1.1 Scaling heuristics and existing literature

One way to have a unified view on the many works on regularization by noise is by a scaling argument; for a similar approach in the Brownian setting and $L^q_t L^p_x$ spaces, see [Reference Beck, Flandoli, Gubinelli and Maurelli8, Section 1.5].

From now on, we sample the perturbation as a fBm $B^H$ with Hurst parameter $H\in (0,+\infty ) \setminus \mathbb {N}$ , which satisfies the scaling relation

(1.5) $$ \begin{align} (B^H_t)_{t\geq0}\overset{\mathrm{law}}{=}(\lambda^{-H}B^H_{\lambda t})_{t\geq 0}, \quad \forall\, \lambda>0. \end{align} $$

Details about the processes $B^H$ are given in Section 1.4 below; let us just briefly recall that $H=1/2$ gives the standard Brownian motion and that this is the only case where $B^H$ is a Markov process. For the values $H=k+1/2$ , $k\in \mathbb {N}_+$ (which we call ‘degenerate Brownian’), the Markovian toolbox is still available since the SDE can be rewritten as a higher-dimensional equation driven by degenerate Brownian noise; see, for example, [Reference Chaudru de Raynal, Honoré and Menozzi24]. For all other choices of H, such tools are unavailable, and the study of the SDE requires a fundamentally different approach. The equation then takes the form

(1.6) $$ \begin{align} X_t = x_0 + \int_0^t b_r ( X_r) \mathrm{d} r + B^H_t. \end{align} $$

In order for the regularising effects of $B^H$ to dominate the irregularities of b, it is natural to require that, when zooming into small scales in a way that keeps the noise strength constant, the nonlinearity vanishes; if this were not the case, and the nonlinearity were dominant, we would expect to see all the same pathologies (e.g., coalescence or branching of solutions) which could manifest in the ODE without noise. Therefore, keeping (1.5) in mind, for a fixed parameter H, we call a space V of functions (or distributions) on $\mathbb {R}_+\times \mathbb {R}^d$ critical (resp. subcritical/supercritical) if for the rescaled drift coefficient

$$ \begin{align*} b^\lambda_t(x)=\lambda^{1-H} b(\lambda t, \lambda^H x), \end{align*} $$

the leading order seminorm (see the examples below for its practical meaning) scales like , for all $\lambda \leq 1$ ,Footnote ¹ with $\gamma =0$ (resp. $\gamma>0$ / $\gamma <0$ ).

We refer to Section 1.5 for more details on the function spaces appearing in the upcoming examples.

Example 1.1. Consider autonomous, inhomogeneous Hölder-Besov spaces $V=B^\alpha _{\infty ,\infty }$ , where b does not depend on the time variable. Here, the leading order seminorm is the associated homogeneous seminorm; namely, we set as defined in [Reference Bahouri, Chemin and Danchin5]; alternatively, for $f\in B^\alpha _{\infty ,\infty }$ and $\alpha \geq 0$ , one can regard it as $\| (-\Delta )^{\alpha /2} f\|_{B^0_{\infty ,\infty }}$ , while for $\alpha <0$ , one can define it by duality with the homogeneous seminorm of $\dot {B}^{-\alpha }_{1,1}$ . Either way, one finds the scaling relation

$$ \begin{align*} \| f(\eta\, \cdot)\|_{\dot{B}^\alpha_{\infty,\infty}} \sim_\alpha \eta^\alpha \| f(\cdot)\|_{\dot{B}^\alpha_{\infty,\infty}} \quad\, \forall\, (\eta,\alpha)\in \mathbb{R}_{>0}\times \mathbb{R}. \end{align*} $$

Combined with our definition of $b^\lambda $ , one finds $\gamma =1-H+\alpha H$ , and so the subcriticality condition reads as

(1.7) $$ \begin{align} \alpha>1-\frac{1}{H}. \end{align} $$

However, even in the classical Brownian case, where one gets the condition $\alpha>-1$ , this remains out of reach. Weak well-posedness is known for $\alpha>-1/2$ [Reference Flandoli, Issoglio and Russo41], and a nonstandard kind of well-posedness (where uniqueness is even weaker than uniqueness in law) is shown for $\alpha>-2/3$ [Reference Cannizzaro and Chouk18, Reference Delarue and Diel35], for special classes of drift b. The classical works [Reference Veretennikov97, Reference Zvonkin104] show strong well-posedness for $V=C^\alpha _x$ and $\alpha \geq 0$ .Footnote ² Interestingly, in the degenerate Brownian case, weak well-posedness is proved in [Reference Chaudru de Raynal, Honoré and Menozzi24] in the full regime $\alpha>(2k-1)/(2k+1)$ , which is precisely the condition (1.7). For strong well-posedness, one requires the more restrictive condition

$$ \begin{align*} \alpha>1-\frac{1}{2H}; \end{align*} $$

see [Reference Chaudru de Raynal, Honoré and Menozzi23, Equation (1.11)]. The same condition is required for strong well-posedness in the non-Markovian case for all $H\in (0,\infty )\setminus \mathbb {N}$ ; cf. [Reference Catellier and Gubinelli20, Reference Galeati and Gubinelli49, Reference Gerencsér53, Reference Nualart and Ouknine81]. After the first version of this manuscript, the work [Reference Butkovsky, Lê and Mytnik16] appeared, where the authors are able to establish (among several results) weak existence of solutions in the full subcritical regime (1.7), under the additional assumption that b is a Radon measure; however, uniqueness is still open.

Example 1.2. Another well-studied case is the mixed Lebesgue space $V=L^q_tL^p_x$ . Here, we can take the seminorm to be $\| \cdot \|_V$ itself; using the scaling relation $\|f(\eta \,\cdot )\|_{L^p_x}= \eta ^{-d/p}$ , one finds $\gamma =1-H-1/q-(Hd)/p$ , and the subcritical regime is

(1.8) $$ \begin{align} \frac{1}{q}+\frac{Hd}{p}<1-H. \end{align} $$

In the classical case $H=1/2$ , equation (1.8) reads as

(1.9) $$ \begin{align} \frac{2}{q}+\frac{d}{p}<1, \end{align} $$

which is precisely the condition from the classical work [Reference Krylov and Röckner68], where strong well-posedness is proved (under the additional constrant $p\geq 2$ ); instead, the critical regime corresponds to the celebrated Ladyzhenskaya–Prodi–Serrin (LPS) condition. This case has then been extensively studied by several authors, allowing also for multiplicative noise with Sobolev diffusion coefficients; see, among others, [Reference Fedrizzi and Flandoli37, Reference Xia, Xie, Zhang and Zhao99, Reference Zhang101, Reference Zhang102]. In recent years, even the critical case has been reached [Reference Krylov65, Reference Röckner and Zhao88] under certain constraints on $d,p,q$ ; the results have been further refined by allowing coefficients in Morrey spaces (cf. [Reference Krylov66, Reference Krylov67]) or form-bounded drifts (cf. [Reference Kinzebulatov and Madou60, Reference Kinzebulatov and Semënov61]) and the references therein. It was recently understood in [Reference Zhang and Zhao103] that one can go beyond condition (1.8), up to imposing additional constraints on $\mathrm {div}\, b$ ; for further progress in this exciting direction, see also [Reference Gräfner and Perkowski55, Reference Hao and Zhang57].

For $H\in (1/2,1)$ , no results are known, and for $H\in (0, 1/2)$ , the main previously known results for weak and strong well-posedness are both from [Reference Lê71], under the stronger conditions

(1.10) $$ \begin{align} \frac{1}{q}+\frac{Hd}{p}<\frac{1}{2},\qquad\frac{1}{q}+\frac{Hd}{p}<\frac{1}{2}-H, \end{align} $$

respectively, with the additional constraint $p\in [2,\infty ]$ , later removed in [Reference Galeati and Gubinelli49]. It is conjectured in [Reference Lê71] that the first condition in (1.10) is enough to guarantee strong well-posedness. One particular corollary of our result is that for $q\in (1,2]$ , even (1.8) is sufficient. Therefore, we propose to update the conjecture of [Reference Lê71] (if $q\in (1,2]$ , now a theorem) to assert strong well-posedness under the scaling condition (1.8). Let us also mention that we have recently learned about an ongoing work [Reference Butkovsky, Lê and Matsuda15] towards improving (1.10).

Example 1.3. A common generalisation of Examples 1.1 and 1.2 is the space $V=L^q_t C^\alpha _x$ , where (adopting the leading seminorm to be the one of $L^q_t \dot {B}^\alpha _{\infty ,\infty }$ , in agreement with both previous casesFootnote ³ ) the scaling works out to be $\gamma =1-H-1/q+\alpha H$ . Therefore, the subcriticality condition reads as

$$ \begin{align*} \alpha>1-\frac{1}{H}+\frac{1}{Hq}=1-\frac{1}{q'H}, \end{align*} $$

where, here and in the rest of the paper, q and $q'$ are conjugate exponents, $1/q+1/q'=1$ . This generality has only been studied recently in [Reference Galeati and Gubinelli49, Reference Galeati, Harang and Mayorcas51], where strong well-posedness is proved under the stronger condition

(1.11) $$ \begin{align} \alpha>1-\frac{1}{2H}+\frac{1}{Hq}, \end{align} $$

with the additional constraints $H\in (0,1/2]$ , $q\in (2,\infty ]$ . Note that by setting $\alpha =-d/p$ , condition (1.11) coincides with the second one in (1.10).

In summary, to the best of our knowledge, weak well-posedness results in a whole subcritical regime are available only in the degenerate Brownian case $H=k+1/2$ , $k\in \mathbb {N}$ , and strong well-posedness only in the standard Brownian case $H=1/2$ .

1.2 Discussion of the main results

In the present paper, we establish strong well-posedness in the full subcritical regime for all $H\in (0,\infty )\setminus \mathbb {N}$ , with coefficients from the class in Example 1.3, under the additional constraint $q\in (1,2]$ . In other terms, our main conditions are summarised by the assumption

(A) $$ \begin{align} H\in(0,\infty) \setminus \mathbb{N},\qquad q\in(1,2],\qquad \alpha\in\Big(1-\frac{1}{q'H},1\Big). \end{align} $$

The solution theory we present in fact goes beyond strong well-posedness. We show existence in the strong sense not only of solutions but also of solution flows, and uniqueness in the path-by-path sense. Furthermore, several further properties of solutions are established such as stability, continuous differentiability of the flow and its inverse, Malliavin differentiability and $\rho $ -irregularity.

Many of these results are even new in the time-independent case: if b is only a function of x and belongs to $C^\alpha _x$ , then the optimal choice to put it in the framework of (A) is to choose $q=2$ , leading to the condition $\alpha>1-1/(2H)$ . This is the classical condition under which strong well-posedness is known [Reference Catellier and Gubinelli20, Reference Gerencsér53, Reference Nualart and Ouknine81], but several of the further properties have not been previously established.

Our main findings are loosely summarised (without aiming for full precision or generality) in the following statement; the corresponding results (often in a somewhat sharper form) can be found throughout the paper in Theorems 4.3, 4.4, 5.5, 5.6 for i), 3.2 for ii), 6.2 for iii), 6.8 for iv), 7.4 for v), 9.3 for vi), 10.4 for vii). For simplicity, we restrict ourselves to the time interval $t\in [0,1]$ , but it is clear that up to rescaling, we could consider any finite $[0,T]$ (up to allowing the hidden constants to depend on T).

The various aspects of the main results are discussed in detail in their respective sections, so here, let us just briefly comment on them.

The notion of path-by-path uniqueness in i), as a strengthening of the classical pathwise uniqueness, was first established in the seminal work [Reference Davie32] by Davie, with a simpler proof that was later provided by Shaposhnikov [Reference Shaposhnikov91]. This kind of result was then generalised to fBm in [Reference Catellier and Gubinelli20], suggesting it is a consequence of the pathwise properties of the trajectories of the driving noise. Such a uniqueness concept requires giving a pathwise interpretation to the SDE, which becomes nontrivial for $\alpha <0$ , where b can be a distribution of negative regularity and not a function anymore. In this case, following [Reference Catellier and Gubinelli20], we will give meaning to (1.6) as a nonlinear Young ODE; see Section 5 for more details.

Stability estimates in the style of ii) are useful to bypass abstract Yamada-Watanabe arguments and get strong existence directly. Among other possible applications, let us mention their importance in numerical schemes with distributional drifts; see, for example, the recent work [Reference Goudenège, Haress and Richard54]. In this paper, stability estimates play a key role when solving McKean-Vlasov equations as in v); see Section 7. The study of stochastic flows iii) for SDEs goes back to the classical work [Reference Kunita69]; see also [Reference Chen and Li26, Reference Fedrizzi and Flandoli37, Reference Menoukeu-Pamen, Meyer-Brandis, Nilssen, Proske and Zhang76] for flows in irregular settings. In iv), we can in fact derive differentiability with respect to perturbations of the noise quite a bit more generally than Cameron-Martin directions (see Remark 6.9), in line with the observations from [Reference Friz and Victoir42, Reference Kusuoka70]. Concerning v), regularisation by fractional noise for distribution dependent SDEs has been investigated in [Reference Galeati, Harang and Mayorcas51] and recently in [Reference Han56]. Above, we only stated the simplest example of the McKean-Vlasov equation for the sake of presentation. Theorem 7.4 below allows for more general dependence on $(X,\mu )$ . The notion of $\rho $ -irregularity in vi) was introduced by [Reference Catellier and Gubinelli20] as a powerful measurement of the averaging properties of paths. Extending $\rho $ -irregularity from Gaussian processes to perturbed Gaussian processes has previously only been achieved efficiently via Girsanov transform. Here, we provide a simple and more robust alternative. Concerning vii), regularisation by noise results for the transport equation were first established for Brownian noise in [Reference Flandoli, Gubinelli and Priola40] and further developed in [Reference Fedrizzi and Flandoli38, Reference Mohammed, Nilssen and Proske78]; see also [Reference Catellier21, Reference Galeati and Gubinelli49, Reference Nilssen79] for further investigations in the fractional case.

The scope of some intermediate estimates we obtain is larger than (A), and therefore, in some regime where we do not obtain strong well-posedness, we still get compactness and therefore existence of weak solutions. To state the result, we need to enforce the following different condition:

(B) $$ \begin{align} H\in(0,1) ,\quad q\in(1,\infty],\quad \alpha>\frac{1}{2}-\frac{1}{2H}, \quad \alpha>1-\frac{1}{H q'}. \end{align} $$

The proof of Theorem 1.5 is presented in Section 8, where we also define rigorously what we mean by weak solution to (1.6) in this case; see Theorem 8.2 for a more precise statement.

Remark 1.9. Theorem 1.4 gives new results also in the classical $H=1/2$ case. Indeed, to solve (1.6) with classical tools, one would require a good solution theory of the corresponding Kolmogorov equation

(1.13) $$ \begin{align} \partial_t u-\tfrac{1}{2}\Delta u=b\cdot\nabla u. \end{align} $$

Suppose that $b\in L^q_t C^\alpha _x$ with $q\in (1,2)$ . Then the naive power counting fails: replacing first u by a smooth function on the right-hand side gives, by Schauder estimates, $u\in L^\infty _t C^\beta _x$ with $\beta =\alpha +2-2/q$ , and so $b\cdot \nabla u\in L^q_t C^{\alpha +1-2/q}_x$ . Since $\alpha +1-2/q<\alpha $ , iterating the procedure implies worse and worse spatial regularity on u, and after finitely many steps, the product $b\cdot \nabla u$ becomes even ill-defined. This is somewhat similar to the issue of the Kolmogorov equation of Lévy SDEs with low stability index, which was circumvented in [Reference Chaudru de Raynal, Menozzi and Priola25]. After this manuscript appeared, Schauder estimates for (1.13) with $b\in L^q_t C^\alpha _x$ with $q\in (1,2)$ were developed in [Reference Wei, Hu and Yuan98].

The rest of the article is structured as follows. In Section 1.3, we present some counterexamples in the supercritical regime, demonstrating that (up to reaching the critical equality) condition (A) cannot be improved; we then conclude the introduction by recalling some fundamental properties of fBm in Section 1.4 and by introducing the main notations used throughout the paper in Section 1.5. In Section 2, we state and prove some fundamental lemmata, including the aforementioned a priori estimates for solutions of (1.6) and the two new forms of the stochastic sewing lemma of [Reference Lê71]. Section 3 contains further estimates for additive functionals of processes, as well as a key stability property of solutions. In Sections 4 and 5, we use these estimates to establish well-posedness of (1.6); we distinguish the cases $\alpha>0$ and $\alpha <0$ cases, which require a different analysis. Along the way, we prove the existence of a solution semiflow, which we upgrade to a flow of diffeomphisms in Section 6. Section 7 contains applications of our stability estimates to McKean-Vlasov equations. In Section 8, we construct weak solutions under condition (B), via a compactness argument enabled by the available a priori estimates. In Section 9, we show $\rho $ -irregularity of solutions and more general perturbations of fractional Brownian motions. Finally, Section 10 contains applications to transport and continuity equations. In the appendices, we collect some useful tools for which we did not find exact references in the literature: Appendix A contains variants of Kolmogorov continuity criterion, and Appendix B gives two basic bounds for solutions of Young differential equations. In Appendix C, we summarise relations of various Sobolev spaces and their use in Girsanov transform for fractional Brownian motions.

1.3 Counterexamples to uniqueness in the supercritical regime

Although the scaling argument is heuristic, one can often construct counterexamples in the supercritical case. The constructions below are motivated by [Reference Chaudru de Raynal22], which gives counterexamples for $q=\infty $ , $\alpha>0$ .

Assume $d\geq 1$ , $H\in (0,1)$ and $(\alpha ,q)\in \mathbb {R}\times (1,\infty )$ satisfy

(1.14) $$ \begin{align} \alpha<1-\frac{1}{H q'},\qquad \alpha>-1; \end{align} $$

let B be an $\mathbb {R}^d$ -valued stochastic process such that $\mathbb {P}$ -almost surely $B\in C^\gamma $ for all $\gamma \in (0,H)$ . We claim that under (1.14), there exists $b\in L^q_t C^\alpha _x$ such that the equation

(1.15) $$ \begin{align} X_t=x+\int_0^t b_s(X_s)\,ds+B_t \end{align} $$

with initial condition $x_0=0$ has at least two solutions whose laws are mutually singular.

We will treat separately the cases $\alpha \in (0,1)$ and $\alpha \in (-1,0)$ .

For $\alpha \in (0,1)$ , the construction is actually one-dimensional and can be extended trivially to higher dimensions by taking $b=(b^i)_{i=1}^d$ with $b^i\equiv 0$ for $i\geq 2$ ; therefore, here we will set $d=1$ . Take $\tilde q>q$ such that $(\alpha ,\tilde q)$ still satisfies (1.14) and define the function

$$ \begin{align*} b_t(x)=t^{-1/\tilde q}\,{\mathrm{sign}}(x)|x|^\alpha; \end{align*} $$

clearly, $b\in L^q_t C^\alpha _x$ . Let $\gamma =1/(\tilde q'(1-\alpha ))$ ; by definition, $\gamma $ satisfies the identity

(1.16) $$ \begin{align} \gamma=1-\frac{1}{\tilde q}+\gamma\alpha, \end{align} $$

and furthermore, $\gamma <H$ thanks to (1.14). Fix furthermore $\delta>0$ small such that $\delta ^\alpha /\gamma>2\delta $ , which exists since $\alpha \in (0,1)$ . Take $x\in (0,1]$ and consider a weak solution $(X^x,B)$ of (1.15), which is well-known to exist due to the spatial continuity and sublinear growth of b. Define the stopping time

$$ \begin{align*} \tilde\tau:=\inf\{t>0:\, |B_t|\geq \delta t^\gamma\}\wedge 1; \end{align*} $$

it is strictly positive $\mathbb {P}$ -almost surely since $\gamma <H$ and $B\in C^{\tilde \gamma }$ with $\tilde \gamma \in (\gamma ,H)$ . Also define

$$ \begin{align*} \tau_x:=\inf\{t\geq 0:\, X_t^x\leq\delta t^\gamma\}\wedge 1. \end{align*} $$

We claim that $\tau _x\geq \tilde \tau $ . Indeed, $\tau _x>0$ since $x>0$ , and for all $t\leq \tau _x$ by (1.15) and our construction, it holds

(1.17) $$ \begin{align} X_t^x>\int_0^ts^{-1/\tilde q}(\delta s^\gamma)^{\alpha}\,ds+B_t= (\delta^\alpha/\gamma) t^\gamma+B_t>\delta t^\gamma+\big(\delta t^\gamma+B_t), \end{align} $$

where in the intermediate passage, we used (1.16). Since $\tau _x\geq \tilde \tau>0 \mathbb {P}$ -a.s., there exist $\rho>0$ independent of $x\in (0,1]$ such that

$$ \begin{align*} \mathbb{P}(\tilde\tau_x>\rho)\geq 3/4. \end{align*} $$

The laws of $(X^x,B)$ on $C([0,1])^2$ are tight, and therefore by Skorohod’s representation theorem, we may assume that for a sequence $x_n\searrow 0$ , the random variables $(X^{x_n},B^{x_n})$ live on the same probability space and converge in $C([0,1])^2 \mathbb {P}$ -a.s. The limit $(X^{0,+},B^{0,+})$ is a solution to (1.15) with initial condition $0$ and satisfies

$$ \begin{align*} \mathbb{P}\big(X^{0,+}_t>0\,\,\forall t\in(0,\rho]\big)&\geq \mathbb{P}\big(X^{0,+}_t\geq \delta t^\gamma\,\,\forall t\in[0,\rho]\big) \\&=\lim_{n\to\infty}\mathbb{P}\big(X^{x_n}_t\geq \delta t^\gamma\,\,\forall t\in[0,\rho]\big)\geq3/4. \end{align*} $$

Since b is odd, we can run the same argument for $y\in [-1,0)$ : if $X^y$ is a solution to (1.15), then $-X^y$ is a solution for $(-y,-B)$ , and the definition of $\tilde \tau $ only depends on $|B|$ . Therefore, for the same choice of $\rho $ , in this case, one finds that

$$ \begin{align*} \mathbb{P}\big(X^y_t \leq -\delta t^\gamma \,\forall t\in(0,\rho]\big) \geq 3/4, \end{align*} $$

and so by considering a sequence $y_n\nearrow 0$ by compactness, one can construct $(X^{0,-}, B^{0,-})$ another weak solution to (1.15) with initial condition $0$ satisfying

$$ \begin{align*} \mathbb{P}\big(X^{0,-}_t<0\,\,\forall t\in(0,\rho]\big)\geq 3/4. \end{align*} $$

This shows that $X^{0,+}$ and $X^{0,-}$ do not have the same law, yielding weak non-uniqueness (we leave it as an exercise to the reader to show that their laws are in fact mutually singular).

In the distributional case $\alpha \in (-1,0)$ , we have to be a bit more careful since the meaning of the SDE becomes unclear if X gets too close to $0$ . To this end, we argue again by stopping times, and the construction we present this time is genuinely d-dimensional. Again, take $\tilde q>q$ such that $(\alpha ,\tilde q)$ still satisfy (1.14) and define a vector field $b=(b^i)_{i=1}^d$ by

$$ \begin{align*} b^1(t,x)=t^{-1/\tilde q}\,{\mathrm{sign}}(x^1)|x|^{\alpha},\qquad b^i(t,x)\equiv 0 \text{ for } i=2,\ldots,d; \end{align*} $$

again, $b\in L^q_t C^\alpha _x$ . Take $x\in (0,1]$ and consider a local-in-time solution $X^x$ of (1.15) with initial condition $x_0=(x,0,\ldots ,0)$ , which is well-known to exist due to the spatial regularity of b locally around $x_0$ . Define $\gamma $ as before, so that $\gamma <H$ and (1.16) holds; let us furthermore take an auxiliary parameter $\delta $ that will be specified later. Define the stopping times

$$ \begin{align*} \tilde\tau:=\inf\{t>0:\, |B_t|\geq \delta t^\gamma\}\wedge 1, \quad \tau_x:=\inf\{t\geq 0:\, (X_t^x)^1\leq\delta t^\gamma\}\wedge 1; \end{align*} $$

as before, $\tilde \tau $ is strictly positive $\mathbb {P}$ -almost surely since $\gamma <H$ . We claim that $\tau _x\geq \tilde \tau $ , for which it suffices to show that for $t\leq \tau _x\wedge \tilde \tau $ , one has $(X_t^x)^1\geq 2\delta t^\gamma $ . If $x>3\delta t^\gamma $ , then by simply using the nonnegativity of the first component of b up to $\tau _x$ and the definition of $\tilde \tau $ , we see that

$$ \begin{align*} (X_t^x)^1\geq x+B_t^1\geq 3\delta t^\gamma-\delta t^\gamma, \end{align*} $$

as required. Suppose now that $x\leq 3\delta t^\gamma $ . Clearly, for $s\leq \tau _x$ , one also has $|X_s^x|\geq \delta s^\gamma $ . Inserting this bound in the equation, we get for $s\leq \tau _x\wedge \tilde \tau $

$$ \begin{align*} (X_s^x)^1&\leq x+\int_0^sr^{-1/\tilde q}(\delta r^\gamma)^{\alpha}\,dr+B_s^1 \\ &=x+ (\delta^{\alpha}/\gamma) s^\gamma+B_s^1 \\ &\leq x+ \big(\delta^{\alpha}/\gamma+\delta\big)s^\gamma; \end{align*} $$

observe that since $\alpha <0$ , we find reversed inequalities compared to the previous case. In particular, if $s\geq t/2$ , then using $x\leq 3\delta t^\gamma $ , we also get

$$ \begin{align*} (X_s^x)^1 \leq \big(3\delta 2^\gamma+\delta+\delta^{\alpha}/\gamma\big)s^\gamma. \end{align*} $$

For $\delta \in (0,1)$ , there exist constants $C',C$ depending only on $d,\alpha ,\gamma $ such that the above bound implies $(X_s^x)^1\leq C'\delta ^{\alpha }s^\gamma $ , as well as $|X_s^x|\leq C \delta ^{\alpha }s^\gamma $ . Using this bound in the equation once more,

$$ \begin{align*} (X_t^x)^1&>\int_{t/2}^ts^{-1/\tilde q}\big(C \delta^{\alpha}s^\gamma)^{\alpha}\,ds+B_t^1 \\ &\geq (1/2)C^{\alpha}\delta^{\alpha^2}t^\gamma -\delta t^\gamma. \end{align*} $$

At this point (using the condition $\alpha>-1$ , so that $\alpha ^2<1$ ), one can choose $\delta $ small enough so that the right-hand side is bounded from below by $2\delta t^\gamma $ . With this, we conclude the proof of the property $\tau _x\geq \tilde \tau $ . In other words, for all $t\leq \tilde \tau $ , for all $x\in (0,1]$ , we have $(X_t^x)^1\geq \delta t^\gamma $ . In a symmetric way, for all $t\leq \tilde \tau $ , for all $y\in [-1,0)$ we have $(X_t^y)^1\leq -\delta t^\gamma $ .

We now want to pass to the $x\to 0$ limit, which we can do by noticing that the laws of $(B,\tilde \tau ,X^x,X^{-x})$ are tight on the space

$$ \begin{align*} \mathcal{S}= C([0,1])\times\{(a,g):\,a\in(0,1],g\in C([0,a])^2\} \end{align*} $$

with the metric

$$ \begin{align*} d\big((f,a,g),(f',a',g')\big)=\|f-f'\|_{C([0,1])}+|a-a'|+\|g-g'\|_{C([0,a\wedge a'])^2}. \end{align*} $$

By Prokhorov’s theorem and Skorohod’s representation, we get a sequence $x_n\to 0$ , and on another probability space, a sequence $(\bar B^{x_n},\bar {\tilde \tau }^{x_n},\bar X^{x_n},\bar X^{-x_n})\overset {\mathrm {law}}{=}(B,\tilde \tau ,X^{x_n},X^{-x_n})$ converging $\mathbb {P}$ -almost surely as random variables taking values $\mathcal {S}$ . The limits $X^{0,+}:=\lim \bar X^{x_n}$ and $X^{0,-}:=\lim \bar X^{-x_n}$ both solve (1.15) with initial condition $0$ and driving noise $B^{0}:=\lim \bar B^{x_n}$ . Moreover, $X^{0,+}_t\geq \delta t^\gamma $ for $t\leq \tilde \tau ^0:=\lim \bar {\tilde \tau }^{x_n}$ and $X^{0,-}_t\leq -\delta t^\gamma $ for $t\leq \tilde \tau ^0$ . Since $\tilde \tau ^0\overset {\mathrm {law}}{=}\tilde \tau $ , it is $\mathbb {P}$ -a.s. positive, and therefore, the laws of $X^{0,+}$ and $X^{0,-}$ are mutually singular (for example, on $C([0,1])$ after extending them as constants after $\tilde \tau ^0$ ).

Remark 1.11. Up to multiplying b by a cutoff function at infinity, by taking $\alpha =-d/(p+\varepsilon )$ for sufficiently small $\varepsilon>0$ , the construction presented in the regime $\alpha <0$ provides non-uniqueness for $b\in L^q_tL^p_x$ , for any pair $(p,q)\in [1,\infty ]^2$ satisfying

(1.18) $$ \begin{align} \frac{1}{q}+\frac{Hd}{p}>1-H,\qquad p>d. \end{align} $$

If $H=1/2$ , then B can be taken as Brownian motion and (1.18) becomes

(1.19) $$ \begin{align} \frac{2}{q}+\frac{d}{p}>1,\qquad p>d; \end{align} $$

in particular, the exponents $p,q$ violate the LPS condition (1.9). It is interesting to compare (1.19) to the result from [Reference Krylov64], where weak existence for the Brownian SDE was established under the condition

(1.20) $$ \begin{align} \frac{1}{q}+\frac{d}{p}\leq 1, \end{align} $$

which is further shown to be optimal by construction of counterexamples in the case $1/q+d/p>1$ . Let us also mention [Reference Galeati47] for a heuristic explanation on why condition (1.20) (as well as (1.21) below) arises naturally when only focusing on weak existence results. Our counterexample shows that under (1.20), uniqueness in law in general does not hold, answering a problem left open in [Reference Krylov64] (see the discussion right above Remark 3.1 therein).

After the completion of this work, it has been further shown in [Reference Butkovsky, Lê and Mytnik16] that in the time-independent case, for $H\in (0,1)$ , there exist $b\in C^\alpha $ with supercritical $\alpha <1-1/H$ for which even weak existence does not hold; see Theorem 2.7 therein. More recently, [Reference Butkovsky and Gallay14] expanded the result from [Reference Krylov64] by establishing weak existence of solutions for $H\in (0,1)$ and $b\in L^q_t L^p_x$ with

(1.21) $$ \begin{align} \frac{1-H}{q}+ \frac{H d}{p} < 1-H. \end{align} $$

Combined with our counterexample, one gets a regime (namely, the intersection of (1.18) and (1.21)) where weak existence holds but uniqueness in law does not.

1.4 Preliminaries on fractional Brownian motion

We recall here several facts about fractional Brownian motion (fBm); for some standard references, we refer to [Reference Nualart80, Reference Picard87].

An $\mathbb {R}^d$ -valued fBm of Hurst parameter H is defined as the unique centered Gaussian process with covariance

$$ \begin{align*} \mathbb{E}(B^H_t\otimes B^H_s)=\tfrac{1}{2}\big(|t|^{2H}+|s|^{2H}-|t-s|^{2H}\big) I_d, \end{align*} $$

where $I_d$ denotes the $d\times d$ identity matrix; in other words, its components are i.i.d. one-dimensional fBms. FBm paths are well-known to be $\mathbb {P}$ -a.s. $(H-\varepsilon )$ -Hölder, but nowhere H-Hölder continuous. FBm admits several representations as a stochastic integral; in particular, given any fBm $B^H$ defined on a probability space, one can construct therein a standard Bm W such that

(1.22) $$ \begin{align} B^H_t=\int_0^t K_H(t,r)\mathrm{d} W_r\quad\forall\, t \geq 0. \end{align} $$

Such Volterra kernel representation is referred as canonical since $B^H$ and W generate the same filtration. The exact formula for the kernels $K_H$ can be found in, for example, [Reference Nualart and Ouknine81]. For our purposes, it is enough to recall that $K_H$ is deterministic and $K_H(t,\cdot )\in L^2([0,t])$ .

Another standard representation of fBm is the one introduced in [Reference Mandelbrot and van Ness74]: given $B^H$ , one can construct a two-sided Bm $\tilde W$ such that

(1.23) $$ \begin{align} B^H_t = \gamma_H \int_{-\infty}^t \big[(t-r)_+^{H-1/2}-(-r)_+^{H-1/2}\big]\, \mathrm{d} \tilde W_r, \end{align} $$

where $\gamma _H=\Gamma (H+1/2)^{-1}$ is a normalizing constant and $x_+$ denotes the positive part.

We will mostly work with representation (1.22), but we invite the reader to keep in mind (1.23) since it is usually easier to manipulate in order to derive key properties of the process, like its local nondeterminism; see (1.24) and the discussion below. Given a filtration $\mathbb {F}$ , we say that $B^H$ is a $\mathbb {F}$ -fBM if the associated W given by (1.22) is a $\mathbb {F}$ -Brownian motion.

FBm of parameter $H=1$ is somewhat trivial or ill-defined (see [Reference Picard87]); however, one can extend the definition to all values $H\in (0,+\infty )\setminus \mathbb {N}$ inductively as in [Reference Perrin, Harba, Berzin-Joseph, Iribarren and Bonami86] by $B^{H+1}_t:=\int _0^t B^H_s\mathrm {d} s$ .

Such definition is consistent with most aforementioned properties: it is still a centered, Gaussian process, with trajectories $\mathbb {P}$ -a.s. in $C^{H-\varepsilon }_t$ but nowhere $C^H_t$ , satisfying the scaling relation (1.5); using stochastic Fubini, one can also easily derive similar representations as (1.22)–(1.23). A key consequence of the last property is that for any $H\in (0,+\infty )\setminus \mathbb {N}$ , there exists a constant $c_H\in (0, +\infty )$ such that

(1.24) $$ \begin{align} \mathrm{Cov} \big(B^H_t - \mathbb{E}_s B^H_t \big) = c_H |t-s|^{2H} I_d \quad\forall \, s\leq t \end{align} $$

(see [Reference Gerencsér53, Proposition 2.1]); here, $\mathbb {E}_s B^H_t:=\mathbb {E}[B^H_t|\mathcal {F}_s]$ , where $\mathcal {F}_s$ can be the natural filtration of $B^H$ or more generally any filtration such that $B^H$ is a $\mathbb {F}$ -fBm. Property (1.24) is a special form of strong local nondeterminism (LND)Footnote ⁴ ; see [Reference Galeati and Gubinelli48, Section 2.4] for a deeper discussion on its relevance on regularisation by noise. Since conditional expectations are also $L^2$ -projections, $B^H_t-\mathbb {E}_s B^H_t$ and $\mathbb {E}_s B^H_t$ are orthogonal Gaussian variables, and thus independent; more generally, $B^H_t-\mathbb {E}_sB^H_t$ is independent of all the history up to time s. Therefore, for any $s\leq t$ , any bounded measurable function $f:\mathbb {R}^d\to \mathbb {R}$ and any other $\mathcal {F}_s$ -measurable random variable X, it holds

(1.25) $$ \begin{align} \mathbb{E}_s f(B^H_t+X)= P_{ \mathrm{Cov}(B^H_t - \mathbb{E}_s B^H_t)} f(\mathbb{E}_s B^H_t+X) = P_{c_H |t-s|^{2H} I_d} f(\mathbb{E}_s B^H_t+X), \end{align} $$

where in the last passage, we applied (1.24); here, given a symmetric nonnegative $\Sigma $ , $P_\Sigma $ denotes the convolution with the Gaussian density $p_\Sigma $ associated to $\mathcal {N}(0,\Sigma )$ . Throughout the paper, we will adopt the convention that $P_{t I_d}=P_t$ , in agreement with the standard notation for heat kernels, and for simplicity, we will drop the constant $c_H$ , so that in expressions like (1.25), only $P_{|t-s|^{2H}}$ will appear.

Remark 1.12. At the price of slightly anticipating some key concepts which will be introduced throughout the paper, let us discuss here how our methods extend to a larger class of random perturbations $B^H$ than just pure fBm. The main requirement we need, relaxing (1.24), is for $B^H$ to be a Gaussian processFootnote ⁵ satisfying a two-sided bound

(1.26) $$ \begin{align} C^{-1} |t-s|^{2H} I_d \leq \mathrm{Cov} \big( B^H_t-\mathbb{E}_s B^H_t \big)\leq C |t-s|^{2H} I_d \end{align} $$

for some $C\in (0,+\infty )$ and for all $s<t$ with $|t-s|$ sufficiently small; here, $\mathcal {F}_t$ is the natural filtration of $B^H$ . More precisely, the upper bound in (1.26) provides a priori estimates in the style of Lemma 2.1, while the lower bound (which is the actual LND property) ensures the regularising effect of $B^H$ and the application of stochastic sewing techniques. Indeed, by using properties of Gaussian convolutions, heat kernel bounds and a relation of the form (1.25), one can still find estimates of the form

$$ \begin{align*} \| \mathbb{E}_s f(B^H_t+X)\|_{L^\infty} & = \| \big( P_{ \mathrm{Cov} (B^H_t-\mathbb{E}_s B^H_t)} f \big) (\mathbb{E}_s B^H_t+X)\|_{L^\infty} \leq \| P_{ \mathrm{Cov} (B^H_t-\mathbb{E}_s B^H_t)} f\|_{L^\infty}\\ & \lesssim \| P_{C^{-1}|t-s|^{2H}} f\|_{L^\infty} \lesssim |t-s|^{\alpha H} \| f\|_{C^\alpha}, \end{align*} $$

for $\alpha \leq 0$ , which are the typical bounds needed throughout the proof. There are some passages where condition (1.26) alone is not enough, and we exploited other properties of fBm. Specifically, the counterexamples in Section 1.3 assume $B^H$ to be $(H-\varepsilon )$ -Hölder continuous and symmetric; the flows constructed in Sections 4–5 need some basic time-continuity $\mathbb {E}|B^H_t-B^H_s|\lesssim |t-s|^{H\wedge 1}$ in order to apply Kolmogorov-type criteria; more substantially, the results from Section 8 rely on a Volterra representation $B^H_t =\int _0^t K(t,s) \mathrm {d} W_s$ . These properties are satisfied by other interesting examples – for instance, type-II fBm and mixed fBm discussed in Remark 1.13 below.

The only section truly specific to fBm is Appendix C, which however, exactly for this reason, is not used throughout the main body of the paper. In this case, ad hoc criteria to check Girsanov transform for fBm are presented; any extension to other processes would require precise knowledge of the associated kernel $K(t,s)$ , and its verification can be very technical; cf. [Reference Nualart and Sönmez82].

1.5 Setup and notation

We provide here in a list all the main notations and conventions adopted throughout the paper.

• • We always work on the time interval $t\in [0,1]$ . Increments of functions f on $[0,1]$ are denoted by $f_{s,t}:=f_t-f_s$ .

• • Whenever considering a filtered probability space $(\Omega ,\mathcal {F},\mathbb {F},\mathbb {P})$ , we will implicitly assume that the filtration $\mathbb {F}=(\mathcal {F}_t)_{t\in [0,1]}$ satisfies the standard assumptions; in particular, $\mathcal {F}_0$ is complete. To denote conditional expectations, we use the shortcut notation $\mathbb {E}_s Y :=\mathbb {E} [Y| \mathcal {F}_s]$ .

• • $L^m$ -norms without further notation are understood with respect to $\omega $ ; that is, $\|Y\|_{L^m}=\big (\mathbb {E}|Y|^m\big )^{1/m}$ for $m<\infty $ and $\|Y\|_{L^\infty }=\mathrm {esssup}_{\omega \in \Omega }|Y(\omega )|$ . For conditional $L^m$ -norms, we use the notation $\|Y\|_{L^m|\mathcal {F}_s}=\big (\mathbb {E}(|Y|^m|\mathcal {F}_s)\big )^{1/m}.$ For any $X,Y\in L^m$ such that Y is $\mathcal {F}_s$ -measurable, by conditional Jensen’s inequality, one has the $\mathbb {P}$ -a.s. bound

  (1.27) $$ \begin{align} \|X-\mathbb{E}_s X\|_{L^m|\mathcal{F}_s} \leq \|X-Y\|_{L^m|\mathcal{F}_s} +\|Y-\mathbb{E}_s X\|_{L^m|\mathcal{F}_s} \leq 2\|X-Y\|_{L^m|\mathcal{F}_s}. \end{align} $$
  Apart from the usual $L^m$ -norms, we also use the norms $\big \|\,\|\cdot \|_{L^m|\mathcal {F}_s}\big \|_{L^n}$ . We will always consider $n\geq m$ , in which case again by conditional Jensen, it holds
  $$ \begin{align*} \| X\|_{L^m} \leq \big\|\,\| X \|_{L^m|\mathcal{F}_s}\big\|_{L^n} \end{align*} $$
  with equality in the case $m=n$ . Such mixed norms still satisfy natural analogues of classical inequalities like Jensen’s, Hölder’s and Minkowski’s, as can be verified using properties of conditional expectation. Moreover, by the tower property, one can see that for $t\geq s$ , $\big \|\,\|\cdot \|_{L^m|\mathcal {F}_t}\big \|_{L^n}$ is stronger than $\big \|\,\|\cdot \|_{L^m|\mathcal {F}_s}\big \|_{L^n}$ .

• • Whenever talking about a weak solution X to the SDE (1.6), we will actually mean a tuple $(X,B^H; \Omega , \mathbb {F}, \mathbb {P})$ such that $(\Omega ,\mathbb {F},\mathbb {P})$ is a filtered probability space as above, X is $\mathbb {F}$ -adapted and $B^H$ is a $\mathbb {F}$ -fBm of parameter H. As usual, X is a strong solution if it is adapted to the (standard augmentation of) the filtration generated by $B^H$ . We say that pathwise uniqueness holds for the SDE if for any two solutions $X^1$ , $X^2$ , defined on the same $(\Omega ,\mathbb {F},\mathbb {P})$ , driven by the same $B^H$ and with same initial condition $x_0$ , it holds $X^1\equiv X^2 \mathbb {P}$ -a.s. We warn the reader to keep in mind that all such concepts are rather classical when b is at least a measurable function, so that (1.6) is meaningful in the Lebesgue sense. In the distributional regime $\alpha <0$ , this is not the case anymore. Therefore, the concept of weak solution becomes less standard. We postpone this discussion to the relevant Section 5, similarly for the concept of path-by-path uniqueness.

• • Function spaces in the variable $x\in \mathbb {R}^d$ will often be denoted by the subscript x. For instance, standard Lebesgue spaces $L^p(\mathbb {R}^d;\mathbb {R}^m)$ with $p\in [1,\infty ]$ will often be denoted, when the target dimension m is clear, simply by $L^p_x$ . For $\alpha \in \mathbb {R}\setminus \mathbb {N}$ , we denote by $C^\alpha _x$ the inhomogeneous Hölder-Besov space $B^{\alpha }_{\infty ,\infty }$ (cf. [Reference Bahouri, Chemin and Danchin5]); instead, for nonnegative integer $\alpha $ , by $C^\alpha _x$ , we mean the space of bounded measurable functions whose all partial weak derivatives up to order $\alpha $ are also essentially bounded and measurable (in other words, $C^\alpha _x=W^{\alpha ,\infty }_x$ Sobolev spaces); note that with this convention, elements of $C^0_x$ are not necessarily continuous. Recall that for $\alpha \in (0,1)$ , the space $C^\alpha _x=B^\alpha _{\infty ,\infty }$ coincides with the usual space of bounded $\alpha $ -Hölder continuous functions. By $C^{\alpha ,\mathrm {loc}}_x$ , we mean the space of functions f such that for all compactly supported smooth g, one has $f g\in C^\alpha _x$ . More quantitative versions of them are the weighted Hölder spaces $C^{\alpha ,\lambda }_x$ , for $\alpha \in (0,1]$ and $\lambda \in \mathbb {R}$ , defined through the (semi)norms

  
  where $B_R$ is the ball of radius R around the origin.

• • Given a Banach space E, we will use the shortcut notation $L^q_t E$ to denote the space $L^q([0,1];E)$ of Bochner measurable function with finite norm $\| f\|_{L^q E}^q=\int _0^1 \| f_t\|_E^q\, \mathrm {d} t$ for any $q\in [1,\infty ]$ (up to the standard essential supremum convention for $q=\infty $ ). We use the shortcut notation $C_t E = C([0,1];E)$ for the space of continuous, E-valued functions with supremum norm; similarly, for $\gamma \in (0,1)$ , $C^\gamma _t E= C^\gamma ([0,1];E)$ is the space of E-valued, bounded and $\gamma $ -Hölder continuous functions. All definitions can be extended classically to Fréchet spaces E (in particular, allowing for $E=C^{\alpha ,\mathrm {loc}}_x$ or $L^{p,\mathrm {loc}}_x$ ) – for instance, in the the case of $L^q_t E$ by requiring the associated countable seminorms $t\mapsto \| f_t \|_k$ to be all $L^q$ -integrable.

• • Given a metric space E and $p\in [1,\infty )$ , we say that a continuous E-valued function f on $[0,1]$ is of finite p-variation, in notation $f\in C^{p-{\mathrm {var}}}_t E$ , if

  
  where the supremum runs over all possible partitions $0=t_0\leq t_1\leq \cdots \leq t_n=1$ of $[0,1]$ . The p-variation seminorm on subintervals $[s,t]\subset [0,1]$ is defined similarly and denoted by . Whenever $E=\mathbb {R}^m$ for some $m\in \mathbb {N}$ , for simplicity, we just drop it and write $C^{p-{\mathrm {var}}}_t$ , , and similarly for $C^\alpha _t$ .

• • All the notations introduced above can be concatenated by considering a different Banach/Fréchet space at each step. The convention we adopt is that, when writing spaces with respect to different variables, this is to be read from left to right; for example, $L^q_t C^\alpha _x L^m$ stands for $L^q\big ([0,1], C^\alpha (\mathbb {R}^d,L^m(\Omega ))\big )$ . Similarly, one can define, for example, $L^m C^{p-{\mathrm {var}}}_t C^{\alpha ,\mathrm {loc}}_x$ , $C^\gamma _t L^\infty _x$ , and so on. Mind in particular that with this convention $C^\alpha _t C^\alpha _x\neq C^\alpha _{t,x}$ , the latter denoting the space of $\alpha $ -Hölder continuous functions in $(t,x)$ .

• • Let us recall some standard heat kernel estimates: for any $\alpha \geq \beta $ , there exists a constant $N=N(d,\alpha ,\beta )$ such that, for all $t\in (0,1]$ , one has the bound

  (1.28) $$ \begin{align} \|P_{t}f\|_{C^\alpha_x}\leq N t^{(\beta-\alpha)/2}\|f\|_{C^\beta_x}; \end{align} $$
  see [Reference Galeati and Gubinelli49, Lemma A.10] and the references therein for a more general statement.

• • For $0\leq S\leq T\leq 1$ , we denote $[S,T]^2_\leq =\{(s,t)\in [S,T]^2:\,s\leq t\}$ and $[S,T]^3_\leq =\{(s,u,t)\in [S,T]^3:\,s\leq u\leq t\}$ . For $(s,t)\in [S,T]^2_\leq $ , define $s_-=s-(t-s)$ . We then set the slightly more restricted sets of pairs/triples as

  $$ \begin{align*} & \overline{[S,T]}^2_\leq:=\{(s,t)\in[S,T]^2_\leq:\,s_-\geq S\},\\ & \overline{[S,T]}^3_\leq=\{(s,u,t)\in[S,T]^3_\leq:\,(u-s)\wedge(t-u)\geq (t-s)/3,\,s_-\geq S\}. \end{align*} $$

• • Given a Frechét space E and a map $A:[S,T]^2_\leq \to E$ , we define $\delta A:[S,T]^3_\leq \to E$ by $\delta A_{s,u,t} = A_{s,t}- A_{s,u}- A_{u,t}$ .

• • We say that a function $w:[0,1]^2_\leq \to \mathbb {R}_+ $ is a control if it is continuous and superadditive (i.e., $w(s,u)+w(u,t)\leq w(s,t)$ for all $(s,u,t)\in [S,T]^3_\leq $ ). The most common controls for us will be of the form

  (1.29) $$ \begin{align} w_{b,\alpha,q}(s,t):=\int_s^t\|b_r\|_{C^\alpha_x}^q\,dr. \end{align} $$
  Recall that for any two controls $w_1,w_2$ and $\theta _1,\theta _2\in [0,\infty )$ such that $\theta _1+\theta _2\geq 1$ , $w=w_1^{\theta _1}w_2^{\theta _2}$ is also a control (see [Reference Friz and Victoir45, Exercises 1.8,1.9]). Note also that if w is a control, $\psi $ is an $\mathbb {R}^m$ -valued path and $\gamma \in (0,1]$ , then
  (1.30) $$ \begin{align} \|\psi\|_{\frac{1}{\gamma}-{\mathrm{var}}}\leq w(0,1)^\gamma\sup_{0\leq s<t\leq 1}\frac{|\psi_{s,t}|}{w(s,t)^\gamma}; \end{align} $$
  conversely, for $p\geq 1$ , if $\psi \in C^{p-{\mathrm {var}}}_t$ , then is a control and $|\psi _{s,t}|\leq w(s,t)^{1/p}$ ; cf. [Reference Friz and Victoir45, Propositions 5.8-5.10].

• • The space of probability measures on $\mathbb {R}^d$ is denoted by $\mathcal {P}(\mathbb {R}^d)$ . The law of a random variable X is denoted by $\mathcal {L}(X)$ . For $p\geq 1$ , we denote the p-Wasserstein distance on $\mathcal {P}(\mathbb {R}^d)$ by $\mathbb {W}_p$ , defined as

  $$ \begin{align*} \mathbb{W}_p(\mu,\nu)^p=\inf_{\gamma\in \Gamma(\mu,\nu)}\int_{\mathbb{R}^{d}\times\mathbb{R}^d}|x-y|^p\gamma(\mathrm{d} x,\mathrm{d} y), \end{align*} $$
  where $\Gamma (\mu ,\nu )$ is the set of all couplings of $\mu $ and $\nu $ (i.e., the probability measures on $\mathbb {R}^{d}\times \mathbb {R}^d$ whose first and second marginals are $\mu $ and $\nu $ , respectively). Note that $\mathbb {W}_p$ can take value $+\infty $ and is defined for any $\mu $ , $\nu $ , without any moment assumption.

• • When a statement contains an estimate with a constant depending on a certain set of parameters, in the proof, we do not carry the constants from line to line. Rather, we write $A\lesssim B$ to denote the existence of a constant N depending on the same set of parameters such that $A\leq N B$ . Whenever such a set of parameters includes a parameter that is a norm (this will typically be the norm of the coefficient b), this dependence is always monotone increasing.