can show that Λ b K (t, y) → Λ b (t, y) in, say, L 2 (Ω) for all t and y. To see this the reader is invited to modify the above proof to see that which converges to zero as K → ∞. In the above C is a constant that is independent of K. Now we have thus proving (4.2). We summarize these considerations.
). Then (4.2) holds for b and we have P -a.s.
Proof. Assume first that b satisfies the assumptions of Lemma 4.6, and let φ ∈ C 1 c (R). Using Λ φb (t, y) = φ(y)Λ b (t, y) as in the above proof we get that if b is time homogeneous, Λ b (t, y) = b(y)∂ y L B (t, y) where L B (t, y) denotes the local time of the fractional Brownian motion (which is well known to be differentiable when H < 1 3 , see [11]). Proposition 4.8. There exists a constant C > 0 such that for all even integers m for an appropriate constant C, where we have used Lemma 4.5.

Derivative free estimates
In this section we assume that b ∈ L ∞ ([0, T ]; C 1 c (R)) and denote by φ · (x) the solution to (4.1). It is well known that φ t (·) continuously differentiable, and we have where φ · (x) is the unique solution of (4.1) driven by b.
ThenP is a probability measure and underP the solution {φ t (x)} t is a fractional Brownian motion starting in x. From (4.5) we get which converges by Stirling's formula.
From Theorem 3.1 we know that we can boundẼ[Z −2 ] by a function depending on b ∞ . The result follows.

Singular SDE's
For this section we shall consider a bounded and measurable b : [0, T ] × R → R and the corresponding SDE (4.1). As indicated above we shall use an approximation b n of b and comparison to generate strong convergence in L 2 (Ω). The technique is somewhat classical, and we refer to [19] for a proof, but let us briefly explain the idea: Let b be bounded and measurable and define for n ∈ N b n (t, x) := n R ρ(n(x − y))b(t, y)dy EJP 25 (2020), paper 34.
where ρ is a non-negative smooth function with compact support in R such that R ρ(y)dy = 1. We letb n,k := k j=n b j , n ≤ k, and B n = ∞ j=n b j , so thatb n,k is Lipschitz. Denote byφ n,k (t, x) the unique solution to (4.1) when we replace b byb n,k . Then one can use comparison to show that where φ n (t, x) solves (4.1) when we replace b by B n . Furthermore, where φ t (x) is a solution to (4.1). For details see [19].
We are ready to prove the main result of the section.
Proof of 4.4. Let U ⊂ R be open and bounded. We know from the discussion above that φ n (t, x) → φ t (x) in L 2 (Ω). From Theorem 4.9 plus elementary bounds we see that φ n (t, ·) is bounded in L 2 (Ω; W 1,2 (U )). Consequently we may extract a subsequence {φ n k (t, ·)} k≥1 converging to an element f t in the weak topology of L 2 (Ω; W 1,2 (U )). Let A ∈ F and η ∈ C ∞ (U ). Using strong convergence coupled with weak convergence we get

Local time of the flow
We now return to the general case of d ≥ 1.
In this section we develop a local time theory for the solutions φ t (x) of (4.1). Assuming we have a solution to φ t (x), the results here will rely only on Girsanov's theorem 3.1 meaning we only use boundedness of b. Let now Q : [0, T ] × R d → R be given and define q = D α Q for some multiindex, α = (α (1) , . . . α (d) ). The main objective of this section is to prove that there exists a random field Λ φ(x),Q α and that the right hand side above can be bounded in terms of Q. Motivated by the previous subsection, we define We denote by Λ Q α (t, y) the random field obtained by choosing B instead of φ(x) in the above definition. Note that from Girsanov's theorem we have for any f such that the above expressions exists and ξ T was defined in Theorem 3.1 and We get a similar result as Lemma 4.5.
Proof. The proof follows the same lines as in the proof of Lemma 4.5. Begin by writing where we have used the independence of the components of B in the second line. Using EJP 25 (2020), paper 34.
Using Theorem 3.1 we get Corollary 4.12. Let Q, H and α be as in the previous lemma. There exists a constant H(d + 2|α|)) + 1 .
If we assume integrability of Q in the spatial variable we see that we can define the stochastic process where C is an increasing function.

Proof. Begin by writing
where C is as in Corollary 4.12. We get which converges as long as H < 1 d+2 by Stirling's formula.
We now proceed to prove stability of the vector field Λ φ(x),Q α in both Q and φ in the following way.
Remark 4.14. We shall need stability of the mapping (φ(x), Q) → R d Λ φ(x),Q (t, z)dz, but we only need continuity in each variable separately. If φ · (x) converges to φ · (x) in, say, Lebesgue measure over [0, T ] and Q is smooth, we immediately get Stability in Q as a mapping L 1 (R d ; L ∞ ([0, T ]; R d )) → L m (Ω) follows from the linearity of the mapping Q → R d Λ φ(x),Q (t, z)dz as well as the bounds from Lemma 4.13.

Convergence in Hölder spaces
With the notation of Proposition 2.13 we shall need a result to ensure convergence of ν(f (φ n · )) is uniform on a set of full measure. Proposition 4.15. Let γ ∈ (0, H), f ∈ C 1 b (R d ; R d ) and ν be a finite signed measure on R d . Then there exists a set Ω γ,ν of full measure such that on this set we have Proof. We begin by showing that ν(f (φ n t )) → ν(f (φ t )) in L 2 (Ω) for every t. To see this, as n → ∞ by dominated convergence, which proves the first claim.

Continuity equation
In this section we study the measure valued rough linear continuity equation with given initial condition µ 0 . The notion of solution is as follows.
If we know that there exists a solution to then for any test function η ∈ C ∞ c (R d ) we have from Lemma 2.12 We integrate the equation w.r.t. µ 0 to see that µ t := (φ t ) µ 0 solves (5.1) if we can use integration by parts for the rough path integral, namely We summarize the above in a lemma.
Then there exists a solution to (5.1) and the solution is given by µ t := (φ t ) µ 0 .
Given the previous sections the reader will not be surprised that we can extend this to when the drift is discontinuous provided we choose the rough path to be the lift of a fractional Brownian motion with low Hurst index.
where we have used the change of variables φ r (y) = x. It is clear from Section 2.2 that ∇η(φ · (y)) can be regarded as a controlled path. However, the terms are not expected to be more than 1 − H(2 + d) regular in time (at least at the current level of knowledge) so we can not invoke Lemma 2.1 and it is not clear how to define the product as a controlled path. In fact this seems to require that also e.g. |∇φ · (y)| is controlled by B and we do not yet know how do this construction.
In its full generality we still cannot show that u defined as above solves the equation, but we provide some examples (d = 1, div b bounded, c = div b and time-homogenuous drift) where we can.
First, let us study the equation when the coefficients and the noise are regular.

Regular case
Assume for a moment that the drift b ∈ L ∞ ([0, T ]; C 1 b (R d )) and we want to study the rough linear transport equation with given initial condition u| t=0 = u 0 . If we assume that X is the geometric lift of a smooth path X ∈ C 1 , we may read (6.2) in a classical way: with initial condition u(0, x) = u 0 (x). To solve this equation, let us define where φ t (x) is the solution to (2.9). Immediately, u(t, φ t (x)) = u 0 (x) exp{− t 0 c(r, φ r (x))dr} Making a change of variables we see that u(t, x) is indeed a solution of (6.2).
Integrating the above w.r.t. t and approximating a rough path X by smooth paths and taking the limit, it is reasonable that we should get provided the solution is such that ∇u(·, x) is controlled by X. Unfortunately, to guarantee that ∇u(t, x) is a controlled path we need higher order differentiability of the solution than the regularization of the fractional noise can provide. To circumvent this we use a spatially weak notion of solution. to (6.1) if for all η ∈ C ∞ c (R d ) the path R d ∇u(·, x)η(x)dx is controlled by X and the following equality holds Existence of such a solution when the drift is nice is relatively straightforward. The proof is a consequence of the discussion in Section 2.3 together with the above computations.
, and X ∈ C γ g . Then there exists a weak solution to (6.1).
Proof. Consider a smooth approximation X of X and let Consider now R ∇u (r, x)η(x)dx as above. Using integration by parts we get c(s, φ s (y))ds |∇φ r (y)|η(φ r (y))dy where we have used a change of variable y = φ ,−1 r (x) in the last equality. From Liouville's formula we get |∇φ r (y)| = exp r 0 div b(s, φ s (y))ds From Section 2.2, if we can show that exp{ r 0 div b(s, φ s (y)) − c(s, φ s (y))ds}η(φ r (y)) converges in D pγ X to exp{ r 0 div b(s, φ s (y)) − c(s, φ s (y))ds}η(φ r (y)), then it follows imme- Towards this goal, we notice that from Lemma 2.1 it is enough to prove that · 0 div b(s, φ s (y)) − c(s, φ s (y))ds converges in C β to · 0 div b(s, φ s (y)) − c(s, φ s (y))ds. From Hölder's inequality we get t r div b(s, φ s (y)) − c(s, φ s (y)) − div b(s, φ s (y)) + c(s, φ s (y))ds The result follows by dominated convergence and continuity (of c and div b) as long as we choose β = pγ < 1.
Convergence of the remaining terms follows by similar considerations.

Singular case
Motivated by the previous section we define our solution via the flow transformation. We go on to prove existence of such a solution for almost all sample paths of the fBm.
Theorem 6.4. Assume we have • There exists smooth functions C k j such that R d sup t∈[0,T ] |C j (t, y) − C k j (t, y)|dy → 0 as k → ∞ for all j, • u 0 is continuous, • |α j | ≤ 1 • B is a fBm with Hurst parameter H < 1 d+2 .
Then there exists a set of full measure, Ω 0 such that for every ω ∈ Ω 0 there exists a local time solution of (6.1).
Proof. The proof is done by approximation of b and then c as in the above assumptions. For notational simplicity we assume J = 1. Let Ω γ,δx be as in Proposition 4.15 where δ x is the Dirac centered at x, so that we have u 0 (φ n (t, x, ω) −1 ) → u 0 (φ(t, x, ω) −1 ). For a fixed k we have by assumption, and thus there exists a subsequence and a set of full measure,Ω such that we have lim k→∞ R d Λ C k ,φ(x) α (t, y)dy = R d Λ C,φ(x) α (t, y)dy onΩ. The result follows when we choose Ω 0 = Ω γ,δx ∩Ω ∩ ∩ k≥1 Ω k . EJP 25 (2020), paper 34.

Divergence of b bounded
When the divergence of b is bounded, we can write |∇φ t (x)| = exp t 0 div b(r, φ r (x))dr , and so the mapping t → |∇φ t (x)| is of bounded variation. Using Lemma 2.1 we can show the following result.