We consider stochastic differential equations (SDEs) driven by a fractional Brownian motion with a drift coefficient that is allowed to be arbitrarily close to criticality in a scaling sense. We develop a comprehensive solution theory that includes strong existence, path-by-path uniqueness, existence of a solution flow of diffeomorphisms, Malliavin differentiability and $\rho$-irregularity.
As a consequence, we can also treat McKean-Vlasov, transport and continuity equations.

§ INTRODUCTION

Given a vector field $b:\R_+\times\R^d\to\R^d$, an initial condition $x_0\in\R^d$, and a function $f:\R_+\to\R^d$, consider the differential equation

\begin{equation}\label{eq:ODE}
X_t = x_0 + \int_0^t b_r ( X_r) \mathd r + f_t.
\end{equation}

When $f$ is chosen according to some random distribution, one obtains a stochastic differential equation (SDE), which often exhibits much better properties than the unperturbed equation ($f\equiv 0$), even at the level of existence and uniqueness of solutions.
This phenomenon is often referred to as regularisation by noise and its study goes back to the works of Zvonkin [94] and Veterennikov [87], see the monograph [38] for a survey in the case of standard  Brownian $f$.

Although there is plenty of evidence [31, 19, 54, 47] that it is the pathwise properties of the perturbation that determine the regularisation effects, the available results are far more abundant in the Brownian, and in general, the Markovian case.

However, a wide variety of applications motivate models with anomalous diffusions with long-range memory, including statistical description of turbulence [56], hydrology [55], anomalous polymer dynamics [74], diffusion in living cells [84], rough volatility models in finance [50].
Such non-Markovian processes are commonly modeled by fractional Brownian motion (fBm).
In this case the lack of Markovian and semimartingale structure renders a large part of a “standard” toolbox (Itô's formula, Kolmogorov equations, Zvonkin transformation, martingale problem) unavailable.
Nevertheless, since fBm paths share many properties with the standard Brownian ones (up to changes in the scaling exponents), one would expect similar regularisation phenomena.

The goal of the present work is twofold.
First, we provide the first well-posedness results in the case of non-Markovian noise under demonstrably sharp conditions on $b$. The optimality follows both from a scaling heuristic (see Section <ref> below) and from rigorous construction of counterexamples (see Section <ref> below).
The second goal is to expand the existing well-posedness theory by studying various properties of the solutions that are well-known (though often nontrivial) in the Brownian case, but much less so for fractional noise.
These include existence, regularity, invertibility of the solution flow, stability with respect to perturbations of the initial condition and/or the nonlinearity, and Malliavin differentiability. The proofs can also be of interest in cases where the results are not new: the methods presented here go beyond not only the Markovian framework, but also the scope of Girsanov's theorem (see Remark <ref> and Appendix <ref>).

At the same time, the idea is quite intuitive: in order to develop a strong solution theory for (<ref>), it is natural to investigate first the solvability of the linearised equation around any given solution $X$, namely to show that

\begin{equation}\label{eq:ODE-linearised}
Y_t= y + \int_0^t \nabla b_r(X_r) Y_r\, \dd r
\end{equation}

has a well-defined, unique solution for any $y\in \R^d$; observe that, due to its additive nature, the perturbation $f$ does not appear in (<ref>). The study of (<ref>) is perfectly in line with the classical setting of a continuously differentiable drift $b$, where (<ref>) can be solved directly and its behaviour matches the Grönwall-type estimates encountered when looking at the difference of any two solutions.
However if $b$ is not assumed to be differentiable, $Db$ a priori doesn't make sense and thus a standard interpretation for (<ref>) is no longer possible. The key idea in order to overcome this difficulty is two-fold:

a) $Db$ in (<ref>) is not evaluated at arbitrary space points, rather along the solution $X$, which can have very special properties inherited from the noise $f$.

b) In order to give meaning to (<ref>) in a Young integral sense, we don't need to define $Db_r(X_r)$ pointwise, instead it suffices to show that the path

\begin{equation}\label{eq:intro-additive}
t\mapsto L_t:=\int_0^t Db_r(X_r) \dd r
\end{equation}

is well-defined and enjoys sufficiently nice time regularity (more precisely, it is of finite $p$-variation for some $p<2$).  In view of a), depending on the structure of the noise $f$, this can be a much more reasonable requirement.

In analogy with the Lipschitz setting, one can then transfer estimates for classical linear Young equations of the form

sup_t∈[0,1] |Y_t|≲e^CL_p-^p |y|

to pathwise bounds for the difference of any two solutions $X$ and $\tilde X$ with different initial conditions, up to replacing $L$ by another process $\hat L=\hat L(X,\tilde X)$ similar in spirit to (<ref>).

In order to rigorously formalise all of the above, it is crucial to identify the correct space of perturbations $\varphi$ such that $X=\varphi+f$ indeed inherits the relevant properties from $f$; these are the a priori estimates given by Lemmas <ref>-<ref>. Correspondingly, we formulate two new versions of the
Stochastic Sewing Lemma (SSL) by Lê [62], cf. Lemmas <ref> and <ref> below, which are tailor-made for our analysis.
Once this setup is in place, it provides exponential moment estimates of certain additive functionals of $X$, like the one defined in (<ref>), turning pathwise bounds like (<ref>) into moment bounds. Finally, once the behaviour of the linearised equation (<ref>) is understood, many further properties (uniqueness, stability, differentiability of the flow) of the ODE follow similarly.

§.§ 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 <cit.>.

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

(B^H_t)_t≥0law=(λ^-HB^H_λt)_t≥0,   ∀  λ>0.

Details about the processes $B^H$ are given in Section <ref> below; let us just briefly recall that $H=1/2$ gives the standard Brownian motion, that this is the only case where $B^H$ is a Markov process.
For the values $H=k+1/2$, $k\in\N_+$ (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 e.g. [23].

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

\begin{equation}\label{eq:SDE}
X_t = x_0 + \int_0^t b_r ( X_r) \mathd r + B^H_t.
\end{equation}

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 weren't 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 (<ref>) in mind, for a fixed parameter $H$ we call a space $V$ of functions (or distributions) on $\R_+\times\R^d$ critical (resp. subcritical/supercritical) if for the rescaled drift coefficient

b^λ_t(x)=λ^1-H b(λt, λ^H x),

the leading order seminorm $\llbracket b^\lambda\rrbracket_V$ (see the examples below for its practical meaning) scales like $\lambda^\gamma \llbracket b\rrbracket_V$, for all $\lambda\leq 1$,[

Specifically, we are interested in understanding how $\llbracket b^\lambda\rrbracket_V$ scales as $\lambda\to 0$, which is related to studying the local behaviour of solutions; instead the scaling of $\llbracket b^\lambda\rrbracket_V$ as $\lambda\to\infty$ reflects a “zoom out” which identifies the dominant term concerning the long-time dynamics.] with $\gamma=0$ (resp. $\gamma>0$/$\gamma<0$).

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

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 $\llbracket f\rrbracket_V:=\| f\|_{\dot{B}^\alpha_{\infty,\infty}}$ as defined in [5]; 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{equation*}
\| f(\eta\, \cdot)\|_{\dot{B}^\alpha_{\infty,\infty}} \sim_\alpha \eta^\alpha \| f(\cdot)\|_{\dot{B}^\alpha_{\infty,\infty}}  \quad\, \forall\, (\eta,\alpha)\in \R_{>0}\times \R.
\end{equation*}

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

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 $\alpha>-1/2$ [40], and a nonstandard kind of well-posedness (where uniqueness is even weaker than uniqueness in law) is shown for $\alpha>-2/3$ [34, 17], for special classes of drift $b$.
The classical works [94, 87] show strong well-posedness for $V=C^\alpha_x$ and $\alpha\geq 0$[Please see our convention on the definition of $C^\alpha_x$ from Section <ref> below, especially for $\alpha\in\N$; in particular, $C^0_x$ is understood as the space bounded and measurable functions, with $L^\infty$-norm.].

Interestingly, in the degenerate Brownian case weak well-posedness is proved in [23] in the full regime $\alpha>(2k-1)/(2k+1)$, which is precisely the condition (<ref>).
For strong well-posedness one requires the more restrictive condition

see <cit.>.
The same condition is required for strong well-posedness in the non-Markovian case for all $H\in(0,\infty)\setminus\N$, cf. [72, 19, 47, 51]. After the first version of this manuscript, the work [16] appeared, where the authors are able to establish (among several results) weak existence of solutions in the full subcritical regime (<ref>), under the additional assumption that $b$ is a Radon measure; however, uniqueness is still open.

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

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

\begin{equation}\label{eq:LPS}
\frac{2}{q}+\frac{d}{p}<1
\end{equation}

which is precisely the condition from the classical work [59], where strong well-posedness is proved (under the additional constrant $p\geq 2$); instead the critical regime corresponds to the celebrated Prodi-Serrin-Ladyzhenskaya (LPS) condition.
This case has then been extensively studied by several authors, allowing also for multiplicative noise with Sobolev diffusion coefficients, see among others [91, 36, 92, 89].
In recent years, even the critical case has been reached [58, 79] under certain constraints on $d,p,q$; let us also mention the recent work [93], which goes beyond condition (<ref>), up to additional constraints on ${\rm div}\, b$.

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 [62], under the stronger conditions

1/q+Hd/p<1/2,    1/q+Hd/p<1/2-H,

respectively, with the additional constraint $p\in[2,\infty]$, later removed in [47].
It is conjectured  in [62] that the first condition in (<ref>) is enough to guarantee strong well-posedness. One particular corollary of our result is that for $q\in(1,2]$ even (<ref>) is sufficient.
Therefore we propose to update the conjecture of [62] (if $q\in(1,2]$, now a theorem) to assert strong well-posedness under the scaling condition (<ref>). Let us also mention that we have recently learned about an ongoing work [15] towards improving (<ref>).

A common generalisation of Examples <ref> and <ref> 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 cases[By Besov embeddings $L^p_x \hookrightarrow B^{-d/p}_{\infty,\infty}$, with homogeneous norms behaving in the same way under rescaling.]) the scaling works out to be $\gamma=1-H-1/q+\alpha H$.
Therefore the subcriticality condition reads as

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 [47, 49], where strong well-posedness is proved under the stronger condition

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

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 $H=k+1/2$, $k\in\N$, and strong well-posedness only in the standard Brownian case $H=1/2$.

§.§ Discussion of the main results

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

H∈(0,∞) ∖ℕ,    q∈(1,2],    α∈(1-1/q'H,1).A

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 (<ref>) 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 [72, 19, 51], 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 <ref>, <ref>, <ref>, <ref> for i), <ref> for ii), <ref> for iii), <ref>  for iv), <ref> for v), <ref> for vi), <ref> for vii).
For simplicity, we restrict ourselves to the time interval $t\in [0,1]$, but it's clear that up to rescaling we could consider any finite $[0,T]$ (up to allowing the hidden constants to depend on $T$).

Assume (<ref>) and let $x_0\in\R^d$, $b\in L^q_t C^\alpha_x$, $m\in[1,\infty)$. Then:

* Strong existence and path-by-path uniqueness holds for (<ref>);

* For any other $\tilde x_0\in\R^d$ and $\tilde b\in L^q_t C^\alpha_x$, the associated solutions $X$ and $\tilde X$ satisfy the stability estimate

b̃_L^q_t C^α-1_x;

* The solutions form a stochastic flow of diffeomorphisms $\Phi_{s\to t}(x)$, whose spatial gradient $\nabla \Phi$ is $\PP$-a.s. continuous in all variables; moreover it holds

\begin{equation*}
\sup_{0\leq s\leq t\leq 1, x\in \R^d} \E\big[| \nabla \Phi_{s\to t} (x) |^m\big] <\infty;
\end{equation*}

* For each $s<t$ and $x\in \R^d$, the random variable $\omega\mapsto \Phi_{s\to t}(x;\omega)$ is Malliavin differentiable; moreover it holds

\begin{equation*}
\sup_{0\leq s\leq t\leq 1, x\in \R^d} \E\big[ \| D \Phi_{s\to t}(x)\|_{\mathcal{H}^H}^m \big]<\infty,
\end{equation*}

where $D$ is the Malliavin derivative and $\cH^H$ the Cameron-Martin space of $B^H$;

* Strong existence and uniqueness holds also for the McKean-Vlasov equation

\begin{equation}\label{eq:MKV}
X_t=x_0+\int_0^t(b_r\ast\mu_r)(X_r)\dd r +B^H_t,\qquad\mu_t=\cL(X_t);
\end{equation}

* Solutions $X$ are $\PP$-a.s. $\rho$-irregular for any $\rho<1/(2H)$;

* If additionally $\alpha>0$, then for any $p>1$ strong existence and path-by-path uniqueness holds for solutions $u\in L^\infty_t W^{1,p}_x$ to the transport equation

∂_t u + b·∇u + Ḃ^H_t·∇u=0

for all initial data $u_0\in W^{1,p}_x$.

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 [31], and later popularised by [82, 19].
Such uniqueness concept requires the give a pathwise interpretation to the SDE, which becomes non-trivial for $\alpha<0$, where $b$ can be a distribution of negative regularity and not a function anymore. In this case, following [19], we will give meaning to (<ref>) as a nonlinear Young ODE, see Section <ref> 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 e.g. the recent work [52].

In this paper, stability estimates play a key role when solving McKean-Vlasov equations as in v), see Section <ref>.
The study of stochastic flows iii) for SDEs goes back to the classical work [60], see also [36, 25, 67] for flows in irregular settings.
In iv), we can in fact derive differentiability with respect to perturbations of the noise in quite a bit more general than Cameron-Martin directions (see Remark <ref>), in line with the observations from [61, 41].
Concerning v), regularisation by fractional noise for distribution dependent SDEs has been investigated in [49] and recently in [53].
Above we only stated the simplest example of McKean-Vlasov equation for the sake of presentation, Theorem <ref> below allows for more general dependence on $(X,\mu)$.
The notion of $\rho$-irregularity in vi) was introduced by [19] 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 [39] and further developed in [69, 37], see also [20, 70, 47] for further investigations in the fractional case.

Finally, let us mention that the scope of some intermediate estimates are larger than (<ref>), and therefore in some regime where we do not obtain strong well-posedness, we still obtain compactness and therefore existence of weak solutions. This is the content of Section <ref>.

One fundamental stochastic analytic tool that still applies in the non-Markovian fBm setting is Girsanov's transform. Indeed, it is heavily used in the seminal works [72, 19] and many subsequent ones.
However, it has its limitations: in our setting it only applies under the additional assumption $1-1/(q'H)<0$ (which in turn may only happen if $H\in(0,1/2)$), see Appendix <ref> for details.

Even in the Brownian case $H=1/2$ our methods yield results beyond the scope of Girsanov's theorem, which is not available for $q<2$, see Remark <ref> below.
Therefore, throughout the article we avoid Girsanov's transform altogether.

Another motivation for a Girsanov-free approach is to develop tools that are robust enough to extend to other classes of process; see [13] for some first results on such equations via stochastic sewing for Lévy-driven SDEs and Remarks <ref>-<ref> below for other classes of Gaussian processes which fit our framework.

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

∂_t u-12Δu=b·∇u.

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 [24]. After this manuscript appeared,  Schauder estimates for (<ref>) with $b\in L^q_t C^\alpha_x$ with $q\in (1,2)$ were developed in [88].

By the embedding $L^p_x\subset C^{-d/p}_x$ our result immediately implies well-posedness of (<ref>) with $L^q_t L^p_x$ drift in the full subcritical regime (with respect to $p$) (<ref>) if $q\in(1,2]$, which can be seen as a fractional analogue of [59].
Note that unlike in [62], $p\in[1,2)$ is also allowed.

The rest of the article is structured as follows. In Section <ref> we present some counterexamples in the supercritical regime, demonstrating that (up to reaching the critical equality) condition (<ref>) cannot be improved; we then conclude the introduction by recalling some fundamental properties of fBm in Section <ref> and by introducing the main notations used throughout the paper in Section <ref>.
In Section <ref> we state and prove some fundamental lemmata, including the aforementioned a priori estimates for solutions of (<ref>) and the two new forms of the stochastic sewing lemma of [62].
Section <ref> contains further estimates for additive functionals of processes, as well as a key stability property of solutions.
In Sections <ref> and <ref> we use these estimates to establish well-posedness of (<ref>); 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 <ref>.
Section <ref> contains applications of our stability estimates to McKean-Vlasov equations.
In Section <ref> we construct weak solutions in some regimes beyond (<ref>), via a compactness argument enabled by the available a priori estimates.
In Section <ref> we show $\rho$-irregularity of solutions and more general perturbations of fractional Brownian motions.
Finally, Section <ref> 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 <ref> contains variants of Kolmogorov continuity criterion, Appendix <ref> gives two basic bounds for solutions of Young differential equations, and in Appendix <ref> we summarise relations of various Sobolev spaces and their use in Girsanov transform for fractional Brownian motions.

§.§ 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 [21], which gives counterexamples for $q=\infty$, $\alpha>0$.

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

α<1-1/H q',    α>-1;

let $B$ be an $\R^d$-valued stochastic process such almost surely $B\in C^\gamma$ for all $\gamma\in(0,H)$.
We claim that under (<ref>) there exists $b\in L^q_t C^\alpha_x$ such that the equation

X_t=x+∫_0^t b_s(X_s) ds+B_t

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 (<ref>) and define the function

b_t(x)=t^-1/q̃ (x)|x|^α;

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

and furthermore $\gamma<H$ thanks to (<ref>). 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 (<ref>), which is well-known to exist due to the spatial continuity and sublinear growth of $b$.
Define the stopping time

τ̃:=inf{t>0:  |B_t|≥δt^γ}∧1;

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

τ_x:=inf{t≥0:  X_t^x≤δt^γ}∧1.

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

X_t^x>∫_0^ts^-1/q̃(δs^γ)^α ds+B_t= (δ^α/γ) t^γ+B_t>δt^γ+(δt^γ+B_t)

where in the intermediate passage we used (<ref>).
Since $\tau_x\geq\tilde\tau>0$ $\PP$-a.s., there exist $\rho>0$ independent of $x\in (0,1]$ such that

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 $C([0,1])^2$ a.s.
The limit $(X^{0,+},B^{0,+})$ is a solution to (<ref>) with initial condition $0$ and satisfies

(X^0,+_t>0  ∀t∈(0,ρ])   ≥(X^0,+_t≥δt^γ  ∀t∈[0,ρ])

=lim_n→∞(X^x_n_t≥δt^γ  ∀t∈[0,ρ])≥3/4.

Since $b$ is odd, we can run the same argument for $y\in [-1,0)$: if $X^y$ is a solution to (<ref>), 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

(X^y_t ≤-δt^γ ∀t∈(0,ρ]) ≥3/4

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

(X^0,-_t<0  ∀t∈(0,ρ])≥3/4.

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 (<ref>) and define a vector field $b=(b^i)_{i=1}^d$ by

b^1(t,x)=t^-1/q̃ (x^1)|x|^α,    b^i(t,x)≡0  for  i=2,…,d;

again $b\in L^q_t C^\alpha_x$. Take $x\in(0,1]$ and consider a local-in-time solution $X^x$ of (<ref>) 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 (<ref>) holds; let us furthermore take an auxiliary parameter $\delta$ that will be specified later. Define the stopping times

τ̃:=inf{t>0:  |B_t|≥δt^γ}∧1,   τ_x:=inf{t≥0:  (X_t^x)^1≤δt^γ}∧1;

as before, $\tilde\tau$ is strictly positive 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

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 $s\leq \tau_x\wedge\tilde \tau$

(X_s^x)^1   ≤x+∫_0^sr^-1/q̃(δr^γ)^α dr+B_s^1

=x+ (δ^α/γ) s^γ+B_s^1

≤x+ (δ^α/γ+δ)s^γ;

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

(X_s^x)^1 ≤(3δ2^γ+δ+δ^α/γ)s^γ.

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:

(X_t^x)^1   >∫_t/2^ts^-1/q̃(C δ^αs^γ)^α ds+B_t^1

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

C([0,1])×{(a,g): a∈(0,1],g∈C([0,a])^2}

with the metric

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 almost surely as random variables taking values $\cS$. The limits $X^{0,+}:=\lim\bar X^{x_n}$ and $X^{0,-}:=\lim\bar X^{-x_n}$ both solve (<ref>) 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 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 $\tilde\tau^0$).

Up to multiplying $b$ by a cutoff function at infinity, by taking $\alpha=-d/(p+\eps)$ for sufficiently small $\eps>0$, 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/q+Hd/p>1-H,    p>d.

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

2/q+d/p>1,    p>d;

in particular, the exponents $p,q$ violate the LPS condition (<ref>). It is interesting to compare (<ref>) to the result from [57], where weak existence for the Brownian SDE was established under the condition

\begin{equation}\label{eq:parameters-weak-existence-brownian}
\frac{1}{q}+\frac{d}{p}\leq 1
\end{equation}

which is further shown to be optimal by construction of counterexamples in the case $1/q+d/p>1$.
Our counterexample shows that under (<ref>) uniqueness in law in general does not hold, answering a problem left open in [57] (see the discussion right above Remark 3.1 therein).

After the completion of this work, it has been further shown in [16] 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 doesn't hold, see Theorem 2.7 therein. More recently, [14] expanded the result from [57] by establishing weak existence of solutions for $H\in (0,1)$ and $b\in L^q_t L^p_x$ with

\begin{equation}\label{eq:parameters-weak-existence}
\frac{1-H}{q}+ \frac{H d}{p} < 1-H.
\end{equation}

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

§.§ Preliminaries on fractional Brownian motion

We recall here several
facts about fractional Brownian motion (fBm); for some standard references we refer to [71, 78].

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

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

where $I_d$ denotes is 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 $\PP$-a.s. $(H-\eps)$-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 such that

B^H_t=∫_0^t K_H(t,r)W_r  ∀  t ≥0.

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 e.g. [72], 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 [65]: given $B^H$, one can construct a two-sided Bm $\tilde W$ such that

\begin{equation}\label{eq:MvN-representation}
B^H_t = \gamma_H \int_{-\infty}^t \big[(t-r)_+^{H-1/2}-(-r)_+^{H-1/2}\big]\, \dd \tilde W_r;
\end{equation}

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

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

FBm of parameter $H=1$ is somewhat trivial or ill-defined, see [78];

however one can extend the definition to all values $H\in (0,+\infty)\setminus\N$ inductively as in [77] by $B^{H+1}_t:=\int_0^t B^H_s\dd s$.

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

\begin{equation}\label{eq:LND-fBm}
{\rm Cov} \big(B^H_t - \E_s B^H_t \big) = c_H |t-s|^{2H} I_d \quad\forall \, s\leq t,
\end{equation}

see <cit.>; here $\E_s B^H_t:=\E[B^H_t|\mathcal{F}_s]$, where $\mathcal{F}_s$ can be the natural filtration of $B^H$ or more generally any filtration such that $B^H$ is a $\bF$-fBm.
Property (<ref>) is a special form of strong local nondeterminism (LND)[In fact, any integral in time of an LND Gaussian process admitting a moving average representation in the style of (<ref>) is still LND, see <cit.>.], see <cit.> for a deeper discussion on its relevance on regularisation by noise.
Since conditional expectations are also $L^2$-projections, $B^H_t-\E_s B^H_t$ and $\E_s B^H_t$ are orthogonal Gaussian variables, thus independent; more generally, $B^H_t-\E_sB^H_t$ is independent of all the history up to time $s$.
Therefore for any $s\leq t$, any bounded measurable function $f:\R^d\to\R$ and any other $\cF_s$-measurable random variable $X$, it holds

_s f(B^H_t+X)= P_ Cov(B^H_t - _s B^H_t) f(_s B^H_t+X) = P_c_H |t-s|^2H I_d f(_s B^H_t+X).

where in the last passage we applied (<ref>); 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 expression like (<ref>) only $P_{|t-s|^{2H}}$ will appear.

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 (<ref>), is for $B^H$ to be a Gaussian process[For non-Gaussian processes one can still find a replacement for (<ref>), for example in the case of Lévy processes see [13]. ] satisfying a two-sided bound

\begin{equation}\label{eq:requirements}
C^{-1} |t-s|^{2H} I_d \leq {\rm Cov} \big( B^H_t-\E_s B^H_t \big)\leq C |t-s|^{2H} I_d
\end{equation}

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 (<ref>) provides a priori estimates in the style of Lemma <ref>, 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 (<ref>), one can still find estimates of the form

\begin{align*}
\| \E_s f(B^H_t+X)\|_{L^\infty}
& = \| \big( P_{ {\rm Cov} (B^H_t-\E_s B^H_t)} f \big) (\E_s B^H_t+X)\|_{L^\infty}
\leq \| P_{ {\rm Cov} (B^H_t-\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 (<ref>) alone is not enough and we exploited other properties of fBm.
Specifically: the counterxamples in Section <ref> assume $B^H$ to be $(H-\eps)$-Hölder continuous and symmetric;
the flows constructed in Sections <ref>-<ref> need some basic time-continuity $\E|B^H_t-B^H_s|\lesssim |t-s|^{H\wedge 1}$ in order to apply Kolmogorov-type criteria;
more substantially, the results from Section <ref> rely on a Volterra representation $B^H_t =\int_0^t K(t,s) \dd W_s$. These properties are satisfied by other interesting examples, e.g. type-II fBm and mixed fBm discussed in Remark <ref> below.

The only section truly specific to fBm is Appendix  <ref>, 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. [73].

Standard examples of processes satisfying (<ref>) are deterministic additive perturbations of fBm (cf. Lemma <ref>), the so called type-II fBm [66] and mixed fBm introduced in [26]; given any $H_1\neq H_2$, the process $B^{H_1}+B^{H_2}$ will satisfy condition (<ref>) with $H=H_1\wedge H_2$, both in the case $B^{H_1}$ and $B^{H_2}$ are sampled independently and the one instead where they are constructed from the same reference Brownian motion. In this case, our results yield a far reaching generalization (also to any $d\geq 2$) of the ones provided in [73], while not requiring highly technical use of Girsanov transform as therein.

Another interesting example is Bifractional Brownian motion of parameters $(H,K)$ (see [81]) which is known to be LND with parameter $HK$ [85]; it is a generalization of fBm ($K=1$), but even in the case $HK=1/2$ is not a semimartingale nor a Dirichlet process, although it scales like standard Bm. Our results show that it has a comparable regularising effect, although not amenable to Markovian/martingale techniques.

Another generalization of fBm is the so called multifractional Brownian motion, in which the Hurst parameter is allowed to vary continuously in time, $H=H(t)$; two non-equivalent definitions for this process are given respectively in [76] (by modifying representation (<ref>) by allowing $H=H(t)$) and in [9] (by a harmonisable representation). In both cases, the process can be shown to be “locally LND around $t$” with parameter $H(t)$ (see [4] in the harmonisable case) and thus we still expect our strategy to yield interesting results, under appropriate modifications. Likely, the admissible range of $\alpha$ here would depend on both the supremum and infimum of $H(t)$; we leave more precise investigations for future research.

Finally, let us mention that for (sufficiently regular) solutions $u(x,t)$ to certain linear stochastic PDEs for any fixed $x$ the process $t\mapsto u(x,t)$ is LND, see e.g. [86]; this fact was exploited crucially in regularisation by noise for nonlinear SPDEs in [3].

§.§ 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 $(\Omega,\cF,\bF,\PP)$, we will implicitly assume that the filtration $\bF=(\cF_t)_{t\in[0,1]}$ satisfies the standard assumptions; in particular, $\cF_0$ 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, $\|Y\|_{L^m}=\big(\E|Y|^m\big)^{1/m}$ for $m<\infty$ and $\|Y\|_{L^\infty}=\mathrm{esssup}_{\omega\in\Omega}|Y(\omega)|$. For conditional $L^m$-norms we use the notation  $\|Y\|_{L^m|\cF_s}=\big(\E(|Y|^m|\cF_s)\big)^{1/m}.$
For any $X,Y\in L^m$ such that $Y$ is $\cF_s$-measurable, by conditional Jensen's inequality one has the $\PP$-a.s. bound

X-_s X_L^m|_s ≤X-Y_L^m|_s +Y-_s X_L^m|_s ≤2X-Y_L^m|_s.

Apart from the usual $L^m$-norms, we also use the norms $\big\|\,\|\cdot\|_{L^m|\cF_s}\big\|_{L^n}$. We will always consider $n\geq m$, in which case again by conditional Jensen it holds

\begin{equation*}
\| X\|_{L^m} \leq \big\|\,\| X \|_{L^m|\cF_s}\big\|_{L^n}
\end{equation*}

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 $t\geq s$, $\big\|\,\|\cdot\|_{L^m|\cF_t}\big\|_{L^n}$ is stronger than $\big\|\,\|\cdot\|_{L^m|\cF_s}\big\|_{L^n}$.

* Whenever talking about a weak solution $X$ to the SDE (<ref>), we will actually mean a tuple $(X,B^H; \Omega, \bF, \PP)$ such that $(\Omega,\bF,\PP)$ is a filtered probability space as above, $X$ is $\bF$-adapted and $B^H$ is a $\bF$-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 $(\Omega,\bF,\PP)$, driven by the same $B^H$ and with same initial condition $x_0$, it holds $X^1\equiv X^2$ $\PP$-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 (<ref>) 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 <ref>, similarly for the concept of path-by-path uniqueness.

* Function spaces in the variable $x\in\R^d$ will often be denoted by the subscript $x$. For instance, standard Lebesgue spaces $L^p(\R^d;\R^m)$ with $p\in[1,\infty]$ will often be denoted, when the target dimension $m$ is clear, simply by $L^p_x$.
For $\alpha\in\R\setminus\N$, we denote by $C^\alpha_x$ the inhomogeneous Hölder-Besov space $B^{\alpha}_{\infty,\infty}$ (cf. [5]); 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 $C^{\alpha,\loc}_x$ 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 $\lambda\in\R$, defined through the (semi)norms

f_C^α,λ_x:=|f(0)|+f_C^α,λ_x:=|f(0)|+sup_R≥1 sup_x≠y∈B_R |f(x)-f(y)|/|x-y|^α  R^λ,

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 $\| f\|_{L^q E}^q=\int_0^1 \| f_t\|_E^q\, \dd t$, 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 $E=C^{\alpha,\loc}_x$ or $L^{p,\loc}_x$), 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 $f\in C^{p-\var}_t E$, if

f_p-,E^p:=sup∑_i=1^n d_E(f_t_i-1,f_t_i)^p<∞,

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 $\llbracket\cdot\rrbracket_{p-\var,E;[s,t]}$. Whenever $E=\R^m$ for some $m\in\N$, for simplicity we just drop it and write $C^{p-\var}_t$, $\llbracket \cdot \rrbracket_{p-\var;[s,t]}$, 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 $L^q\big([0,1], C^\alpha(\R^d,L^m(\Omega))\big)$. Similarly one can define e.g. $L^m C^{p-\var}_t C^{\alpha,\loc}_x$, $C^\gamma_t L^\infty_x$ 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

P_tf_C^α_x≤N t^(β-α)/2f_C^β_x;

see <cit.> 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
$\delta A_{s,u,t} = \delta A_{s,t}-\delta A_{s,u}-\delta A_{u,t}$.

* We say that a function $w:[0,1]^2_\leq\to\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

w_b,α,q(s,t):=∫_s^tb_r_C^α_x^q dr.

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 <cit.>).

Note also that if $w$ is a control, $\psi$ is an $\R^m$-valued path and $\gamma\in(0,1]$, then

conversely, for $p\geq 1$, if $\psi\in C^{p-\var}_t$ then $w(s,t)=\llbracket \psi\rrbracket_{p-\var; [s,t]}^p$ is a control and $|\psi_{s,t}|\leq w(s,t)^{1/p}$, cf. <cit.>.

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

where $\Gamma(\mu,\nu)$ is the set of all couplings of $\mu$ and $\nu$, i.e. the probability measures on $\R^{d}\times\R^d$ 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 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.

§ A PRIORI ESTIMATES AND STOCHASTIC SEWING

The key consequence of the subcriticality condition (<ref>) is that in terms of local nondeterminism, drifts of solutions are more regular than the noise; in particular, the solution decomposes as $X=\varphi+B^H$, where $\varphi$ plays the role of a slow variable, while $B^H$ is the highly oscillating component[In the regularisation by noise literature, to the best of our knowledge this concept originates from [19], where a similar pathwise solution ansatz leads to the formalism of nonlinear Young integrals, based on deterministic sewing. Here, also inspired by the works [62, 43, 51, 13], we take a step further and readapt the concept to a more probabilistic setup, where a combination of (<ref>), LND and stochastic sewing yields sharper results.].
This can be formulated as a precise quantitative bound, by looking at the best conditional error committed by predicting the process $\varphi_t$, given the history up to time $s$; more precisely, we look for estimates of the form

\begin{equation}\label{eq:key-quantity}
\big\|\|\varphi_t-\E_s\varphi_t\|_{L^m|\cF_s}\big\|_{L^\infty}\leq w(s,t)^{1/q}|t-s|^{1/q'+\alpha H}\quad \forall\, (s,t)\in [0,1]^2_{\leq},
\end{equation}

where $m\in [1,\infty)$, $w$ is a suitable control and $(q,\alpha,H)$ are the parameters related to $b$, $B^H$.

The subcritical regime $\alpha>1-1/(q' H)$ correponds to the exponent $1/q'+\alpha H$ appearing in (<ref>) being greater than $H$; this is in stark contrast with the lower bound provided by the LND property of fBm (<ref>), which tells us that such an estimate cannot hold for $\varphi$ replaced by $B^H$, justifying the slow-fast heuristic above.

It is also worth pointing out that $1/q'+\alpha H$ is allowed to exceed $1$ (this is indeed always the case for $H>1$), which will be used crucially in the following; in this case, the same bound could not hold if in (<ref>) $\E_s \varphi_t$ were replaced by $\varphi_s$, as one can easily check that the only processes satisfying the corresponding condition are the constant ones.

It will become clear in the sequel why (<ref>) is exactly the right condition needed in our analysis; for the moment, let us show that solutions to SDEs naturally enjoy (<ref>).

Lemma <ref> below is based on a readaption of <cit.>, <cit.> to our setting.
Note that in the statement, while we enforce the subcritical condition $\alpha>1-1/(q'H)$, the restriction $q\leq 2$ is not necessary; we do however restrict to $\alpha\geq 0$ first. For distributional drifts, similar bounds will be derived from stochastic sewing, see Lemma <ref> below.

Let $H\in(0,\infty)\setminus\N$, $q\in[1,\infty)$, and $\alpha\in[0,1]$ satisfy $\alpha>1-1/(q'H)$; let $b\in L^q_t C^\alpha_x$ and consider any weak solution $X$ of (<ref>). Then it holds $X=\varphi+B^H$ for

\begin{equation*}
\varphi_t := x_0 + \int_0^t b_r(X_r) \dd r
\end{equation*}

for any $m\in [1,\infty)$, there exists a constant $N=N(d,H,\alpha,m,\| b\|_{L^q_t C^\alpha_x})$ such that (<ref>) holds for the choice

w(s,t)=N w_b,α,q(s,t)= N ∫_s^t  b_r_C^α^q r.

First assume that, for some given $\beta \geq 0$, the bound (<ref>) holds with $w$ as above and exponent $\beta$ in place of $1/q'+\alpha H$.
This is definitely the case with $\beta=1/q'$, as one can see from

\begin{equation}\label{eq:drift-regularity-proof}\begin{split}
\big\|\|\varphi_t-\E_s\varphi_t\|_{L^m|\cF_s}\big\|_{L^\infty}
& \leq 2\big\|\|\varphi_t-\varphi_s\|_{L^m|\cF_s}\big\|_{L^\infty}\\
& \leq 2 \int_s^t \| b_r\|_{C^0}\, \dd r
\leq 2 w_{b,\alpha,q}(s,t)^{1/q} |t-s|^{1/q'};
\end{split}\end{equation}

in the above passages, we applied (<ref>), the definition of $\varphi$ and lastly Hölder's inequality.

Assuming we already have the bound for a generic $\beta\geq 1/q'$, we can then apply (<ref>) for the choice $Y=\varphi_s + \int_s^t b_r(\E_s X_r) \dd r$, together with the definition of $\varphi$, to find

\begin{align*}
\| \varphi_t   -\mathbb{E}_s \varphi_t \|_{L^m|\cF_s}
& \leq 2 \Big\| \varphi_t - \varphi_s - \int_s^t b_r ( \mathbb{E}_s \varphi_r +\E_s B^H_r) \mathd r \Big\|_{L^m|\cF_s}\\
& \leq 2 \int_s^t \big\| b_r (\varphi_r + B^H_r) - b_r (\mathbb{E}_s \varphi_r +\E_s B^H_r) \big\|_{L^m|\cF_s} \,\mathd r\\
& \leq 2 \int_s^t \| b_r \|_{C^{\alpha}_x} \big\| \varphi_r -\mathbb{E}_s \varphi_r + B^H_r -\E_s B^H_r \big\|_{L^m|\cF_s}^{\alpha}\,\dd r\\
& \leq 2 \int_s^t \| b_r \|_{C^{\alpha}_x} \big( \| \varphi_r -\mathbb{E}_s \varphi_r\|_{L^m|\cF_s}^{\alpha} + \| B^H_r -\E_s B^H_r \|_{L^m|\cF_s}^{\alpha}\big)\,\dd r;
\end{align*}

in the above estimates we used multiple times basic properties of conditional norms like Jensen's and Minkowski's inequality.
By the properties of fBm recalled in Section <ref> and the independence of $B_r^H-\E_s B^H_r$ from $\cF_s$, we have the bound

\begin{equation*}
\big\| \|B^H_r -\E_s B^H_r \|_{L^m|\cF_s} \big\|_{L^\infty}\lesssim |r-s|^H \quad \forall s\leq r.
\end{equation*}

Combined with our standing assumption on $\varphi$, by taking $L^\infty$-norms on both sides and using Minkowski's and Hölder's inequalities for the integral, we get

\begin{align*}
\big\|\| \varphi_t  -\mathbb{E}_s \varphi_t \|_{L^m|\cF_s}\big\|_{L^\infty}
& \lesssim \int_s^t \| b_r\|_{C^\alpha_x} \Big(  \big\|\| \varphi_r  -\mathbb{E}_s \varphi_r \|_{L^m|\cF_s}\big\|_{L^\infty}^\alpha + |r-s|^{\alpha H} \Big) \dd r\\
& \lesssim \int_s^t \| b_r\|_{C^\alpha_x} \Big(  w_{\alpha,b,q}(s,r)^{\alpha/q}|r-s|^{\alpha\beta} + |r-s|^{\alpha H} \Big) \dd r\\
& \lesssim w_{\alpha,b,q}(s,t)^{1/q}\Big(w_{\alpha,b,q}(s,t)^{\alpha/q}|t-s|^{\alpha\beta+1/q'} + |t-s|^{\alpha H+1/q'}\Big).
\end{align*}

In other terms, if $\varphi$ satisfies (<ref>) with $1/q'+\alpha H$ replaced by $\beta$, then it does so also with $\tilde{\beta}=f(\beta)=\alpha (\beta \wedge H)+1/q'$ (up to a change in the generic constant $N$).

From here, the argument is identical to the one from <cit.>: by iterating, we can define a sequence $\{\beta^n\}_n$ by $\beta^{n+1}= f(\beta^n)$ with $\beta_0=1/q'$; it remains to note that the condition $\alpha>1-1/(q'H)$ guarantees that the only fixed point $\bar{\beta}$ of the map $\tilde{f}(\beta)= \alpha \beta+1/q'$ is strictly larger than $H$ and is attracting exponentially fast any orbit defined by $\tilde{\beta}_{n+1}=\tilde{f}(\tilde \beta_n)$. Given that the sequences $\{\beta_n\}_n$ and $\{\tilde\beta_n\}_n$ coincide as long as the first one doesn't exceed $H$, this necessarily implies that the first one stabilizes to $\beta=\alpha H+ 1/q'$ after a finite number of iterations $\bar{n}$ (i.e. $\beta_n=\alpha H+ 1/q'$ for all $n\geq \bar{n}$).

The case $m=\infty$ can be handled with an appropriate stopping argument, see <cit.>. This can be used to derive similar bounds for processes that are not exact solutions (for example Picard iterates), but we do not need this generality.

The next ingredient is an a priori estimate for $\alpha<0$, analogous to Lemma <ref>.
Recall that for any adapted process $\varphi$, by (<ref>) one has

\begin{equation*}
\big\|\|\varphi_t-\E_s\varphi_t\|_{L^m|\cF_s}\big\|_{L^\infty}\leq 2\big\|\|\varphi_{s,t}\|_{L^m|\cF_s}\big\|_{L^\infty};
\end{equation*}

in the distributional case, we will directly bound the latter quantity.
Unlike Lemma <ref>, here we cannot allow for any $q\in(2,\infty]$ and subcritical $\alpha$, rather we impose the following stronger assumption:

\begin{equation}\label{eq:condition-existence}
%b\in L^q_t C^\alpha_x
,\quad q\in(1,\infty],\quad \alpha>\frac{1}{2}-\frac{1}{2H}, \quad \alpha>1-\frac{1}{H q'}.\tag{B}
\end{equation}

As mentioned, condition (<ref>) will be only used for treating $\alpha<0$, so $H<1$ is not a real restriction as it follows from the first condition on $\alpha$. Note further that in the case $q\in (1,2]$, (<ref>) reduces to (<ref>).
In the case $q\in(2,\infty)$, the a priori estimate below will be also relevant in Section <ref>, where we establish existence of weak solutions in a regime where uniqueness is not known.

Contrary to Lemma <ref>, the proof of Lemma <ref> will rely on stochastic sewing techniques. We could use the upcoming very general (but quite technical) Lemma <ref> for this task; but in order to help the intuition, we prefer first to invoke the result from [43], whose statement is simpler, and postpone the application of Lemma <ref> to where it is truly needed (e.g. Lemma <ref>).

Assume (<ref>) and in addition $\alpha<0$.
Let $b\in L^q_tC^1_x$ and let $X$ be the unique strong solution to (<ref>) for some initial condition $x_0\in\R^d$; set $w:=w_{b,\alpha,q}$ and $\varphi=X-B^H$.
Then for any $m\in [1,\infty)$ there exists a constant $N=N(m,d,\alpha,q,H,\| b\|_{L^q_t C^\alpha_x})$ such that for all $(s,t)\in[0,1]_\leq^2$ one has the bound

\begin{equation}\label{eq:apriori-estim}
\big\|\| \varphi_{s,t}\|_{L^m\vert \mathcal{F}_s}\big\|_{L^\infty} \leq N w(s,t)^{1/q} |t-s|^{\alpha H + 1/q'}.
\end{equation}

Up to shifting, we can assume without loss of generality $x_0=0$; moreover we only need to deal with $m\in [2,\infty)$, since $\| \cdot\|_{L^m\vert \cF_s} \leq \| \cdot\|_{L^2\vert \cF_s}$ otherwise.
Fix $m\in [2,\infty)$, set the shorthand $\beta:=\alpha H +1/q'$; recall that by (<ref>), one has $\beta>H$.

Let us first assume that (<ref>) holds with $w$ replaced by another control $\tilde{w}$; this is definitely the case for $\tilde w = w_{b,0,q}$, arguing as in (<ref>).
Given such $\tilde w$ and any closed subinterval $I\subset [0,1]$, define

\begin{equation*}
\llbracket \varphi\rrbracket_{\beta,\tilde w, I} := \sup_{s,t\in I, s<t} \frac{ \big\|\| \varphi_{s,t}\|_{L^m\vert \mathcal{F}_s} \big\|_{L^\infty}}{|t-s|^\beta \tilde w(s,t)^{1/q}}
\end{equation*}

with the convention $0/0=0$.

Fix $(s,t)\in[0,1]_\leq^2$ and, for any $(s',t')\in[s,t]_\leq^2$, set

A_{s',t'}:=\E_{s'} \int_{s'}^{t'} b_r(\varphi_{s'}+B^H_r)\dd r
= \int_{s'}^{t'} P_{|r-s'|^{2H}} b_r (\varphi_{s'}+\E_{s'} B^H_r) \dd r

where in the second passage we used conditional Fubini and property (<ref>) (please remember our convention about not writing explicitly the constant $c_H$ or the matrix $I_d$).

Our aim is to apply the stochastic sewing lemma (in the version given by <cit.>) to $A$ in order to find a closed estimate for $\llbracket \varphi\rrbracket_{\beta,w,I}$. By the heat kernel estimates (<ref>), we have almost surely

| A_s',t'|
≤∫_s'^t'  P_|r-s'|^2H b_r_C^0 r
≲∫_s'^t' |r-s'|^αH b_r_C^α r

where in the last passage we applied Hölder's inequality and the $L^{q'}$-integrability of $|r-s|^{\alpha H}$ follows from (<ref>).
Similarly, we have the $\PP$-a.s. bound

\begin{align*}
| \E_{s'} \delta A_{s',u',t'} |
& = \bigg| \E_{s'} \E_{u'} \int_{u'}^{t'} b_r(\varphi_{s'}+B_r)- b_r(\varphi_{u'} + B_r) \dd r \bigg|\\
& \lesssim \bigg| \int_{u'}^{t'} |r-u'|^{H(\alpha-1)} \| b_r\|_{C^\alpha} \dd r\, \E_{s'}| \varphi_{s',u'} | \bigg|\\
& \lesssim |t'-s'|^{2\beta-H}  w(s',t')^{1/q} \tilde w(s',t')^{1/q} \llbracket \varphi \rrbracket_{\beta,\tilde w, [s,t]}.
\end{align*}

The integrability of the power follows again from (<ref>), as do the inequalities $\beta+1/q>1/2$, $2\beta-H +2/q>1$ (we remark that it is only the latter for which the additional condition in (<ref>) was introduced).
Therefore the stochastic sewing lemma <cit.> applies and allows us to derive estimates for the sewing $\cA$ associated to $A$. However, one can easily identify $\cA_\cdot$; indeed, by the spatial regularity of $b$, we have the bound

φ_s',t'-A_s',t'_L^m≲|t'-s'|^  w_b,1,1(s',t')

for some $\eps>0$, which allows to conclude that $\cA_{\cdot}=\varphi_{s,\cdot}$ again by <cit.>.
Overall, we deduce that there exists a constant $N_0=N_0(m,d,\alpha,q,H)$ such that

\begin{equation*}
\big\| \|\varphi_{s',t'}\|_{L^m\vert \mathcal{F}_{s'}} \big\|_\infty
\leq N_0 |t'-s'|^\beta w(s',t')^{1/q} \Big(1 + |t'-s'|^{\beta-H} \tilde{w}(s',t')^{1/q} \llbracket\varphi\rrbracket_{\beta,\tilde w,[s',t']} \Big).
\end{equation*}

Diving both sides by $|t'-s'|^\beta w^{1/q}(s',t')$, taking supremum over $[s',t']\subset [s,t]$ and using the fact that all our estimates are on $[s,t]\subset [0,1]$, we obtain

\begin{equation}\label{eq:proof-existence-0}
\llbracket \varphi \rrbracket_{\beta,w,[s,t]} \leq N_0  \Big(1 + |t-s|^{\beta-H} \tilde{w}(s,t)^{1/q}\llbracket\varphi\rrbracket_{\beta,\tilde w,[s,t]} \Big).
\end{equation}

In particular, (<ref>) shows that $\llbracket \varphi \rrbracket_{\beta,w,[s,t]}$ is finite;
we can then go again through the whole argument, with $\tilde{w}$ replaced by $w$, to find

\begin{equation}\label{eq:proof-existence-1}
\llbracket \varphi \rrbracket_{\beta,w,[s,t]} \leq N_0  \Big(1 + |t-s|^{\beta-H} w(s,t)^{1/q} \llbracket\varphi\rrbracket_{\beta,w,[s,t]} \Big)
\end{equation}

which readily yields a closed estimate for $\llbracket \varphi \rrbracket_{\beta,w,[s,t]}$, at least for $[s,t]$ sufficiently small.

Our last task is to remove the smallness condition on $[s,t]$ in order to achieve a global bound.
To this end, define a new control $w_\ast$ by $w_\ast(s,t)^{1/q+\beta-H}=w(s,t)^{1/q} |t-s|^{\beta-H}$ and an increasing sequence $\{t_n\}_n$ by $t_0=0$ and $w_\ast(t_n,t_{n+1})^{1/q+\beta-H}=(2N_0)^{-1}$. Applying (<ref>) for $[s,t]=[t_n,t_{n+1}]$, by construction one finds $\llbracket \varphi \rrbracket_{\beta,w,[t_n,t_{n+1}]} \leq 2N_0$.

If $t_1=1$, this immediately yields the conclusion. Suppose this is not the case, then for any pair $s<t$ which do not belong to the same subinterval $[t_n,t_{n+1}]$, there exist $\ell,m\in\N$ such that $t_{\ell-1}< s\leq t_\ell\leq \ldots \leq t_m\leq t<t_{m+1}$. Set $\tau_{\ell-1}=s$, $\tau_i=t_i$ for $i=\ell,\ldots, m$ and $\tau_{m+1}=t$. It holds

\begin{align*}
\big\| \| \varphi_{s,t}\|_{L^m\vert \mathcal{F}_s}\big\|_{L^\infty}
& \leq \sum_{i=\ell-1}^{m} \big\|\| \varphi_{\tau_i,\tau_{i+1}}\|_{L^m\vert \mathcal{F}_s} \big\|_{L^\infty}
\leq \sum_{i=\ell-1}^{m} \big\|\| \varphi_{\tau_i,\tau_{i+1}}\|_{L^m\vert \mathcal{F}_{\tau_i}} \big\|_{L^\infty}\\
& \lesssim_{N_0} \sum_{i=\ell-1}^{m} w(\tau_i,\tau_{i+1})^{1/q} |\tau_i-\tau_{i+1}|^\beta \\
& \leq (m+1-\ell)^{-\alpha H} \Big( \sum_{i=\ell-1}^{m} \big [w(\tau_i,\tau_{i+1})^{1/q} |\tau_i-\tau_{i+1}|^\beta\big]^{\frac{1}{1+\alpha H}} \Big)^{1+\alpha H}\\
& \leq (m+1-\ell)^{-\alpha H} w(s,t)^{1/q} |t-s|^\beta
\end{align*}

where in the last two passages we used the fact that $\beta+1/q=1+\alpha H\in (0,1)$, Jensen's inequality and the superadditivity of the control $[w(s,t)^{1/q} |t-s|^\beta]^{\frac{1}{1+\alpha H}}$. Observe that $m+1-\ell$ is less or equal to the overall amount of intervals $[t_n,t_{n+1}]$. In turn, by their definition and subadditivity of $w_\ast$, this is bounded by a multiple of

\[ w_\ast (0,1)=w(0,1)^{\frac{(\alpha H + 1-H)^{-1}}{q}}=\| b\|_{L^q C^\alpha}^{(\alpha H + 1-H)^{-1}}\]

which finally yields the conclusion.

Next we formulate two appropriate versions of the stochastic sewing lemma (SSL).
After its introduction by Lê [62], in recent years the SSL has seen many variations.
Our first SSL combines three modifications: it incorporates shifting (as in [51]), as well as controls and general $\big\|\,\|\cdot\|_{L^m|\cF_s}\big\|_{L^n}$ norms (as in [43, 63]).
Let us remark that this combination is not completely obvious and comes with a price: due to the shifting, we need a nontrivial “time component” $|t-s|^\eps$ in our estimates, which does not appear in [43, 63].
Nontheless, the resulting statement is well suited for our applications, where such “time component” always appears naturally.

Recall the notations from Section <ref>, concerning $[0,1]_\leq$, $\overline{[S,T]}^2_\leq$, $s_{-}$ and so on.

Let $w_1,w_2$ be controls, and let $m,n$ satisfy $2\leq m\leq n\leq \infty$ and $m<\infty$. Let $(S,T)\in[0,1]_\leq$. Assume that $(A_{s,t})_{(s,t)\in\overline{[S,T]}^2_\leq}$ is a continuous mapping from $\overline{[S,T]}^2_\leq$ to $L^m$ such that for all $(s,t)\in\overline{[S,T]}^2_\leq$, $A_{s,t}$ is $\cF_t$-measurable.
Suppose that there exist constants $\eps_1,\eps_2>0$ such that the bounds

A_s,t_L^m|_s_L^n   ≤w_1(s_-,t)^1/2|t-s|^_1,

_s_-δA_s,u,t_L^n   ≤w_2(s_-,t)|t-s|^_2

hold for all $(s,u,t)\in\overline{[S,T]}^3_\leq$.
Then for all $S<s\leq t\leq T$ the Riemann sums

∑_j=0^2^ℓ-1 A_s+j2^-ℓ(t-s),s+(j+1)2^-ℓ(t-s)

converge as $\ell\to\infty$ in $L^m$, to the increments $\cA_t-\cA_s$ of
an adapted stochastic process $(\cA_t)_{t\in[S,T]}$ that is continuous as a mapping from $[S,T]$ to $L^m$ and $\cA_S=0$.
Moreover $\cA$ is the unique such process that satisfies the bounds

_t-_s-A_s,t_L^m|_s_L^n   ≤K_1 w_1(s_-,t)^1/2|t-s|^_1+K_2 w_2(s_-,t)|t-s|^_2,

_s_-(_t-_s-A_s,t)_L^n   ≤K_2 w_2(s_-,t)|t-s|^_2,

with some $K_1,K_2$ for all $(s,u,t)\in\overline{[S,T]}^3_\leq$. Furthermore, there exist a constant $K$ depending only on $\eps_1,\eps_2,m,n,d$ such that the bounds (<ref>)-(<ref>) hold with $K_1=K_2=K$, and moreover the bound

_t-_s_L^m|_s_L^n≤K( w_1(s,t)^1/2|t-s|^_1+w_2(s,t)|t-s|^_2)

holds for all $(s,t)\in[S,T]^2_\leq$.

Since by the time of the present work there is an abundance of SSLs in the recent literature, we do not aim to give a fully self-contained proof. We only provide the details as long as the combination of the arguments of [51] and [43, 63] is nontrivial.

Step 1 (convergence along dyadic partitions).

Let $(s,t)\in\overline{[S,T]}_\leq^2$ and for each $k=0,1,\ldots$ define $\cD_k=\{t_0^k,t_1^k,\ldots,t_{2^k}^k\}$, where $t_i^k=s+i2^{-k}(t-s)$, and

We claim that $\cA^k_{s,t}$ converges and its limit $\tilde\cA_{s,t}$ satisfies the bounds (<ref>)-(<ref>) with $K=K_1=K_2$ when replacing $\cA_t-\cA_s$ by it.

In particular, this would also imply the bound

_s,t_L^m|_s_L^n≤K( w_1(s_-,t)^1/2|t-s|^_1+w_2(s_-,t)|t-s|^_2)

for all $(s,t)\in\overline [S,T]^2_\leq$.
The claim clearly follows from the following two bounds:

^k-1 _s,t -^k _s,t   _L^m|_s_L^n      ≲w_1(s_-,t)^1/2|t-s|^_12^-k_1+w_2(s_-,t)|t-s|^_22^-k_2,

_s_- (   ^k-1 _s,t -^k _s,t  )   _L^n    ≲w_2(s_-,t)|t-s|^_22^-k_2.

It is no loss of generality to assume $k\geq 2$ (otherwise the trivial bounds below suffice), in which case we write

For the first term we used the conditions (<ref>)-(<ref>) in a trivial way:

δA_t_0^k,t_1^k,t_2^k_L^m|_s_L^n  ≲w_1(t_0^k-(t_2^k-t_0^k),t_2^k)^1/2|t_2^k-t_0^k|^_1≲w_1(s_-,t)^1/2|t-s|^_12^-k_1,

_s_- δA_t_0^k,t_1^k,t_2^k_L^n  ≤w_2(t_0^k-(t_2^k-t_0^k),t_2^k)|t_2^k-t_0^k|^_2≲w_2(s_-,t)|t-s|^_22^-k_2.

For the sum in (<ref>)
we write

∑_j=1^2^k-1-1δA_t_2j^k,t_2j+1^k,t_2j+2^k   =

where the term $\delta A_{t_{2^k}^k,t_{2^k+1}^k,t_{2^k+2}^k}$ is defined to be $0$.
The point of this unaesthetic decomposition is twofold. First, since $t^k_{2j-2}=t^k_{2j}-(t^k_{2j+2}-t^k_{2j})$, in the terms in the first sum there is sufficient shifting in the conditioning so that they can be estimated via the assumed bound (<ref>). Second, for each $\ell=0,1$, the inner sum above is one of martingale differences.

Therefore, we first estimate by the triangle inequality

I_1_L^m|_s_L^n   ≤∑_j=1^2^k-1-1_t_2j-2^kδA_t_2j^k,t_2j+1^k,t_2j+2^k_L^m|_s_L^n

≤∑_j=1^2^k-1-1 w_2(t_2j-2^k,t_2j+2^k)|t_2j+2^k-t_2j^k|^_2

using the superadditivity of $w_2$ in the last line.
Similarly, but replacing the triangle inequality by the Burkholder-Davis-Gundy and Minkowski inequalities (e.g. in the form given in <cit.> for $\mathfrak{p}=2$),

we have

I_2_L^m|_s_L^n   ≲∑_ℓ=0^1(∑_j=0^2^k-2δA_t_4j+2ℓ+2^k,t_4j+2ℓ+3^k,t_4j+2ℓ+4^k_L^m|_s_L^n^2)^1/2

≲2^-k_1∑_ℓ=0^1  (∑_j=0^2^k-2w_1(t_4j+2ℓ^k,t_4j+2ℓ+4^k))^1/2

This proves (<ref>). As for (<ref>), it is only easier: noting that

_s∑_j=1^2^k-1-1δA_t_2j^k,t_2j+1^k,t_2j+2^k=_s I_1,

we can bound $\|\E_sI_1\|_{L^n}\leq\|I_1\|_{L^n}$ just as in (<ref>). This concludes the proof of (<ref>)-(<ref>).

Step 2 (convergence along regular partitions). Let us say that a partition $\pi=\{s=t_0<t_1<\cdots<t_n=t\}$ is regular, if $|\pi|:=\max(t_i-t_{i-1})\leq 2\min(t_i-t_{i-1})$. For any partition we can define

^π_s,t=∑_i=1^n A_t_i-1,t_i.

Very similarly to Step 1, we get that for any sequence of regular partitions $(\pi_n)_{n\in\N}$ with $|\pi_n|\to 0$, $\cA^{\pi}_{s,t}$ converges (for details see <cit.>). Therefore on one hand this limit has to coincide with $\tilde\cA_{s,t}$, on the other hand, this limit is clearly additive.
Moreover notice that by construction $\tilde\cA_{s,t}$ is $\cF_t$-measurable for all $(s,t)\in\overline{[S,T]}_\leq^2$, and since it vanishes in $L^m$, the additivity implies that it is continuous in both arguments as a two-parameter process with values in $L^m$.

Step 3 (the process $\cA$ and its bounds). For any $t\in(S,T]$ we set $t_i:=S+2^{-i}(t-S)$. We then claim that the series

converges. Indeed, since $(s_i,s_{i-1})\in\overline{[S,T]}^2_\leq$, we may use the bound (<ref>).
By the trivial bounds $w((s_i)_-,s_{i-1})\leq w(S,t)$ and $|s_{i-1}-s_i|\leq 2^{-i}\one_{t-S\geq 2^{-i}}$, we get not only the convergence of the series but also the bound

_t_L^m|_S_L^n≤K( w_1(S,t)^1/2|t-S|^_1+w_2(S,t)|t-S|^_2).

This is precisely (<ref>) with $s=S$. The case for general $(s,t)\in[S,T]_{\leq}^2$ follows in the same way.
It is also clear that $\cA_0=0$, and by the remarks in Step 2, that $\cA$ is adapted and continuous in $L^m$. Therefore $\cA$ satisfies all of the claimed properties.

Step 4 (Uniqueness). The proof of this is standard and can be found in e.g. [63].

The other version of SSL that we use seems to be new.
In Lemma <ref> one can transfer $L^m$ bounds from $A$ to $\cA$ if $m<\infty$.
The $m=\infty$ case is a bit different: $L^\infty$ bounds on $A$ imply Gaussian moment bounds on $\cA$.
An alternative way to obtain Gaussian moment bounds via stochastic sewing is presented in [12] (see e.g. Theorem 3.3. and Lemma 4.6. therein), but the conditions herein are easier to verify.
The proof relies on a conditional version of Azuma–Hoeffding inequality, see Lemma <ref> in Appendix <ref>.

Let the conditions of Lemma <ref> hold with $m=n=\infty$.
Then there exists positive constants $\mu$ and $K$ depending only on $\eps_1,\eps_2,d$ such that the bound

[exp(μ |_t-_s|^2/(w_1(s,t)^1/2|t-s|^_1+w_2(s,t)|t-s|^_2)^2)|ℱ_s]≤K

holds for all $(s,t)\in[S,T]_{\leq}^2$.

We continue using the notation of the proof of Lemma <ref>.
Let $(s,t)\in\overline{[S,T]}_\leq^2$ and $k=0,1,\ldots$, and let us bound $\cA_{s,t}^{k+1}-\cA_{s,t}^k$.
The first term on the right-hand side of (<ref>) is trivially bounded by $2w_1(s_-,t)^{1/2}|t-s|^{\eps_1}2^{-k\eps_1}$ with probability $1$.
Decomposing the second term into $I_1$ and $I_2$ as in (<ref>), a simple use of triangle inequality as in (<ref>) yields the almost sure bound

As for $I_2$, recalling that it is the sum of two martingales, for each we may use the Azuma-Hoeffding inequality. The role of $\delta_j$ as in Lemma <ref> is played by $4w_1(t_{4j+2\ell}^k,t_{4j+2\ell+4}^k)^{1/2}$, so similarly to the calculation as in (<ref>), we get

Λ:=∑_i δ_i^2 ≲2^-2k_1|t-s|^2_1w_1(s,t).

Therefore by (<ref>), combined with the aforementioned almost sure bounds, we get that with some $\mu_1>0$, $K_1$

Since one can write

we get by conditional Jensen's inequality

Using again the assumed bounds on $A_{s,t}$, we get with some other constant $K_2$

It only remains to remove the shifts in the denominator and substitute $\mathcal{F}_S$ with $\mathcal{F}_s$, which can be done just as in Step 3 of the proof of Lemma <ref>, and therefore we obtain (<ref>).

§ STABILITY

The use of the tools from Section <ref> is illustrated by the following lemma, which will play a key role in our analysis.
Let us emphasise the important feature of the statement that although $h$ is assumed to have $\delta$ spatial regularity, in the estimate only its $\alpha-1$ norm is used.

Assume (<ref>) and let $(S,T)\in[0,1]_\leq^2$.

Suppose that $h\in L^q_t C^\delta_x$ for some $\delta>0$ and let $\varphi$ be an adapted process satisfying (<ref>) with $m=1$ and some control $w$.
For $t\in [S,T]$, define the process

ψ_t=∫_S^t h_r(B^H_r+φ_r)r

and set $\eps=1/q'+(\alpha-1)H$.
Then there exists positive constants $\mu$ and $K$, depending only on $H$, $q$, $\alpha$, and $d$, such that for all $(s,t)\in[S,T]^2_\leq$ one has the bound

[ exp(μ |ψ_t-ψ_s|^2/w_h,α-1,q(s,t)^2/q|t-s|^2 (1+w(s,t)^1/q|t-s|^)^2)|ℱ_s]≤K.

As a consequence, for any $\widetilde m\in[1,\infty)$ there exists a constant $\tilde K$, depending only on
$\tilde m$, $H$, $q$, $\alpha$, and $d$, such that for all $(s,t)\in[S,T]^2_\leq$ one has the bound

ψ_t-ψ_s_L^m|_s_L^∞≤K̃ w_h,α-1,q(s,t)^1/q|t-s|^(1+w(s,t)^1/q|t-s|^).

Note that thanks to the condition (<ref>), $\eps>0$.
For $(s,t)\in\overline{[S,T]}^2_\leq$ let us set

A_s,t=_s-(t-s)∫_s^t h_r(B^H_r+_s-(t-s)φ_r)r,

and verify the conditions of Lemma <ref> (namely those of Lemma <ref> with $m=n=\infty$).

Fix $(s,u,t)\in\overline{[S,T]}_\leq^3$ and denote $s_1=s-(t-s)$, $s_2=s-(u-s)$, $s_3=u-(t-u)$, $s_4=s$, $s_5=u$, $s_6=t$. These points are almost ordered according to their indices, except $s_3$ and $s_4$, for which $s_4\leq s_3$ may happen, but this plays no role whatsoever.
First, by property (<ref>) we have

A_s,t=∫_s^t P_|r-s_1|^2Hh_r(_s_1(B^H_r+φ_r))r.

Therefore, by (<ref>) and Hölder's inequality, it holds

|A_s,t|≤∫_s^t P_|r-s_1|^2Hh_r_C^0_xr≲   ∫_s^t|r-s_1|^(α-1) Hh_r_C^α-1_xr

≲|t-s|^1/q'+(α-1) Hw_h,α-1,q(s,t)^1/q.

Since $q\leq 2$, by the definition of $\eps$, (<ref>) is satisfied with $\eps_1=\eps$ and $w_1=N w_{h,\alpha-1,q}^{2/q}$.

Next, we need to bound $\E_{s-(t-s)}\delta A_{s,u,t}=\E_{s_1}\delta A_{s_4,s_5,s_6}$.
After an elementary rearrangement we get

_s_1δA_s_4,s_5,s_6=I+J:   =_s_1_s_2∫_s_4^s_5 h_r(B^H_r+_s_1φ_r)-h_r(B^H_r+_s_2φ_r)r

+_s_1_s_3∫_s_5^s_6 h(B^H_r+_s_1φ_r)-h_r(B^H_r+_s_3φ_r)r.

The two terms are treated in exactly the same way, so we only detail $I$.
We use (<ref>) similarly as before to get

|I|   ≤_s_1∫_s_4^s_5|P_|r-s_2|^2Hh_r(_s_2B^H_r+_s_1φ_r)-P_|r-s_2|^2Hh_r(_s_2B^H_r+_s_2φ_r)|r

≤_s_1∫_s_4^s_5P_|r-s_2|^2H h_r_C^1_x|_s_1φ_r-_s_2φ_r|r

By Jensen's inequality and the assumption on $\varphi$ we have the almost sure bound

Also note that $r\mapsto|r-s_2|^{(\alpha-2)H}\in L^{q'}([s_4,s_5])$ because of the shifted basepoint, in general this would not be true with $s_2$ replaced by $s_4$. Therefore, by Hölder's inequality

Note that the exponent of $|t-s|$ is simply $2\eps$.
Using again that $q\leq 2$, we see that condition (<ref>) is satisfied with $\eps_2=2\eps$ and $w_2=N w_{h,\alpha-1,q}(s,t)^{1/q}w(s_1,t)^{1/q}$.

It remains to verify that the process $\cA$ of Lemma <ref> is given by $\psi$. Since $\psi_0=0$, it suffices to show that

for all $(s,t)\in\overline{[S,T]}^2_{\leq}$, with some control $\tilde w$ and some $\kappa>0$.
This follows from three easy bounds: first,

ψ_t-ψ_s-∫_s^t h_r(B^H_r+_s_-φ_r)r_L^1

≤∫_s^th_r_C^δ_x w(s_-,r)^δ/qr

∫_s^t    h_r(B^H_r+_s_-φ_r)r-∫_s^t h_r(_s_-B^H_r+_s_-φ_r)r_L^1

≤∫_s^th_r_C^δ_x |r-s_-|^δHr

and third,

∫_s^t    h_r(_s_-B^H_r+_s_-φ_r)r-A_s,t_L^1

≤∫_s^th_r-P_|r-s_-|^2Hh_r_C^0_x r

Hence we can conclude $\psi=\cA$ and (<ref>) follows from (<ref>).

We will often consider (<ref>) with nonzero initial time. If $b$ is a function, a solution of (<ref>) on some interval $[S,T]\subset[0,1]$ with initial condition $X_S$ is a process $X$ satisfying

X_t=X_S+∫_S^t b_r(X_r)r+B^H_t-B^H_S

for all $t\in[S,T]$.
Our main stability estimate for solutions is then formulated as follows.

Assume (<ref>). Let $\delta>0$.
Let $[S,T]\subset [0,1]$ and
for $i=1,2$ let $X^i$ be adapted continuous processes satisfying (<ref>) on $[S,T]$ with initial conditions $X^i_S$ and drifts $b^i\in L^q_t C^{1+\delta}_x$.
Denote $M=\max_{i=1,2}\|b^i\|_{L^q_t C^\alpha_x}$.
Then for any $m\in[2,\infty)$ there exists a positive constant $N=N(m,M,H,\alpha,q,d)$, such that one has the almost sure bound

Moreover, if $b^1=b^2$, then one also has the almost sure bound

sup_t∈[S,T](|X^1_t-X^2_t|^-1)_L^m|_s≤N |X_S^1-X_S^2|^-1.

As usual, we denote $\varphi^1=X^1-B^H$ and $\varphi^2=X^2-B^H$.
For $t\in [S,T]$, we write

X^1_t-X^2_t   =X^1_S-X^2_S+∫_S^t(∫_0^1∇b^1_r(B^H_r+λφ^1_r+(1-λ)φ^2_r)λ)·(X^1_r-X^2_r)r

Note that $\nabla b^1\in L^q_tC^\delta_x$, and therefore the process

is well defined.
Define furthermore

We then apply Lemma <ref> with $\varphi=\lambda\varphi^1_r+(1-\lambda)\varphi^2_r$ and $h=\nabla b^1$, as well as with $\varphi=\varphi^2$ and $h=b^1-b^2$.
Since $\varphi^1$ and $\varphi^2$ are the drift parts of solutions, by Lemma <ref> the processes $\varphi=\lambda\varphi^1+(1-\lambda)\varphi^2$ satisfy the bound (<ref>) with control $w=w_{b^1,\alpha,q}+w_{b^2,\alpha,q}$,
and so Lemma <ref> indeed applies.
Combining the bound (<ref>) with Lemma <ref>, we get that there exist random variables $\eta_A,\eta_z$ with Gaussian moments[Note that in terms of the coefficients, the moments of $\eta_A$ depend on $w_{b^1,\alpha,q}+w_{b^2,\alpha,q}$, while the moments of $z$ depend only on $w_{b^2,\alpha,q}$.] conditionally on $\cF_S$, as well as $\delta>0$ and $p\in(1,2)$, such that

A_p-;[S,T]   ≤w_b^1,α,q(S,T)^1/qsup_S≤s<t≤T|A_t-A_s|/w_b^1,α,q(s,t)^1/q|t-s|^δ

z_p-;[S,T]   ≤w_b^1-b^2,α-1,q(S,T)^1/qsup_S≤s<t≤T|z_t-z_s|/w_b^1-b^2,α-1,q(s,t)^1/q|t-s|^δ

We can rewrite  (<ref>) as

(X^1_t-X^2_t)= A_t (X^1_t-X^2_t)+z_t,   (X^1_t-X_t^2)|_t=S=X^1_S-X^2_S,

meaning that we are interpreting (<ref>) as an affine Young differential equation; see also Appendix <ref> for more details.
By applying Lemma <ref> for $x=X^1-X^2$ and $\tilde{p}=p$, we get

Recall that $\eta_A$ satisfies $\E_S [e^{\mu \eta_A^2}]\lesssim 1$ for some $\mu>0$, thus also $\E_S[e^{K \eta_A^p}] \lesssim_{K,p} 1$ for all $K>0$ since $p<2$. Therefore we obtain

_S[ sup_t∈[S,T]|X_t^1-X_t^2|^m ]
≲_S[e^mC A_p-; [S,T]^p ] |X^1_S-X^2_S|^m

+ _S[ e^mC A_p-;[S,T]^p  z_p-;[S,T]^m ]

≲|X^1_S-X^2_S|^m + w_b^1-b^2,α-1,q(S,T)^m/q,

using conditional Hölder's inequality to get the last line.
This gives (<ref>).

In case $b^1=b^2$, we have $z=0$ and the Young equation (<ref>) becomes homogeneous. Moreover, note that Young equations allow time-reversal: if we fix $\tau\in[S,T]$, write $\tilde A_t=A_{\tau-t}$, and

Y_t=Ã_t Y_t,  Y_t|_t=0=X^1_τ-X^2_τ,

then $Y_{\tau-S}=X_S^1-X_S^2$. Therefore by Lemma <ref> we also have the pathwise estimate

Of course $\|\tilde A\|_{p-\var;[0,\tau-S]}^p=\| A\|_{p-\var;[S,\tau]}^p \leq \| A \|_{p-\var;[S,T]}^p$, so after rearranging for the inverses, taking supremum in $\tau\in[S,T]$, and taking $L^m|\cF_S$ norms, we get (<ref>).

§ STRONG WELL-POSEDNESS FOR FUNCTIONAL DRIFT

We first apply the stability estimate to establish existence and uniqueness of solutions of (<ref>) with $\alpha>0$.
In this case the meaning of solutions is unambiguous, but we will also need the following stronger concepts of solutions.

(i) Assume $b\in L^1_t C^{\loc}_x$ and let $\gamma:[0,1]\to\R^d$ be bounded and measurable.
A semiflow associated to the ODE

y_t = y_0 +  ∫_0^t b_s (y_s) s + γ_t

is a jointly measurable map $\Phi:[0,1]^2_\leq\times\R^d\to\R^d$ such that

* for all $(s,x)\in[0,1]\times\R^d$ and all $t\in[s,1]$ one has

Φ_s→t(x)=x+∫_s^t b_r(Φ_s→r(x))r+γ_t-γ_s;

* for all $(s,r,t,x)\in[0,1]^3_\leq\times \R^d$ one has $\Phi_{s\to t}(x)=\Phi_{r\to t}\big(\Phi_{s\to r}(x)\big)$.

(ii) A flow is a semiflow such that for all $(s,t)\in\times[0,1]^2_\leq$ the map $x\mapsto \Phi_{s\to t}(x)$ is a homeomorphism of $\R^d$.

(iii) If $\gamma$ is a stochastic process, a random (semi)flow  is a jointly measurable map $\Phi:\Omega\times[0,1]^2_\leq\times\R^d\to\R^d$ such that for almost all $\omega\in\Omega$, the map $\Phi^\omega:[0,1]^2_\leq\times\R^d\to\R^d$ is a (semi)flow associated to (<ref>) with $\gamma=\gamma(\omega)$.

(iv) We say that a random (semi)flow is adapted if for all $(s,t,x)\in[0,1]^2_\leq\times\R^d$, the random variable $\Phi_{s\to t}(x)$ is $\cF_t$-measurable.

(v) Given $\beta\in(0,1)$, we say that a (semi)flow is locally $\beta$-Hölder continuous if for all $K$ there exists a constant $N$ such that for all $(s,t,x,y)\in[0,1]_\leq^2\times B_K^2$ one has $|\Phi_{s\to t}(x)-\Phi_{s\to t}(y)|\leq N|x-y|^\beta$.

Definition <ref> is based on Kunita's classical one, cf. <cit.>; it is slightly different (in fact, stronger) from other definitions proposed in the literature, like <cit.>, due to the ordering of the quantifiers. One can draw a nice analogy between this kind of difference and the one between so called crude and perfect random dynamical systems, cf. <cit.>.

Assume (<ref>), $\alpha>0$, and let $b\in L^q_t C^\alpha_x$. Then there exists an adapted random semiflow of solutions to (<ref>) that is furthermore almost surely locally $\beta$-Hölder continuous for all $\beta\in(0,1)$.

Let $m\in[2,\infty)$, to be specified later.
Take a sequence of functions $(b^{n})_{n\in\N}$ such that $b^{n}\in L^q_tC^{2}_x$ and $\|b^{n}\|_{L^q_t C^\alpha_x}\leq \|b\|_{L^q_t C^\alpha_x}$ for all $n\in\N$,
and $\|b^{n}-b\|_{L^q_t C^{\alpha-1}_x}\to 0$ as $n\to\infty$.
Replacing $b$ by $b^{n}$ in (<ref>), the equation clearly admits an adapted random semiflow which we denote by $\Phi^{n}$.
For fixed $(s,t)\in[0,1]^2_\leq$, $x\in\R^d$, and $n,n'\in\N$, we may apply Theorem <ref> to obtain the bound

Φ_s→t^n(x)-Φ_s→t^n'(x)_L^m≲b^n-b^n'_L^q_t C^α-1_x.

Here and below the only important feature of the hidden proportionality constant in $\lesssim$ is that it is independent of $n,n'$.
Next, let $(s,s',t),(s,s',t')\in[0,1]^3_\leq$, $x,x'\in\R^d$, and $n\in\N$.
Then from applying Theorem <ref> again we get

by a trivial estimate we get

and using the semigroup property and Theorem <ref> once more we have
Φ_s→t^n(x)-Φ_s'→t^n(x)_L^m   =Φ_s'→t^n(Φ_s→s'^n(x))-Φ_s'→t^n(x)_L^m

We therefore get that the sequence $\big(\Phi^{n}\big)_{n\in\N}$ is on the one hand Cauchy in $C_{s,t,x} L^m_\omega$, and on the other hand, bounded in $C_{s,t}C^1_xL^m_\omega\cap C_x C^{H\wedge (1/q')}_{s,t}L^m_\omega$.
This implies that for some random field $\Phi$,
one has $\Phi^{n}\to \Phi$ in $C_{s,t}C^{1-\kappa}_xL^m_\omega\cap C_x C^{H\wedge (1/q')-\kappa}_{s,t} L^m_\omega$, where $\kappa>0$ is arbitrary.
By Kolmogorov's continuity theorem, for sufficiently large $m$, the convergence also holds in $L^m_\omega C_{s,t}C^{1-2\kappa,\loc}_x\cap L^m_\omega C^{\loc}_x C^{H\wedge (1/q')-2\kappa}_{s,t}$.
This yields the claimed spatial regularity of $\Phi$; the fact that $\Phi$ is indeed a semiflow for (<ref>) instead follows from the locally uniform convergence of $\Phi^n$ to $\Phi$, $\Phi^n$ being semiflows, and the spatial continuity of the drift $b$.

Assume (<ref>), $\alpha>0$, and let $b\in L^q_t C^\alpha_x$. Then there exists an event $\tilde\Omega$ of full probability such that for all $\omega\in\tilde \Omega$, for all $(S,T)\in[0,1]^2_\leq$, $x\in\R^d$, there exists only one solution to (<ref>) on $[S,T]$ with initial condition $x$.

The theorem will follow immediately from Theorem <ref> and the following lemma, which is a refinement of the technique illustrated in <cit.>.

Let $\gamma:[0,1]\to \R^d$ be bounded and measurable, $b\in L^1_t C^{\alpha,\loc}_x$ and consider the ODE (<ref>).

that it admits a locally $\beta$-Hölder continuous semiflow $\Phi$ with

\begin{equation}\label{eq:shaposhnikov-condition}
\beta(1+\alpha)>1.
\end{equation}

Then for any $(S,T)\in[0,1]_\leq^2$ and $y\in \R^d$ there exists a unique solution to the ODE on the interval $[S,T]$ with initial condition $y$, given by $\Phi_{S\to \cdot}(y)$.

Suppose that there exists another solution to the ODE, given by $(z_t)_{t\in[S,T]}$.
Since both $z$ and $\Phi_{S\to\cdot}(y)$ are bounded, we may and will assume $b\in L^1_t C^{\alpha}_x$
and that $\Phi$ is globally $\beta$-Hölder continuous. Define the control $w=w_{b,\alpha,1}$.

Now let us fix $\tau\in [S,T]$ and define the map $f_t:= \Phi_{t\to \tau} (z_t) - \Phi_{S\to \tau}(y)$. If we are able to show that $f$ is constant in time, then $f \equiv f_0=0$, which implies $\Phi_{t\to\tau}(z_t)=\Phi_{S\to\tau}(y)$ and in turn by choosing $t=\tau$ gives $z_\tau=\Phi_{\tau\to\tau}(z_\tau)=\Phi_{S\to\tau}(y)$. In particular, if we above argument holds for any $\tau\in [S,T]$, we reach the conclusion.

It remains to prove that $f$ is constant on $[S,\tau]$. To this end, first observe that for any $S\leq s\leq t\leq\tau$ it holds

[eq:sha trick]

Next, by definition of flow it holds

\[ \Phi_{s\to t}(z_s)-z_t=\int_s^t [b_r(\Phi_{s\to r} (z_s))-b_r(z_r)] \dd r \]

which immediately implies $|\Phi_{s\to t}(z_s)-z_t|\lesssim w(s,t)$; we can improve the estimate by recursively inserting it in the above identity:

\begin{align*}
|\Phi_{s\to t}(z_s)-z_t|
& \leq \int_s^t |b_r(\Phi_{s\to r} (z_s))-b_r(z_r)| \dd r\\
& \leq \int_s^t \| b_r\|_{C^\alpha} |\Phi_{s\to r} (z_s))-z_r|^\alpha \dd r
\leq w(s,t)^{1+\alpha}.
\end{align*}

Inserting the above in estimate (<ref>), we can conclude that

\[ |f_{s,t}| \lesssim |\Phi_{s\to t}(z_s)-z_t|^\beta \lesssim w(s,t)^{\beta (1+\alpha)}.\]

Since $\beta(1+\alpha)>1$ and $w$ is a control, $f$ must be necessarily constant.

Path-by-path uniqueness clearly implies pathwise uniqueness, which in turn implies uniqueness in law by <cit.>.

The statement of Lemma <ref> is given for deterministic initial data $y$ and semiflow $\Phi$, but immediately extends to random ones: if $X_0$ is a $\cF_0$-measurable random variable, then $(\Phi_{0\to t}(X_0)\big)_{t\in[0,1]}$ is clearly the unique adapted solution with initial condition $X_0$.

§ STRONG WELL-POSEDNESS FOR DISTRIBUTIONAL DRIFT

When $\alpha<0$, the very first question one has to address is the meaning of the equation, more precisely the meaning of the integral in (<ref>).
We start by some consequences of Lemma <ref>.
Denote by $\overline{C^\alpha}$ the closure of $C^1$ in $C^\alpha$. Recall that for any $\alpha<\alpha'$ one has $C^{\alpha'}\subset\overline{C^\alpha}$.

Assume (<ref>) and $\alpha<0$, and take $\delta>0$.
Define the linear map $T^{B^H}:L^q_tC^{1+\delta}_x\to L^\infty_\omega C_tC^\delta_x$ by

(T^B^Hh)_t(x)=∫_0^t h_r(B^H_r+x)r.

Denote $w=w_{h,\alpha,q}$.
Then, for any $m\in[2,\infty)$, there exists a constant $K=K(m,H,\alpha,q,d,w(0,1))$ such that for all $(s,t)\in[0,1]^2_\leq$ and $x,y\in\R^d$ one has the bound

Moreover, for any $\kappa\in(0,1)$ sufficiently small there exists a constant $K=K(m,H,\alpha,q,d,w(0,1),\kappa)$ such that one has the bound

Consequently with $p=\big(1+(\alpha-1)H\big)^{-1}\in(1,2)$,  the mapping $h\mapsto T^{B^H} h$ takes values in

$L^m_\omega C^{(p+\kappa)-\var}_tC^{1-\kappa,2\kappa}_x$ and as such, it
extends continuously to $L^q_t \overline{C^\alpha_x}$. This extension also satisfies the bounds (<ref>)-(<ref>).

Applying Lemma <ref> with $t,z\mapsto(x-y)\cdot\int_0^1\nabla h_t(z+\theta x+(1-\theta)y)\dd \theta$ in place of $h$ yields (<ref>).
The bound (<ref>) follow from (<ref>) and (<ref>) by Kolmogorov's continuity theorem in the form of Corollary <ref>.

Corollary <ref> motivates introducing some temporary notation. Given (<ref>), set $p_{\alpha,H}=\big(\big(1+(\alpha-1)H\big)^{-1}+2\big)/2\in(1,2)$ and for any $h\in L^q_t C^\alpha_x$ we define the event

\begin{equation*}
\Omega_h:=\Big\{\omega\in \Omega: \, T^{B^H}h(\omega)\in C^{p_{\alpha,H}-\var}_tC^{1-\kappa,2\kappa}_x\,\,\forall\kappa>0\Big\}
\end{equation*}

which is therefore of full probability.

The regularity of $T^{B^H}$ obtained from Corollary <ref> is sufficient to define a notion of solution via nonlinear Young formalism. For the proof of the next statement we refer to [45], which can be readily readapted to the $p$-variation framework, see also [2].

Let $A:[0,1]\times\R^d\to \R^n$ and $x:[0,1]\to\R^d$ satisfy $A\in C_t^{p-\var}C^{\eta,\loc}_x$ and $x\in C^{\zeta-\var}_t$ such that the exponents $p,\zeta\in[1,\infty)$, $\eta\in (0,1]$ satisfy

Then the nonlinear Young integral

y_t=∫_0^t A_s(x_s):=lim_ℓ→∞∑_j=0^2^ℓ-1A_j2^-ℓt,(j+1)2^-ℓt(x_j2^-ℓt)

is well-defined. If $A\in C_t^{p-\var}C^{\eta}_x$, then for all $(s,t)\in[0,1]_\leq^2$ $y$ satisfies the bound

|y_s,t-A_s,t(x_s)|≤N A_p-,C^η_x;[s,t]x_ζ-;[s,t]^η,

where the constant $N$ depends only on $1/p+\eta/\zeta$.

Assume (<ref>), $\alpha<0$ and $b\in L^q_t C^\alpha_x$.
Given $\omega\in \Omega_b$, we say that a path $x$ is an $\omega$-path solution to (<ref>) if $x=\varphi+B^H(\omega)$, $\varphi\in C^{\zeta-\var}_t$ for some $\gamma$ satisfying $1/p_{\alpha,H} + 1/\zeta>1$ and the equality

φ_t=φ_0 + ∫_0^t (T^B^Hb(ω))_s(φ_s)

holds for all $t\in[0,1]$, the integral being understood in the nonlinear Young sense.
We say that a stochastic process $X$ is a path-by-path solution to (<ref>)  if, for $\PP$-a.e. $\omega\in \Omega_b$, $X(\omega)$ is an $\omega$-path solution in the above sense.
Given this formulation of the SDE, the concepts of strong and weak solutions are analogous to the classical ones, see Section <ref> above.

Typically we encounter more special cases of nonlinear Young integrals than the generality that Lemma <ref> allows. First of all, the spatial growth of $A$ is often quantified (as in e.g. Corollary <ref>). Secondly, whenever $\varphi$ is a solution to a nonlinear Young equation, it is automatically  of $p$-variation and its temporal regularity can be often controlled by that of $A$ (see e.g. <cit.> in the Hölder case or Lemma <ref> in Appendix <ref>).

Let $A\in C_t^{p-\var} C^{\alpha,\lambda}_x$ with parameters $\alpha\in(0,1]$, $\lambda>0$, $p\in[1,2)$ satisfyingThis lemma is not actually needed, (<ref>) is enough to conclude the PBP argument. So I'm not sure, maybe we can remove it

\begin{equation}\label{eq:parameters-nonlinear-young}
\frac{1+\alpha}{p}>1,\quad \alpha+\lambda \leq 1.
\end{equation}

Denote $w(s,t)=\llbracket A\rrbracket_{p-\var,C^{\alpha,\lambda}_x;[s,t]}$
Then for any $y$ solving the equation

y_t=y_0+∫_0^t A_s(y_s)

on $[0,1]$, one has the bounds

|y_t| ≤N e^N w(0,1) (1+ |y_0|),    |y_t-A_s,t(y_s)| ≤N w(s,t)^1+α/p

for all $(s,t)\in[0,1]_\leq^2$,
where the constant $C$ only depends on the exponents.The $1$ in the first bound depends on some kind of $C^0$ norm of $A$ I suppose
As a consequence, $y\in C^{p-\var}$.

Maybe a short sketch

We can then define the notion of flows similarly to Definition <ref>. In fact, the following definition extends the previous one: for functional drifts, taking $A=T^\gamma b$, using the Riemann sums characterization of the nonlinear Young integral one can easily verify that

\begin{equation*}
\int_0^t (T^\gamma b)_{\dd s} (\varphi_s) = \int_0^t b_s(\varphi_s+\gamma_s) \dd s \quad \forall\, t\in [0,1].
\end{equation*}

Therefore in the functional case Definitions <ref> and <ref> coincide via the change of variables

\begin{equation}\label{eq:change-variables}
\Psi_{s\to t}(x)=\Phi_{s\to t}(x+\gamma_s)-\gamma_t.
\end{equation}

Assume $A\in C_t^{p-\var} C^{\eta,\loc}_x$ for some $\eta\in(0,1]$, $p\in[1,2)$ satisfying $(1+\eta)/p>1$.

A semiflow associated to the nonlinear Young equation

y_t = y_0 + ∫_0^t A_s (y_s)

is a jointly measurable map $\Psi:[0,1]^2_\leq\times\R^d\to\R^d$ such that

* for all $(s,x)\in[0,1]\times\R^d$ one has $\Psi_{s\to \cdot}(x)\in C^{p-\var}_t$ and for all $t\in[s,1]$ one has the equality

Ψ_s→t(x)=x+∫_s^t A_r(Ψ_s→r(x));

* for all $(s,r,t,x)\in\times[0,1]^3_\leq\times \R^d$ one has $\Psi_{s\to t}(x)=\Psi_{r\to t}\big(\Psi_{s\to r}(x)\big)$.

The definitions of flow, random (semi)flow, adaptedness, and Hölder continuity are then exactly as in Definition <ref>.

We are now in the position to state and prove our existence and uniqueness theorems in the case of distributional drift.

Assume (<ref>), $\alpha<0$, and let $b\in L^q_t C^\alpha_x$. Then there exists an adapted random semiflow of solutions to (<ref>) that is furthermore locally $\beta$-Hölder continuous almost surely for all $\beta\in(0,1)$.

By sacrificing a small regularity, we may and will assume $b\in L^q_t \overline{C^\alpha_x}$.
The proof follows similar steps as that of Theorem <ref>.
We take $m\in[2,\infty)$, to be chosen large enough later as well a sequence of functions $(b^n)_{n\in\N}$ such that $b^n\in L^q_tC^{2}_x$ and $\|b^n\|_{L^q_t C^\alpha_x}\leq \|b\|_{L^q_t C^\alpha_x}$ for all $n\in\N$,
and $\|b^n-b\|_{L^q_t C^{\alpha-1}_x}\to 0$ as $n\to\infty$.
Replacing $b$ by $b^n$ in (<ref>), the equation clearly admits an adapted random semiflow $\Psi^n_{s\to t}$.
For fixed $(s,t)\in[0,1]^2_\leq$, $x\in\R^d$, and $n,n'\in\N$, by Theorem <ref> one has the bound

Ψ_s→t^n(x)-Ψ_s→t^n'(x)_L^m≲b^n-b^n'_L^q_t C^α-1_x.

Similarly, for $(s,t)\in[0,1]^2_\leq$, $x,x'\in\R^d$, and $n\in\N$, Theorem <ref> yields

The temporal regularity is obtained from Lemma <ref>: in our present notation we get

with $\tilde w$ defined by the above equality.
Regularity in the $s$ variable is obtained precisely as in (<ref>).
From these estimates we obtain the convergence

Ψ^n→Ψ    in L^m_ωC_s,tC^1-κ,_x∩L^m_ωC^_x C^p_α,H-_s,t

to a limit $\Psi$ just as in of Theorem <ref> with all the required properties shown in the same way, except for the fact that $\Psi_{s\to \cdot}(x)$ solves the equation on $[s,1]$ with initial condition $x$ in the nonlinear Young sense.
Since at this point $s$ and $x$ are fixed, we assume for simplicity $s=0, x=0$ and denote $\Psi^{n}_{0\to t}(0)=\psi^{n}_t$, $\Psi_{0\to t}(0)=\psi_t$.
It is sufficient to show the convergence

∫_0^t (T^B^Hb^n)_s(ψ^n_s)→∫_0^t(T^B^Hb)_s(ψ_s)

in probability for each $t\in[0,1]$. Recall that by Corollary <ref> we have that

T^B^H(b^n-b)→0     in  C^p_α,H-_tC^1-κ,_x

in probability. From the above, we have that $\psi^{n}$ converges to $\psi$ (and in particular is bounded) in $C^{p_{\alpha,H}-\var}_t$ in probability. Therefore if we take an auxiliary $\ell\in\N$ and write

∫_0^t    (T^B^Hb^n)_s(ψ^n_s)- ∫_0^t(T^B^Hb)_s(ψ_s)

=∫_0^t (T^B^Hb^ℓ)_s(ψ^n_s)- ∫_0^t(T^B^Hb^ℓ)_s(ψ_s)

-∫_0^t (T^B^H(b^ℓ-b^n))_s(ψ^n_s)+ ∫_0^t (T^B^H(b^ℓ-b))_s(ψ_s),

then we can first choose $\ell$ and $n$ large enough to make the third and fourth integrals small, and then we can keep the same $\ell$ and increase $n$ further to make the difference of the first two terms small, using the Lipschitzness of $b^{\ell}$. This concludes the proof.

A similar statement can be given in the nonlinear Young setting with $p$-variation, which we need in order to handle the case of distributional drifts. Let me shortly recall a “natural” setting (in the sense of one which should be satisfied fairly easily satisfied under our assumptions).
Consider a drift $A\in C^{p-var}_t C^{\alpha,\lambda}_x$ (our case of interest will be $A=T^W b$), in the sense that there exists a continuous control $w:[0,T]^2\to [0,+\infty)$ such that

\[ \| A_{s,t}\|_{\alpha,\lambda} \leq w(s,t)^{1/p}\quad \forall\, s\leq t\]

where the weighted Hölder space $C^{\alpha,\lambda}_x$ is defined by

\[
\| f\|_{\alpha,\lambda} = |f(0)| + \sup_{R\geq 1} \sup_{x\neq y\in B_R(0)} \frac{|f(x)-f(y)|}{|x-y|^\alpha\, R^\lambda}.
\]

Further assume the parameter $p,\,\alpha,\,\lambda$ to satisfy $\alpha\in (0,1],\, \lambda>0,\, p\in[1,2)$ and

\begin{equation}\label{eq:parameters-nonlinear-young}
\frac{1+\alpha}{p}>1,\quad \alpha+\lambda \leq 1.
\end{equation}

Under these conditions, arguing similarly to Proposition 3.7 from [45], one can show that any solution to the nonlinear Young equation $\dd y_t = A_{\dd t}(y_t)$ must satisfy the a priori estimate (the exponents should be correct, will check)

\begin{equation}\label{eq:priori-estim-young}
\sup_{t\in [0,T]} |y_t| \leq C e^{C \| A \|_{p-var}^{p}} (1+ |y_0|).
\end{equation}

Similarly, existence of solutions is known to hold, and solutions can be characterized by the property that

\[
|y_t-A_{s,t}(y_s)| \leq C w(s,t)^{\frac{1+\alpha}{p}}.
\]

Assume (<ref>), $\alpha<0$, and let $b\in L^q_t C^\alpha_x$. Then there exists an event $\tilde\Omega$ of full probability such that for all $\omega\in\tilde \Omega$, for all $(S,T)\in[0,1]^2_\leq$, $x\in\R^d$, there exists only one $\omega$-path solution to (<ref>) on $[S,T]$ with initial condition $x$; in other words, path-by-path uniqueness holds.

In analogy to Remark <ref>, the strong form of uniqueness coming from Theorem <ref> readily implies pathwise uniqueness of solutions defined on random time intervals (e.g. stopping times) as well as uniqueness in law of weak solutions. In fact, it gives us uniqueness in a larger class of possibly non-adapted pathwise solutions, since the nonlinear Young formalism does not require adaptability of the processes in consideration. On the other hand, Theorem <ref> tells us that the unique solution is in fact a strong one.

Theorem <ref> follows from a version of Lemma <ref> in the nonlinear Young setting, which is a generalization of Theorem 5.1 from [45].

Let $A\in C^{p-\var}_t C^{\eta,\loc}_x$ for

some $\eta\in(0,1]$, $p\in[1,2)$ satisfying $(1+\eta)/p>1$. Suppose that the nonlinear YDE

\begin{equation*}
x_t = \int_0^t A_{\dd s}(x_s)
\end{equation*}

admits a locally $\beta$-Hölder continuous semiflow $\Psi$ with any $\beta\in(0,1)$.

Then for any $(S,T)\in[0,1]_\leq^2$ and $y\in\R^d$ there exists a unique solution to the nonlinear YDE on $[S,T]$, which is given by $\Psi_{S\to \cdot}(y)$.

The proof is very similar to that of Lemma <ref>, so we will mostly sketch it.
Let $z$ be a solution on $[S,T]$ starting from $y$, which by definition belongs to $C^{q-\var}_t$ with some $q$ such that $1/p+\eta/q>1$.
Thus $z$ is bounded, and in particular after localizing the argument we may assume that $\Psi$ is globally $\beta$-Hölder and that $A\in C^{p-\var}_t C^{\eta}_x$; furthermore, since the inequalities involving $(\eta,p,q)$ are strict, we can assume $\eta\in (0,1)$.

Set $w(s,t):=\llbracket A\rrbracket_{p-\var,C^{\eta}_x;[s,t]}^p$; an application of Lemma <ref> readily informs us that

\begin{equation}\label{eq:shaposhnikov-young-proof}
|\Psi_{s\to t}(x)-x-A_{s,t}(x)| \lesssim w(s,t)^{\frac{1+\eta}{p}}
\end{equation}

uniformly in $(s,t)\in [0,1]_\leq^2$ and $x\in\R^d$ (the hidden constant can depend on $w(0,1)$); a similar bound also holds for $\Psi_{s\to t}(x)$ replaced by $z_t$.

As before, we fix $\tau\in [S,T]$ and set $f_t:= \Psi_{t\to\tau}(z_t)-\Psi_{S\to \tau}(y)$; in order to conclude, it suffices to show that $f$ is constant.
As in (<ref>), we have $|f_{s,t}|\lesssim |\Psi_{s\to t}(z_s)-z_t|^\beta$.
Moreover by definition of solution to the YDE and estimate (<ref>), it holds that

\begin{align*}
|\Psi_{s\to t}(z_s)-z_t|
= \big|\Psi_{s\to t}(z_s)-z_s - A_{s,t}(z_s) - (z_t-z_s-A_{s,t}(z_s))\big|
\lesssim w(s,t)^{\frac{1+\eta}{p}}.
\end{align*}

Combining the two estimates, we get

\begin{align*}
|f_{s,t}|\lesssim w(s,t)^{\frac{\beta(1+\eta)}{p}};
\end{align*}

by assumption, we can choose $\beta$ close enough to $1$ so that $\beta(1+\eta)/p$ is bigger that $1$, implying the conclusion.

§ FLOW REGULARITY AND MALLIAVIN DIFFERENTIABILITY

So far we have established the existence of a random Hölder continuous semiflow $\Phi_{s\to t}(x)$; the aim of this section is to strengthen this result, by establishing better properties for $\Phi$.
We will start by showing that $\Phi$ is a random flow, in the sense that for each fixed $s<t$ the maps $x\mapsto\Phi_{s\to t}(x)$ are invertible, see Theorem <ref> below.
The main body of the section is devoted to the proof of Theorem <ref>, showing that both $\Phi_{s\to t}$ and its spatial inverse $\Phi_{s\leftarrow t}$ admit continuous derivatives. We conclude the section by showing that the random variables $\Phi_{s\to t}(x)$ possess a rather strong form of Malliavin differentiability, see Theorem <ref> below.

From now on, we will use both $\Phi_{s\to t}(x)$ and $\Phi_{s\to t}(x;\omega)$ to denote the semiflow, so to stress the dependence on the fixed element $\omega\in \Omega$ whenever needed; we start with the promised invertibility.

Let (<ref>) hold, $b\in L^q_tC^\alpha_x$, and denote by $\Phi_{s\to t}(x;\omega)$ the semiflow of solutions constructed in Theorems <ref> and <ref>. Then there exists an event $\tilde{\Omega}$ of full probability such that, for all $\omega\in \tilde{\Omega}$ and all $(s,t)\in [0,1]^2_\leq$, the map $x\mapsto \Phi_{s\to t}(x;\omega)$ is a bijection.

We follow closely the classical arguments by Kunita, cf. <cit.>, as they are completely independent from the driving noise being Brownian.

First, let us define the family of random variables

\begin{equation*}
\eta_{s,t}(x,y) := |\Phi_{s\to t}(x) - \Phi_{s\to t}(y)|^{-1}
\end{equation*}

Set $\gamma=H\wedge 1/q'$ for $\alpha \geq 0$, $\gamma = \alpha H + 1/q'$ in the case $\alpha<0$.
Recall that the estimates in the proof of Theorem <ref>, respectively Theorem <ref>, overall yield

\begin{equation}\label{eq:flow-estim-1}
\| \Phi_{s\to t}(x) - \Phi_{s'\to t'}(y)\|_{L^m} \lesssim |s-s'|^\gamma + |t-t'|^\gamma + |x-y|;
\end{equation}

moreover, by taking expectation in (<ref>), we have

\begin{equation}\label{eq:flow-estim-2}
\| |\Phi_{s\to t}(x) - \Phi_{s\to t}(y)|^{-1} \|_{L^m} \lesssim |x-y|^{-1}.
\end{equation}

Fix any $\delta>0$; we can combine estimates (<ref>) and (<ref>) and argue as in <cit.> to deduce that, for any $s<t$ and any $x$, $x'$, $y$, $y'$ satisfying $|x-y|>\delta$, $|x'-y'|>\delta$, it holds

\begin{equation}\label{eq:flow-estim-3}\begin{split}
\| & \eta_{s,t} (x,y)-\eta_{s',t'}(x',y')\|_{L^m}\\
& \lesssim \delta^{-2}  \Big[ |x-x'|+|y-y'|+ (1+|x|+|x'|+|y|+|y'|)(|t-t'|^\gamma + |s-s'|^\gamma) \Big].
\end{split}\end{equation}

From (<ref>), one can apply Kolmogorov's continuity theorem to deduce that the map $(s,t,x,y)\mapsto \eta_{s,t}(x,y;\omega)$ is continuous on the domain $\{s<t, |x-y|>\delta\}$ for $\PP$-a.e. $\omega$.
As the argument works for any $\delta>0$, we can find an event $\tilde{\Omega}$ of full probability such that, for all $\omega\in\tilde{\Omega}$, the map $\eta_{s,t}(x,y;\omega)$ is continuous on $\{s<t, |x-y|\neq 0\}$, which implies that it must also be finite for all $s<t, x\neq y$. This clearly implies injectivity of $x\mapsto \Phi_{s,t}(x;\omega)$ for all $s<t$ and $\omega\in\tilde{\Omega}$.

We move to proving surjectivity, which this time is closely based on <cit.>, having established the key inequalities (<ref>) and (<ref>).
Let $\hat{\R}^d=\R^d\cup\{\infty\}$ be the one-point compactification of $\R^d$; set $\hat{x}=x/|x|^2$ for $x\in \R^d\setminus\{0\}$ and $\hat{x}=\infty$ for $x=0$. Define

\begin{equation*}
\tilde \eta_{s,t}(\hat{x}) =
\begin{cases} (1+ |\Phi_{s\to t}(x)|)^{-1}\quad & \text{if } \hat{x}\in \R^d\\
0 & \text{if } \hat{x}=0
\end{cases}
\end{equation*}

Arguing as in <cit.> we find

\begin{equation}\label{eq:flow-estim-4}
\| \tilde\eta_{s,t}(\hat{x}) -\tilde\eta_{s',t'}(\hat{y})\|_{L^m} \lesssim |\hat{x}-\hat{y}| + |t-t'|^\gamma + |s-s'|^\gamma;
\end{equation}

by Kolmogorov's theorem, we can find an event of full probability, which we still denote by $\tilde{\Omega}$, such that $\tilde \eta_{s,t}(\hat{x};\omega)$ is continuous at $\hat{x}=0$ and so that $\Phi_{s,t}(\cdot;\omega)$ can be extended to a continuous map from $\hat{\R}^d$ to itself for any $s<t$ and $\omega\in\tilde\Omega$.
This extension, denoted by $\tilde{\Phi}_{s\to t}(x;\omega)$, is continuous in $(s,t,x)$ for every $\omega\in\tilde\Omega$ and thus $\Phi_{s\to t}(\cdot\,; \omega)$ is homotopic to the identity map $\tilde{\Phi}_{s\to s}(\cdot\,;\omega)$, making it surjective.
Its original restriction $\Phi_{s\to t}(\cdot\,; \omega)$ must then be surjective as well, from which we can conclude that $x\mapsto \Phi_{s\to t}(x;\omega)$ is surjective for all $s<t$ and $\omega\in\tilde{\Omega}$.

Our next goal is to establish that $\Phi$ is in fact a random flow of diffeomorphisms; by this we mean that, in addition to the map $(s,t,x,\omega)\mapsto \Phi_{s\to t}(x;\omega)$ satisfying all the properties listed in Definition <ref>, there exists an event of full probability $\tilde{\Omega}$ such that $x\mapsto \Phi_{s\to t}(x;\omega)$ is a diffeomorphism for all $s<t$ and $\omega\in\tilde{\Omega}$.
We will in fact prove a little bit more:

Let (<ref>) hold, $b\in L^q_tC^\alpha_x$, and $\Phi$ be the associated random flow. Then there exists a constant $\delta(\alpha,H)>0$ and an event $\tilde\Omega$ of full probability such that for any $\omega\in\tilde{\Omega}$ and any $s<t$, the map $x\mapsto \Phi_{s\to t}(x;\omega)$ and its inverse are both $C^{1+\delta,\loc}_x$.

In order to prove Theorem <ref>, we will first assume $b$ to be sufficiently smooth ($b\in L^q_t C^{1+\kappa}_x$ would suffice), so that the associated $\Phi$ is already known to be a flow of diffeomorphism, and derive estimates which only depend on $\| b\|_{L^q_t C^\alpha_x}$ (cf. Lemma <ref> and Proposition <ref> below). Establishing the result rigorously for general $b$ is then accomplished by standard approximation procedures, in the style of Theorems <ref>, <ref>.
We will frequently use the exponent $\eps=(\alpha-1) H+1/q'$ from Lemma <ref>, recall that (<ref>) is equivalent to $\eps>0$.

Recall that, for regular $b$, the Jacobian of the flow, namely the matrix $J_{s\to t}^x := \nabla \Phi_{s\to t}(x)\in \R^{d\times d}$, is known to satisfy the variational equation

\begin{equation}\label{eq:variational-equation}
J_{s\to t}^x = I + \int_s^t \nabla b_r(\Phi_{s\to r}(x)) J_{s\to r}^x \dd r.
\end{equation}

Already from this fact we can deduce useful moment estimates for $J^x_{s\to t}$.

Assume (<ref>) and let $b\in L^q_t C^2_x$.
Then there exists $p(\alpha,H)<2$ with the following property: for any $m\in [1,\infty)$, there exists a constant $N=N(m,p,H,\alpha,q,d,\| b\|_{L^q_t C^\alpha_x})$ such that, for all $x\in \R^d$ and $s\in [0,1]$, it holds

\begin{equation}\label{eq:moments-jacobian}
\Big\| \sup_{t\in [s,1]} |J^x_{s\to t}|\, \Big\|_{L^m} + \big\| \llbracket J^x_{s\to \cdot}\rrbracket_{p-\var;[s,1]} \big\|_{L^m} \leq N;
\end{equation}

moreover, for fixed $\delta<\eps$, for any $x\in \R^d$ and $s \leq t \leq t'$ it holds

\begin{equation}\label{eq:moments-jacobian-2}
\| J^x_{s\to t} - J^x_{s\to t'} \|_{L^m} \lesssim |t-t'|^\delta.
\end{equation}

For fixed $s\in [0,1]$ and $x\in \R^d$, setting $A_{s,t}:= \int_s^t \nabla b_r(\Phi_{s\to r}(x)) \dd r$, equation (<ref>) can be regarded as a linear Young differential equation. Arguing as in the proof of Theorem <ref>, one can show that $A$ has finite $p$-variation for some $p<2$ and that in fact there exists $\mu>0$ (depending on the usual parameters and $\| b\|_{L^q_t C^\alpha_x}$, but not on $x$ nor $s$) such that

\begin{equation}\label{eq:jacobian-proof1}
\E\bigg[  \exp\bigg( \mu \bigg| \sup_{s\leq t<t'\leq 1} \frac{|A_{t,t'}|}{w_{b,\alpha,q}(t,t')^{1/q} |t-t'|^\delta} \bigg|^2\bigg)\bigg] <\infty
\end{equation}

Lemma <ref> in Appendix <ref> (with $\tilde{p}=p$) implies the pathwise estimate

\begin{equation*}
\sup_{t\in [s,1]} |J^x_{s,t}| + \llbracket J^x_{s\to \cdot}\rrbracket_{p-\var;[s,1]} \leq C \exp \big( C \llbracket A \rrbracket_{p-\var; [s,1]}^p \big).
\end{equation*}

Claim (<ref>) then follows by taking $L^m$-norms on both sides and observing (as in the proof of Theorem <ref>) that (<ref>) implies $\E[\exp(\lambda \llbracket A \rrbracket_{p-\var}^p)]<\infty$ for all $\lambda>0$. Similarly, claim (<ref>) also follows from Lemma <ref> (this time applying estimate (<ref>) therein) combined with (<ref>).

The next step in the proof of Theorem <ref> is given by the following key estimate.

Let $b$ be a regular drift, define $J^x_{s\to t}$ as above;
set $\eps=(\alpha-1)H + 1/q'$. Then there exists $\gamma\in (0,1)$ such that, for any $m\in [1,\infty)$, there exists $N=N(m,\gamma,H,\alpha,q,d,\| b\|_{L^q_t C^\alpha_x})$ such that

\begin{equation}\label{eq:flow-diffeo-goal}
\| J^x_{s\to t} - J^y_{s'\to t'}\|_{L^m} \leq N\big[ |x-y|^{\gamma} + |t-t'|^{\eps \gamma} + |s-s'|^{\eps \gamma} \big].
\end{equation}

for all $(s,t), (s',t')\in [0,1]^2_\leq $ and $x,y\in \R^d$.

The proof requires the following technical refinement of Lemma <ref>.

Assume (<ref>), $h\in L^q_t C^1_x$, and let $\varphi^i$, $i=1,2$, be two processes satisfying the assumptions of Lemma <ref> for the same control $w$; define $\eps$ as therein and set $\psi^i_t=\int_S^t h_r(B^H_r+\varphi^i_r) \dd r$.
Then for $\gamma\in (0,1)$ satisfying

-γH>0,   (2-γ)-γH>0,   (2-γ)-γH + (2-γ)/q>1,

and any $m\in [2,\infty)$, there exists $N=N(m,\gamma,H,\alpha,q,d,\| h\|_{L^q_t C^{\alpha-1}_x})$ such that

\begin{equation*}
\| (\psi^1-\psi^2)_{s,t} \|_{L^m} \leq N |t-s|^{\eps-\gamma H} w_{h,\alpha-1,q}(s,t)^{\frac{1}{q}} \big(1+w(s,t)\big) \sup_{r\in [S,T]} \| \varphi^1_r-\varphi^2_r\|_{L^m}^\gamma.
\end{equation*}

The conditions in (<ref>) should be understood as “$\gamma$ small enough”. Indeed, note that all three conditions are upper bounds on $\gamma$ and under condition (<ref>) we can always find $\gamma>0$ satisfying (<ref>): as $\gamma\downarrow 0$, the three conditions become respectively $\eps>0$, $2\eps>0$, and $2\eps+2/q>1$, all of which are trivial since $q\leq 2$.

The proof is very similar to that of Lemma 	<ref>, so we will mostly sketch it; the main differences are just the use of Lemma <ref> with $n=m$ and some interpolation arguments.

Define $A^i_{s,t} = \E_{s-(t-s)}\int_s^t h_r (B^H_r + \E_{s-(t-s)}\varphi_r) \dd r$, so that $\psi^1-\psi^2$ is the stochastic sewing of $A^1-A^2$.
Arguing similarly as in Lemma <ref>, we have the estimate

\begin{align*}
\|A_{s,t}\|_{L^m}
& \leq \bigg\| \int_s^t \| P_{|r-s_1|^{2H}} h_r\|_{C^\gamma_x}\, |\E_{s_1} \varphi^1_r-\E_{s_1} \varphi^2_r|^\gamma \dd r \bigg\|_{L^m}\\
& \lesssim |t-s|^{\eps-\gamma H} w_{h,\alpha-1,q}(s,t)^{1/q} \sup_{r\in [S,T]} \| \varphi^1_r-\varphi^2_r\|_{L^m}^\gamma;
\end{align*}

the first condition of Lemma <ref> is verified, since $\eps-\gamma H>0$ and $1/q \geq 1/2$.
To control $\E_{s_1} \delta A_{s,u,t}=\E_{s_1} \delta A^1_{s,u,t}-\E_{s_1} \delta A^2_{s,u,t}$, we can decompose it as $\E_{s_1} \delta A_{s,u,t} = I^1-I^2+J^1-J^2$, similarly to Lemma <ref>. Estimating each one of them separately as therein yields

\begin{align*}
\sup_i \{|I^i|,|J^i|\}\lesssim |t-s|^{2\eps}w_{h,\alpha-1,q}(s,t)^{1/q}w(s_1,t)^{1/q};
\end{align*}

on the other hand, we have

\begin{align*}
\| I^1-I^2\|_{L^m}
& \leq \bigg\| \int_{s_4}^{s_5}\big|P_{|r-s_2|^{2H}}h_r(\E_{s_2}B^H_r+\E_{s_1}\varphi^1_r)-P_{|r-s_2|^{2H}}h_r(\E_{s_2}B^H_r+\E_{s_2}\varphi^1_r)\big|\dd r \\
& \quad -\int_{s_4}^{s_5}\big|P_{|r-s_2|^{2H}}h_r(\E_{s_2}B^H_r+\E_{s_1}\varphi^2_r)-P_{|r-s_2|^{2H}}h_r(\E_{s_2}B^H_r+\E_{s_2}\varphi^2_r)\big|\dd r\bigg\|_{L^m}\\
& \leq \int_{s_4}^{s_5} \| P_{|r-s_2|^{2H}}h_r\|_{C^1_x} \big( \| \E_{s_1} \varphi^1_r-\E_{s_1} \varphi^2_r\|_{L^m} + \| \E_{s_2} \varphi^1_r-\E_{s_2} \varphi^2_r\|_{L^m}\big) \dd r\\
& \lesssim |t-s|^{(\alpha-2)H + 1/q'} w_{h,\alpha-1,q}(s,t)^{1/q} \sup_{r\in [S,T]} \| \varphi_r^1-\varphi^2_r\|_{L^m},
\end{align*}

similarly for $\| J^1-J^2\|_{L^m}$. Interpolating the two bounds together overall yields

\begin{align*}
\| \E_{s_1} \delta A_{s,u,t}\|_{L^m}
\lesssim |t-s|^{\eps(2-\gamma)-\gamma H} w_{h,\alpha-1,q}(s,t)^{1/q} w(s_1,t)^{\frac{1-\gamma}{q}}
\sup_{r\in [S,T]} \| \varphi^1_r-\varphi^2_r\|_{L^m}^\gamma.
\end{align*}

By the hypothesis (<ref>), the power of $|t-s|$ is positive and the total power of all the controls is greater than $1$.
The conclusion then follows from Lemma <ref>.

As usual, we can split estimate (<ref>) into three subestimates, with two of the three parameters $(s,t,x)$ fixed and only one varying. From now on we will fix $\gamma\in (0,1)$ satisfying condition (<ref>).

Step 1: $(s,x)$ fixed, $t<t'$. In this case the desired estimate is just (<ref>) from Lemma <ref>, for the choice $\delta=\gamma \eps < \eps$.

Step 2: $(s,t)$ fixed, $x\neq y$.  The difference  process $v_t:=J^x_{s,t}-J^{y}_{s,t}$ satisfies an affine Young equation of the form $ \dd v_t = \dd A_t\, v_t + \dd z_t$, $v_s=0$, for

\begin{equation*}
A_t = \int_s^t \nabla b_r(\Phi_{s\to r}(x)) \dd r, \quad z_t = \int_s^t \big[ \nabla b_r(\Phi_{s\to r}(x)) - \nabla b_r(\Phi_{s\to r}(y))\big] J^y_{s\to r} \dd r;
\end{equation*}

invoking as usual Lemma <ref> (for $\tilde{p}=1/2$) and applying estimate (<ref>), one ends up with

\[ \| J^x_{s,t}-J^y_{s,t}\|_{L^m} \lesssim \big\| \llbracket z\rrbracket_{2-\var} \big\|_{L^m}.\]

Observe that $z$ itself can be interpreted as a Young integral: $z_t= \int_s^t \dd \tilde{A}_r J^y_{s\to r}$ for

\[
\tilde{A}_u:=\int_s^u \big[ \nabla b_r(\Phi_{s\to r}(x)) - \nabla b_r(\Phi_{s\to r}(y))\big] \dd r
\]

Standard properties of Young integral, together with Cauchy's inequality, then yield

\begin{align*}
\big\| \llbracket z\rrbracket_{2-\var} \big\|_{L^m}
\lesssim \big\| \llbracket \tilde{A}\rrbracket_{2-\var}\, \llbracket J^y_{s\to \cdot}\rrbracket_{p-\var} \big\|_{L^m}
\lesssim \big\| \llbracket \tilde{A}\rrbracket_{2-\var}\|_{L^{2m}} \big\| \llbracket J^y_{s\to \cdot}\rrbracket_{p-\var} \big\|_{L^{2m}};
\end{align*}

by estimate (<ref>), it only remains to find a bound for $\llbracket \tilde{A}\rrbracket_{2-\var}$.
Recall that by construction $\Phi_{s\to r}(x) = \varphi_{s\to r}(x) + B^H_r$, where the process $\varphi_{s\to \cdot}(x)$ satisfies condition (<ref>) (or even (<ref>) for $\alpha<0$) for $w=w_{b,\alpha,q}$. We can apply Lemma <ref> with the choice $h=\nabla b$, $\varphi^1_r=\varphi_{s\to r}(x)$, $\varphi^2_r=\varphi_{s\to r}(y)$ to obtain, for all $s\leq r<u\leq 1$ and all $m\in [1,\infty)$,

\begin{align*}
\| \tilde{A}_{r,u}\|_{L^m}
&\lesssim |r-u|^{\eps-\gamma H} w(r,u)^{1/q} (1+ \| b\|_{L^q_t C^\alpha_x}^q) \sup_{r\in [s,1]} \| \varphi^1_r - \varphi^2_r\|_{L^m}^\gamma\\
& \lesssim |r-u|^{\eps-\gamma H} w(r,u)^{1/q} |x-y|^\gamma
\end{align*}

where in the second inequality we used estimate (<ref>). By Lemma <ref> in Appendix <ref> we deduce that, for any $m\in [1,\infty)$ and $\delta<\eps-\gamma H$, it holds

\begin{align*}
\big\| \llbracket \tilde{A}\rrbracket_{2-\var}\|_{L^{2m}} \lesssim \bigg\| \sup_{r<u} \frac{|\tilde A_{r,u}|}{|r-u|^{\delta} w(r,u)^{1/q}}\bigg\|_{L^{2m}} \lesssim |x-y|^\gamma.
\end{align*}

Combining all the above estimates yields the conclusion in this case.

Step 3: $(t,x)$ fixed, $s<s'$. This step is mostly a variation on the arguments presented in the previous cases, so we only sketch it. We can write

\[
J^x_{s,t}= J^x_{s,s'} + \int_{s'}^t \nabla b(\Phi_{s\to t}(x))  J^x_{s,r} \dd r
\]

so that the difference $v_t= J^x_{s,t} - J^x_{s',t}$ can be regarded as the solution to an affine Young equation on $[s',t]$, for $A$ and $z$ defined similarly as in Step 2; the only difference is that now $v_{s'} = J^x_{s,s'}-I$ and $z_t = \int_{s'}^t \dd \tilde A_r J^z_{s'\to r}$ for the choice

\begin{equation*}
\tilde{A}_u:=\int_{s'}^u \big[ \nabla b_r(\Phi_{s\to r}(x)) - \nabla b_r(\Phi_{s'\to r}(x))\big] \dd r.
\end{equation*}

From here, the estimates are almost identical to those of Step 2, relying on a combination of Lemmas <ref>, <ref> and <ref>; however in this case an application of Step 1 and estimate (<ref>) gives us

\begin{equation*}
\| J^x_{s'\to s}-I\|_{L^m} \lesssim |s-s'|^{\eps \gamma}, \quad \sup_{r\in [s',1]} \| \Phi_{s\to r}(x)-\Phi_{s'\to r}(x)\|_{L^m}^\gamma \lesssim |s-s'|^{\eps \gamma}. \qedhere
\end{equation*}

We are now finally ready to complete the

The argument is based on Theorem II.4.4 from [60]; assume first $b$ to be a regular field.
It is clear from (<ref>) that, for any $\delta <\eps \gamma$, the map $(s,t,x)\mapsto \nabla J_{s\to t}^x$ is $\PP$-a.s. locally $\delta$-Hölder continuous, suitable moment estimates depending only on $\| b\|_{L^q_t C^\alpha_x}$.
Furthermore, letting $K_{s\to t}^x$ denote the inverse of $J_{s\to t}^x$ in the sense of matrices, it is well-known that it solves the linear equation

K_s→t^x = I - ∫_s^t K_s→r^x  ∇b_r(Φ^x_s→r(x)) 	r;

arguing as in the proof of Proposition <ref>, one can prove that

\begin{equation*}
\| K^x_{s\to t} - K^y_{s'\to t'}\|_{L^m} \lesssim |x-y|^\gamma + |t-t'|^{\eps \gamma} + |s-s'|^{\eps \gamma}
\end{equation*}

and so that it is $\PP$-a.s. $\delta$-Hölder continuous as well.

In the case of general $b\in L^q_t C^\alpha_x$, we can consider a sequence $b^n$ of regular functions such that $b^n\to b$ in $L^q_t C^\alpha_x$ (up to sacrificing a little bit of spatial regularity as usual), in which case we already know that the associated flows $\Phi^n$ converge to $\Phi$ in $L^m_\omega C_{s,t} C^{\delta,\loc}_x$; combined with the aforementioned moments estimates, one can then upgrade it to convergence in $L^m_\omega C_{s,t} C^{1+\delta,\loc}_x$. In particular, the fields $J^{x,n}_{s\to t}=\nabla \Phi^n_{s\to t}(x)$ and $K^{x,n}_{s\to t}=(\nabla \Phi^n_{s\to t}(x))^{-1}$ converge respectively to $J^x_{s\to t}$ and $K^x_{s\to t}$; by the limiting procedure, there exists an event $\tilde{\Omega}$ of full probability such that, for all $\omega\in\tilde\Omega$, it holds $J^x_{s\to t}(\omega)=\nabla \Phi_{s\to t}(x;\omega)$ and and $J^x_{s\to t}(\omega) K^x_{s\to t}(\omega)=I$ for all $s<t$ and $x\in \R^d$, as well as $J(\omega), K(\omega)\in C_{s,t} C^{\delta,\loc}_x$.

Overall, for every $\omega\in\tilde{\Omega}$, the map $(s,t,x)\mapsto \Phi_{s\to t}(x;\omega)$ has regularity $C_{s,t} C^{1+\delta,\loc}_x$ and its Jacobian admits a continuous inverse $K^x_{s\to t}(\omega)$. But this implies that, for any $s<t$, $\nabla \Phi_{s\to t}(x;\omega)$ is a nondegenerate matrix for all $x\in \R^d$, which by the implicit function theorem readily implies that the inverse of $x\mapsto \Phi_{s\to t}(x;\omega)$ must belong to $C^{1+\delta,\loc}_x$ as well. This concludes the proof.

It is well known in the regular case that the Jacobian of the flow and the Malliavin derivative satisfy the same type of linear equation.
Therefore, as the last main result of the section, we show Malliavin differentiability of the random variables $X^x_{s\to t}(\omega):= \Phi_{s\to t}(x;\omega)$. To this end, we start with a simple, yet powerful lemma, showing that deterministic perturbations of the driving noise $B^H$ do not affect our solution theory.

Assume (<ref>), $b\in L^q_t C^\alpha_x$, and $h: [0,1]\to \R^d$ be a deterministic, measurable function; then for any $s\in [0,1]$ and any $x\in \R^d$, there exists a pathwise unique strong solution to the perturbed SDE

\begin{equation}\label{eq:perturbed-sde}
X_t = x + \int_s^t b_r(X_r) \dd r + B^H_{s,t} + h_{s,t} \quad \forall\, t\in [s,1]
\end{equation}

which we denote by $X_{s\to \cdot}(x;h)$; in the distributional case $\alpha<0$, eq. (<ref>) must be interpreted in the sense of Definition <ref>.

We give two short alternative arguments to verify the claim. On one hand, carefully going through the proofs of Sections <ref>-<ref>, the only key properties needed on the process $B^H$ (cf. also Remark <ref>) are its Gaussianity and the two-sided bounds

\begin{equation*}
\E[ |B^H_t - \E_s B^H_t|^2] \sim  |t-s|^{2H}
\end{equation*}

which are clearly still true for $\tilde{B}^H=B^H+h$, due to $h$ being deterministic.

Alternatively, if we define $\tilde{b}_t(z):= b_r(z+h_r)$, $y=x+h_s$, then any solution $X$ to (<ref>) must be in a $1$-$1$ correspondence with a solution $Y:=X+h$ to the unperturbed SDE

\[
Y_t = y + \int_s^t \tilde{b}_r(Y_r)\dd r + B^H_{s,t}
\]

and it is clear that $\tilde{b}$ still satisfies condition (<ref>), thus implying its well-posedness.

We are now ready to verify Malliavin differentiability of $X^x_{s\to t}$.
To this end, let us recall that if we denote by $\mathcal{H}^H$ the Cameron-Martin space associated to $B^H$, then the Malliavin derivative $D X^x_{s\to t}$, when it exists, can be identified as the (random) linear bounded operator from $\mathcal{H}^H$ to $\R^d$ given by

\begin{equation*}
h\mapsto \partial_h X^x_{s\to t}, \quad \text{where} \quad\partial_h X^x_{s\to t}:= \lim_{\eps\to 0} \frac{X_{s\to t}(x; \eps h) - X_{s\to t}(x ; 0)}{\eps}.
\end{equation*}

We will use $\| D X^x_{s\to t}\|_{\mathcal{H}^H}$ to denote the operator norm of $D X^x_{s\to t} $ as a linear operator from $\mathcal{H}^H$ to $\R^d$.

Assume (<ref>) and $b\in L^q_t C^\alpha_x$. Then $\PP$-a.s. the random variables $\partial_h X^x_{s\to t}$ exist for all $h\in C^{2-\var}_t$ and define a (random) linear map $h\mapsto \partial_h X^x_{s,t}$. Moreover for any $m\in [1,\infty)$ it holds

\begin{equation}\label{eq:generalized-malliavin-moment-estim}
\sup_{s\in [0,1],x\in \R^d} \Big\| \sup_{t\in [s,1]} \| h\mapsto \partial_h X^x_{s,t}\|_{\mathcal{L}(C^{2-\var};\R^d)} \Big\|_{L^m}<\infty.
\end{equation}

In particular, $X^x_{s\to t}$ is Malliavin differentiable and for any $m\in [1,\infty)$ it holds

\begin{equation}\label{eq:malliavin-moment-estim}
\sup_{s\in [0,1], x\in \R^d} \Big\| \sup_{t\in [s,1]} \| D X^x_{s\to t} \|_{\mathcal{H}^H} \Big\|_{L^m} <\infty.
\end{equation}

For simplicity, we give the proof in the case where $b$ is smooth, so that all the computations are rigorous, but keeping track that the estimate (<ref>) only depends on $\| b\|_{L^q_t C^\alpha_x}$.
The general case then follows by standard (but a bit tedious) approximation arguments, similar to those of Theorems <ref>-<ref>; for estimate (<ref>), one can alternatively invoke <cit.>.

For smooth $b$, $\partial_h X^x_{s\to t}$ is classically characterized as the unique solution to the affine equation

\begin{equation}\label{eq:variational-malliavin}
\partial_h X^x_{s\to t} = \int_s^t \nabla b_r(X^x_{s\to t}) \partial_h X^x_{s\to r} \dd r  + h_{s,t}.
\end{equation}

Consider the process $A_t:= \int_s^t \nabla b_r(X^x_{s\to r}) \dd r$ as usual, which satisfies (<ref>), so that it has $\PP$-a.s. finite $p$-variation for some $p<2$ and moreover

\begin{equation}\label{eq:malliavin-proof-1}
\E[\exp(\lambda \llbracket A\rrbracket_{p-\var;[s,1]}^p)]<\infty
\end{equation}

for all $\lambda\in \R$, where the estimate only depends on $\| b\|_{L^q_t C^\alpha_x}$ and does not depend on $x$ or $s$.
Interpreting (<ref>) as an affine Young equation and applying Lemma <ref> from Appendix <ref> with $\tilde{p}=2$, we then find $C>0$ such that

\begin{equation*}
|\partial_h X^x_{s\to t}| \leq C e^{C \llbracket A\rrbracket_{p-\var;[s,t]}^p} \llbracket h\rrbracket_{2-\var;[s,t]};
\end{equation*}

taking first supremum over $h\in C^{2-\var}$ with $\| h\|_{2-\var}=1$ and then over $t\in [s,1]$, we arrive at the pathwise $\PP$-a.s. inequality

\begin{equation*}
\sup_{t\in [s,1]} \| h\mapsto \partial_h X^x_{s\to t} \|_{\mathcal{L}(C^{2-\var};\R^d)} \leq e^{C \llbracket A\rrbracket_{p-\var;[s,1]}^p}.
\end{equation*}

Taking the $L^m$-norm on both sides, using (<ref>), then readily yields (<ref>).

Estimate (<ref>) then follows from the characterization of $D X^x_{s,t}$ as a random linear operator from $\mathcal{H}^H$ to $\R^d$, combined with the functional embedding $\mathcal{H}^H\hookrightarrow C^{2-\var}_t$, see Lemma <ref> in Appendix <ref> for $H\in(0,1/2)$ and recall that $\cH^H\hookrightarrow C^{1-\var}_t$ for $H\geq 1/2$.

Results on differentiability beyond the usual Malliavin sense, in the sense of the existence of $\partial_h X^x_{s,t}$ for $h$ belonging to a larger class than $\mathcal{H}^H$, were already observed for standard SDEs in [61] and have natural explanations in rough path theory, cf. [18, 41];
in these works however only $h\in C^{\tilde p-\var}_t$ for some $\tilde p<2$ are allowed.
Here instead, not only are we able to reach $C^{2-\var}_t$, but the result can be further strengthened to allow for some $\tilde p>2$: indeed, the key point is a combination of estimate (<ref>) and Lemma <ref>, which works as long as the condition $1/\tilde{p}>1-1/p$ is satisfied.

§ MCKEAN-VLASOV EQUATIONS

Armed with the stability estimate (<ref>), we can now solve distribution dependent SDEs (henceforth DDSDEs) of the form

\begin{equation}\label{eq:ddsde-general}
X_t = X_0 + \int_0^t F_s(X_s,\mu_s)\dd s + B^H_t, \quad \mu_t=\mathcal{L}(X_t).
\end{equation}

The initial condition $X_0$ is assumed to be $\cF_0$-measurable, in particular, independent of $B^H$.
The idea that estimates of the form (<ref>), where the difference of two drifts only appears in the weaker norm of $L^q_t C^{\alpha-1}_x$, can be exploited to solve DDSDEs was first introduced in [49]; the results presented here can be regarded as a natural extension, requiring less time regularity on the drift and allowing to cover $H>1$ as well. In particular, as in the previous sections, we will not need to exploit Girsanov transform, which instead played a prominent role in [49].

Since our analysis also includes the case of distributional drifts $F$, we provide a meaningful definition of solution;  observe that in the case $F$ is actually continuous in the space variable (i..e $\alpha>0$), it reduces to the classical one.

Let $H\in (0,\infty)\setminus \N$ and $F:[0,1]\times \mathcal{P}(\R^d)\to C^\alpha_x$ be a measurable function. We say that a tuple $(\Omega,\bF,\PP; X,B^H)$ is a weak solution to (<ref>) if:

i) $B^H$ is an $\bF$-fBm of parameter $H$ and $X$ is $\bF$-adapted;

ii) setting $b^X_t(\cdot):=F_t(\cdot,\mathcal{L}(X_t))$, it holds $b^X\in L^q_t C^\alpha_x$ for some $(q,\alpha)$ satisfying (<ref>);

iii) $X$ solves the SDE associated to $b^X$, in the sense of Section <ref>.

Similarly to Definition <ref>, one can immediately extend the concepts of strong existence, pathwise uniqueness and uniqueness in law to the DDSDE (<ref>).
With a slight abuse, we will use the terminology input data of the DDSDE (<ref>) to indicate both the pair $(X_0,B^H)$ (when discussing strong existence and/or pathwise uniqueness of solutions) and the pair $(\xi,\mu^H)=(\mathcal{L}(X_0),\mathcal{L}(B^H))$ (when discussing uniqueness in law).
We are now ready to formulate our main assumptions on the drift $F$.

Let $H\in (0,\infty)\setminus \N$ fixed, $F:[0,1]\times \cP(\R^d)\to C^\alpha_x$ be a measurable function; we assume that there exist parameters $(\alpha,q)$ satisfying (<ref>) and $h\in L^q_t$ such that:

i) for all $t\in [0,1],\, \mu\in \cP(\R^d)$, it holds $\| F_t(\cdot,\mu)\|_{C^\alpha} \leq h_t$;

ii) for all $t\in [0,1],\, \mu,\nu\in \cP(\R^d)$, it holds $\| F_t(\cdot,\mu)-F_t(\cdot,\nu)\|_{C^{\alpha-1}} \leq h_t \mathbb{W}_1(\mu,\nu)$;

Basic examples of $F$ satisfying Assumption (<ref>) include the following (for their verification, we refer to Section 2.1 from [49]):

i) The true McKean–Vlasov case $F_t(\cdot,\mu)=f_t(\cdot)+(g_t\ast \mu)(\cdot)$ for $f,g\in L^q_t C^\alpha_x$;

ii) Mean-dependence of the form $F_t(\cdot,\mu)=f_t(\cdot\,-\langle \mu\rangle)$, where $\langle \mu\rangle:=\int y\,\mu(\dd y)$;