# Weak solutions for singular Lévy SDEs in the rough regime

Helena Kremp  Nicolas Perkowski

Rough weak solutions for singular Lévy SDEs
Helena Kremp111Technische Universität Wien<EMAIL_ADDRESS>Nicolas Perkowski222Freie Universität Berlin<EMAIL_ADDRESS>

We introduce a weak solution concept (called ”rough weak solutions”) for
singular SDEs with additive $\alpha$-stable Lévy noise (including the Brownian
noise case) and prove its equivalence to martingale solutions from [KP22] in
the Young and rough regularity regime. In the rough regime this leads to the
construction of certain rough stochastic sewing integrals involved. For rough
weak solutions, we can then prove a generalized Itô formula. Furthermore, we
show that canonical weak solutions are wellposed in the Young case (and
equivalent to rough weak solutions), while ill-posed in the rough case. For
the latter, we construct a counterexample for uniqueness in law.
Keywords: fractional Laplace operator, singular Lévy SDEs, equivalence of weak
solution concepts, paracontrolled distributions, rough stochastic integrals
MSC2020: 60H10, 60G51, 60L20, 60L40, 60K37.

##  1 Introduction

We study the weak well-posedness of SDEs

$dX_{t}=V(t,X_{t})dt+dL_{t},\qquad X_{0}=x\in\mathbb{R}^{d},$ (1.1)

driven by non-degenerate symmetric $\alpha$-stable Lévy noise $L$ for
$\alpha\in(1,2]$ and with drift $V(t,\cdot)$ that is a Besov distribution in
the space variable.

The special case where $L$ is a Brownian motion has received lots of attention
in recent years, since such singular diffusions arise as models for stochastic
processes in random media. Examples are random directed polymers [AKQ14, DD16,
CSZ17], self-attracting Brownian motion in a random medium [CC18], or a
continuum analogue of Sinai’s random walk in random environment (Brox
diffusion, [Bro86]). Singular diffusions also arise as “stochastic
characteristics” of singular SPDEs, for example the KPZ equation, cf. [GP17],
or the parabolic Anderson model, cf. [CC18].

SDEs with distributional drifts were first considered in [BC01, FRW03] in the
one-dimensional time-homogeneous setting. Of course, for distributional $V$
the point evaluation $V(t,X_{t})$ is not meaningful, so a priori it is not
clear how to make sense of (1.1). The appropriate perspective is not to
consider $V(t,X_{t})$ at fixed time $t$, but rather to work with the integral
$\int_{0}^{t}V(s,X_{s})ds$. The intuition is that, because of small scale
oscillations of $X$ induced by the oscillations of $L$, we only “see an
averaged version” of $V$ and this gives rise to some regularization, at least
for a Brownian motion or a sufficiently “wild” Lévy jump process. On the other
hand we would not expect any regularization from a Poisson process. In the
Brownian case, this intuition can be made rigorous in different ways. For
example via a Zvonkin transform which removes the drift, cf. [Zvo74, Ver81,
BC01, KR05, FGP10, FIR17], by considering the associated martingale problem
and by constructing a domain for the singular generator, cf. [FRW03, DD16,
CC18], or by Dirichlet forms as in [Mat94]. In the one-dimensional case it is
also possible to apply an Itô-McKean construction based on space and time
transformations, cf. [Bro86].

In [KP22] we followed the martingale problem approach in the spirit of [DD16,
CC18] who considered the Brownian case. Formally, $X$ solves (1.1) if and only
if it solves the martingale problem for the generator
$\mathcal{G}^{V}=\partial_{t}-\mathfrak{L}^{\alpha}_{\nu}+V\cdot\nabla$, where
the fractional Laplacian $(-\mathfrak{L}^{\alpha}_{\nu})$ is the generator of
$L$. That is, for all functions $u$ in the domain of $\mathcal{G}^{V}$, the
process $u(t,X_{t})-u(0,x)-\int_{0}^{t}\mathcal{G}^{V}u(s,X_{s})ds$,
$t\geqslant 0$, is a martingale. One difficulty is that the domain of
$\mathcal{G}^{V}$ necessarily has a trivial intersection with the set of
smooth functions: If $u$ is smooth, then
$(\partial_{t}-\mathfrak{L}^{\alpha}_{\nu})u$ is smooth as well, while for
non-constant $u$ the product $V\cdot\nabla u$ is only a distribution, but not
a continuous function. If we want $\mathcal{G}^{V}u$ to be a continuous
function, then $u$ has to be non-smooth, so that
$(\partial_{t}-\mathfrak{L}^{\alpha}_{\nu})u$ is also a distribution which has
appropriate cancellations with $V\cdot\nabla u$.

We can find such $u$ by solving the Kolmogorov backward equation for
continuous right-hand sides and regular terminal conditions, such that
$\mathcal{G}^{\mathscr{V}}u=f$ by construction, as carried out in [KP22].

There have been several results on singular Lévy SDEs in the Young regime in
recent years. [ABM20] consider the time-homogeneous one-dimensional case and
construct weak solutions via a Zvonkin transform. They additionally establish
strong uniqueness and existence by a Yamada-Watanabe type argument, which in
particular is restricted to $d=1$. Related to that, but for Hölder continuous
drift of regularity at least $1-\alpha/2$, [Pri12] proves pathwise uniqueness
in the multidimensional, time-homogeneous case. Two nearly simultaneous
articles, [LZ19] and [dRM19], consider the multidimensional case (time-
homogeneous, respectively time-inhomogeneous) and prove existence and
uniqueness for the martingale problem. They consider $V\in
C([0,T],B_{p,q}^{\beta})$ for general $p,q$ (subject to suitable conditions),
where for $p=q=\infty$ they require the Young regime, $\beta>(1-\alpha)/2$.
Let us also mention [HL20], who prove pathwise regularization by noise results
for SDEs driven by (very irregular) fractional Lévy noise, based on the
methods of [CG16, HP21].

We treat the multidimensional time-inhomogeneous case with drifts in the rough
regime. However, we concentrate on $B_{\infty,\infty}$ Besov spaces and do not
consider $B_{p,q}$ for general $p,q$.

The main contribution of this paper is the derivation of a well-posed rough
weak solution concept. In the case of bounded and measurable coefficients the
equivalence of probabilitic weak solutions and solutions of the martingale
problem is by now classical, cf. [SV06] (in the Brownian noise case) and
[KC11] (in the Lévy noise case). It has so far been an open problem (in both
the Brownian and Lévy noise case), whether these results can be generalized to
distributional drifts in the rough regime. In the Young regime with time-
independent drift [ABM20] introduces a canonical weak solution concept,
replacing the singular drift by the limit of smooth drift terms. The same
concept can be considered in multiple dimensions and with time-depending
drift. A canonical weak solution is then a tuple of stochastic processes
$(X,L)$ on some probability base space, such that $L$ is a symmetric
$\alpha$-stable Lévy process and $X$ is given by

$\displaystyle X=x+Z+L,$ (1.2)

for a continuous drift process $Z$ that is given as a limiting object (in
probability)

$\displaystyle
Z=\lim_{n\to\infty}\int_{0}^{\cdot}V^{n}(r,X_{r})dr=:\lim_{n\to\infty}Z^{n}$

for smooth $(V^{n})$ approximating $V$. Furthermore, $Z$ shall satisfy the
following Hölder-type bound: there exists $C>0$, such that for all $0\leqslant
s<t\leqslant T$,

$\displaystyle\mathds{E}[\lvert Z_{t}-Z_{s}\rvert^{2}]\leqslant C\lvert
t-s\rvert^{2(\alpha+\beta)/\alpha}.$ (1.3)

The particular Hölder-regularity in (1.3) originates from the time regularity
of the solution of the Kolmogorov backward equation. This implies that the
solution $X$ is a Dirichlet process and Itô formulas are available.
The recent article [IR22] consideres the multidimensional Brownian noise case
and proves equivalence of martingale solutions and so-called $B$-solutions
from [IR22, Definition 6.2] with $B=C_{T}\mathscr{C}^{\beta}_{\mathbb{R}^{d}}$
for $\beta\in(-1/2,0)$, that additionally satisfy the “reinforced local time
property” (cf. [IR22, Definition 6.7]). The reinforced local time property
boils down to stability of the integral

$\displaystyle\int_{0}^{t}\nabla u(s,X_{s})dZ_{s},$ (1.4)

when approximating the drift term by $(Z^{n})$ and $u$ solving
$\mathcal{G}^{V}u=f$, by $(u^{n})\subset
C^{1,2}_{b}([0,T]\times\mathbb{R}^{d})$.
In the Young case we show that the regularity of the PDE solution, stability
of the PDE solution map and the bound (1.3) together yield stability for the
stochastic integral (1.4) using the stochastic sewing lemma by [Lê20].
However, this fails in the rough case as the PDE solution is too irregular. In
the setting of general dimension, time-depending drift and $\alpha$-stable
noise for $\alpha\in(1,2]$, we prove that canonical weak solutions are
equivalent to martingale solutions in the Young regime. In the rough case, we
prove that canonical weak solutions are in general non-unique in law.
Heuristically, this is due to the fact that the canonical weak solution does
not uniquely determine the enhancement $\mathscr{V}$ of $V$. In the one-
dimensional Brownian case the situation becomes particularily interesting.
While for any smooth approximating sequence $(V^{n})$ of the singular drift
$V$, the strong solutions $X^{n}$ of

$\displaystyle dX_{t}^{n}=V^{n}(t,X_{t}^{n})dt+dB_{t}$

converge in distribution to the same limit, given by the solution of the
$\mathcal{G}^{\cal V}$\- martingale problem, it is however not the case that
there exists a unique canonical weak solution in the above sense.
Our approach to obtain a well-posed weak solution concept in the rough regime
is to impose further assumptions that ensure the uniqueness of the extension
of the integral (1.4). To that aim, we use ideas from rough paths theory,
specifically the construction of rough stochastic integrals from [FHL21].
Lifting $Z$ to a rough stochastic integrator process $(Z,\mathbb{Z}^{V})$
enables to extend the integral from smooth integrands $\nabla u^{n}$ in a
stable manner to (para-)controlled integrands $\nabla u$. The lift is chosen
in such a way as to correct for the most irregular terms in the paracontrolled
decomposition of $\nabla u$. We then define rough weak solutions accordingly.
That is, we require a rough weak solution $X$ to satisfy the conditions of a
canonical weak solution and to be furthermore such that the iterated integrals
$\mathbb{Z}^{V}$ are well-defined and satisfy suitable Hölder-moment-bounds.
Here $\mathbb{Z}^{V}$ formally corresponds to the resonant product component
in the enhancement $\mathscr{V}$. We say that $(Z,\mathbb{Z}^{V})$ is a so-
called rough stochastic integrator, while the process $(\nabla u(t,X_{t}))$
for the solution $u$ of the Kolmogorov equation for regular right-hand side,
is stochastically controlled.
The terms “stochastically controlled” and “rough (stochastic) integrator” are
motivated by [FHL21], but our definitions and the integral we construct
differ. The difficulty here does not arise due to a low regularity integrator,
but due to a integrand of low regularity, that is however controlled. Further
difficulties arise due to the integrability issue in the pure stable noise
case, which means that we can construct the integral in $L^{2}(\mathds{P})$,
but possibly not in $L^{p}(\mathds{P})$ for $p>2$. Summarizing, for a rough
weak solution $X$ we obtain a stable rough stochastic integral

$\displaystyle\int_{0}^{t}\nabla u(s,X_{s})d(Z,\mathbb{Z}^{V})_{s}$ (1.5)

in $L^{2}(\mathds{P})$. For regular integrands the rough stochastic integral
(1.5) coincides with the stochastic integral against $Z$ in (1.4). More
generally, we construct the rough stochastic integral
$\int_{0}^{\cdot}f_{t}d(Z,\mathbb{Z}^{A})_{t}$, for rough stochastic
integrators $(Z,\mathbb{Z}^{A})$ for a given stochastic process $A$ (formally
$\mathbb{Z}^{A}_{st}=\int_{s}^{t}A_{s,r}dZ_{r}$) and an integrand $f$, that is
stochastically controlled by $A$.
The stability of the rough stochastic integral (1.5) then enables to prove
that a rough weak solution is indeed a solution of the
$\mathcal{G}^{\mathscr{V}}$-martingale problem, in particular unique in law.
To prove that a martingale solution is a rough weak solution we need to show
the existence of the iterated integrals $\mathbb{Z}^{V}$ satisfying suitable
bounds. We show that the bounds on $\mathbb{Z}^{V}$ are implied by regularity
properties and fortunate cancellations between the solutions of Kolmogorov
backward equations for certain singular paracontrolled terminal conditions,
whose existence theory we carried out in [KP23].

The paper is structured as follows. Section 3 introduces the rough weak
solution concept and states the main Theorems 3.10 and 3.11. In Section 4 we
construct the general rough stochastic integral. In Section 5 we prove in
Theorems 5.7 and 5.10 equivalence of the weak solution concepts, which
together imply Theorem 3.10. Furthermore we prove a generalized Itô formula
for rough weak solutions in Theorem 5.11. Section 6 investigates the canonical
weak solution concept. We prove well-posedness in the Young case and ill-
posedness in the rough case.

##  2 Preliminaries

Let $(p_{j})_{j\geqslant-1}$ be a smooth dyadic partition of unity, i.e. a
family of functions $p_{j}\in C^{\infty}_{c}(\mathbb{R}^{d})$ for
$j\geqslant-1$, such that

  1. 1.)

$p_{-1}$ and $p_{0}$ are non-negative radial functions (they just depend on
the absolute value of $x\in\mathbb{R}^{d}$), such that the support of $p_{-1}$
is contained in a ball and the support of $p_{0}$ is contained in an annulus;

  2. 2.)

$p_{j}(x):=p_{0}(2^{-j}x)$, $x\in\mathbb{R}^{d}$, $j\geqslant 0$;

  3. 3.)

$\sum_{j=-1}^{\infty}p_{j}(x)=1$ for every $x\in\mathbb{R}^{d}$; and

  4. 4.)

$\operatorname{supp}(p_{i})\cap\operatorname{supp}(p_{j})=\emptyset$ for all
$\lvert i-j\rvert>1$.

We then define the Besov spaces for $p,q\in[1,\infty]$,

$\displaystyle B^{\theta}_{p,q}:=\\{u\in\mathcal{S}^{\prime}:\lVert
u\rVert_{B^{\theta}_{p,q}}=\big{\lVert}(2^{j\theta}\lVert\Delta_{j}u\rVert_{L^{p}})_{j\geqslant-1}\big{\rVert}_{\ell^{q}}<\infty\\},$
(2.1)

where $\Delta_{j}u=\mathcal{F}^{-1}(p_{j}\mathcal{F}u)$ are the Littlewood-
Paley blocks, and the Fourier transform is defined with the normalization
$\hat{\varphi}(y):=\mathcal{F}\varphi(y):=\int_{\mathbb{R}^{d}}\varphi(x)e^{-2\pi
i\langle x,y\rangle}dx$ (and $\mathcal{F}^{-1}\varphi(x)=\hat{\varphi}(-x)$);
moreover, $\mathcal{S}$ are the Schwartz functions and $\mathcal{S}^{\prime}$
are the Schwartz distributions.
Let $C^{\infty}_{b}=C^{\infty}_{b}(\mathbb{R}^{d},\mathbb{R})$ denote the
space of bounded and smooth functions with bounded partial derivatives. For
$q=\infty$, the space $B^{\theta}_{p,\infty}$ has the unpleasant property that
$C^{\infty}_{b}\subset B^{\theta}_{p,\infty}$ is not dense. Therefore, we
rather work with the following space:

$\displaystyle
B^{\theta}_{p,\infty}:=\\{u\in\mathcal{S}^{\prime}\mid\lim_{j\to\infty}2^{j\theta}\lVert\Delta_{j}u\rVert_{L^{p}}=0\\},$
(2.2)

for which $C^{\infty}_{b}$ is a dense subset (cf. [BCD11, Remark 2.75]). We
also use the notation
$\mathscr{C}^{\theta}_{\mathbb{R}^{d}}:=(\mathscr{C}^{\theta})^{d}=\mathscr{C}^{\theta}(\mathbb{R}^{d},\mathbb{R}^{d})$,
$\mathscr{C}^{\theta-}:=\bigcap_{\gamma<\theta}\mathscr{C}^{\gamma}$ and
$\mathscr{C}^{\theta+}=\bigcup_{\gamma>\theta}\mathscr{C}^{\gamma}$.
Furthermore, we introduce the notation
$\mathscr{C}^{\theta}_{p}:=B^{\theta}_{p,\infty}$ for $\theta\in\mathbb{R}$
and $p\in[1,\infty]$, where
$\mathscr{C}^{\theta}:=\mathscr{C}^{\theta}_{\infty}$ with norm denoted by
$\lVert\cdot\rVert_{\theta}:=\lVert\cdot\rVert_{\mathscr{C}^{\theta}}$.
For $1\leqslant p_{1}\leqslant p_{2}\leqslant\infty$, $1\leqslant
q_{1}\leqslant q_{2}\leqslant\infty$ and $s\in\mathbb{R}$, the Besov space
$B^{s}_{p_{1},q_{1}}$ is continuously embedded in
$B_{p_{2},q_{2}}^{s-d(1/p_{1}-1/p_{2})}$ (cf. [BCD11, Proposition 2.71]).
Furthermore, we will use that for $u\in B^{s}_{p,q}$ and a multi-index
$n\in\mathbb{N}^{d}$, $\lVert\partial^{n}u\rVert_{B^{s-\lvert
n\rvert}_{p,q}}\lesssim\lVert u\rVert_{B^{s}_{p,q}}$, which follows from the
more general multiplier result from [BCD11, Proposition 2.78].
We recall from Bony’s paraproduct theory (cf. [BCD11, Section 2]) that in
general for $u\in\mathscr{C}^{\theta}$ and $v\in\mathscr{C}^{\beta}$ with
$\theta,\beta\in\mathbb{R}$, the product $uv:=u\varolessthan
v+u\varogreaterthan v+u\varodot v$ , is well defined in
$\mathscr{C}^{\min(\theta,\beta,\theta+\beta)}$ if and only if
$\theta+\beta>0$. Denoting $S_{i}u=\sum_{j=-1}^{i-1}\Delta_{j}u$, the
paraproducts are defined as follows

$\displaystyle u\varolessthan v:=\sum_{i\geqslant-1}S_{i-1}u\Delta_{i}v,\quad
u\varogreaterthan v:=v\varolessthan u,\quad u\varodot v:=\sum_{\lvert
i-j\rvert\leqslant 1}\Delta_{i}u\Delta_{j}v.$

Here, we use the notation of [MP19, MW17] for the para- and resonant products
$\varolessthan,\varogreaterthan$ and $\varodot$.
In estimates we often use the notation $a\lesssim b$, which means, that there
exists a constant $C>0$, such that $a\leqslant Cb$. In the case that we want
to stress the dependence of the constant $C(d)$ in the estimate on a parameter
$d$, we write $a\lesssim_{d}b$.
The paraproducts satisfy the following estimates for
$p,p_{1},p_{2}\in[1,\infty]$ with
$\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}\leqslant 1$ and
$\theta,\beta\in\mathbb{R}$ (cf. [PvZ22, Theorem A.1] and [BCD11, Theorem
2.82, Theorem 2.85])

$\displaystyle\lVert u\varodot v\rVert_{\mathscr{C}^{\theta+\beta}_{p}}$
$\displaystyle\lesssim\lVert u\rVert_{\mathscr{C}^{\theta}_{p_{1}}}\lVert
v\rVert_{\mathscr{C}^{\beta}_{p_{2}}},\qquad\text{if }\theta+\beta>0,$ (2.3)
$\displaystyle\lVert u\varolessthan
v\rVert_{\mathscr{C}^{\beta}_{p}}\lesssim\lVert u\rVert_{L^{p_{1}}}\lVert
v\rVert_{\mathscr{C}^{\beta}_{p_{2}}}$ $\displaystyle\lesssim\lVert
u\rVert_{\mathscr{C}^{\theta}_{p_{1}}}\lVert
v\rVert_{\mathscr{C}^{\beta}_{p_{2}}},\qquad\text{if }\theta>0,$
$\displaystyle\lVert u\varolessthan v\rVert_{\mathscr{C}^{\beta+\theta}_{p}}$
$\displaystyle\lesssim\lVert u\rVert_{\mathscr{C}^{\theta}_{p_{1}}}\lVert
v\rVert_{\mathscr{C}^{\beta}_{p_{2}}},\qquad\text{if }\theta<0.$

So if $\theta+\beta>0$, we have $\lVert
uv\rVert_{\mathscr{C}^{\gamma}_{p}}\lesssim\lVert
u\rVert_{\mathscr{C}^{\theta}_{p_{1}}}\lVert
v\rVert_{\mathscr{C}^{\beta}_{p_{2}}}$ for
$\gamma:=\min(\theta,\beta,\theta+\beta)$.
Next, we collect some facts about $\alpha$-stable Lévy processes and their
generators and semigroups. For $\alpha\in(0,2]$, a symmetric $\alpha$-stable
Lévy process $L$ is a Lévy process, that moreover satisfies the scaling
property $(L_{kt})_{t\geqslant 0}\stackrel{{\scriptstyle
d}}{{=}}k^{1/\alpha}(L_{t})_{t\geqslant 0}$ for any $k>0$ and
$L\stackrel{{\scriptstyle d}}{{=}}-L$, where $\stackrel{{\scriptstyle
d}}{{=}}$ denotes equality in law. These properties determine the jump measure
$\mu$ of $L$, see [Sat99, Theorem 14.3]. That is, if $\alpha\in(0,2)$, the
Lévy jump measure $\mu$ of $L$ is given by

$\displaystyle\mu(A):=\mathds{E}\bigg{[}\sum_{0\leqslant t\leqslant
1}\mathbf{1}_{A}(\Delta
L_{t})\bigg{]}=\int_{S}\int_{\mathbb{R}^{+}}\mathbf{1}_{A}(k\xi)\frac{1}{k^{1+\alpha}}dk\tilde{\nu}(d\xi),\quad
A\in\mathcal{B}(\mathbb{R}^{d}\setminus\\{0\\}),$ (2.4)

where $\tilde{\nu}$ is a finite, symmetric, non-zero measure on the unit
sphere $S\subset\mathbb{R}^{d}$. Furthermore, we also define for
$A\in\mathcal{B}(\mathbb{R}^{d}\setminus\\{0\\})$ and $t\geqslant 0$ the
Poisson random measure

$\displaystyle\pi(A\times[0,t])=\sum_{0\leqslant s\leqslant
t}\mathbf{1}_{A}(\Delta L_{s}),$

with intensity measure $dt\mu(dy)$. Denote the compensated Poisson random
measure of $L$ by $\hat{\pi}(dr,dy):=\pi(dr,dy)-dr\mu(dy)$. We refer to the
book by Peszat and Zabczyk [PZ07] for the integration theory against Poisson
random measures and for the Burkholder-Davis-Gundy inequality [PZ07, Lemma
8.21 and 8.22], which we will both use in the sequel. The generator $A$ of $L$
satisfies $C_{b}^{\infty}(\mathbb{R}^{d})\subset\operatorname{dom}(A)$ and is
given by

$\displaystyle
A\varphi(x)=\int_{\mathbb{R}^{d}}\big{(}\varphi(x+y)-\varphi(x)-\mathbf{1}_{\\{\lvert
y\rvert\leqslant 1\\}}(y)\nabla\varphi(x)\cdot y\big{)}\mu(dy)\qquad\text{for
}\varphi\in C_{b}^{\infty}(\mathbb{R}^{d}).$ (2.5)

If $(P_{t})_{t\geqslant 0}$ denotes the semigroup of $L$, the convergence
$t^{-1}(P_{t}f(x)-f(x))\to Af(x)$ is uniform in $x\in\mathbb{R}^{d}$ (see
[PZ07, Theorem 5.4]).

Next, we recall the equivalent Fourier definition of the operator $A$ and the
Schauder and commutator estimates for its semigroup from [KP22, KP23].

######  Definition 2.1.

Let $\alpha\in(0,2)$ and let $\nu$ be a symmetric (i.e. $\nu(A)=\nu(-A)$),
finite and non-zero measure on the unit sphere $S\subset\mathbb{R}^{d}$. We
define the operator $\mathfrak{L}^{\alpha}_{\nu}$ as

$\displaystyle\mathfrak{L}^{\alpha}_{\nu}\mathcal{F}^{-1}\varphi=\mathcal{F}^{-1}(\psi^{\alpha}_{\nu}\varphi)\qquad\text{for
$\varphi\in C^{\infty}_{b}$,}$ (2.6)

where $\psi^{\alpha}_{\nu}(z):=\int_{S}\lvert\langle
z,\xi\rangle\rvert^{\alpha}\nu(d\xi).$ For $\alpha=2$, we set
$\mathfrak{L}^{\alpha}_{\nu}:=-\frac{1}{2}\Delta$.

######  Remark 2.2.

If we take $\nu$ as a suitable multiple of the Lebesgue measure on the sphere,
then $\psi^{\alpha}_{\nu}(z)=|2\pi z|^{\alpha}$ and thus
$\mathfrak{L}^{\alpha}_{\nu}$ is the fractional Laplace operator
$(-\Delta)^{\alpha/2}$.

######  Lemma 2.3.

333[KP22, Lemma 2.3]

Let $\alpha\in(0,2)$ and let again $\nu$ be a symmetric, finite and non-zero
measure on the unit sphere $S\subset\mathbb{R}^{d}$. Then for $\varphi\in
C^{\infty}_{b}$ we have $-\mathfrak{L}^{\alpha}_{\nu}\varphi=A\varphi$, where
$A$ is the generator of the symmetric, $\alpha$-stable Lévy process $L$ with
characteristic exponent $\mathds{E}[\exp(2\pi i\langle
z,L_{t}\rangle)]=\exp(-t\psi^{\alpha}_{\nu}(z))$. The process $L$ has the jump
measure $\mu$ as defined in Equation 2.4, with $\tilde{\nu}=C\nu$ for some
constant $C>0$.

If $\alpha=2$, then the generator of the symmetric, $\alpha$-stable process
coincides with $\sum_{i,j}C(i,j)\partial_{x_{i}}\partial_{x_{j}}$ for an
explicit covariance matrix $C$ (cf. [Sat99, Theorem 14.2]), that is, the
generator of $\sqrt{2C}B$ for a standard Brownian motion $B$. To ease
notation, we consider here $C=\frac{1}{2}\operatorname{Id}_{d\times d}$ and
whenever we refer to the case $\alpha=2$, we mean the standard Brownian motion
noise case and $\mathfrak{L}^{\alpha}_{\nu}=-\frac{1}{2}\Delta$.

######  Assumption 2.4.

Throughout the paper, we assume that the measure $\nu$ from Definition 2.1 has
$d$-dimensional support, in the sense that the linear span of its support is
$\mathbb{R}^{d}$. This means that the process $L$ can reach every open set in
$\mathbb{R}^{d}$ with positive probability.

An $\alpha$-stable, symmetric Lévy process, that satisfies 2.4, we also call
non-degenerate.

######  Proposition 2.5 (Continuity of the operator
$\mathfrak{L}^{\alpha}_{\nu}$).

444[KP23, Proposition 2.4]

Let $\alpha\in(0,2]$. Then for $\beta\in\mathbb{R}$ and $u\in C^{\infty}_{b}$,
$p\in[1,\infty]$, we have

$\displaystyle\lVert\mathfrak{L}^{\alpha}_{\nu}u\rVert_{\mathscr{C}^{\beta-\alpha}_{p}}\lesssim\lVert
u\rVert_{\mathscr{C}^{\beta}_{p}}.$

In particular, $\mathfrak{L}^{\alpha}_{\nu}$ can be uniquely extended to a
continuous operator from $\mathscr{C}^{\beta}_{p}$ to
$\mathscr{C}^{\beta-\alpha}_{p}$.

######  Remark 2.6.

One can show that the operators $A$ and $-\mathfrak{L}^{\alpha}_{\nu}$ agree
on $\bigcup_{\varepsilon>0}\mathscr{C}^{2+\varepsilon}$.

The formal generator associated to the diffusion (1.1) is thus given by

$\displaystyle\mathcal{G}^{V}=\partial_{t}-\mathfrak{L}^{\alpha}_{\nu}+V\cdot\nabla.$
(2.7)

The connection between the generator $\mathcal{G}^{V}$ and the solution $X$ is
made rigorous in [KP22] by solving the associated martingale problem, that we
recall below. Solving the martingale problem is based on solving the singular
Kolmogorov backward PDE via the paracontrolled approach (in the rough
regularity regime). Before stating the result from [KP23], let us introduce
the notation

$\displaystyle\mathcal{K}(\eta_{1},\eta_{2}):=\big{[}\mathring{\Delta}_{T}\ni(s,t)\mapsto\sum_{j=1}^{d}P_{t-s}\partial_{j}\eta_{1}^{i}(t)\varodot\eta_{2}^{j}(s)\big{]},$

for $\eta_{1},\eta_{2}\in C_{T}C^{\infty}_{b}(\mathbb{R}^{d},\mathbb{R}^{d})$
and $\mathring{\Delta}_{T}:=\\{(s,t)\in[0,T]^{2}\mid s<t\\}$. Furthermore, we
set $\mathcal{K}(\eta):=\mathcal{K}(\eta,\eta)$. Let also
$\Delta_{T}:=\\{(s,t)\in[0,T]^{2}\mid s\leqslant t\\}$.
For a Banach space $X$ and $\gamma\in(0,1)$, we define the blow-up space

$\displaystyle\mathcal{M}_{\mathring{\Delta}_{T}}^{\gamma}X:=\\{g:\mathring{\Delta}_{T}\to
X\mid\sup_{0\leqslant s<t\leqslant T}(t-s)^{\gamma}\lVert
g(s,t)\rVert_{X}<\infty\\}.$

Moreover, for a sequence $(a_{m,n})_{m,n\in\mathbb{N}}$ in a Banach space, for
which the convergence

$\displaystyle\lim_{m\to\infty}\lim_{n\to\infty}a_{m,n}=a=\lim_{n\to\infty}\lim_{m\to\infty}a_{m,n}$

holds, we use the short-hand notation $a^{m,n}\to a$ for $m,n\to\infty$ or
$\lim_{m,n\to\infty}a^{m,n}=a$.
Next, we define the space of enhanced distributions
$\mathcal{X}^{\beta,\gamma}$ as follows. Comparing with [KP23, Definition
4.2], we furthermore assume that for
$(\mathcal{V}_{1},\mathcal{V}_{2})\in\mathcal{X}^{\beta,\gamma}$, the mixed
resonant products $(\mathcal{K}(V^{n},V^{m}))$ converge to $\mathscr{V}_{2}$.

######  Definition 2.7 (Enhancement).

Let $T>0$. For $\beta\in(\frac{2-2\alpha}{3},\frac{1-\alpha}{2})$ and
$\gamma\in[\frac{2\beta+2\alpha-1}{\alpha},1)$, we define
$\mathcal{X}^{\beta,\gamma}$ as the closure of

$\displaystyle\\{(\eta_{1},\mathcal{K}(\eta_{1},\eta_{2}))\mid\eta_{1},\eta_{2}\in
C_{T}C^{\infty}_{b}(\mathbb{R}^{d},\mathbb{R}^{d})\\}$

in
$C_{T}\mathscr{C}^{\beta+(1-\gamma)\alpha}\times\mathcal{M}_{\mathring{\Delta}_{T}}\mathscr{C}^{2\beta+\alpha-1}_{\mathbb{R}^{d\times
d}}$. We say that $\mathscr{V}$ is a lift or enhancement of $V$ if
$\mathcal{V}_{1}=V$ and also shortly write $V\in\mathcal{X}^{\beta,\gamma}$.
For $\beta\in(\frac{1-\alpha}{2},0)$ and
$\gamma\in[\frac{\beta-1}{\alpha},1)$, we set
$\mathcal{X}^{\beta,\gamma}=C_{T}\mathscr{C}^{\beta+(1-\gamma)\alpha}$.

######  Remark 2.8.

$V\in\mathcal{X}^{\beta,\gamma}$ in the sense of Definition 2.7 implies that
$V$ is also enhanced in the sense of [KP23, Definition 4.2] and in the sense
of [KP22, Definition 3.5].

Let us define the function spaces, that we encounter in the sequel. For
$\gamma\in(0,1)$, $T>0$ and $\overline{T}\in(0,T]$, and a Banach space $X$, we
define the blow-up space

$\displaystyle\mathcal{M}_{\overline{T},T}^{\gamma}X:=\\{u:[T-\overline{T},T)\to
X\mid t\mapsto(T-t)^{\gamma}u_{t}\in C([T-\overline{T},T),X)\\},$

with $\lVert
u\rVert_{\mathcal{M}_{\overline{T},T}^{\gamma}X}:=\sup_{t\in[T-\overline{T},T)}(T-t)^{\gamma}\lVert
u_{t}\rVert_{X}$ and
$\mathcal{M}_{\overline{T},T}^{0}X:=C([T-\overline{T},T),X)$. For
$\overline{T}=T$, we use the notation
$\mathcal{M}^{\gamma}_{T}X:=\mathcal{M}_{T,T}^{\gamma}X$. For
$\vartheta\in(0,1]$, $\gamma\in(0,1)$, we furthermore define

$\displaystyle
C_{\overline{T},T}^{\gamma,\vartheta}X:=\biggl{\\{}u:[T-\overline{T},T)\to
X\biggm{|}\lVert f\rVert_{C_{T}^{\gamma,\vartheta}X}:=\sup_{0\leqslant
s<t<T}\frac{(T-t)^{\gamma}\lVert f_{t}-f_{s}\rVert_{X}}{\lvert
t-s\rvert^{\vartheta}}<\infty\biggr{\\}}$

and $C_{T}^{\gamma,\vartheta}X:=C_{T,T}^{\gamma,\vartheta}X$. Let us also
define for $\vartheta\in(0,1]$, $\overline{T}\in(0,T]$, the space of
$\vartheta$-Hölder continuous functions on $[T-\overline{T},T]$ with values in
$X$,

$\displaystyle
C_{\overline{T},T}^{\vartheta}X:=\biggl{\\{}u:[T-\overline{T},T]\to
X\biggm{|}\lVert u\rVert_{C_{T}^{\vartheta}X}:=\sup_{T-\overline{T}\leqslant
s<t\leqslant T}\frac{\lVert u_{t}-u_{s}\rVert_{X}}{\lvert
t-s\rvert^{\vartheta}}<\infty\biggr{\\}}$

and $C_{T}^{\vartheta}X:=C_{T,T}^{\vartheta}X$. We set
$C_{\overline{T},T}^{0,\vartheta}X:=C^{\vartheta}([T-\overline{T},T),X)$.
Then we define for $\gamma\in(0,1)$ and $\theta\in\mathbb{R}$,
$p\in[1,\infty]$, the space

$\displaystyle\mathcal{L}_{T}^{\gamma,\theta}:=\mathcal{M}_{T}^{\gamma}\mathscr{C}^{\theta}_{p}\cap
C_{T}^{1-\gamma}\mathscr{C}^{\theta-\alpha}_{p}\cap
C_{T}^{\gamma,1}\mathscr{C}^{\theta-\alpha}_{p}.$ (2.8)

We moreover define for $\gamma=0$,

$\displaystyle\mathcal{L}_{T}^{0,\theta}:=C_{T}^{1}\mathscr{C}^{\theta-\alpha}_{p}\cap
C_{T}\mathscr{C}^{\theta}_{p},$ (2.9)

where $C^{1}_{T}X$ denotes the space of $1$-Hölder or Lipschitz functions with
values in $X$.
For $\overline{T}\in(0,T)$, we define
$\mathcal{L}_{\overline{T},T}^{\gamma,\theta}:=\mathcal{M}_{\overline{T},T}^{\gamma}\mathscr{C}^{\theta}_{p}\cap
C_{\overline{T},T}^{1-\gamma}\mathscr{C}^{\theta-\alpha}_{p}\cap
C_{\overline{T},T}^{\gamma,1}\mathscr{C}^{\theta-\alpha}_{p}$ and similarly
$\mathcal{L}_{\overline{T},T}^{0,\theta}$.
The spaces $\mathcal{L}_{T}^{\gamma,\theta}$ are Banach spaces equipped with
the norm

$\displaystyle\lVert u\rVert_{\mathcal{L}_{T}^{\gamma,\theta}}$
$\displaystyle:=\lVert
u\rVert_{\mathcal{M}_{T}^{\gamma}\mathscr{C}^{\theta}_{p}}+\lVert
u\rVert_{C_{T}^{1-\gamma}\mathscr{C}^{\theta-\alpha}_{p}}+\lVert
u\rVert_{C_{T}^{\gamma,1}\mathscr{C}^{\theta-\alpha}_{p}}$
$\displaystyle=\mbox{$\displaystyle\sup_{t\in[0,T)}(T-t)^{\gamma}\lVert
u_{t}\rVert_{\mathscr{C}^{\theta}_{p}}+\sup_{0\leqslant s<t\leqslant
T}\frac{\lVert u_{t}-u_{s}\rVert_{\mathscr{C}^{\theta-\alpha}_{p}}}{\lvert
t-s\rvert^{1-\gamma}}+\sup_{0\leqslant s<t<T}\frac{(T-t)^{\gamma}\lVert
u_{t}-u_{s}\rVert_{\mathscr{C}^{\theta-\alpha}_{p}}}{\lvert t-s\rvert}.$}$

We can finally recall the result from [KP23, Theorem 4.1, Theorem 4.7,
Corollary 4.10 and Theorem 4.12], which treat singular paracontrolled terminal
conditions.

######  Theorem 2.9.

Let $T>0$, $\alpha\in(1,2]$, $p\in[1,\infty]$ and
$\beta\in(\frac{2-2\alpha}{3},0)$ and
$\mathcal{V}\in\mathscr{X}^{\beta,\gamma^{\prime}}$.
For
$\beta\in(\frac{1-\alpha}{2},0),\gamma^{\prime}\in[\frac{\beta-1}{\alpha},1)$,
let $f\in\mathcal{L}_{T}^{\beta,\gamma^{\prime}}$ and
$u^{T}\in\mathscr{C}_{p}^{(1-\gamma^{\prime})\alpha+\beta}$.
For
$\beta\in(\frac{2-2\alpha}{3},\frac{1-\alpha}{2}],\gamma^{\prime}\in[\frac{2\beta+2\alpha-1}{\alpha},1)$,
let

$\displaystyle f=f^{\sharp}+f^{\prime}\varolessthan V$

for $f^{\sharp}\in\mathcal{L}_{T}^{\gamma^{\prime},\alpha+2\beta-1}$,
$f^{\prime}\in(\mathcal{L}_{T}^{\gamma^{\prime},\alpha+\beta-1})^{d}$ and

$\displaystyle u^{T}=u^{T,\sharp}+u^{T,\prime}\varolessthan V_{T}$

for $u^{T,\sharp}\in\mathscr{C}^{(2-\gamma^{\prime})\alpha+2\beta-1}_{p}$,
$u^{T,\prime}\in(\mathscr{C}^{\alpha+\beta-1}_{p})^{d}$.
Then there exists a unique mild solution
$u\in\mathcal{L}_{T}^{\gamma,\alpha+\beta}$ for
$\gamma\in(\gamma^{\prime},\frac{\alpha}{2-\alpha-3\beta}\gamma^{\prime})$ to
the singular Kolmogorov backward PDE

$\displaystyle\mathcal{G}^{\mathcal{V}}u=f,\qquad u(T,\cdot)=u^{T}.$

and if $\beta\in(\frac{2-2\alpha}{3},\frac{1-\alpha}{2}]$, $u$ has the
paracontrolled structure

$\displaystyle u=u^{\sharp}+\nabla u\varolessthan
J^{T}(V)+u^{T,\prime}\varolessthan P_{T-\cdot}V_{T}$

for $u^{\sharp}\in\mathcal{L}_{T}^{\gamma,2\alpha+2\beta-1}$.
Furthermore in case of regular terminal condition, that is
$f^{\prime}\in\mathcal{L}_{T}^{0,\alpha+\beta-1},f^{\sharp}\in\mathcal{L}_{T}^{0,\alpha+2\beta-1]}$
and $u^{T}=u^{T,\sharp}\in\mathscr{C}_{p}^{2\alpha+2\beta-1}$, the solution
satisfies
$u\in\mathcal{L}_{T}^{0,\alpha+\beta},u^{\sharp}\in\mathcal{L}_{T}^{0,2\alpha+2\beta-1}$.
Moreover, the solution map
$(u^{T},f,\mathscr{V})\mapsto(u,u^{\sharp})\in\mathcal{L}_{T}^{\gamma,\alpha+\beta}\times\mathcal{L}_{T}^{\gamma,2\alpha+2\beta-1}$
is locally Lipschitz continuous.

We use the definition of a solution to the singular martingale problem from
[KP22, Definition 4.1], cf. also [CC18].

######  Definition 2.10 (Martingale problem).

Let $\alpha\in(1,2]$ and $\beta\in(\frac{2-2\alpha}{3},0)$, and let $T>0$ and
$V\in\mathcal{X}^{\beta,\gamma}$. Then, we call a probability measure
$\mathds{P}$ on the Skorokhod space $(\Omega,\mathcal{F})$ a solution of the
martingale problem for $(\mathcal{G}^{V},\delta_{x})$, if

  1. 1.)

$\mathds{P}(X_{0}\equiv x)=1$ (i.e. $\mathds{P}^{X_{0}}=\delta_{x}$), and

  2. 2.)

for all $f\in C_{T}\mathscr{C}^{\varepsilon}$ with $\varepsilon>2-\alpha$ and
for all $u^{T}\in\mathcal{C}^{3}$, the process $M=(M_{t})_{t\in[0,T]}$ is a
martingale under $\mathds{P}$ with respect to $(\mathcal{F}_{t})$, where

$\displaystyle M_{t}=u(t,X_{t})-u(0,x)-\int_{0}^{t}f(s,X_{s})ds$ (2.10)

and where $u$ solves the Kolmogorov backward equation $\mathcal{G}^{V}u=f$
with terminal condition $u(T,\cdot)=u^{T}$.

######  Remark 2.11.

To solve the martingale problem, we do not need to solve the Kolmogorov
equation for singular terminal conditions. Instead it suffices to consider
regular terminal conditions $u^{T}\in\mathcal{C}^{3}$, cf. also [KP22]. To
prove the equivalence of rough weak solutions and martingale solutions, we
will employ Theorem 2.9 in its full generality.

We state a version of [KP22, Theorem 4.2] about the existence and uniqueness
of martingale solutions and [KP22, Corollary 4.5] concerning moment estimates
for the drift term.

######  Theorem 2.12.

555The statement of the theorem holds also under the weaker enhancement
assumption on $V$ from [KP22, Definition 3.5], cf. [KP22, Theorem 4.2].

Let $\alpha\in(1,2]$ and $L$ be a symmetric, $\alpha$-stable Lévy process,
such that the measure $\nu$ satisfies 2.4. Let $T>0$ and
$\beta\in((2-2\alpha)/3,0)$ and let $V\in\mathscr{X}^{\beta,\gamma}$. Then for
all $x\in\mathbb{R}^{d}$, there exists a unique solution $\mathbb{Q}$ on
$(\Omega,\mathcal{F})$ of the martingale problem for
$(\mathcal{G}^{V},\delta_{x})$. Under $\mathbb{Q}$ the canonical process is a
strong Markov process.

######  Lemma 2.13.

In the setting of Theorem 2.12, let $(V^{n})_{n\in\mathbb{N}}\subset
C_{T}C^{\infty}_{b}(\mathbb{R}^{d},\mathbb{R}^{d})$ be a smooth approximation
with $(V^{n},\mathcal{K}(V^{n}))\to\mathcal{V}$ in
$\mathscr{X}^{\beta,\gamma}$. Let $(X^{n}_{t})_{t\in[0,T]}$ be the strong
solution of the SDE

$\displaystyle dX_{t}^{n}=V^{n}(t,X^{n}_{t})dt+dL_{t},\qquad
X_{0}=x\in\mathbb{R}^{d}.$

Let666In [KP22, Corollary 4.5], the result was proven for
$\theta<\alpha+\beta$, with $\theta$ being the regularity of the PDE solution
from [KP22, section 3]. Due to Theorem 2.9, we can indeed generalize to
$\theta=\alpha+\beta$. $\theta=\alpha+\beta$ and $\rho\in 2\mathbb{N}$.
Then, there exists $N\in\mathbb{N}$, such that uniformly in $0\leqslant
r\leqslant t\leqslant T$:

$\displaystyle\sup_{n\geqslant
N}\mathds{E}\bigg{[}\bigg{\lvert}\int_{r}^{t}V^{n}(s,X^{n}_{s})ds\bigg{\rvert}^{\rho}\bigg{]}\lesssim\lvert
t-r\rvert^{\theta\rho/\alpha}.$ (2.11)

Let us end the preliminaries by stating the stochastic sewing lemma from
[Lê20, Theorem 2.1], that will be a key tool in this paper.
Here and in the following, we use the notation $Z_{st}=Z_{s,t}=Z_{t}-Z_{s}$,
$s\leqslant t$, for the increment of a stochastic process $(Z_{t})$ and
$\mathds{E}_{s}[\cdot]:=\mathds{E}[\cdot\mid\mathcal{F}_{s}]$ for the
conditional expectation. A stochastic process
$(\Xi_{st})_{(s,t)\in\Delta_{T}}$ indexed by $\Delta_{T}$ we call adapted to
$(\mathcal{F}_{t})$ if $\Xi_{st}$ is $\mathcal{F}_{t}$-measurable for all
$(s,t)\in\Delta_{T}$. For $p\in[2,\infty]$, $m\in\mathbb{N}$ and an adapted
$\mathbb{R}^{m}$-valued stochastic process $(\Xi_{st})_{(s,t)\in\Delta_{T}}$
indexed by $\Delta_{T}$, we define for $\theta\in(0,1)$,

$\displaystyle\lVert\Xi\rVert_{\theta,p}:=\sup_{(s,t)\in\Delta_{T}}\frac{\lVert\Xi_{st}\rVert_{L^{p}(\mathds{P},\mathbb{R}^{m})}}{\lvert
t-s\rvert^{\theta}},\quad\lVert\Xi\mid\mathcal{F}\rVert_{\theta,p}:=\sup_{(s,t)\in\Delta_{T}}\frac{\lVert\mathds{E}_{s}[\Xi_{st}]\rVert_{L^{p}(\mathds{P},\mathbb{R}^{m})}}{\lvert
t-s\rvert^{\theta}}.$ (2.12)

If $(Z_{t})_{t\in[0,T]}$ is an adapted process indexed by $[0,T]$, we use the
same notation (2.12) for $\Xi_{st}=Z_{st}=Z_{t}-Z_{s}$.

######  Lemma 2.14 (Stochastic sewing lemma).

Let $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\in[0,T]},\mathds{P})$ be a
complete probability space. Let $(\Xi_{s,t})_{0\leqslant s\leqslant t\leqslant
T}$ be a two-parameter stochastic process with values in $\mathbb{R}^{d}$,
that is adapted ($\Xi_{s,t}$ being $\mathcal{F}_{t}$-measurable for
$s\leqslant t$) and $L^{2}(\mathds{P})$-integrable. Let
$\delta\Xi_{s,u,t}:=\Xi_{st}-\Xi_{su}-\Xi_{ut}$, $0\leqslant s\leqslant
u\leqslant t\leqslant T$. Suppose that there are constants
$\Gamma_{1},\Gamma_{2},\varepsilon_{1},\varepsilon_{2}>0$, such that for all
$0\leqslant s\leqslant u\leqslant t\leqslant T$,

$\displaystyle\lVert\mathds{E}[\delta\Xi_{s,u,t}\mid\mathcal{F}_{s}]\rVert_{L^{2}(\mathds{P})}\leqslant\Gamma_{1}\lvert
t-s\rvert^{1+\varepsilon_{1}},\quad\lVert\delta\Xi_{s,u,t}\rVert_{L^{2}(\mathds{P})}\leqslant\Gamma_{2}\lvert
t-s\rvert^{\frac{1}{2}+\varepsilon_{2}}.$ (2.13)

Then, there exists a unique (up to modifications) stochastic process
$(I_{t})_{t\in[0,T]}:=(I_{t}(\Xi))_{t\in[0,T]}$ with values in
$\mathbb{R}^{d}$ satisfying the following properties

  1. •

$I_{0}=0$, $(I_{t})_{t\in[0,T]}$ is $(\mathcal{F}_{t})$-adapted and
$L^{2}(\mathds{P})$-integrable and

  2. •

there exist constants $C_{1}=C(\varepsilon_{1}),C_{2}=C(\varepsilon_{2})>0$,
such that for all $0\leqslant s\leqslant t\leqslant T$,

$\displaystyle\lVert I_{t}-I_{s}-\Xi_{s,t}\rVert_{L^{2}(\mathds{P})}\leqslant
C_{1}\Gamma_{1}\lvert t-s\rvert^{1+\varepsilon_{1}}+C_{2}\Gamma_{2}\lvert
t-s\rvert^{\frac{1}{2}+\varepsilon_{2}},$
$\displaystyle\lVert\mathds{E}[I_{t}-I_{s}-\Xi_{s,t}\mid\mathcal{F}_{s}]\rVert_{L^{2}(\mathds{P})}\leqslant
C_{1}\Gamma_{1}\lvert t-s\rvert^{1+\varepsilon_{1}}.$ (2.14)

Furthermore, for every $t\in[0,T]$ and any partition
$\Pi=\\{0=t_{0}<t_{1}<\dots<t_{N}=T\\}$, the Riemann sums
$I^{\Pi}_{t}=\sum_{i=0}^{N-1}\Xi_{t_{i},t_{i+1}}$ converge to $I_{t}$ in
$L^{2}(\mathds{P})$ for vanishing mesh size $\lvert\Pi\rvert:=\max_{i}\lvert
t_{i+1}-t_{i}\rvert\to 0$.

######  Remark 2.15.

We can apply the stochastic sewing lemma to a germ
$\Xi_{st}:=f_{s}Y_{st}=f_{s}(Y_{t}-Y_{s})$ for stochastic processes
$(f_{t}),(Y_{t})$. Then, typically the constants $\Gamma_{1},\Gamma_{2}$
depend linearily on the Hölder-type moment bounds of the stochastic processes
$f$ and $Y$. The bounds from Lemma 2.14 then imply that $\sup_{0\leqslant
s<t\leqslant T}\lVert I_{t}-I_{s}-\Xi_{s,t}\rVert_{L^{2}(\mathds{P})}\lvert
t-s\rvert^{-(1/2+\varepsilon_{2})}\lesssim
T^{1/2+\varepsilon_{1}-\varepsilon_{2}}\Gamma_{1}+\Gamma_{2}$ (assuming
$\varepsilon_{2}\in(0,1/2)$), which yields the stability of the stochastic
sewing integral.

##  3 Weak rough-path-type solutions

In this section, we define a (rough) weak solution to the singular SDE

$\displaystyle dX_{t}=V(t,X_{t})dt+dL_{t},\quad X_{0}=x\in\mathbb{R}^{d}$
(3.1)

for an enhanced Besov drift $V\in\mathcal{X}^{\beta,\gamma}$ with
$\beta\in(\frac{2-2\alpha}{3},0)$ and
$\gamma\in[\frac{2\beta+2\alpha-1}{\alpha},1)$ in the rough case, respectively
$\gamma\in(\frac{1-\beta}{\alpha},1)$ in the Young case (cf. Definition 2.7).
We furthermore state our main Theorem 3.10, that proves equivalence of rough
weak solutions and martingale solutions Definition 2.10. The proof of Theorem
3.10 follows from Theorem 5.7 and Theorem 5.10 in Section 5.
In the case of (locally) bounded drifts $V$ the equivalence of solutions to
the martingale problem and weak solutions is known by [SV06, Theorem 8.1.1] in
the Brownian noise case and by [KC11, Theorem 1.1] in the Lévy noise setting.
The first attempt to define weak solutions in the singular drift case, is to
replace the singular drift by the limit $Z$ of the drift terms
$\int_{0}^{\cdot}V^{n}(t,X_{t})dt$ for a smooth sequence $(V^{n})$
approximating $V$. In this way, one obtains a canonical weak solution concept.
In the Young case, this yields a well-posed solution concept (if moreover
requiring regularity bounds on the drift $Z$). However, it turns out that
canonical weak solutions are in general non-unique in the rough regime (cf.
Section 6 below). In this section, we thus adapt the canonical weak solution
concept, imposing additional assumptions to ensure well-posedness. The idea
for a (rough) weak solution is to impose rough-paths-type assumptions on
certain iterated integrals $\mathbb{Z}^{V}$, that formally correspont to the
resonant component in the enhancement $\mathscr{V}$. This motivates Definition
3.1 below.

We call a filtered probability space
$(\Omega,\mathcal{F},(\mathcal{F})_{t\in[0,T]},\mathds{P})$ a stochastic
basis, if $(\Omega,\mathcal{F},(\mathcal{F})_{t\in[0,T]},\mathds{P})$ is
complete and the filtration $(\mathcal{F}_{t})$ is right-continuous. We call a
process $L$ a $(\mathcal{F}_{t})$-Lévy process, if $L$ is adapted to
$(\mathcal{F}_{t})$ with $L_{0}=0$, $L_{t}-L_{s}$ being independent of
$\mathcal{F}_{s}$ and $L_{t}-L_{s}\stackrel{{\scriptstyle d}}{{=}}L_{t-s}$ for
all $0\leqslant s<t\leqslant T$.

######  Definition 3.1 (Rough weak solution).

Let $V\in\mathscr{X}^{\beta,\gamma}$ for $\beta\in(\frac{2-2\alpha}{3},0)$
(with $\gamma\in[\frac{2\beta+2\alpha-1}{\alpha},1)$ in the rough case,
respectively $\gamma\in[\frac{1-\beta}{\alpha},1)$ in the Young case). Let
$x\in\mathbb{R}^{d}$. We call a triple $(X,L,\mathbb{Z}^{V})$ a rough weak
solution to the SDE (3.1) starting at $X_{0}=x\in\mathbb{R}^{d}$, if there
exists a stochastic basis $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geqslant
0},\mathds{P})$, such that $L$ is an $\alpha$-stable symmetric non-degenerate
$(\mathcal{F}_{t})$-Lévy process and almost surely

$\displaystyle X=x+Z+L,$

where $Z$ is a continuous and $(\mathcal{F}_{t})$-adapted process with the
property that

$\displaystyle\lVert Z\rVert_{\frac{\alpha+\beta}{\alpha},2}+\lVert
Z\mid\mathcal{F}\rVert_{\frac{\alpha+\beta}{\alpha},\infty}<\infty.$ (3.2)

Moreover $(\mathbb{Z}^{V}_{st})_{(s,t)\in\Delta_{T}}$ is a continuous,
$(\mathcal{F}_{t})$-adapted, $\mathbb{R}^{d\times d}$-valued stochastic
process indexed by $\Delta_{T}$ with

$\displaystyle\lVert\mathbb{Z}^{V}\rVert_{\frac{\alpha+\beta}{\alpha},2}+\lVert\mathbb{Z}^{V}\mid\mathcal{F}\rVert_{\frac{2\alpha+2\beta-1}{\alpha},\infty}<\infty.$
(3.3)

Furthermore, $Z$ and $\mathbb{Z}^{V}$ are given as follows. There exists a
sequence $(V^{n})\subset C_{T}(C^{\infty}_{b})^{d}$ with
$(V^{n},\mathcal{K}(V^{m},V^{n}))\to(V,\mathcal{V}_{2})$ in
$\mathscr{X}^{\beta,\gamma}$ for $m,n\to\infty$ with the following properties:

  1. 1.)

$Z^{n}:=\int_{0}^{\cdot}V^{n}(s,X_{s})ds$ converges to $Z$ in the sense that

$\displaystyle\lim_{n\to\infty}[\lVert
Z^{n}-Z\rVert_{\frac{\alpha+\beta}{\alpha},2}+\lVert
Z^{n}-Z\mid\mathcal{F}\rVert_{\frac{\alpha+\beta}{\alpha},\infty}]=0$ (3.4)

and

  2. 2.)

$\mathbb{Z}^{m,n}_{st}=(\mathbb{Z}^{m,n}_{st}(i,j))_{i,j}=\big{(}\int_{s}^{t}[J^{T}(\partial_{i}V^{m,j})(r,X_{r})-J^{T}(\partial_{i}V^{m,j})(s,X_{s})]dZ_{r}^{n,i}\big{)}_{i,j}$
converges to $\mathbb{Z}^{V}$ in the sense that

$\displaystyle\lim_{m,n\to\infty}[\lVert\mathbb{Z}^{m,n}-\mathbb{Z}^{V}\rVert_{\frac{\alpha+\beta}{\alpha},2}+\lVert\mathbb{Z}^{m,n}-\mathbb{Z}^{V}\mid\mathcal{F}\rVert_{\frac{2\alpha+2\beta-1}{\alpha},\infty}]=0.$
(3.5)

We also call $X$ a (rough) weak solution, if there exists a stochastic basis,
an $\alpha$-stable symmetric non-degenerate Lévy process $L$ and a stochastic
process $\mathbb{Z}^{V}$, such that $(X,L,\mathbb{Z}^{V})$ is a rough weak
solution.

######  Remark 3.2 (Notation).

We also use the following abbreviations. Let

$\displaystyle\mathbb{Z}^{m,n}_{st}(i,j)$
$\displaystyle=\int_{s}^{t}J^{T}(\partial_{i}V^{m,j})(r,X_{r})dZ_{r}^{n,i}-J^{T}(\partial_{i}V^{m,j})(s,X_{s})Z^{n,i}_{st}$
$\displaystyle=:\mathbb{A}^{m,n}_{st}(i,j)-J^{T}(\partial_{i}V^{m,j})(s,X_{s})Z^{n,i}_{st}.$

Let furthermore

$\displaystyle\mathbb{A}^{i}_{st}:=\lim_{m,n\to\infty}(\mathbb{A}^{m,n}_{st}(i,j))_{j=1,\dots,d}=:\lim_{m,n\to\infty}\mathbb{A}^{m,n,i}_{st}$

for $i=1,\dots,d$. Here, the convergence with respect to
$\lVert\cdot\rVert_{\frac{\alpha+\beta}{\alpha},2}$ follows from the
convergences $1.)$ and $2.)$ and that $J^{T}(\partial_{i}V^{m,j})\to
J^{T}(\partial_{i}V^{j})$ in $C_{T}L^{\infty}$. Moreover,
$(\mathbb{A}^{m,n})_{m,n},\mathbb{A}$ satisfy the same bounds as $Z$ in (3.2).
We write

$\displaystyle\mathbb{Z}^{V,i}_{st}:=\lim_{m,n\to\infty}(\mathbb{Z}^{m,n}_{st}(i,j))_{j=1,\dots,d}=:\lim_{m,n\to\infty}\mathbb{Z}^{m,n,i}_{st}$

for $i=1,\dots,d$. And for fixed $m\in\mathbb{N}$, we define
$\mathbb{A}^{m,\infty,i}_{st}:=\int_{s}^{t}J^{T}(\partial_{i}V^{m})(r,X_{r})dZ_{r}^{i}$,
and analogously we define
$\mathbb{Z}^{m,\infty,i}_{st}:=\int_{s}^{t}[J^{T}(\partial_{i}V^{m})(r,X_{r})-J^{T}(\partial_{i}V^{m})(s,X_{s})]dZ_{r}^{i}$.

######  Remark 3.3 ($X$ is a Dirichlet process).

By the $L^{2}$-moment bound on $Z$ from (3.2), $Z$ has zero quadratic
variation as $2(\alpha+\beta)/\alpha>1$. Thus, $X=x+Z+L$ is a Dirichlet
process, i.e. the sum of a local martingale and a zero quadratic variation
process, cf. [CJMS06, Definition 2.4]. In particular, for $F\in
C^{1,2}_{b}([0,T]\times\mathbb{R}^{d},\mathbb{R})$, the Itô-formula for
$F(t,X_{t})$ from [CJMS06, Theorem 3.1] holds, cf. also the classical article
[Föl81], which can be extended to time depending $F$, that are $C^{1}$ in
time. Notice that $Z$ is continuous and thus in the Itô-formula the terms
involving the quadratic variation and the pure jump part of $Z$ vanish. In
particular, the stochastic integral $\int_{0}^{\cdot}\nabla F(s,X_{s})\cdot
dZ_{s}$ is the limit, in probability, of the classical Riemann sums.

######  Remark 3.4 (One and all sequences $(V^{n})$).

If $(X,L,\mathbb{Z}^{V})$ is a weak solution and
$\mathscr{V}\in\mathscr{X}^{\beta,\gamma}$, then the convergences in $1.)$ and
$2.)$ are true for all sequences $(V^{n})$ with
$(V^{n},\mathcal{K}(V^{n},V^{m}))\to\mathscr{V}$ in
$\mathscr{X}^{\beta,\gamma}$. This means, that $(Z^{n})$, $(\mathbb{Z}^{m,n})$
converge and the limit is the same for any such sequence $(V^{n})$. Indeed,
this will follow from the equivalence proof of the solution concepts below
(specifically Theorem 5.10).

The following lemma is an application of Lemma 2.14. Lemma 3.5 will in
particular be used to prove (together with Theorem 3.10 and Proposition 5.8
below), that in the Young case rough weak solutions are equivalent to
canonical weak solutions, defined in Section 6.

######  Lemma 3.5.

Let $(X,L,\mathbb{Z}^{V})$ be a weak solution and let $f\in
C_{T}^{\theta/\alpha}L^{\infty}\cap C_{T}\mathscr{C}^{\theta}$ for
$\theta\in(0,1)$ with $(\theta+\alpha+\beta)/\alpha>1$. Let
$i\in\\{1,\dots,d\\}$.
Then, for every $t\in[0,T]$, the stochastic sewing integral

$\displaystyle I(f,Z^{i})_{t}:=\lim_{\lvert\Pi\rvert\to
0}\sum_{s,r\in\Pi}f(s,X_{s})(Z_{r}^{i}-Z_{s}^{i})\in L^{2}(\mathds{P})$

for finite partitions $\Pi$ of $[0,t]$ with
$\lvert\Pi\rvert=\max_{s,r\in\Pi}\lvert r-s\rvert\to 0$, is well-defined and
allows for the bound,

$\displaystyle\lVert
I(f,Z^{i})_{t}-I(f,Z^{i})_{s}-f(s,X_{s})Z_{s,t}^{i}\rVert_{L^{2}(\mathds{P})}$
$\displaystyle\quad\lesssim\lVert f\rVert_{C_{T}L^{\infty}}\lVert
Z\rVert_{\frac{\alpha+\beta}{\alpha},2}\lvert
t-s\rvert^{(\alpha+\beta)/\alpha}$ $\displaystyle\qquad\qquad+\lVert
f\rVert_{C_{T}^{\theta/\alpha}L^{\infty}\cap
C_{T}\mathscr{C}^{\theta}}(1+\lVert
Z\rVert_{\frac{\alpha+\beta}{\alpha},2}^{\theta})\lVert
Z\mid\mathcal{F}\rVert_{\frac{\alpha+\beta}{\alpha},\infty}\lvert
t-s\rvert^{(\theta+\alpha+\beta)/\alpha}.$

In particular, for $\beta\in(\frac{1-\alpha}{2},0)$ (Young regime), the
existence of the integral

$\displaystyle\mathbb{A}_{0,t}=\Big{(}\int_{0}^{t}J^{T}(\partial_{i}V^{j})(s,X_{s})dZ^{i}_{s}\Big{)}_{i,j}\in
L^{2}(\mathds{P},\mathbb{R}^{d\times d}),\quad t\in[0,T]$

and the bound $\lVert\mathbb{Z}^{V}\rVert_{(\alpha+\beta)/\alpha,2}<\infty$,
as well as the convergence
$\lVert\mathbb{Z}^{m,n}-\mathbb{Z}^{V}\rVert_{\frac{\alpha+\beta}{\alpha},2}\to
0$ in $2.)$ follow. Furthermore, the convergence
$\lVert\mathbb{Z}^{m,n}-\mathbb{Z}^{V}\mid\mathcal{F}\rVert_{\frac{2\alpha+2\beta-1}{\alpha},2}\to
0$ follows.

######  Remark 3.6.

Lemma 3.5 implies well-definedness of $\mathbb{A}^{m,\infty}_{0,t}(i,j)$ and
$\mathbb{A}^{m,n}_{0,t}(i,j)$ as the limit of the Riemann sums
$\sum_{s,r\in\Pi}J^{T}(\partial_{i}V^{m,j})(s,X_{s})Z^{i}_{s,r}$, analogously
for $\mathbb{A}^{m,n}_{0,t}(i,j)$. In particular, the stability of the
stochastic sewing integral yields that, for fixed $m\in\mathbb{N}$,
$\lim_{n\to\infty}\mathbb{A}^{m,n}_{st}=\mathbb{A}^{m,\infty}_{st}$ in
$L^{2}(\mathds{P})$, uniformly in $(s,t)\in\Delta_{T}$.
Furthermore, the lemma shows that in the Young case
$f=J^{T}(\partial_{i}V^{j})\in C_{T}^{(\alpha+\beta-1)/\alpha}L^{\infty}\cap
C_{T}\mathscr{C}^{\alpha+\beta-1}$ satisfies the assumptions for
$\theta=\alpha+\beta-1$. Thus, existence of the integral
$\mathbb{A}_{st}(i,j)$ follows. In the rough case, the regularity of
$f=J^{T}(\partial_{i}V^{j})$ does not suffice, because
$2(\alpha+\beta)-1\leqslant\alpha$ if $\beta\leqslant(1-\alpha)/2$. Thus, the
bounds (3.3) on $\mathbb{Z}^{V}$ and the convergence in $2.)$ are non-trivial
requirements on a weak solution in the rough case.

######  Remark 3.7.

Notice that the lemma yields the convergence
$\lVert\mathbb{Z}^{m,n}-\mathbb{Z}^{V}\rVert_{\frac{2\alpha+2\beta-1}{\alpha},2}\to
0$ in the Young case, but not
$\lVert\mathbb{Z}^{m,n}-\mathbb{Z}^{V}\rVert_{\frac{2\alpha+2\beta-1}{\alpha},\infty}\to
0$. In fact the latter also holds true and we refer to Proposition 5.8 below.

###### Proof of Lemma 3.5.

We apply the stochastic sewing lemma, Lemma 2.14, to the germ

$\displaystyle\Xi_{st}=f(s,X_{s})(Z^{i}_{t}-Z^{i}_{s}).$

We have that

$\displaystyle\delta\Xi_{srt}=\Xi_{st}-\Xi_{sr}-\Xi_{rt}=[f(r,X_{r})-f(s,X_{s})]Z^{i}_{rt}.$

To prove the bound on the expectation in the stochastic sewing lemma, we will
use the trivial estimate $\lvert f(r,X_{r})-f(s,X_{s})\rvert\leqslant 2\lVert
f\rVert_{C_{T}L^{\infty}}\lesssim 2\lVert f\rVert_{C_{T}\mathscr{C}^{\theta}}$
as $\theta>0$ and the bound on $Z$ from (3.2), such that

$\displaystyle\mathds{E}[\lvert\delta\Xi_{srt}\rvert^{2}]\leqslant\lVert
f\rVert_{C_{T}L^{\infty}}^{2}\mathds{E}[\lvert
Z^{i}_{rt}\rvert^{2}]\leqslant\lVert
f\rVert_{C_{T}\mathscr{C}^{\theta}}^{2}\lVert
Z\rVert_{\frac{\alpha+\beta}{\alpha},2}^{2}\lvert
t-s\rvert^{2(\alpha+\beta)/\alpha}$

with $(\alpha+\beta)/\alpha>1/2$. For bound on the conditional expectation, we
utilize the bounds (3.2) on $Z$ and the time and space regularity of $f$, such
that

$\displaystyle\lvert f(r,X_{r})-f(s,X_{s})\rvert$
$\displaystyle\leqslant\lvert f(r,X_{r})-f(s,X_{r})\rvert+\lvert
f(s,X_{s})-f(s,X_{r})\rvert$ $\displaystyle\lesssim\lVert
f\rVert_{C_{T}^{\theta/\alpha}L^{\infty}}\lvert
r-s\rvert^{\theta/\alpha}+\lVert f\rVert_{C_{T}\mathscr{C}^{\theta}}\lvert
X_{r}-X_{s}\rvert^{\theta}$ $\displaystyle\lesssim\lVert
f\rVert_{C_{T}^{\theta/\alpha}L^{\infty}\cap C_{T}\mathscr{C}^{\theta}}[\lvert
r-s\rvert^{\theta/\alpha}+\lvert X_{r}-X_{s}\rvert^{\theta}],$

where we used that the norm in $\mathscr{C}^{\theta}$ is equivalent to the
norm in the Hölder space $C^{\theta}_{b}$ of bounded, $\theta$-Hölder
continuous functions for $\theta\in(0,1)$ (cf. [BCD11, Section 2.7,
Examples]). Hence, with $\lvert X_{r}-X_{s}\rvert\leqslant\lvert
Z_{r}-Z_{s}\rvert+\lvert L_{r}-L_{s}\rvert$, we can estimate

$\displaystyle\mathds{E}[\lvert\mathds{E}_{s}[\delta\Xi_{srt}]\rvert^{2}]$
$\displaystyle=\mathds{E}[\lvert\mathds{E}_{s}[f(r,X_{r})-f(s,X_{s}))\mathds{E}_{r}[Z_{rt}^{i}]]\rvert^{2}]$
$\displaystyle\leqslant\mathds{E}[\mathds{E}_{s}[\lvert
f(r,X_{r})-f(s,X_{s})\rvert\lvert\mathds{E}_{r}[Z_{rt}^{i}]\rvert]^{2}]$
$\displaystyle\leqslant\lVert\mathds{E}_{r}[Z_{rt}^{i}]\rVert_{L^{\infty}(\mathds{P})}^{2}\mathds{E}[\mathds{E}_{s}[\lvert
f(r,X_{r})-f(s,X_{s})\rvert]^{2}]$ $\displaystyle\lesssim\lVert
Z\mid\mathcal{F}\rVert_{\frac{\alpha+\beta}{\alpha},\infty}^{2}\lvert
t-r\rvert^{2(\alpha+\beta)/\alpha}\lVert
f\rVert_{C_{T}^{\theta/\alpha}L^{\infty}\cap
C_{T}\mathscr{C}^{\theta}}^{2}\times$ $\displaystyle\qquad\qquad\big{(}\lvert
r-s\rvert^{2\theta/\alpha}+\mathds{E}[\lvert
Z_{r}-Z_{s}\rvert^{2\theta}]+\mathds{E}[\mathds{E}_{s}[\lvert
L_{r}-L_{s}\rvert^{\theta}]^{2}]\big{)}$ $\displaystyle\lesssim_{T}\lVert
Z\mid\mathcal{F}\rVert_{\frac{\alpha+\beta}{\alpha},\infty}^{2}(1+\lVert
Z\rVert_{\frac{\alpha+\beta}{\alpha},2}^{2\theta})\lVert
f\rVert_{C_{T}^{\theta/\alpha}L^{\infty}\cap
C_{T}\mathscr{C}^{\theta}}^{2}\lvert
t-s\rvert^{2(\theta+\alpha+\beta)/\alpha}$

In the last estimate above, we used stationarity, scaling and independence of
the increment $L_{r}-L_{s}$ of $\mathcal{F}_{s}$ for $s\leqslant r$, such that
almost surely

$\displaystyle\mathds{E}_{s}[\lvert
L_{r}-L_{s}\rvert^{\theta}]=\mathds{E}[\lvert
L_{r}-L_{s}\rvert^{\theta}]=\lvert r-s\rvert^{\theta/\alpha}\mathds{E}[\lvert
L_{1}\rvert^{\theta}]\lesssim\lvert r-s\rvert^{\theta/\alpha}$

and the expectation is finite due to $\theta<\alpha$. Furthermore, as
$\theta\in(0,1)$, we used above Jensen’s inequality and the following estimate
on $Z$:

$\displaystyle\mathds{E}[\lvert
Z_{r}-Z_{s}\rvert^{2\theta}]\leqslant\mathds{E}[\lvert
Z_{r}-Z_{s}\rvert^{2}]^{\theta}$ $\displaystyle\leqslant\lVert
Z\rVert_{\frac{\alpha+\beta}{\alpha},2}^{2\theta}\lvert
r-s\rvert^{2\theta(\alpha+\beta)/\alpha}\lesssim_{T}\lVert
Z\rVert_{\frac{\alpha+\beta}{\alpha},2}^{2\theta}\lvert
r-s\rvert^{2\theta/\alpha}.$

Due to $(\theta+\alpha+\beta)/\alpha>1$, Lemma 2.14 applies and yields the
bound for $I(f,Z^{i})$.
In the Young case, $\beta>(1-\alpha)/2$, we can take
$f=J^{T}(\partial_{i}V^{j})\in C_{T}^{(\alpha+\beta-1)/\alpha}L^{\infty}\cap
C_{T}\mathscr{C}^{\alpha+\beta-1}$ with $(2\alpha+2\beta-1)/\alpha>1$, which
yields existence of the integral $I(\Xi)_{t}=\mathbb{A}_{0,t}(i,j)$. Let
$I(\Xi^{m,n})$ be the sewing integral with germ

$\displaystyle\Xi_{sr}^{m,n}=J^{T}(\partial_{i}V^{m,j})(s,X_{s})(Z^{n,i}_{r}-Z^{n,i}_{s})$

and let $f^{m}:=J^{T}(\partial_{i}V^{m,j})$. By the Schauder and interpolation
estimates we furthermore have that $f^{m}\to f$ in
$\mathcal{L}_{T}^{0,\alpha+\beta-1}$ and in
$C_{T}^{(\alpha+\beta-1)/\alpha}L^{\infty}\cap
C_{T}\mathscr{C}^{\alpha+\beta-1}$.
Let $\mathbb{Z}^{V}_{st}=\mathbb{A}_{st}-\Xi_{st}=I(\Xi)_{st}-\Xi_{st}$. Then,
Lemma 2.14 yields that
$\lVert\mathbb{Z}^{V}\rVert_{\frac{\alpha+\beta}{\alpha},2}<\infty$ and
$\lVert\mathbb{Z}^{V}\mid\mathcal{F}\rVert_{\frac{2\alpha+2\beta-1}{\alpha},2}<\infty$.
The stability of the sewing integral implies the convergence
$\lVert\mathbb{Z}^{m,n}-\mathbb{Z}^{V}\rVert_{\frac{\alpha+\beta}{\alpha},2}\to
0$, since by the estimates above we obtain for $(s,t)\in\Delta_{T}$ that

$\displaystyle\lVert\mathbb{Z}^{m,n}_{st}-\mathbb{Z}^{V}_{st}\rVert_{L^{2}}$
$\displaystyle\leqslant\lVert
I(\Xi^{m,n})_{st}-\Xi^{m,n}_{s,t}-(I(\Xi)_{st}-\Xi_{s,t})\rVert_{L^{2}}$
$\displaystyle\lesssim_{T}\lvert
t-s\rvert^{(\alpha+\beta)/\alpha}\Big{(}\sup_{n}\lVert
Z^{n}\mid\mathcal{F}\rVert_{\frac{\alpha+\beta}{\alpha},\infty}(1+\lVert
Z\rVert_{\frac{\alpha+\beta}{\alpha},2}^{\theta})\lVert
f^{m}-f\rVert_{\mathcal{L}_{T}^{0,\alpha+\beta-1}}$
$\displaystyle\qquad\qquad\hskip 30.00005pt+\lVert
Z^{n}-Z\mid\mathcal{F}\rVert_{\frac{\alpha+\beta}{\alpha},\infty}(1+\lVert
Z\rVert_{\frac{\alpha+\beta}{\alpha},2}^{\theta})\lVert
f\rVert_{\mathcal{L}_{T}^{0,\alpha+\beta-1}}$ $\displaystyle\qquad\qquad\hskip
30.00005pt+\lVert
f^{m}-f\rVert_{\mathcal{L}_{T}^{0,\alpha+\beta-1}}\sup_{n}\lVert
Z^{n}\rVert_{\frac{\alpha+\beta}{\alpha},2}$ $\displaystyle\qquad\qquad\hskip
30.00005pt+\lVert f\rVert_{\mathcal{L}_{T}^{0,\alpha+\beta-1}}\lVert
Z^{n}-Z\rVert_{\frac{\alpha+\beta}{\alpha},2}\Big{)}$ $\displaystyle\to 0$

for $n,m\to\infty$. Analogously we can show that
$\lVert\mathbb{Z}^{m,n}-\mathbb{Z}^{V}\mid\mathcal{F}\rVert_{\frac{2\alpha+2\beta-1}{\alpha},2}\to
0$. ∎

Let $\mathscr{V}=(V,\mathscr{V}_{2})\in\mathscr{X}^{\beta,\gamma}$ and let
$(X,L,\mathbb{Z}^{V})$ be a weak solution. Then we expect (for a proof see
Theorem 5.10 below) the following representations of $Z$ and $\mathbb{A}$ (and
thus of $\mathbb{Z}^{V}$):

$\displaystyle\mathds{E}_{s}[Z_{s,t}^{i}]=\mathds{E}_{s}[u^{t,i}(t,X_{t})-u^{t,i}(s,X_{s})]\,\text{
and
}\,\mathds{E}_{s}[\mathbb{A}_{s,t}(i,j)]=\mathds{E}_{s}[v^{t,i,j}(t,X_{t})-v^{t,i,j}(s,X_{s})]$

for the solutions $u^{t}=(u^{t,i})_{i=1,\dots,d}$,
$v^{t}=(v^{t,i,j})_{i,j=1,\dots,d}$ of the backward PDEs

$\displaystyle\mathcal{G}^{\mathscr{V}}u^{t,i}=V^{i},\quad
u^{t,i}(t,\cdot)=0.$ (3.6)

and

$\displaystyle\mathcal{G}^{\mathscr{V}}v^{t,i,j}$
$\displaystyle=J^{T}(\partial_{i}V^{j})\cdot V^{i}$
$\displaystyle=\mathscr{V}_{2}(i,j)+J^{T}(\partial_{i}V^{j})\varolessthan
V^{i}+J^{T}(\partial_{i}V^{j})\varogreaterthan V^{i},\quad
v^{t,i,j}(t,\cdot)=0.$ (3.7)

Let for $i\in\\{1,\dots,d\\}$, $v^{t,i}:=(v^{t,i,j})_{j=1,\dots,d}$.
Together with the bound
$\lVert\mathbb{Z}^{V}\mid\mathcal{F}\rVert_{\frac{2\alpha+2\beta-1}{\alpha},\infty}<\infty$
from (3.3), this motivates the definition of the class of processes
$\mathcal{K}^{\vartheta}=\mathcal{K}^{\vartheta}(\mathscr{V})$.

######  Definition 3.8 (Class $\mathcal{K}^{\vartheta}$).

Let $\beta\in(\frac{2-2\alpha}{3},0)$,
$\mathscr{V}\in\mathscr{X}^{\beta,\gamma}$ and
$\vartheta\in((\alpha+\beta)/\alpha,1]$. Let for $t\in(0,T]$, $v^{t}$ be the
solution of the PDE (3) and $u^{t}$ be the solution of (3.6).
An adapted càdlàg stochastic process $(X_{t})_{t\in[0,T]}$ is said to be of
class $\mathcal{K}^{\vartheta}(\mathscr{V})$ if for all $i=1,\dots,d$:

$\displaystyle\lVert\mathds{E}_{s}[(v^{t,i}(t,X_{t})-v^{t,i}(s,X_{s}))-J^{T}(\partial_{i}V)(s,X_{s})(u^{t,i}(t,X_{t})-u^{t,i}(s,X_{s}))]\rVert_{(L^{\infty}(\mathds{P}))^{d}}$
$\displaystyle=\max_{j=1,\dots,d}\lVert\mathds{E}_{s}[(v^{t,i,j}(t,X_{t})-v^{t,i,j}(s,X_{s}))-J^{T}(\partial_{i}V^{j})(s,X_{s})(u^{t,i}(t,X_{t})-u^{t,i}(s,X_{s}))]\rVert_{L^{\infty}(\mathds{P})}$
$\displaystyle\lesssim\lvert t-s\rvert^{\vartheta}.$ (3.8)

######  Remark 3.9.

Using $v^{t,i,j},u^{t,i}\in C_{t}^{(\alpha+\beta)/\alpha}L^{\infty}$ with norm
uniformly bounded for $t\in[0,T]$ by [KP23, Corollary 4.14], a trivial
estimate yields (3.8) for $\vartheta=(\alpha+\beta)/\alpha$. We are interested
in the case $\vartheta>(\alpha+\beta)/\alpha$.

As an intermediate step in the equivalence proof of the solution concepts we
will prove (cf. Proposition 5.8) that the solution of the martingale problem
for $\mathcal{G}^{\mathscr{V}}$ is a process of class
$\mathcal{K}^{\vartheta}(\mathscr{V})$ for
$\vartheta=(2\alpha+2\beta-1)/\alpha$.
The following result is the main theorem of this paper. The proof follows
directly from Theorems 5.7 and 5.10 in Section 5.

######  Theorem 3.10.

Let $\alpha\in(1,2]$, $\beta\in(\frac{2-2\alpha}{3},0)$ and
$V\in\mathscr{X}^{\beta,\gamma}$. Let $x\in\mathbb{R}^{d}$. Then, $X$ is a
rough weak solution in the sense of Definition 3.1 if and only if $X$ solves
the $(\mathcal{G}^{\mathscr{V}},\delta_{x})$-martingale problem. In
particular, rough weak solutions are unique in law.

The second main theorem of this work generalizes Itô’s formula for rough weak
solutions. We state an informal version of Theorem 5.11 in Theorem 3.11 below.

######  Theorem 3.11.

Let $\alpha\in(1,2]$, $\beta\in(\frac{2-2\alpha}{3},0)$ and
$V\in\mathscr{X}^{\beta,\gamma}$. Let $(X,L,\mathbb{Z}^{V})$ be a rough weak
solution and let $u\in C_{T}\mathscr{C}^{\alpha+\beta}\cap
C^{1}([0,T],\mathscr{C}^{\beta})$ be such that
$(\partial_{t}-\mathfrak{L}^{\alpha}_{\nu})u$ is paracontrolled by $V$. Then,
if $\alpha\in(1,2)$, the following Itô-formula holds true:

$\displaystyle u(t,X_{t})$
$\displaystyle=u(0,x)+\int_{0}^{t}(\partial_{s}-\mathfrak{L}^{\alpha}_{\nu})u(s,X_{s})ds+\int_{0}^{t}\nabla
u(s,X_{s})\cdot d(Z,\mathbb{Z}^{V})_{s}$
$\displaystyle\qquad+\int_{0}^{t}\int_{\mathbb{R}^{d}\setminus\\{0\\}}[u(s,X_{s-}+y)-u(s,X_{s-})]\hat{\pi}(ds,dy),$

where $Z=X-x-L$ and $\hat{\pi}$ denotes the compensated Poisson random measure
of $L$. If $\alpha=2$ and $L=B$ for a Brownian motion $B$, the martingale is
replaced by $\int_{0}^{t}\nabla u(s,X_{s})\cdot dB_{s}$.

To prove Theorem 3.10, it turns out that for a weak solution $X$ we need to
define the limit of the stochastic integrals

$\displaystyle\int_{0}^{t}\nabla u^{m}(s,X_{s})\cdot
dZ_{s}=\sum_{i=1}^{d}\int_{0}^{t}\partial_{i}u^{m}(s,X_{s})dZ_{s}^{i},$

as $m\to\infty$, where $u^{m}\in C^{1,2}_{b}([0,T]\times\mathbb{R}^{d})$ with
$u^{m}\to u$ and $u$ solves the Kolmogorov backward equation (for regular data
$f,u^{T}$). In the rough regime, $\partial_{i}u$ won’t have enough regularity,
such that we can define the integral of $\partial_{i}u(s,X_{s})$ against $Z$
in a stable manner utilizing Lemma 3.5. Indeed, a regularity counting argument
yields that the time regularity of $\partial_{i}u$, i.e. $\partial_{i}u\in
C_{T}^{(\alpha+\beta-1)/\alpha}L^{\infty}$, together with the
$(\alpha+\beta)/\alpha$-Hölder regularity of $Z$ in $L^{2}(\mathds{P})$ sum up
to $(2\alpha+2\beta-1)/\alpha\leqslant 1$ for $\beta\leqslant(1-\alpha)/2$.
The idea is thus to enhance $Z$ by $\mathbb{Z}^{V}$ from Definition 3.1 in
order to correct for the irregular terms in $\partial_{i}u$ and to define the
stable rough stochastic integral

$\displaystyle\int_{0}^{t}\nabla u(s,X_{s})\cdot
d(Z,\mathbb{Z}^{V})=\sum_{i=1}^{d}\int_{0}^{t}\partial_{i}u(s,X_{s})d(Z^{i},\mathbb{Z}^{V,i})_{s}.$

For regular integrands $\nabla u^{m}$ the stochastic integral against $Z$ and
the rough stochastic integral against $(Z,\mathbb{Z}^{V})$ coincide. This
motivates the general theory in the next section.

##  4 A rough stochastic sewing integral

In this section, we construct in Theorem 4.4 a rough stochastic integral using
the stochastic sewing Lemma 2.14. The theory is inspired by [FHL21].
Nonetheless, the results from [FHL21] do not apply in our setting and our
rough stochastic integral differs from the one constructed there. Instead of
considering a $\gamma$-rough path as an integrator, we consider a lifted
stochastic process $(Z,\mathbb{Z}^{A})$ with bounds with respect to the semi-
norms $\lVert\cdot\rVert_{\theta,2}$ and
$\lVert\cdot\mid\mathcal{F}\rVert_{\theta,\infty}$, cf. (2.12). The Hölder
exponent $\sigma$ of $Z$ is assumed to satisfy $\sigma>1/2$. That Hölder-
regularity is satisfied in the case of the drift term we are interested in. In
this sense the integrator in good. However the integrand is rough and we need
to assume that it is stochastically controlled by a given stochastic process
$A$. We construct the integral in $L^{2}(\mathds{P})$, but replacing $2$ by
$p\geqslant 2$ in the bounds below, we could more generally construct the
integral in $L^{p}(\mathds{P})$. We will not bother doing so, as we aim for
the situation where the integrand is given by a $\vartheta$-Hölder function of
the $\alpha$-stable process for $\vartheta\in(0,\alpha/2)$, which lacks
arbitrarily high moments, i.e. $\mathds{E}[\lvert
L_{1}\rvert^{2\vartheta}]<\infty$ as $2\vartheta<\alpha$, but we may not
necessarily have $\mathds{E}[\lvert L_{1}\rvert^{p\vartheta}]<\infty$ for
$p>2$.
Let us start with the definition of stochastically controlled processes. Let
here and below,
$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\in[0,T]},\mathds{P})$ be a complete
filtered probability space.

######  Definition 4.1 (Stochastically controlled processes).

Let $(A_{t})_{t\in[0,T]}$ be an $\mathbb{R}^{d}$-valued adapted stochastic
process with $\sup_{t\in[0,T]}\lVert
A_{t}\rVert_{L^{\infty}(\mathds{P},\mathbb{R}^{d})}<\infty$ and $\lVert
A\rVert_{\varsigma,2}<\infty$ for $\varsigma\in(0,1)$. We call an adapted
stochastic process $(f_{t},f^{\prime}_{t})_{t\in[0,T]}$ with
$\sup_{t\in[0,T]}[\lVert
f_{t}\rVert_{L^{\infty}(\mathds{P},\mathbb{R})}+\lVert
f^{\prime}_{t}\rVert_{L^{\infty}(\mathds{P},\mathbb{R}^{d})}]<\infty$
stochastically controlled by $A$, if for
$\varsigma,\varsigma^{\prime}\in(0,1)$ the following bounds hold true

$\displaystyle\lVert f\rVert_{\varsigma,2}+\lVert
f^{\prime}\rVert_{\varsigma^{\prime},2}<\infty$

and the remainder $(R^{f}_{st})_{(s,t)\in\Delta_{T}}$ defined by

$\displaystyle R^{f}_{st}:=f_{st}-f^{\prime}_{s}\cdot A_{st},$

satisfies

$\displaystyle\lVert R^{f}\rVert_{\varsigma+\varsigma^{\prime},2}<\infty.$

The remainder is then an adapted process indexed by $\Delta_{T}$. We denote
the space of all such $(f,f^{\prime})$, that are stochastically controlled by
$A$ by $D^{\varsigma,\varsigma^{\prime}}_{T}(A)$ and we define the complete
norm

$\displaystyle\lVert f\rVert_{D^{\varsigma,\varsigma^{\prime}}_{T}(A)}$
$\displaystyle:=\sup_{t\in[0,T]}[\lVert
f_{t}\rVert_{L^{\infty}(\mathds{P})}+\lVert
f^{\prime}_{t}\rVert_{L^{\infty}(\mathds{P},\mathbb{R}^{d})}]$
$\displaystyle\qquad+\lVert f\rVert_{\varsigma,2}+\lVert
f^{\prime}\rVert_{\varsigma^{\prime},2}+\lVert
R^{f}\rVert_{\varsigma+\varsigma^{\prime},2}.$

######  Definition 4.2 (Rough stochastic integrator).

Let $(A_{t})_{t\in[0,T]}$ be an adapted $\mathbb{R}^{d}$-valued stochastic
process, which satisfies $\sup_{t\in[0,T]}\lVert
A_{t}\rVert_{L^{\infty}(\mathds{P},\mathbb{R}^{d})}<\infty$ and $\lVert
A\rVert_{\varsigma,2}<\infty$ for $\varsigma\in(0,1)$. Let $\sigma\in(1/2,1)$.
Then we call $(Z,\mathbb{Z}^{A})$ a rough integrator, if
$(Z_{t})_{t\in[0,T]}$, $(\mathbb{Z}^{A}_{st})_{(s,t)\in\Delta_{T}}$ are
adapted stochastic processes with $Z$ being $\mathbb{R}$-valued and
$\mathbb{Z}^{A}$ being $\mathbb{R}^{d}$-valued, such that $Z_{0}=0$ and for
all $0\leqslant s\leqslant l\leqslant t\leqslant T$, the following algebraic
relation holds

$\displaystyle\mathbb{Z}_{st}^{A}=\mathbb{Z}^{A}_{sl}+\mathbb{Z}^{A}_{lt}+A_{sl}Z_{lt}.$

Furthermore the following Hölder-type moment bounds hold:

$\displaystyle\lVert(Z,\mathbb{Z}^{A})\rVert_{\mathcal{R}^{\sigma}_{T}(A)}:=\lVert
Z\rVert_{\sigma,2}+\lVert
Z\mid\mathcal{F}\rVert_{\sigma,\infty}+\lVert\mathbb{Z}^{A}\rVert_{\sigma,2}+\lVert\mathbb{Z}^{A}\mid\mathcal{F}\rVert_{\sigma+\varsigma,\infty}<\infty.$

We call the space of stochastic processes, that are rough integrators
$\mathcal{R}^{\sigma}_{T}(A)$. Equipped with the norm

$\displaystyle\lVert(Z,\mathbb{Z}^{A})\rVert_{\mathcal{R}_{T}^{\sigma}(A)}$
$\displaystyle:=\lVert Z\rVert_{\sigma,2}+\lVert
Z-W\mid\mathcal{F}\rVert_{\sigma,\infty}$
$\displaystyle\qquad+\lVert\mathbb{Z}^{A}\rVert_{\sigma,2}+\lVert\mathbb{Z}^{A}\mid\mathcal{F}\rVert_{\sigma+\varsigma,\infty}$

the space becomes a Banach space.

######  Remark 4.3.

We refer to [FH20] for the connection to rough paths. The space of rough
stochastic integrators is a vector space, because the algebraic relation here
is linear. Furthermore, we assume that $Z_{0}=0$ to obtain a norm.

######  Theorem 4.4.

Let $(A_{t})_{t\in[0,T]}$ be an adapted $\mathbb{R}^{d}$-valued stochastic
process with $A\in C_{T}L^{\infty}(\mathds{P},\mathbb{R}^{d})$ and let $f$ be
stochastically controlled by $A$. Let $(Z,\mathbb{Z}^{A})$ be a rough
integrator. Let the parameters be such that

$\displaystyle\sigma+\varsigma^{\prime}+\varsigma>1.$

Then for $t\in(0,T]$, the rough stochatic sewing integral

$\displaystyle
I(f,Z)_{t}:=\int_{0}^{t}f_{s}\,d(Z,\mathbb{Z}^{A})_{s}=\lim_{\lvert\Pi\rvert\to
0}\sum_{r,l\in\Pi}[f_{r}Z_{rl}+f^{\prime}_{r}\cdot\mathbb{Z}^{A}_{rl}],$

exists in $L^{2}(\mathds{P})$, where the limit ranges over partitions $\Pi$ of
$[0,t]$ with mesh size $\lvert\Pi\rvert\to 0$. Moreover, the following
Lipschitz bound holds:

$\displaystyle\sup_{(s,t)\in\Delta_{T}}\frac{\lVert
I(f,Z)_{s,t}\rVert_{L^{2}(\mathds{P})}}{\lvert
t-s\rvert^{\sigma}}\lesssim\lVert
f\rVert_{D^{\varsigma,\varsigma^{\prime}}_{T}(A)}\lVert(Z,\mathbb{Z}^{A})\rVert_{\mathcal{R}^{\sigma}_{T}(A)}.$

Furthermore, if $\sigma+\varsigma>1$, then the rough stochastic integral
$I(f,Z)_{t}$ almost surely agrees with the integral
$\tilde{I}(f,Z)_{t}=\lim_{\lvert\Pi\rvert\to 0}\sum_{r,l\in\Pi}f_{r}Z_{rl}\in
L^{2}(\mathds{P})$.

###### Proof.

Define for $r<s$,
$\Xi_{rs}:=f_{r}Z_{rs}+f_{r}^{\prime}\cdot\mathbb{Z}^{A}_{rs}$. Then we have
that for $r<l<s$, by the algebraic relation of $Z,\mathbb{Z}^{A}$ and the
definition of the remainder $R=R^{f}$,

$\displaystyle\delta\Xi_{r,l,s}=\Xi_{rs}-\Xi_{rl}-\Xi_{ls}$
$\displaystyle=-R_{r,l}Z_{l,s}-f_{r,l}^{\prime}\cdot\mathbb{Z}^{A}_{l,s}.$

To apply the stochastic sewing Lemma 2.14, we need to show the bounds on the
expectation and conditional expectation. We start with the bound on the
expectation. As $\sigma>1/2$ it suffices to trivially estimate the
$L^{2}$-norm using

$\displaystyle\sup_{r,l\in\Delta_{T}}\lVert
R_{r,l}\rVert_{L^{\infty}}\leqslant 2\sup_{t\in[0,T]}[\lVert
f_{t}\rVert_{L^{\infty}(\mathds{P})}+\lVert
f^{\prime}_{t}\rVert_{L^{\infty}(\mathds{P},\mathbb{R}^{d})}\lVert
A_{t}\rVert_{L^{\infty}(\mathds{P},\mathbb{R}^{d})}]<\infty,$

such that

$\displaystyle\mathds{E}[\lvert\delta\Xi_{r,l,s}\rvert^{2}]^{1/2}$
$\displaystyle\lesssim\sup_{r,l\in\Delta_{T}}\lVert
R_{r,l}\rVert_{L^{\infty}}\lVert Z_{l,s}\rVert_{L^{2}}+\sup_{t\in[0,T]}\lVert
f^{\prime}_{t}\rVert_{L^{\infty}(\mathds{P},\mathbb{R}^{d})}\lVert\mathbb{Z}^{A}_{l,s}\rVert_{L^{2}}$
$\displaystyle\lesssim\lVert
f\rVert_{D_{T}^{\varsigma,\varsigma^{\prime}}(A)}\lVert(Z,\mathbb{Z}^{A})\rVert_{\mathcal{R}_{T}^{\sigma}(A)}\lvert
s-r\rvert^{\sigma}.$

For the conditional expectation, we have

$\displaystyle\mathds{E}[\lvert\mathds{E}_{r}[\delta\Xi_{r,l,s}]\rvert^{2}]$
$\displaystyle\leqslant\mathds{E}\bigl{[}\mathds{E}_{r}[\lvert
R_{r,l}\mathds{E}_{l}[Z_{l,s}]\rvert]^{2}\bigr{]}+\mathds{E}\bigl{[}\mathds{E}_{r}[\lvert
f^{\prime}_{r,l}\cdot\mathds{E}_{l}[\mathbb{Z}^{A}_{l,s}]\rvert]^{2}\bigr{]}$
$\displaystyle\leqslant\lVert
R_{r,l}\rVert_{L^{2}(\mathds{P})}^{2}\lVert\mathds{E}_{l}[Z_{l,s}]\rVert_{L^{\infty}}^{2}+\lVert
f^{\prime}_{r,l}\rVert_{L^{2}(\mathds{P},\mathbb{R}^{d})}^{2}\lVert\mathds{E}_{l}[\mathbb{Z}^{A}_{l,s}]\rVert_{L^{\infty}(\mathds{P},\mathbb{R}^{d})}^{2}$
$\displaystyle\leqslant\lVert
f\rVert_{D_{T}^{\varsigma,\varsigma^{\prime}}(A)}\lVert(Z,\mathbb{Z}^{A})\rVert_{\mathcal{R}_{T}^{\sigma}(A)}\lvert
s-r\rvert^{2(\varsigma^{\prime}+\sigma+\varsigma)},$

with $\varsigma^{\prime}+\sigma+\varsigma>1$ by assumption. Thus Lemma 2.14
applies for the existence of $I(f,Z)_{t}$, $t\in[0,T]$. The prescribed
Lipschitz bound follows from the bound on $\lVert
I(f,Z)_{st}-\Xi_{st}\rVert_{L^{2}}$ from Lemma 2.14 and
$\lVert\Xi_{st}\rVert_{L^{2}}\leqslant\lVert
f\rVert_{D_{T}^{\varsigma,\varsigma^{\prime}}(A)}[\lVert
Z\rVert_{\sigma,2}+\lVert\mathbb{Z}^{A}\rVert_{\sigma,2}]\lvert
t-s\rvert^{\sigma}$.
If $\sigma+\varsigma>1$, then by the uniqueness of the stochastic sewing
integral

$\displaystyle\tilde{I}(f,Z)_{t}=\lim_{\lvert\Pi\rvert\to
0}\sum_{r<l\in\Pi}f_{r}Z_{rl}\in L^{2}(\mathds{P}),$

we obtain that $\tilde{I}(f,Z)_{t}=I(f,Z)_{t}$ almost surely. Indeed, this
uses that $I$ satisfies the bounds for the germ $\tilde{\Xi}_{st}=f_{s}Z_{st}$
of $\tilde{I}$, that is

$\displaystyle\lVert I(f,Z)_{st}-f_{s}Z_{st}\rVert_{L^{2}}$
$\displaystyle\leqslant\lVert I(f,Z)_{st}\rVert_{L^{2}}+\lVert
f_{s}Z_{st}\rVert_{L^{2}}\lesssim\lvert t-s\rvert^{\sigma},$

with $\sigma>1/2$ and using the bound of $I(f,Z)$ from Lemma 2.14,

$\displaystyle\lVert\mathds{E}_{s}[I(f,Z)_{s,t}-f_{s}Z_{st}]\rVert_{L^{2}}$
$\displaystyle\leqslant\lVert\mathds{E}_{s}[I(f,Z)_{s,t}-f_{s}Z_{st}-f^{\prime}_{s}\mathbb{Z}^{A}_{st}]\rVert_{L^{2}}+\lVert\mathds{E}_{s}[f^{\prime}_{s}\mathbb{Z}^{A}_{st}]\rVert_{L^{2}}$
$\displaystyle\leqslant\lVert\mathds{E}_{s}[I(f,Z)_{s,t}-f_{s}Z_{st}-f^{\prime}_{s}\mathbb{Z}^{A}_{st}]\rVert_{L^{2}}+\lVert\mathds{E}_{s}[f^{\prime}_{s}\mathbb{Z}^{A}_{st}]\rVert_{L^{\infty}}$
$\displaystyle\lesssim\lvert t-s\rvert^{\sigma+\varsigma},$

where $\sigma+\varsigma>1$ by assumption. ∎

##  5 Equivalence of weak solution concepts

In this section, we prove in Theorems 5.7 and 5.10 showing that the weak
solution concept from Definition 3.1 yields an equivalent notion of solution
to the martingale solutions. Furthermore, we show in Theorem 5.11 an extension
of the Itô formula for rough weak solutions.
To prove that a weak solution solves the martingale problem we employ the
stability of the rough stochastic integral. In functionalanalytic terms, the
rough stochastic integral is the unique extension of the integral of regular
integrands $u(s,X_{s})$, i.e. for $u\in C^{1,2}_{b}$, against $Z$ to
paracontrolled integrands:

$\displaystyle
D_{T}(V)\times\mathcal{R}_{T}(V)\ni(u,(Z,\mathbb{Z}^{V}))\mapsto\int_{0}^{t}\nabla
u(s,X_{s})\cdot d(Z,\mathbb{Z}^{V})_{s}\in L^{2}(\mathds{P}).$ (5.1)

Here $D_{T}(V)$ denotes the space of paracontrolled distributions defined by

$\displaystyle
D_{T}(V)=\bigl{\\{}(u,u^{\prime})\in\mathcal{L}_{T}^{0,\alpha+\beta}\times\mathcal{L}_{T}^{0,\alpha+\beta-1}\bigm{|}u^{\sharp}:=u-u^{\prime}\varolessthan
J^{T}(V)\in\mathcal{L}_{T}^{0,2(\alpha+\beta)-1}\bigr{\\}}$ (5.2)

and

$\displaystyle\mathcal{R}_{T}(V):=\bigtimes_{i=1}^{d}\mathcal{R}_{T}^{(\alpha+\beta)/\alpha}(A^{i}),$

for $A^{i}=(J^{T}(\partial_{i}V)(t,X_{t}))_{t\in[0,T]}$ and
$\mathcal{R}_{T}^{\sigma}(A)$ from Definition 4.2 in Section 4.
To obtain stability of the rough integral (5.1), we apply Theorem 4.4, for any
$i=1,\dots,d$, to $f=(\partial_{i}u(t,X_{t}))_{t\in[0,T]}$ and the rough
integrator $(Z^{i},\mathbb{Z}^{V,i})$ that is given by the definition of a
weak solution (cf. Definition 3.1).
Lemma 5.4 below shows that, if $u\in D_{T}(V)$, then
$(\partial_{i}u(t,X_{t}))$ is stochastically controlled by
$(A_{t}^{i})=(J^{T}(\partial_{i}V)(t,X_{t}))$ for
$\varsigma=\varsigma^{\prime}=(\alpha+\beta-1)/\alpha$. The following lemma
verifies that $(Z^{i},\mathbb{Z}^{V,i})$ is a rough stochastic integrator in
the sense of Definition 4.2.

######  Lemma 5.1.

Let $\beta\in(\frac{2-2\alpha}{3},0)$ and $V\in\mathscr{X}^{\beta,\gamma}$.
Let $(X,L,\mathbb{Z}^{V})$ be a weak solution. Then,
$(Z,\mathbb{Z}^{V})\in\mathcal{R}_{T}(V)$, i.e.
$(Z^{i},\mathbb{Z}^{V,i})\in\mathcal{R}_{T}^{(\alpha+\beta)/\alpha}(A^{i})$
for $A^{i}=(J^{T}(\partial_{i}V)(t,X_{t}))_{t\in[0,T]}$ and $i=1,\dots,d$.
Moreover, the following convergence holds for $n,m\to\infty$

$\displaystyle\lVert(Z^{n},\mathbb{Z}^{m,n})-(Z,\mathbb{Z}^{V})\rVert_{\mathcal{R}_{T}(V)}\to
0.$

######  Remark 5.2.

Notice that we use the notation
$\lVert(Z^{n},\mathbb{Z}^{m,n})-(Z,\mathbb{Z}^{V})\rVert_{\mathcal{R}_{T}(V)}$
despite the fact, that $(Z^{n},\mathbb{Z}^{m,n})$ and $(Z,\mathbb{Z}^{V})$ do
not live in the same space. The notation means that the four semi-norms in
Definition 4.2 converge.

###### Proof.

Let $i=1,\dots,d$. To prove that $(Z^{i},\mathbb{Z}^{V,i})$ is a rough
integrator, notice that for
$\mathbb{A}^{i}=\lim_{n,m\to\infty}\mathbb{A}^{m,n,i}$ the following
additivity holds:
$\mathbb{A}^{i}_{st}=\mathbb{A}^{i}_{sl}+\mathbb{A}^{i}_{lt}$ (using that
additivity holds for $\mathbb{A}^{m,n,i}$). Thus for
$\mathbb{Z}^{V,i}_{st}=\mathbb{A}^{i}_{st}-J^{T}(\partial_{i}V)(s,X_{s})Z^{i}_{st}$,
we obtain the algebraic relation

$\displaystyle\mathbb{Z}^{V,i}_{st}=\mathbb{Z}^{V,i}_{sl}+\mathbb{Z}^{V,i}_{lt}+[J^{T}(\partial_{i}V)(l,X_{l})-J^{T}(\partial_{i}V)(s,X_{s})]Z^{i}_{lt}.$

Furthermore, (3.2) and (3.3) yield the bounds on $Z^{i}$ and
$\mathbb{Z}^{V,i}$ and hence
$(Z^{i},\mathbb{Z}^{V,i})\in\mathcal{R}_{T}(V^{i})$. We have that
$\mathbb{Z}^{m,n,i}_{st}=\mathbb{A}^{m,n,i}_{st}-J^{T}(\partial_{i}V^{m})(s,X_{s})Z^{n,i}_{st}$.
Due to bounds (3.2) and (3.3) and the convergence in $2.)$, it follows that
$(Z^{n,i},\mathbb{Z}^{m,n,i})\in\mathcal{R}_{T}(V^{m,i}):=\mathcal{R}_{T}^{(\alpha+\beta)/\alpha}((J^{T}(\partial_{i}V^{m})(t,X_{t}))_{t})$.
The convergence with respect to $\lVert\cdot\rVert_{\mathcal{R}_{T}(V)}$
follows directly from the convergence in $2.)$. ∎

To prove that $(\partial_{i}u(t,X_{t}))_{t}$ is stochastically controlled by
$A=(J^{T}(\partial_{i}V)(t,X_{t}))_{t}$, we will need the following auxillary
lemma, which is also of independent interest. Its proof only relies on
regularity properties of the solution of the Kolmogorov backward equation. The
lemma proves the bound (5.3) on the time-space differences of the
paracontrolled remainder. The proof of the lemma can be found in Appendix A.

######  Lemma 5.3.

Let $\alpha\in(1,2]$, $\beta\in(\frac{2-2\alpha}{3},0)$,
$\mathscr{V}\in\mathscr{X}^{\beta,\gamma}$ and $u\in D_{T}(V)$ (cf. (5.2)).
Then, we have the following time-space Hölder bound:

$\displaystyle\lvert\partial_{i}u(r,x)-\partial_{i}u(s,y)-\nabla
u(s,y)\cdot(J^{T}(\partial_{i}V)(r,x)-J^{T}(\partial_{i}V)(s,y))\rvert$
$\displaystyle\lesssim\lVert
u\rVert_{D_{T}(V)}(1+\lVert\mathscr{V}\rVert_{\mathscr{X}^{\beta,\gamma}})[\lvert
r-s\rvert^{(2(\alpha+\beta)-2)/\alpha}+\lvert x-y\rvert^{2(\alpha+\beta)-2}],$
(5.3)

for $i=1,\dots,d$.

######  Lemma 5.4.

Let $\beta\in((2-2\alpha)/3,(1-\alpha)/2]$, $\alpha\in(1,2]$. Let $u\in
D_{T}(V)$ and let $(X,L,\mathbb{Z}^{V})$ be a weak solution. Then, for every
$i=1,\dots,d$, $(\partial_{i}u(t,X_{t}))_{t\in[0,T]}$ is stochastically
controlled by $(J^{T}(\partial_{i}V)(t,X_{t}))_{t\in[0,T]}$ with
$\varsigma=\varsigma^{\prime}=(\alpha+\beta-1)/\alpha$.

######  Remark 5.5.

In the pure stable noise case, $\alpha\in(1,2)$, the proof of Lemma 5.4 does
not apply for $\beta\in((1-\alpha)/2,0)$ (i.e. in the Young case), while for
$\alpha=2$, the statement of the lemma is also valid in the Young regime. The
reason is the integrability issue for the $\alpha$-stable process. Indeed, in
the proof we need that $4(\alpha+\beta)-4<\alpha$, that is
$\beta<(4-3\alpha)/4$. If $\alpha<2$ and $\beta\leqslant(1-\alpha)/2$, then in
particular $\beta<(4-3\alpha)/4$. However, the latter doesn’t need to be
satisfied in the Young regime, unless $\alpha<4/3$.

###### Proof of Lemma 5.4.

By Lemma 5.3, we obtain that

$\displaystyle\lvert
R_{st}\rvert:=\lvert\partial_{i}u(t,X_{t})-\partial_{i}u(s,X_{s})-\nabla
u_{s}\cdot[J^{T}(\partial_{i}V)(t,X_{t})-J^{T}(\partial_{i}V)(s,X_{s})]\rvert$
$\displaystyle\lesssim\lVert
u\rVert_{D_{T}}(1+\lVert\mathscr{V}\rVert_{\mathscr{X}^{\beta,\gamma}})[\lvert
t-s\rvert^{(2(\alpha+\beta)-2)/\alpha}+\lvert
X_{t}-X_{s}\rvert^{2(\alpha+\beta)-2}].$ (5.4)

Using the triangle inequality and $u\in\mathcal{L}_{T}^{0,\alpha+\beta}$ as
well as the interpolation estimate [KP23, Lemma 3.7, (3.11)], we can bound

$\displaystyle\mathds{E}[\lvert\partial_{i}u(t,X_{t})-\partial_{i}u(s,X_{s})\rvert^{2}]$
$\displaystyle\lesssim\lVert\partial_{i}u\rVert_{C_{T}^{(\alpha+\beta-1)/\alpha}L^{\infty}}^{2}\lvert
t-s\rvert^{2(\alpha+\beta-1)/\alpha}+\lVert\partial_{i}u\rVert_{C_{T}\mathscr{C}^{\alpha+\beta-1}}^{2}\mathds{E}[\lvert
X_{t}-X_{s}\rvert^{2(\alpha+\beta-1)}]$ $\displaystyle\lesssim_{T}\lVert
u\rVert_{D_{T}(V)}^{2}\lvert t-s\rvert^{2(\alpha+\beta-1)/\alpha},$

and the same bound is also valid for $\nabla u$. In the last estimate above,
we used that due to the bound (3.2) on $Z$ in $L^{2}$,

$\displaystyle\mathds{E}[\lvert X_{t}-X_{s}\rvert^{2(\alpha+\beta-1)}]$
$\displaystyle\leqslant\mathds{E}[\lvert
Z_{t}-Z_{s}\rvert^{2(\alpha+\beta-1)}]+\mathds{E}[\lvert
L_{t}-L_{s}\rvert^{2(\alpha+\beta-1)}]$
$\displaystyle\leqslant\mathds{E}[\lvert
Z_{t}-Z_{s}\rvert^{2}]^{(\alpha+\beta-1)}+\mathds{E}[\lvert
L_{t}-L_{s}\rvert^{2(\alpha+\beta-1)}]$ $\displaystyle\leqslant\lVert
Z\rVert_{\frac{\alpha+\beta}{\alpha},2}^{2(\alpha+\beta-1)}\lvert
t-s\rvert^{2(\alpha+\beta)(\alpha+\beta-1)/\alpha}+\lvert
t-s\rvert^{2(\alpha+\beta-1)/\alpha}\mathds{E}[\lvert
L_{1}\rvert^{2(\alpha+\beta-1)}]$ $\displaystyle\lesssim_{T}\lvert
t-s\rvert^{2(\alpha+\beta-1)/\alpha},$

due to Jensen’s inequality with $\alpha+\beta-1\in(0,1)$ and due to
$(L_{t}-L_{s})\stackrel{{\scriptstyle d}}{{=}}(t-s)^{1/\alpha}L_{1}$ for
$s\leqslant t$, and $2(\alpha+\beta-1)<\alpha$, such that $\mathds{E}[\lvert
L_{1}\rvert^{2(\alpha+\beta-1)}]<\infty$. For the remainder, we employ the
bound (5) to obtain

$\displaystyle\mathds{E}[\lvert R_{st}\rvert^{2}]$
$\displaystyle\lesssim\lVert u\rVert_{D_{T}}^{2}\big{(}\lvert
t-s\rvert^{(4(\alpha+\beta)-4)/\alpha}+\mathds{E}[\lvert
X_{t}-X_{s}\rvert^{4(\alpha+\beta)-4}]\big{)}$
$\displaystyle\lesssim_{T}\lVert u\rVert_{D_{T}}^{2}\lvert
t-s\rvert^{4\varsigma/\alpha},$

using an analogoue argument to estimate $\mathds{E}[\lvert
X_{t}-X_{s}\rvert^{4(\alpha+\beta)-4}]$ as above and $\mathds{E}[\lvert
L_{1}\rvert^{4(\alpha+\beta)-4}]<\infty$, since $4(\alpha+\beta)-4<\alpha$ in
the case of $\alpha<2$, as $\beta\leqslant(1-\alpha)/2$. In the case of
$\alpha=2$, we have all moments on the Brownian motion $B_{1}=L_{1}$. Thus, we
obtain that $(\partial_{i}u(t,X_{t}))_{t}$ is stochastically controlled by
$(J^{T}(\partial_{i}V)(t,X_{t}))_{t}$ with
$\varsigma=\varsigma^{\prime}=(\alpha+\beta-1)/\alpha$ and
$(\partial_{i}u)^{\prime}=\nabla u$. ∎

######  Proposition 5.6.

Let $\beta\in(\frac{2-2\alpha}{3},\frac{1-\alpha}{2}]$ and
$V\in\mathscr{X}^{\beta,\gamma}$. Let $(X,L,\mathbb{Z}^{V})$ be a weak
solution.
Then for all $0\leqslant t\leqslant T$ and for
$(Z,\mathbb{Z}^{V})\in\mathcal{R}_{T}(V)$ given by Definition 3.1 and for
$u\in D_{T}(V)$, the rough stochastic integral

$\displaystyle\int_{0}^{t}\nabla u(s,X_{s})\cdot
d(Z,\mathbb{Z}^{V})_{s}:=\sum_{i=1}^{d}\lim_{\lvert\Pi\rvert\to
0}\sum_{r,l\in\Pi}[\partial_{i}u(r,X_{r})Z^{i}_{rl}+\nabla
u(r,X_{r})\cdot\mathbb{Z}^{V,i}_{rl}]\in L^{2}(\mathds{P}),$

where the limit ranges over all partitions $\Pi$ of $[0,t]\subset[0,T]$ with
mesh-size $\lvert\Pi\rvert:=\max_{r,s\in\Pi}\lvert r-s\rvert\to 0$, is well-
defined and allows for the bound

$\displaystyle\bigg{\lVert}\int_{s}^{t}\nabla u(r,X_{r})\cdot
d(Z,\mathbb{Z}^{V})_{r}\bigg{\rVert}_{L^{2}(\mathds{P})}\lesssim_{T}\lvert
t-s\rvert^{(\alpha+\beta)/\alpha}\lVert
u\rVert_{D_{T}(V)}\lVert(Z,\mathbb{Z}^{V})\rVert_{\mathcal{R}_{T}(V)}.$

In particular, for the sequence $(V^{n})$ from Definition 3.1 and
$(Z^{n},\mathbb{Z}^{m,n}),(Z,\mathbb{Z}^{m,\infty})\in\mathcal{R}_{T}^{\sigma}(V^{m})$
and for a sequence $(u^{m})_{m}\subset D_{T}(V^{m})$ with $\lVert
u^{m}-u\rVert_{D_{T}(V)}\to 0$, it follows, that almost surely

$\displaystyle\int_{0}^{t}\nabla u^{m}(r,X_{r})\cdot
d(Z^{n},\mathbb{Z}^{m,n})_{r}$ $\displaystyle=\int_{0}^{t}\nabla
u^{m}(r,X_{r})\cdot dZ^{n}_{r},$ $\displaystyle\int_{0}^{t}\nabla
u^{m}(r,X_{r})\cdot d(Z,\mathbb{Z}^{m,\infty})_{r}$
$\displaystyle=\int_{0}^{t}\nabla u^{m}(r,X_{r})\cdot dZ_{r},$

where the stochastic integrals on the right-hand side are defined as the
$L^{2}(\mathds{P})$-limit of the classical Riemann sums. And the following
convergence in $L^{2}(\mathds{P})$, uniformly in $t\in[0,T]$, holds for
$n,m\to\infty$, respectively $m\to\infty$,

$\displaystyle\int_{0}^{t}\nabla u^{m}(r,X_{r})\cdot
d(Z^{n},\mathbb{Z}^{m,n})_{r}$ $\displaystyle\to\int_{0}^{t}\nabla
u(r,X_{r})\cdot d(Z,\mathbb{Z}^{V})_{r},$ $\displaystyle\int_{0}^{t}\nabla
u^{m}(r,X_{r})\cdot d(Z,\mathbb{Z}^{m,\infty})_{r}$
$\displaystyle\to\int_{0}^{t}\nabla u(r,X_{r})\cdot d(Z,\mathbb{Z}^{V})_{r}.$

###### Proof.

Recall that $u^{m}$ is paracontrolled by $J^{T}(V^{m})$ (i.e. $u^{m}\in
D_{T}(V^{m})$),

$\displaystyle\mathbb{Z}^{m,n,i}_{st}$
$\displaystyle=\int_{s}^{t}[J^{T}(\partial_{i}V^{m})(r,X_{r})-J^{T}(\partial_{i}V^{m})(s,X_{s})]dZ^{n,i}_{r}\quad\text{
and }$ $\displaystyle\mathbb{Z}^{m,\infty,i}_{st}$
$\displaystyle=\int_{s}^{t}[J^{T}(\partial_{i}V^{m})(r,X_{r})-J^{T}(\partial_{i}V^{m})(s,X_{s})]dZ^{i}_{r}.$

Then the proof follows from Theorem 4.4, Lemma 5.4 and Lemma 5.1. ∎

Proposition 5.6 extendes the stochastic integral to the stable rough
stochastic integral, which finally enables to prove the following theorem.

######  Theorem 5.7.

Let $\mathscr{V}\in\mathscr{X}^{\beta,\gamma}$ and
$\beta\in((2-2\alpha)/3,0)$. Let $(X,L,\mathbb{Z}^{V})$ be a weak solution,
starting at $x\in\mathbb{R}^{d}$. Then $X$ solves the martingale problem for
the generator $\mathcal{G}^{\mathscr{V}}$, starting at $x\in\mathbb{R}^{d}$.

###### Proof.

Let $f\in C_{T}\mathscr{C}^{\varepsilon}$, $\varepsilon>2-\alpha$,
$u^{T}\in\mathscr{C}^{3}$ and $u$ be the solution of
$\mathscr{G}^{\mathscr{V}}u=f$, $u(T,\cdot)=u^{T}$. Let $(X,L,\mathbb{Z}^{V})$
be a weak solution starting at $x$ on the stochastic basis
$(\Omega,\mathcal{F},(\mathcal{F}_{t}),\mathds{P})$. Then the goal is to show
that the process $(M_{t})_{t\in[0,T]}$ with

$\displaystyle M_{t}=u(t,X_{t})-u(0,x)-\int_{0}^{t}f(s,X_{s})ds$ (5.5)

is a martingale with respect to
$(\mathcal{F}^{X}_{t})\subset(\mathcal{F}_{t})$ under $\mathds{P}$, where
$\mathcal{F}^{X}_{t}:=\sigma(X_{s}\mid s\leqslant t)$ is the canonical
filtration. Because $X$ is a weak solution, there exists a sequence
$(V^{n})\subset C_{T}(C^{\infty}_{b})^{d}$ satisfying
$(V^{n},\mathcal{K}(V^{m},V^{n}))\to\mathscr{V}$ in
$\mathscr{X}^{\beta,\gamma}$ and such that the convergences from $1.),2.)$
from Definition 3.1 hold. Consider the solution $u^{n}$ of
$\mathcal{G}^{V^{n}}u^{n}=f$, $u^{n}(T,\cdot)=u^{T}$, which converges to $u$
in $D_{T}(V)$ by the continuity of the solution map from Theorem 2.9. Then as
$u^{n}\in C^{1,2}_{b}([0,T]\times\mathbb{R}^{d})$ and $X$ is a Dirichlet
process, we can apply the Itô formula from [CJMS06, Theorem 3.1] to
$u^{n}(t,X_{t})$ (see also Remark 3.3), such that for $n\in\mathbb{N}$,

$\displaystyle u^{n}(t,X_{t})-u^{n}(0,x)$
$\displaystyle=\int_{0}^{t}(\partial_{t}-\mathfrak{L}^{\alpha}_{\nu})u^{n}(s,X_{s})ds+\int_{0}^{t}\nabla
u^{n}(s,X_{s})\cdot dZ_{s}+M^{n}_{t}$
$\displaystyle=\int_{0}^{t}f(s,X_{s})ds+M_{t}^{n}$
$\displaystyle\quad+\bigg{(}\int_{0}^{t}\nabla u^{n}(s,X_{s})\cdot
dZ_{s}-\int_{0}^{t}\nabla u^{n}(s,X_{s})\cdot V^{n}(s,X_{s})ds\bigg{)},$ (5.6)

where we furthermore used the equation for $u^{n}$ and abbreviate the
martingale, in the case $\alpha\in(1,2)$, by

$\displaystyle
M_{t}^{n}:=\int_{0}^{t}\int_{\mathbb{R}^{d}\setminus\\{0\\}}(u^{n}(s,X_{s-}+y)-u^{n}(s,X_{s-}))\hat{\pi}(ds,dy).$
(5.7)

In the Brownian noise case, $\alpha=2$, we have that
$M_{t}^{n}=\int_{0}^{t}\nabla u^{n}(s,X_{s})\cdot dB_{s}$. It follows that
$M^{n}$ is a $(\mathcal{F}_{t})$-martingale (cf. the argument in the proof of
Theorem 2.12) and it is $(\mathcal{F}_{t}^{X})$-adapted by (5.5). Thus $M^{n}$
is a $(\mathcal{F}^{X}_{t})$-martingale. We claim that $(M^{n})_{n}$ converges
in $L^{2}(\mathds{P})$, uniformly in $t\in[0,T]$, to a martingale $\tilde{M}$
given by the expression (5.7) for $u^{n}$ replaced by $u$. Indeed, this uses
the Burkholder-Davis-Gundy-type inequality from [PZ07, Lemma 8.21] and an
analogoue argument as in the proof of [KP22, Lemma 4.4].
Since $u^{n}\to u$ in $C_{T}L^{\infty}$, we thus obtain that the process $M$
in (5.5) is a martingale with $M=\tilde{M}$, provided that the remainder in
(5.6) vanishes in $L^{2}(\mathds{P})$. That is, for
$Z^{n}=\int_{0}^{\cdot}V^{n}(s,X_{s})ds$, we claim that when $n\to\infty$,

$\displaystyle\sup_{t\in[0,T]}\bigg{\lVert}\int_{0}^{t}\nabla
u^{n}(s,X_{s-})\cdot dZ^{n}_{t}-\int_{0}^{t}\nabla u^{n}(s,X_{s-})\cdot
dZ_{s}\bigg{\rVert}_{L^{2}(\mathds{P})}\to 0.$ (5.8)

The stochastic integrals are given by the limit of the classical Riemann sums
in $L^{2}(\mathds{P})$.
If $\beta>(1-\alpha)/2$ (i.e. in the Young case), the convergence follows from
the stability of the stochastic sewing integral from Lemma 3.5 applied for
$f=\partial_{i}u$ and using only that $\lVert
Z^{n}-Z\rVert_{\frac{\alpha+\beta}{\alpha},2}\to 0$, $\lVert
Z^{n}-Z\mid\mathcal{F}\rVert_{\frac{\alpha+\beta}{\alpha},\infty}\to 0$ and
$u^{n}\to u$ in $D_{T}(V)$. Thus, in the Young case, the remainder vanishes.
If $\beta\in((2-2\alpha)/3,(1-\alpha)/2]$, then an application of Proposition
5.6 yields (5.8). ∎

In what follows, we prove the reverse implication: a martingale solution is a
(rough) weak solution in the sense of Definition 3.1. The first step is to
prove that a martingale solution is of class $\mathcal{K}^{\vartheta}$ for
$\vartheta=(2(\alpha+\beta)-1)/\alpha$, cf. Definition 3.8. With that, the
bound on $\mathbb{Z}^{V}$ in (3.3) follows after identifying $Z$ and
$\mathbb{A}$ with the respective solutions $u^{t},v^{t}$ of the backward PDEs
in (3.6), (3). The latter will be established in the following proposition.

######  Proposition 5.8.

Let $V\in\mathcal{X}^{\beta,\gamma}$ for $\beta\in((2-2\alpha)/3,0)$. Let $X$
be the solution of the martingale problem for the generator
$\mathcal{G}^{\mathscr{V}}$, starting at $x\in\mathbb{R}^{d}$. Then $X$ is of
class $\mathcal{K}^{\vartheta}(\cal V)$ (cf. Definition 3.8) for
$\vartheta=(2(\alpha+\beta)-1)/\alpha$.

######  Remark 5.9.

For a martingale solution $X$, we prove below that

$\displaystyle\mathbb{Z}^{V,i}_{st}$
$\displaystyle:=\int_{s}^{t}[J^{T}(\partial_{i}V)(r,X_{r})-J^{T}(\partial_{i}V)(s,X_{s})]dZ^{i}_{s}$
$\displaystyle=v^{t,i}(t,X_{t})-v^{t,i}(s,X_{s})-J^{T}(\partial_{i}V)(s,X_{s})(u^{t,i}(t,X_{t})-u^{t,i}(s,X_{s}))+M^{i}_{st}$
(5.9)

for martingale differences $M^{i}_{st}$ and the solutions $v^{t,i},u^{t,i}$
from (3), (3.6). Then, Proposition 5.8 shows
$\lVert\mathbb{Z}^{V}\mid\mathcal{F}\rVert_{\vartheta,\infty}<\infty$ for
$\vartheta=(2(\alpha+\beta)-1)/\alpha$. It remains open, if one can prove the
bound $\lVert\mathbb{Z}^{V}\rVert_{\vartheta,2}<\infty$, which is a stronger
bound than $\lVert\mathbb{Z}^{V}\rVert_{(\alpha+\beta)/\alpha,2}<\infty$ (the
latter can be inferred from the time regularity of $v^{t},u^{t}$). This is
also the reason why we employ the stochastic sewing lemma instead of the
classical sewing lemma in $L^{2}(\mathds{P})$.
It is not straightforward to adjust the proof of Proposition 5.8 to show
$\lVert\mathbb{Z}^{V}\rVert_{\vartheta,2}<\infty$. Indeed, the proof exploits
the conditional expectation in two ways. On the one hand, it removes the
martingale differences, for which the estimates would be more involved. On the
other hand, the conditional expectation enables to use the Markov property of
$X$ and thus to transform the claim into a question on Schauder and commutator
estimates for the semigroup $(T_{s,r})_{s\leqslant r}$ of $X$. In the proof
below, we infer those estimates on $(T_{s,r})_{s\leqslant r}$ from regularity
properties of the solutions of the generator PDE with singular terminal
conditons from Theorem 2.9.

###### Proof of Proposition 5.8.

Let $0\leqslant s<t\leqslant T$. By Theorem 3.10 the martingale solution $X$
for $\mathcal{G}^{\mathscr{V}}$ is a strong Markov process. Replacing
$v^{t,i}$ by the solution $v^{n,i}$ of
$\mathcal{G}^{(V,\mathscr{V}_{2})}v^{n,i}=J^{T}(\partial_{i}V)\cdot V^{n,i}$,
$v^{n,i}(t,\cdot)=0$ and $u^{t,i}$ by $u^{n,i}$ with
$\mathcal{G}^{(V,\mathscr{V}_{2})}u^{n,i}=V^{n,i}$, $u^{n,i}(t,\cdot)=0$ and
rewriting the conditional expectation in (3.8) with the semigroup
$(T_{r,l})_{0\leqslant r\leqslant l\leqslant T}$ of the (time-inhomogeneous)
Markov process $X$, we obtain

$\displaystyle\mathds{E}_{s}[v^{t,i}(t,X_{t})-v^{t,i}(s,X_{s})-J^{T}(\partial_{i}V)(s,X_{s})(u^{t,i}(t,X_{t})-u^{t,i}(s,X_{s}))]$
$\displaystyle=\mbox{$\displaystyle\lim_{n\to\infty}\mathds{E}_{s}[v^{n,i}(t,X_{t})-v^{n,i}(s,X_{s})-J^{T}(\partial_{i}V)(s,X_{s})(u^{n,i}(t,X_{t})-u^{n,i}(s,X_{s}))]$}$
(5.10)
$\displaystyle=\lim_{n\to\infty}\mathds{E}_{s}\bigg{[}\int_{s}^{t}(J^{T}(\partial_{i}V)(r,X_{r})-J^{T}(\partial_{i}V)(s,X_{s}))\cdot
V^{n,i}(r,X_{r})dr\bigg{]}$ (5.11)
$\displaystyle=\lim_{n\to\infty}\int_{s}^{t}T_{s,r}(J^{T}(\partial_{i}V)\cdot
V^{n,i})_{r}(X_{s})-J^{T}(\partial_{i}V)(s,X_{s})T_{s,r}(V^{n,i}_{r})(X_{s})dr\allowdisplaybreaks$
$\displaystyle=\lim_{n\to\infty}\int_{s}^{t}T_{s,r}(J^{T}(\partial_{i}V)\cdot
V^{n,i})_{r}(X_{s})-J^{T}(\partial_{i}V)(r,X_{s})T_{s,r}(V^{n,i}_{r})(X_{s})dr$
$\displaystyle\qquad+\lim_{n\to\infty}\int_{s}^{t}(J^{T}(\partial_{i}V)(s,X_{s})-J^{T}(\partial_{i}V)(r,X_{s}))T_{s,r}V^{n,i}_{r}(X_{s})dr$
$\displaystyle=\int_{s}^{t}w^{r,i}_{s}(X_{s})-J^{T}(\partial_{i}V)(r,X_{s})y^{r,i}_{s}(X_{s})dr$
(5.12)
$\displaystyle\qquad+\int_{s}^{t}(J^{T}(\partial_{i}V)(s,X_{s})-J^{T}(\partial_{i}V)(r,X_{s}))y^{r,i}_{s}(X_{s})dr.$
(5.13)

The convergence above is in $L^{\infty}(\mathds{P})$, uniformly in $s,t$. The
convergence in (5.10) follows from $\sup_{t\in[0,T]}\lVert
v^{n,i}-v^{t,i}\rVert_{C_{t}L^{\infty}_{\mathbb{R}^{d}}}\to 0$ and
$\sup_{t\in[0,T]}\lVert u^{n,i}-u^{t,i}\rVert_{C_{t}L^{\infty}}\to 0$ by
[KP23, Corollary 4.14] (taking $f^{t,n}=V^{n,i}_{\mid[0,t]}$ and
$y^{n,t}=u^{n,i}(t,\cdot)=0$). For the equality (5.11), we used that $X$
solves the martingale problem for $\mathcal{G}^{\mathscr{V}}$, such that

$\displaystyle
u^{n,i}(t,X_{t})-u^{n,i}(s,X_{s})=\int_{s}^{t}V^{n,i}(r,X_{r})dr+M_{r,t}^{n,i},$

for martingale differences $M^{n,i}_{r,t}$, that vanishes after taking the
conditional expectation, analogously for $v^{n,i}$. Moreover, $y^{r},w^{r}$
are defined as follows. Let

$\displaystyle T_{s,r}V_{r}^{n,i}=:y^{r,n,i}_{s},\quad
T_{s,r}(J^{T}(\partial_{i}V)\cdot V_{r}^{n,i})=:w^{r,n,i}_{s},\quad
s\in[0,r].$ (5.14)

Then we have that $y^{r,n,i}$ solves $\mathcal{G}^{\mathscr{V}}y^{r,n,i}=0$
with singular terminal condition $y^{n,r}_{r}=V_{r}^{n,i}$ at time $r$ and
$w^{r,n,i}=(w^{r,n,i,j})_{j=1,\dots,d}$ solves
$\mathcal{G}^{\mathscr{V}}w^{n,r,i,j}=0$ with
$w^{r,n,i,j}_{r}=J^{T}(\partial_{i}V^{j})_{r}\cdot V_{r}^{n,i}$. By the
continuity of the PDE solution map (cf. Theorem 2.9), we conclude that

$\displaystyle(y^{r,n,i},y^{r,n,i,\sharp})\to(y^{r,i},y^{r,i,\sharp}),$
$\displaystyle(w^{r,n,i,j},w^{r,n,i,j,\sharp})\to(w^{r,i,j},w^{r,i,j,\sharp})\in\mathcal{L}_{r}^{\gamma,\alpha+\beta}\times\mathcal{L}_{r}^{\gamma,2(\alpha+\beta)-1},$

for $n\to\infty$, where $y^{r,i},w^{r,i}$ solve
$\mathcal{G}^{\mathscr{V}}y^{r,i}=0$, $\mathcal{G}^{\mathscr{V}}w^{r,i,j}=0$
with terminal conditions $y^{r}_{r}=V^{i}_{r}$, respectively
$w^{r,i,j}_{r}=J^{T}(\partial_{i}V^{j})_{r}\cdot V^{i}_{r}$, and
$\gamma\in(0,1)$. The convergence is uniform in $r\in[0,T]$ by [KP23,
Corollary 4.14]. In particular, it follows that $y^{r,n,i}\to y^{r,i}$ and
$w^{r,n,i,j}\to w^{r,i,j}$ in $\mathcal{M}_{r}^{\gamma}L^{\infty}$, uniformly
in $r\in[0,T]$, which implies the convergence of the integrals (5.12) and
(5.13) in $L^{\infty}(\mathds{P})$, uniformly in $0\leqslant s<t\leqslant T$.
Our goal is now to estimate both integrals (5.12) and (5.13) in
$L^{\infty}(\mathds{P})$ by $\lvert t-s\rvert^{\vartheta}$.
To estimate the integral (5.12), we apply again [KP23, Corollary 4.14] for the
solutions $(y^{r,i})$, $(w^{r,i,j})$. The uniform bound from the corollary
together with the interpolation bound (3.13) from [KP23, Lemma 3.7] for
$\tilde{\theta}=\alpha+\beta\gamma^{-1}$ (note $\beta(1-\gamma^{-1})>0$) ,
$\theta=\alpha+\beta$ and the embedding
$\mathscr{C}^{(1-\gamma^{\prime})\alpha}\hookrightarrow L^{\infty}$ yield that

$\displaystyle\sup_{r\in[0,T]}\lVert
y^{r,i}\rVert_{\mathcal{M}_{r}^{-\beta/\alpha}L^{\infty}}\lesssim\sup_{r\in[0,T]}\lVert
y^{r,i}\rVert_{\mathcal{M}_{r}^{-\beta/\alpha}\mathscr{C}^{\beta(1-\gamma^{-1})}}\lesssim\sup_{r\in[0,T]}\lVert
y^{r,i}\rVert_{\mathcal{L}_{r}^{\gamma,\alpha+\beta}}\lesssim_{T,\mathscr{V}}1.$

Thus, we obtain

$\displaystyle\bigg{\lVert}\int_{s}^{t}(J^{T}(\partial_{i}V)_{s}-J^{T}(\partial_{i}V)_{r})y^{r,i}_{s}dr\bigg{\rVert}_{L^{\infty}}$
$\displaystyle\lesssim\sup_{r\in[0,T]}\lVert
y^{r,i}\rVert_{\mathcal{M}_{r}^{-\beta/\alpha}L^{\infty}}\lVert
J^{T}(\partial_{i}V)\rVert_{C_{T}^{(\alpha+\beta-1)/\alpha}L^{\infty}_{\mathbb{R}^{d}}}\int_{s}^{t}\lvert
r-s\rvert^{(\alpha+2\beta-1)/\alpha}dr$ $\displaystyle\lesssim_{T}\lVert
V\rVert_{C_{T}\mathscr{C}^{\beta}_{\mathbb{R}^{d}}}\lvert
t-s\rvert^{(2\alpha+2\beta-1)/\alpha}$ $\displaystyle=\lVert
V\rVert_{C_{T}\mathscr{C}^{\beta}_{\mathbb{R}^{d}}}\lvert
t-s\rvert^{\vartheta},$

using that $2\alpha+2\beta-1>0$ and $J^{T}(\partial_{i}V)\in
C_{T}^{(\alpha+\beta-1)/\alpha}L^{\infty}_{\mathbb{R}^{d}}$.
To estimate the term in (5.13), we use cancellations between the solution
$w^{r,i}$ and $y^{r,i}$. For the argument, we need to distinguish between the
cases $\beta\in((1-\alpha)/2,0)$ and $\beta\in((2-2\alpha)/3,(1-\alpha)/2]$.

First, we consider the Young case, $\beta\in((1-\alpha)/2,0)$. Then, we have
that $\alpha+2\beta-1>0$. We can write the difference of the solutions as
follows:

$\displaystyle w^{r,i}_{s}-J^{T}(\partial_{i}V)_{r}y^{r,i}_{s}$
$\displaystyle=[P_{r-s}(J^{T}(\partial_{i}V)_{r}\cdot
V^{i}_{r})-J^{T}(\partial_{i}V)_{r}\cdot P_{r-s}V_{r}^{i}]$
$\displaystyle\qquad+J^{r}_{s}(\nabla w^{r,i}\cdot V)-J^{r}_{s}(\nabla
y^{r,i}\cdot V)$
$\displaystyle=(P_{r-s}-\operatorname{Id})(J^{T}(\partial_{i}V)_{r}\varodot
V^{i}_{r}+J^{T}(\partial_{i}V)_{r}\varogreaterthan V^{i}_{r})$ (5.15)
$\displaystyle\qquad-J^{T}(\partial_{i}V)_{r}\varodot(P_{r-s}-\operatorname{Id})V_{r}^{i}+J^{T}(\partial_{i}V)_{r}\varogreaterthan(P_{r-s}-\operatorname{Id})V_{r}^{i}$
(5.16) $\displaystyle\qquad+[P_{r-s}(J^{T}(\partial_{i}V)_{r}\varolessthan
V^{i}_{r})-J^{T}(\partial_{i}V)_{r}\varolessthan P_{r-s}V_{r}^{i}]$ (5.17)
$\displaystyle\qquad+J^{r}_{s}(\nabla w^{r,i}\cdot
V)-J^{T}(\partial_{i}V)_{r}J^{r}_{s}(\nabla y^{r,i}\cdot V).$ (5.18)

Above and below we use the notation
$J^{t}_{r}(v):=J^{t}(v)_{r}=J^{t}(v)(r,\cdot)$ for $r\leqslant t$. The term in
(5.15), we estimate with the semigroup estimate (2.8) from [KP23, Lemma 2.5],
using that $\alpha+2\beta-1\in(0,1)$,

$\displaystyle\lVert(P_{r-s}-\operatorname{Id})(J^{T}(\partial_{i}V)_{r}\varodot
V^{i}_{r}+J^{T}(\partial_{i}V)_{r}\varogreaterthan
V^{i}_{r})\rVert_{L^{\infty}_{\mathbb{R}^{d}}}$ $\displaystyle\lesssim\lvert
r-s\rvert^{(\alpha+2\beta-1)/\alpha}\lVert J^{T}(\partial_{i}V)_{r}\varodot
V^{i}_{r}+J^{T}(\partial_{i}V)_{r}\varogreaterthan
V^{i}_{r})\rVert_{\mathscr{C}^{\alpha+2\beta-1}_{\mathbb{R}^{d}}}$
$\displaystyle\lesssim\lvert r-s\rvert^{(\alpha+2\beta-1)/\alpha}\lVert
V\rVert_{C_{T}\mathscr{C}^{\beta}_{\mathbb{R}^{d}}}^{2}.$

The terms in (5.16), we also estimate with the semigroup and Schauder
estimates using that $V\in
C_{T}(\mathscr{C}^{\beta+(1-\gamma^{\prime})\alpha})^{d}$ for
$\gamma^{\prime}\in[(1-\beta)/\alpha,1)$ and again that
$\alpha+2\beta-1\in(0,1]$ and the embedding
$\mathscr{C}^{(1-\gamma^{\prime})\alpha}\hookrightarrow L^{\infty}$,

$\displaystyle\lVert
J^{T}(\partial_{i}V)_{r}\varogreaterthan(P_{r-s}-\operatorname{Id})V_{r}^{i}\rVert_{L^{\infty}}$
$\displaystyle\lesssim\lVert
J^{T}(\partial_{i}V)_{r}\varogreaterthan(P_{r-s}-\operatorname{Id})V_{r}^{i}\rVert_{\mathscr{C}^{(1-\gamma^{\prime})\alpha}_{\mathbb{R}^{d}}}$
$\displaystyle\lesssim\lVert
J^{T}(\partial_{i}V)_{r}\rVert_{\mathscr{C}^{(2-\gamma^{\prime})\alpha+\beta-1}_{\mathbb{R}^{d}}}\lVert(P_{r-s}-\operatorname{Id})V_{r}^{i}\rVert_{\mathscr{C}^{-(\alpha+\beta-1)}}$
$\displaystyle\lesssim\lVert
V\rVert_{C_{T}(\mathscr{C}^{\beta+(1-\gamma^{\prime})\alpha})^{d}}\lvert
r-s\rvert^{(\alpha+2\beta-1)/\alpha}\lVert
V\rVert_{C_{T}(\mathscr{C}^{\beta})^{d}}.$

The argument for the $\varodot$-product is analogous. Moreover, the term in
(5.17) equals the semigroup commutator from [KP23, Lemma 2.7] and using that
$\alpha+2\beta-1<\alpha$, we obtain

$\displaystyle\lVert[P_{r-s}(J^{T}(\partial_{i}V)_{r}\varolessthan
V^{i}_{r})-J^{T}(\partial_{i}V)_{r}\varolessthan
P_{r-s}V_{r}^{i}]\rVert_{L^{\infty}}$
$\displaystyle\lesssim\lVert[P_{r-s}(J^{T}(\partial_{i}V)_{r}\varolessthan
V^{i}_{r})-J^{T}(\partial_{i}V)_{r}\varolessthan
P_{r-s}V_{r}^{i}]\rVert_{\mathscr{C}^{(1-\gamma^{\prime})\alpha}_{\mathbb{R}^{d}}}$
$\displaystyle\lesssim\lvert r-s\rvert^{(\alpha+2\beta-1)/\alpha}\lVert
V\rVert_{C_{T}(\mathscr{C}^{\beta+(1-\gamma^{\prime})\alpha})^{d}}\lVert
V\rVert_{C_{T}(\mathscr{C}^{\beta})^{d}}.$

For the last term (5.18), we give the argument for the term involving
$y^{r,i}$ and the argument for the term with $w^{r,i}$ is analogue. We
decompose

$\displaystyle J^{r}_{s}(\nabla y^{r,i}\cdot V)$
$\displaystyle=J^{r}_{s}(\nabla y^{r,i}\varodot V+\nabla
y^{r,i}\varogreaterthan V)+\nabla y^{r,i}_{s}\varolessthan J^{r}_{s}(V)$
$\displaystyle\qquad+[J^{r}_{s}(\nabla y^{r,i}\varolessthan V)-\nabla
y^{r,i}_{s}\varolessthan J^{r}_{s}(V)].$

Due to the interpolation bound (3.13) from [KP23, Lemma 3.7], we obtain that

$\displaystyle\nabla
y^{r,i}\in\mathcal{M}_{r}^{(1-\beta)/\alpha}\mathscr{C}^{(1-\beta)(\frac{\alpha}{\gamma^{\prime}}-1)}_{\mathbb{R}^{d}}\hookrightarrow\mathcal{M}_{r}^{(1-\beta)/\alpha}L^{\infty}_{\mathbb{R}^{d}}$

as $\gamma^{\prime}<1\leqslant\alpha$. This yields together with [KP23, Lemma
2.5] that

$\displaystyle\lVert\nabla y^{r,i}_{s}\varolessthan
J^{r}_{s}(V)\rVert_{L^{\infty}_{\mathbb{R}^{d}}}$
$\displaystyle\lesssim\lVert\nabla
y^{r,i}\rVert_{\mathcal{M}_{r}^{(1-\beta)/\alpha}L^{\infty}_{\mathbb{R}^{d}}}\lvert
r-s\rvert^{(\beta-1)/\alpha}\lVert
J^{r}_{s}(V)\rVert_{L^{\infty}_{\mathbb{R}^{d}}}$
$\displaystyle\lesssim\lVert\nabla
y^{r,i}\rVert_{\mathcal{M}_{r}^{(1-\beta)/\alpha}L^{\infty}_{\mathbb{R}^{d}}}\lvert
r-s\rvert^{(\beta-1)/\alpha}\lVert
J^{r}_{s}(V)\rVert_{\mathscr{C}^{(1-\gamma^{\prime})\alpha}_{\mathbb{R}^{d}}}$
$\displaystyle\lesssim\lvert r-s\rvert^{(\alpha+2\beta-1)/\alpha}\lVert\nabla
y^{r,i}\rVert_{\mathcal{M}_{r}^{(1-\beta)/\alpha}L^{\infty}_{\mathbb{R}^{d}}}\lVert
V\rVert_{C_{T}\mathscr{C}^{\beta+(1-\gamma^{\prime})\alpha}_{\mathbb{R}^{d}}}.$
(5.19)

[KP23, Corollary 3.2] furthermore yields that $J^{r}(\nabla y^{r,i}\varodot
V)\in\mathcal{L}_{r}^{\gamma,2\beta+\alpha-1+(2-\gamma^{\prime})\alpha}$ using
that $V\in C_{T}(\mathscr{C}^{\beta+(1-\gamma^{\prime})\alpha})^{d}$, which
implies by the interpolation bound [KP23, Lemma 3.7, (3.11)] and
$J^{r}_{r}(v)=0$ that $J^{r}(\nabla y^{r,i}\varodot V)\in
C_{r}^{(2\beta+\alpha-1+(2-\gamma^{\prime})\alpha-\gamma\alpha)/\alpha}L^{\infty}$
(and analogously for the $\varogreaterthan$-product). Thus, as
$J^{r}_{r}(v)=0$ and $2-\gamma^{\prime}-\gamma>0$ as $\gamma\in(0,1)$,
$\gamma^{\prime}\in(0,1)$ , we obtain that

$\displaystyle\lVert J^{r}_{s}(\nabla y^{r,i}\varodot V+\nabla
y^{r,i}\varogreaterthan V)\rVert_{L^{\infty}}\lesssim\lvert
r-s\rvert^{(\alpha+2\beta-1)/\alpha}\lVert
y^{r,i}\rVert_{\mathcal{L}_{r}^{\gamma,\alpha+\beta}}\lVert
V\rVert_{C_{T}(\mathscr{C}^{\beta+(1-\gamma^{\prime})\alpha})^{d}}.$

Due to the commutator [KP23, Lemma 3.4], we obtain that

$\displaystyle[J^{r}(\nabla y^{r,i}\varolessthan V)-\nabla
y^{r,i}\varolessthan
J^{r}(V)]\in\mathcal{L}_{r}^{\gamma,2\beta+\alpha-1+(2-\gamma^{\prime})\alpha}$

and by an interpolation argument as above we thus find

$\displaystyle\lVert J^{r}_{s}(\nabla y^{r,i}\varolessthan V)-\nabla
y^{r,i}_{s}\varolessthan J^{r}_{s}(V)\rVert_{L^{\infty}}\lesssim\lvert
r-s\rvert^{(\alpha+2\beta-1)/\alpha}\lVert
y^{r,i}\rVert_{\mathcal{L}_{r}^{\gamma,\alpha+\beta}}\lVert
V\rVert_{C_{T}(\mathscr{C}^{\beta+(1-\gamma^{\prime})\alpha})^{d}}.$

Together, we obtain for the term (5.18):

$\displaystyle\lVert J^{T}(\partial_{i}V)_{r}J^{r}_{s}(\nabla y^{r,i}\cdot
V)\rVert_{L^{\infty}}\lesssim\lvert r-s\rvert^{(\alpha+2\beta-1)/\alpha}\lVert
y^{r}\rVert_{\mathcal{L}_{r}^{\gamma,\alpha+\beta}}\lVert
V\rVert_{C_{T}(\mathscr{C}^{\beta+(1-\gamma^{\prime})\alpha})^{d}}^{2}.$

Together with the uniform bound of the norm of $y^{r},w^{r}$ from [KP23,
Corollary 4.14] the above estimates yield

$\displaystyle\int_{s}^{t}\lVert
w^{r,i}_{s}-J^{T}(\partial_{i}V)_{r}y^{r,i}_{s}\rVert_{L^{\infty}_{\mathbb{R}^{d}}}dr\lesssim\lvert
t-s\rvert^{(2\alpha+2\beta-1)/\alpha},$

which, together with the estimate for the integral (5.12) yields the claim in
the Young case.

If $\beta\in((2-2\alpha)/3,(1-\alpha)/2]$, we need to estimate the integral
(5.13) differently. In order to rewrite the difference of the solutions
$w^{r,i}_{s}-J^{T}(\partial_{i}V)y^{r,i}_{s}$ in a way so that we can prove
the claimed bound, we first specify the paracontrolled structure of
$y^{r},w^{r}$.
The terminal condition of $w^{r,i}$ has the paracontrolled structure

$\displaystyle w_{r}^{r,i,j}=J^{T}(\partial_{i}V^{j})_{r}\cdot V^{i}_{r}$
$\displaystyle=J^{T}(\partial_{i}V^{j})_{r}\varodot
V^{i}_{r}+J^{T}(\partial_{i}V^{j})_{r}\varogreaterthan
V^{i}_{r}+J^{T}(\partial_{i}V^{j})_{r}\varolessthan V^{i}_{r}$
$\displaystyle=:w^{R,\sharp,j}+w^{R,\prime,j}\varolessthan V_{r}$

with $w^{R,\sharp,j}\in\mathscr{C}^{(2-\gamma^{\prime})\alpha+2\beta-1}$ (due
to $V\in C_{T}\mathscr{C}^{\beta+(1-\gamma^{\prime})\alpha}_{\mathbb{R}^{d}}$)
and
$w^{R,\prime,j}=J^{T}(\partial_{i}V^{j})_{r}e_{i}\in\mathscr{C}^{\alpha+\beta-1}_{\mathbb{R}^{d}}$
($e_{i}$ denoting the $i$-th unit vector). By Theorem 2.9, the solution has
the following paracontrolled structure

$\displaystyle w^{r,i,j}_{s}=w^{r,\sharp,i,j}_{s}+\nabla
w^{r,i,j}_{s}\varolessthan
J^{r}(V)_{s}+J^{T}(\partial_{i}V^{j})_{r}e_{i}\varolessthan P_{r-s}V_{r}$
(5.20)

with $w^{r,\sharp,i,j}\in\mathcal{L}_{r}^{\gamma,2(\alpha+\beta)-1}$. Then,
again by the interpolation estimate [KP23, Lemma 3.7, (3.13)] applied for
$\theta=\tilde{\theta}=2\alpha+2\beta-1\in(0,\alpha)$, since
$\beta\leqslant(1-\alpha)/2$, and for
$\theta=\tilde{\theta}=\alpha+\beta-1\in(0,\alpha)$ we obtain together with
[KP23, Corollary 4.14] the uniform bound

$\displaystyle\sup_{r\in[0,T]}[\lVert
w^{r,\sharp,i}\rVert_{\mathcal{M}_{r}^{(1-\alpha-2\beta)/\alpha}L^{\infty}_{\mathbb{R}^{d}}}+\lVert\nabla
w^{r,i}\rVert_{\mathcal{M}_{r}^{(1-\beta)/\alpha}L^{\infty}_{\mathbb{R}^{d\times
d}}}]$ $\displaystyle\lesssim\sup_{r\in[0,T]}[\lVert
w^{r,\sharp,i}\rVert_{(\mathcal{L}_{r}^{\gamma,2(\alpha+\beta)-1})^{d}}+\lVert
w^{r,i}\rVert_{(\mathcal{L}_{r}^{\gamma,\alpha+\beta})^{d}}]\lesssim_{T,\mathscr{V}}1.$
(5.21)

Thus, using an estimate as for (5), we can estimate

$\displaystyle\lVert\nabla w^{r,i,j}_{s}\varolessthan
J^{r}(V)_{s}\rVert_{L^{\infty}}\lesssim\lvert
r-s\rvert^{(\alpha+2\beta-1)/\alpha}\lVert
V\rVert_{C_{T}\mathscr{C}^{\beta+(1-\gamma^{\prime})\alpha}_{\mathbb{R}^{d}}}\sup_{r\in[0,T]}\lVert\nabla
w^{r,i}\rVert_{\mathcal{M}_{r}^{(1-\beta)/\alpha}L^{\infty}_{\mathbb{R}^{d\times
d}}}.$ (5.22)

Furthermore, (5) implies that

$\displaystyle\int_{s}^{t}\lVert w^{r,\sharp,i,j}_{s}\rVert_{L^{\infty}}dr$
$\displaystyle\lesssim\sup_{r\in[0,T]}\lVert
w^{r,\sharp,i}\rVert_{\mathcal{M}_{r}^{(1-\alpha-2\beta)/\alpha}L^{\infty}_{\mathbb{R}^{d}}}\,\int_{s}^{t}\lvert
r-s\rvert^{(2\beta+\alpha-1)/\alpha}dr$ $\displaystyle\lesssim\lvert
t-s\rvert^{(2\alpha+2\beta-1)/\alpha}\sup_{r\in[0,T]}\lVert
w^{r,\sharp,i}\rVert_{\mathcal{M}_{r}^{(1-\alpha-2\beta)/\alpha}L^{\infty}_{\mathbb{R}^{d}}},$
(5.23)

since $(2\alpha+2\beta-1)/\alpha>0$.
Moreover, for the solution $y^{r,i}$ from above, we have the paracontrolled
structure

$\displaystyle y^{r,i}_{s}=y_{s}^{r,\sharp,i}+e_{i}\varolessthan
P_{r-s}V_{r}+\nabla y^{r,i}_{s}\varolessthan J^{r}(V)_{s}.$

The bounds (5.22), (5) hold analogously for $\nabla w^{i,j}$,
$w^{r,\sharp,i,j}$ replaced by $\nabla y^{r,i}$, $y^{r,i,\sharp}$.
Furthermore, the interpolation estimate [KP23, Lemma 3.7, (3.13)] (again for
$\theta=\alpha+\beta$, $\tilde{\theta}=2\alpha+2\beta-1\in(0,\alpha)$ as
$\beta\leqslant(1-\alpha)/2$) yields that

$\displaystyle\sup_{r\in[0,T]}[\lVert
y^{r,i}\rVert_{\mathcal{M}_{r}^{(1-\alpha-2\beta)/\alpha}\mathscr{C}^{-(\alpha+\beta-1)}}+\lVert
y^{r,\sharp,i}\rVert_{\mathcal{M}_{r}^{(1-\alpha-2\beta)/\alpha}L^{\infty}}]\lesssim_{T,\mathscr{V}}1.$

With the latter bound, we can estimate,

$\displaystyle\lVert J^{T}(\partial_{i}V)_{r}\varodot
y^{r,i}_{s}+J^{T}(\partial_{i}V)_{r}\varogreaterthan
y^{r,i}_{s}\rVert_{L^{\infty}_{\mathbb{R}^{d}}}$ $\displaystyle\lesssim\lVert
J^{T}(\partial_{i}V)_{r}\varodot
y^{r,i}_{s}+J^{T}(\partial_{i}V)_{r}\varogreaterthan
y^{r,i}_{s}\rVert_{\mathscr{C}^{(1-\gamma)\alpha}_{\mathbb{R}^{d}}}$
$\displaystyle\lesssim\lVert
J^{T}(\partial_{i}V)_{r}\rVert_{(\mathscr{C}^{\alpha+\beta+(1-\gamma^{\prime})\alpha-1})^{d}}\lVert
y^{r,i}_{s}\rVert_{\mathscr{C}^{-(\alpha+\beta-1)}}$
$\displaystyle\lesssim\lvert r-s\rvert^{(2\beta+\alpha-1)/\alpha}\lVert
y^{r,i}\rVert_{\mathcal{M}_{r}^{(1-\alpha-2\beta)/\alpha}\mathscr{C}^{-(\alpha+\beta-1)}}\lVert
V\rVert_{C_{T}(\mathscr{C}^{\beta+(1-\gamma^{\prime})\alpha})^{d}}.$

Using the paracontrolled structure of the solutions $w^{r,i,j},y^{r,i}$, we
obtain for $j=1,\dots,d$,

$\displaystyle w^{r,i,j}_{s}-J^{T}(\partial_{i}V^{j})_{r}\varolessthan
y^{r,i}_{s}$
$\displaystyle=w^{r,\sharp,i,j}_{s}+J^{T}(\partial_{i}V^{j})_{r}\varolessthan
y^{r,\sharp,i}_{s}+J^{T}(\partial_{i}V^{j})_{r}\varolessthan(\nabla
y^{r,i}\varolessthan J^{r}(V))_{s}$ $\displaystyle\qquad+(\nabla
w^{r,i,j}\varolessthan J^{r}(V))_{s}.$ (5.24)

Finally, we can estimate the integral (5.13) using the bounds derived above
and (5):

$\displaystyle\bigg{\lVert}\bigg{(}\int_{s}^{t}(w^{r,i,j}_{s}-J^{T}(\partial_{i}V^{j})_{r}\cdot
y^{r,i}_{s})dr\bigg{)}_{j}\bigg{\rVert}_{L^{\infty}_{\mathbb{R}^{d}}}$
$\displaystyle\leqslant\bigg{\lVert}\bigg{(}\int_{s}^{t}(w^{r,i,j}_{s}-J^{T}(\partial_{i}V^{j})_{r}\varolessthan
y^{r,i}_{s})dr\bigg{)}_{j}\bigg{\rVert}_{L^{\infty}_{\mathbb{R}^{d}}}$
$\displaystyle\qquad\qquad+\bigg{\lVert}\bigg{(}\int_{s}^{t}[J^{T}(\partial_{i}V^{j})_{r}\varodot
y^{r,i}_{s}+J^{T}(\partial_{i}V^{j})_{r}\varogreaterthan
y^{r,i}_{s}]dr\bigg{)}_{j}\bigg{\rVert}_{L^{\infty}_{\mathbb{R}^{d}}}\allowdisplaybreaks$
$\displaystyle\leqslant\bigg{\lVert}\int_{s}^{t}w^{r,\sharp,i}_{s}dr\bigg{\rVert}_{L^{\infty}_{\mathbb{R}^{d}}}+\lVert
J^{T}(\partial_{i}V)\rVert_{L^{\infty}_{\mathbb{R}^{d}}}\bigg{\lVert}\int_{s}^{t}y^{r,\sharp,i}_{s}dr\bigg{\rVert}_{L^{\infty}}$
$\displaystyle\qquad\qquad+\lVert
J^{T}(\partial_{i}V)\rVert_{L^{\infty}_{\mathbb{R}^{d}}}\bigg{\lVert}\int_{s}^{t}(\nabla
y^{r,i}\varolessthan J^{r}(V))_{s}dr\bigg{\rVert}_{L^{\infty}}$
$\displaystyle\qquad\qquad+\bigg{\lVert}\bigg{(}\int_{s}^{t}(\nabla
w^{r,i,j}\varolessthan
J^{r}(V))_{s}dr\bigg{)}_{j}\bigg{\rVert}_{L^{\infty}_{\mathbb{R}^{d}}}$
$\displaystyle\qquad\qquad+\bigg{\lVert}\bigg{(}\int_{s}^{t}[J^{T}(\partial_{i}V^{j})_{r}\varodot
y^{r,i}_{s}+J^{T}(\partial_{i}V^{j})_{r}\varogreaterthan
y^{r,i}_{s}]dr\bigg{)}_{j}\bigg{\rVert}_{L^{\infty}_{\mathbb{R}^{d}}}$
$\displaystyle\lesssim\lvert
t-s\rvert^{\vartheta}\sup_{r\in[0,T]}\Big{[}\lVert
w^{r,i}\rVert_{(\mathcal{L}_{r}^{\gamma,\alpha+\beta})^{d}}+\lVert
w^{r,i,\sharp}\rVert_{(\mathcal{L}_{r}^{\gamma,2(\alpha+\beta)-1})^{d}}$
$\displaystyle\qquad\qquad\qquad\qquad+\lVert\mathscr{V}\rVert_{\mathscr{X}^{\beta,\gamma^{\prime}}}(\lVert
y^{r,i}\rVert_{\mathcal{L}_{r}^{\gamma,\alpha+\beta}}+\lVert
y^{r,i,\sharp}\rVert_{\mathcal{L}_{r}^{\gamma,2(\alpha+\beta)-1}})\Big{]}.$
(5.25)

In the last estimate, we used also that
$\vartheta=(2\alpha+2\beta-1)/\alpha>0$. Together with the bound for (5.12),
this yields the claim in the rough case. ∎

######  Theorem 5.10.

Let $\mathscr{V}\in\mathscr{X}^{\beta,\gamma}$ for
$\beta\in((2-2\alpha)/3,0)$. Let $X$ be the solution of the martingale problem
for the generator $\mathcal{G}^{\mathscr{V}}$, starting at
$x\in\mathbb{R}^{d}$.
Then, there exists a stochastic basis
$(\Omega,\mathcal{F},(\mathcal{F}_{t}),\mathds{P})$ and an $\alpha$-stable
symmtric non-degenerate $(\mathcal{F}_{t})$-Lévy process $L$, such that
$(X,L,\mathbb{Z}^{V})$ is a weak solution starting at $x\in\mathbb{R}^{d}$.
Furthermore, the following representations for $Z=X-x-L$ and $\mathbb{Z}^{V}$
follow for $(s,t)\in\Delta_{T}$

$\displaystyle Z_{st}=\mathds{E}_{s}[u^{t}(t,X_{t})-u^{t}(s,X_{s})],$
$\displaystyle\mathbb{Z}^{V,i}_{st}=\mathds{E}_{s}[v^{t,i}(t,X_{t})-v^{t,i}(s,X_{s})-J^{T}(\partial_{i}V)(s,X_{s})(u^{t,i}(t,X_{t})-u^{t,i}(s,X_{s}))]$

almost surely, for the solutions $v^{t,i},u^{t,i}$ from (3), (3.6).

###### Proof.

Let $(V^{n})\subset C_{T}C^{\infty}_{b}(\mathbb{R}^{d},\mathbb{R}^{d})$ with
$(V^{n},\mathcal{K}(V^{n},V^{m}))\to\mathscr{V}$ in
$\mathscr{X}^{\beta,\gamma}$ for $n,m\to\infty$ (existence by Definition 2.7).
Let $X$ be the solution of the
$(\mathcal{G}^{\mathscr{V}},\delta_{x})$-martingale problem. Let, as in the
proof of Theorem 2.12, $X^{n}$ be the strong solution of

$\displaystyle X^{n}=x+\int_{0}^{\cdot}V^{n}(s,X_{s}^{n})ds+L=:x+Z^{n,n}+L$

and let for $l,m,n\in\mathbb{N}$,

$\displaystyle Z^{m,n}_{t}$
$\displaystyle:=\int_{0}^{t}V^{m}(s,X_{s}^{n})ds,\quad t\in[0,T]$
$\displaystyle\mathbb{Z}^{l,m,n}_{st}$
$\displaystyle:=\bigg{(}\int_{s}^{t}[J^{T}(\partial_{i}V^{j,l})(r,X_{r}^{n})-J^{T}(\partial_{i}V^{j,l})(s,X_{s}^{n})]dZ_{r}^{i,m,n}\bigg{)}_{i,j},\quad(s,t)\in\Delta_{T}.$

Theorem 2.12 proves the distributional convergence
$(X^{n},Z^{n,n})\Rightarrow(X,Z)$, where $Z$ is a continuous process. Let
$u^{t,m,n}=(u^{t,m,n,i})_{i=1,\dots,d}$ and
$v^{t,l,m,n}=(v^{t,l,m,n,i,j})_{i,j=1,\dots,d}$ solve

$\displaystyle\mathcal{G}^{V^{n}}u^{t,m,n,i}=V^{m,i},\quad\mathcal{G}^{V^{n}}v^{t,l,m,n,i,j}=J^{T}(\partial_{i}V^{l,j})\cdot
V^{m,i}$

with zero terminal condition at time $t\in[0,T]$. By convergence of the mixed
resonant products, i.e. $(V^{n},\mathcal{K}(V^{n},V^{m}))\to\mathscr{V}$ in
$\mathscr{X}^{\beta,\gamma}$, and by continuity of the PDE solution map
(Theorem 2.9), we obtain that $u^{t,m,n,i}\to u^{t,i}$ and $v^{t,l,m,n,i,j}\to
v^{t,i,j}$ in $D_{t}$, where $u^{t},v^{t}$ solve the PDEs (3.6), (3). An
application of Itô’s formula (cf. in the proof of Theorem 2.12) then yields
the representations

$\displaystyle
Z^{m,n}_{st}=u^{t,m,n}(t,X_{t}^{n})-u^{t,m,n}(s,X_{s}^{n})+M_{st}^{u^{t,m,n}}$
(5.26)

for the martingale differences $M^{u^{t,m,n}}_{st}$ defined as

$\displaystyle
M^{u^{t,m,n}}_{st}:=\int_{s}^{t}\int_{\mathbb{R}^{d}\setminus\\{0\\}}[u^{t,m,n}(r,X_{r-}^{n}+y)-u^{t,m,n}(r,X_{r-}^{n})]\hat{\pi}(dr,dy)$

for the compensated Poisson random measure $\hat{\pi}$, respectively
$M_{st}^{u^{t,m,n}}:=\int_{s}^{t}\nabla u^{t,m,n}(r,X_{r}^{n})dB_{r}$ in the
case $\alpha=2$. Moreover we have that

$\displaystyle\mathbb{Z}^{l,m,n}_{st}(i,j)$
$\displaystyle=v^{t,l,m,n,i,j}(t,X_{t}^{n})-v^{t,l,m,n,i,j}(s,X_{s}^{n})$
$\displaystyle\qquad-J^{T}(\partial_{i}V^{j,l})(s,X_{s}^{n})\big{(}u^{t,m,n,i}(t,X_{t}^{n})-u^{t,m,n,i}(s,X_{s}^{n})\big{)}$
$\displaystyle\qquad+M_{st}^{l,m,n}(i,j)$ (5.27)

for the martingale differences defined by

$\displaystyle
M^{l,m,n}_{st}(i,j):=M_{st}^{v^{t,l,m,n}}-J^{T}(\partial_{i}V^{j,l})(s,X_{s}^{n})M_{st}^{u^{t,m,n}}.$
(5.28)

Notice that $\mathds{E}_{s}[M^{l,m,n}_{st}]=0$. By convergence of the PDE
solutions, we obtain that for all large enough $l,m,n$,

$\displaystyle\sup_{t\in[0,T]}[\lVert u^{t,m,n}\rVert_{D_{t}^{d}}+\lVert
v^{t,l,m,n}\rVert_{D_{t}^{d\times d}}]\leqslant 2\sup_{t\in[0,T]}[\lVert
u^{t}\rVert_{D_{t}^{d}}+\lVert v^{t}\rVert_{D_{t}^{d\times d}}]<\infty,$
(5.29)

where the right-hand side is finite due to [KP23, Corollary 4.14].
By (5.29), $J^{T}(\partial_{i}V^{j})\in C_{T}L^{\infty}$ and [KP22, Lemma 4.4]
applied to both $M^{u^{t,m,n}}_{st},M^{v^{t,l,m,n}}_{st}$ we can estimate
$\lVert(s,t)\mapsto M_{st}^{l,m,n}\rVert_{(\alpha+\beta)/\alpha,p}$ for $p\in
2\mathbb{N}$. Together with Lemma 2.13 we thus obtain, for any $p\in
2\mathbb{N}$ the following uniform bound for large enough $l,n,m$:

$\displaystyle\sup_{l,m,n}[\lVert
Z^{m,n}\rVert_{(\alpha+\beta)/\alpha,p}+\lVert\mathbb{Z}^{l,m,n}\rVert_{(\alpha+\beta)/\alpha,p}]<\infty.$
(5.30)

Kolmogorov’s continuity criterion and (5.30) yield tightness in
$C(\Delta_{T},\mathbb{R}^{d+d\times d})$ of the laws of the processes
$(Z^{m,n},\mathbb{Z}^{l,m,n})_{l,m,n}$. Furthermore, the uniform bound

$\displaystyle\sup_{m,n}\lVert
Z^{m,n}\mid\mathcal{F}\rVert_{(\alpha+\beta)/\alpha,\infty}<\infty$ (5.31)

follows from the representation (5.26) and
$u^{t,m,n}(t,X_{t}^{n})=0=u^{t,m,n}(t,X_{s}^{n})$ and thus for $m,n$ large
enough,

$\displaystyle\lVert\mathds{E}_{s}[Z^{m,n}_{st}]\rVert_{L^{\infty}(\mathds{P})}$
$\displaystyle\leqslant\sup_{t\in[0,T]}\lVert
u^{t,m,n}\rVert_{C_{t}^{(\alpha+\beta)/\alpha}L^{\infty}_{\mathbb{R}^{d}}}\lvert
t-s\rvert^{(\alpha+\beta)/\alpha}$ $\displaystyle\leqslant
2\sup_{t\in[0,T]}\lVert
u^{t}\rVert_{(\mathcal{L}_{t}^{0,\alpha+\beta})^{d}}\lvert
t-s\rvert^{(\alpha+\beta)/\alpha}$

and the $\mathcal{L}_{t}^{0,\alpha+\beta}$-norm of $u^{t}$ is bounded in
$t\in[0,T]$ due to [KP23, Corollary 4.14]. By Proposition 5.8, $X$ is of class
$\mathcal{K}^{\vartheta}$. The bound from that proposition applied to the
solutions $u^{t,m,n},v^{t,l,m,n}$ yields together with (5.29), that for large
enough $l,m,n$

$\displaystyle\sup_{l,m,n}\lVert\mathbb{Z}^{l,m,n}\mid\mathcal{F}\rVert_{(2(\alpha+\beta)-1)/\alpha,\infty}<\infty.$
(5.32)

By tightness of $(Z^{m,n},\mathbb{Z}^{l,m,n})_{l,m,n}$, uniqueness of the
limit, since $X$ is the unique solution of the martingale problem (cf. Theorem
2.12), and continuity of $(Z,\mathbb{Z}^{V})$ we can thus deduce the
distributional convergence

$\displaystyle(Z^{m,n},\mathbb{Z}^{l,m,n},L)\stackrel{{\scriptstyle
l,m,n\to\infty}}{{\Longrightarrow}}(Z,\mathbb{Z}^{V},L)\qquad\text{ in }\quad
C(\Delta_{T},\mathbb{R}^{d+d\times d})\times D([0,T],\mathbb{R}^{d}).$

Hence, we obtain the distributional convergence

$\displaystyle(X^{n},Z^{m,n},\mathbb{Z}^{l,m,n})\stackrel{{\scriptstyle
l,m,n\to\infty}}{{\Longrightarrow}}(X,Z,\mathbb{Z}^{V})\qquad\text{ in }\quad
D([0,T],\mathbb{R}^{d})\times C(\Delta_{T},\mathbb{R}^{d+d\times d})$

as $X^{n}=x+Z^{n,n}+L$, i.e. $X^{n}$ is given by a continuous map of
$(Z^{n,n},L)$ (using [JS03, Proposition VI.1.23]).
An application of Skorokhods representation theorem (cf. [Bil99, Theorem 6.7])
then yields that there exists a probability space
$(\Omega,\mathcal{F},\mathds{P})$ and random variables

$\displaystyle(Y^{n},W^{m,n},\mathbb{W}^{l,m,n})_{l,m,n}\quad\text{and}\quad(Y,W,\mathbb{W})$

on $(\Omega,\mathcal{F},\mathds{P})$ with
$\text{Law}((Y^{n},W^{m,n},\mathbb{W}^{l,m,n}))=\text{Law}((X^{n},Z^{m,n},\mathbb{Z}^{l,m,n}))$
for $l,m,n\in\mathbb{N}$ and
$\text{Law}((Y,W,\mathbb{W}))=\text{Law}((X,Z,\mathbb{Z}^{V}))$, such that the
convergence

$\displaystyle(Y^{n},W^{m,n},\mathbb{W}^{l,m,n})\stackrel{{\scriptstyle
l,m,n\to\infty}}{{\longrightarrow}}(Y,W,\mathbb{W})$ (5.33)

holds almost surely with respect to the topology on
$D([0,T],\mathbb{R}^{d})\times C(\Delta_{T},\mathbb{R}^{d+d\times d})$ (i.e.
$J_{1}$-topology on the Skorokhod space and uniform topology on the space of
continuous functions on $\Delta_{T}$).
We define the filtration as the completion of the canonical filtration of $Y$,
$(\mathcal{F}_{t}):=(\mathcal{F}_{t}^{Y})\subset\mathcal{F}$. The filtration
is right-continuous, since $Y$ is càdlàg and by construction complete. It
follows that $L:=Y-x-W$ is an $\alpha$-stable symmetric non-degenerate
$(\mathcal{F}_{t}^{(Y,W)})$-Lévy process, because
$\text{Law}(X,Z)=\text{Law}(Y,W)$. Below, we show that $W$ is the almost sure
limit of $(\mathcal{F}^{Y}_{t})$-adapted processes $(W^{m})_{m}$. This then
implies, using also completeness of the filtration, that $L$ is
$(\mathcal{F}^{Y}_{t})$-measurable and thus also a
$(\mathcal{F}_{t}^{Y})$-Lévy process.
Moreover, we have that
$0=Z^{m,n}-\int_{0}^{\cdot}V^{m}(r,X^{n}_{r})dr\stackrel{{\scriptstyle
d}}{{=}}W^{m,n}-\int_{0}^{\cdot}V^{m}(r,Y^{n}_{r})dr$. This implies that
$W^{m,n}=\int_{0}^{\cdot}V^{m}(r,Y^{n}_{r})dr$ almost surely. Analogously, we
deduce the representation
$\mathbb{W}^{l,m,n}_{st}(i,j)=\int_{s}^{t}[J^{T}(\partial_{i}V^{j,l})(r,Y_{r}^{n})-J^{T}(\partial_{i}V^{j,l})(s,Y_{s}^{n})]dW_{r}^{i,m,n}$.
Let $W^{m}$, $\mathbb{W}^{l,m}$ be defined analogously with $Y^{n}$ replaced
by $Y$. This yields the represenations (5.26), (5) for
$W^{m,n},\mathbb{W}^{l,m,n}$. By almost sure convergence of (5.33), letting
first $n\to\infty$, we obtain that for $(s,t)\in\Delta_{T}$,

$\displaystyle W_{st}$
$\displaystyle=u^{t}(t,Y_{t})-u^{t}(s,Y_{s})+M_{st}^{u^{t}},$ $\displaystyle
W^{m}_{st}$
$\displaystyle=u^{t,m}(t,Y_{t})-u^{t,m}(s,Y_{s})+M_{st}^{u^{t,m}},$

and

$\displaystyle\mathbb{W}_{st}(i,j)$
$\displaystyle=v^{t,i,j}(t,Y_{t})-v^{t,i,j}(s,Y_{s})$
$\displaystyle\qquad-J^{T}(\partial_{i}V^{j,l})(s,Y_{s})\big{(}u^{t,i}(t,Y_{t})-u^{t,i}(s,Y_{s})\big{)}$
$\displaystyle\qquad+M_{st}^{v^{t,i,j}}-J^{T}(\partial_{i}V^{j,l})(s,Y_{s})M_{st}^{u^{t,i}},$

and

$\displaystyle\mathbb{W}_{st}^{l,m}(i,j)$
$\displaystyle=v^{t,l,m,i,j}(t,Y_{t})-v^{t,l,m,i,j}(s,Y_{s})$
$\displaystyle\qquad-J^{T}(\partial_{i}V^{j,l})(s,Y_{s})\big{(}u^{t,m,i}(t,Y_{t})-u^{t,m,i}(s,Y_{s})\big{)}$
$\displaystyle\qquad+M_{st}^{v^{t,l,m,i,j}}-J^{T}(\partial_{i}V^{j,l})(s,Y_{s})M_{st}^{u^{t,m,i}}.$

Herein, we used that convergence in the $J_{1}$-topology implies in particular
convergence Lebesgue almost everywhere in $[0,T]$ and that $Y$ almost surely
does not jump at fixed times $t$ (cf. in the proof of Theorem 2.12). The
differences
$M^{u^{t}}_{st},M^{u^{t,m}}_{st},M^{v^{t}}_{st},M^{v^{t,l,m}}_{st}$ are
defined analogously as above and $u^{t,m}=(u^{t,m,i})_{i=1,\dots,d}$ and
$v^{t,l,m}=(v^{t,l,m,i,j})_{i,j=1,\dots,d}$ solve

$\displaystyle\mathcal{G}^{\mathscr{V}}u^{t,m,i}=V^{m,i},\quad\mathcal{G}^{\mathscr{V}}v^{t,l,m,i,j}=J^{T}(\partial_{i}V^{l,j})\cdot
V^{m,i}$

with zero terminal condition at $t\in[0,T]$.

It remains to prove that for $l,m\to\infty$

$\displaystyle\lVert W^{m}-W\rVert_{(\alpha+\beta)/\alpha,2}+\lVert
W^{m}-W\mid\mathcal{F}\rVert_{(\alpha+\beta)/\alpha,\infty}\to 0$ (5.34)

and

$\displaystyle\lVert\mathbb{W}^{l,m}-\mathbb{W}\rVert_{(\alpha+\beta)/\alpha,2}+\lVert\mathbb{W}^{l,m}-\mathbb{W}\mid\mathcal{F}\rVert_{(2(\alpha+\beta)-1)/\alpha,\infty}\to
0.$ (5.35)

The bounds (3.2), (3.3) for the limits $W,\mathbb{W}$ then follow from the
convergences (5.34) and (5.35) and the uniform bounds (5.30), (5.31) and
(5.32) above.
The proof of Lemma 2.13 shows the bound

$\displaystyle\lVert W^{m}-W\rVert_{(\alpha+\beta)/\alpha,2}+\lVert
W^{m}-W\mid\mathcal{F}\rVert_{(\alpha+\beta)/\alpha,\infty}\lesssim\sup_{t\in[0,T]}\lVert
u^{t,m}-u^{t}\rVert_{(\mathcal{L}_{t}^{0,\alpha+\beta})^{d}}.$

The right-hand side vanishes if $m\to\infty$ by the uniform Lipschitz
continuity from [KP23, Corollary 4.14]. Together this yields (5.34).
Analogously, we can argue for the convergence of
$\lVert\mathbb{W}^{l,m}-\mathbb{W}\rVert_{(\alpha+\beta)/\alpha,2}$, using
[KP23, Corollary 4.14] and that

$\displaystyle\lVert\mathbb{W}^{l,m}-\mathbb{W}\rVert_{(\alpha+\beta)/\alpha,2}\lesssim\sup_{t\in[0,T]}[\lVert
v^{t,l,m}-v^{t}\rVert_{(\mathcal{L}_{t}^{0,\alpha+\beta})^{d\times d}}+\lVert
V\rVert_{C_{T}\mathscr{C}^{\beta}_{\mathbb{R}^{d}}}\lVert
u^{t,m}-u^{t}\rVert_{(\mathcal{L}_{t}^{0,\alpha+\beta})^{d}}].$

Moreover, the estimate (5) from the proof of Proposition 5.8 yields the bound

$\displaystyle\lVert\mathbb{W}^{l,m}-\mathbb{W}\mid\mathcal{F}\rVert_{(2(\alpha+\beta)-1)/\alpha,\infty}$
$\displaystyle\lesssim\sup_{r\in[0,T]}\Big{[}\lVert
w^{r,l,m}-w^{r}\rVert_{(\mathcal{L}_{r}^{\gamma,\alpha+\beta})^{d\times
d}}+\lVert
w^{r,l,m,\sharp}-w^{r,\sharp}\rVert_{(\mathcal{L}_{r}^{\gamma,2(\alpha+\beta)-1})^{d\times
d}}$
$\displaystyle\qquad\qquad+\lVert\mathscr{V}\rVert_{\mathscr{X}^{\beta,\gamma^{\prime}}}(\lVert
y^{r,m}-y^{r}\rVert_{(\mathcal{L}_{r}^{\gamma,\alpha+\beta})^{d}}+\lVert
y^{r,m}-y^{r}\rVert_{(\mathcal{L}_{r}^{\gamma,2(\alpha+\beta)-1})^{d}})\Big{]},$

where $w^{r}=(w^{r,i,j})_{i,j}$ denotes the solution of
$\mathcal{G}^{\mathscr{V}}w^{r,i,j}=0$ with terminal condition
$w^{r,i,j}_{r}=J^{T}(\partial_{i}V^{j})_{r}\cdot V^{i}_{r}$ at time $r$ and
$y^{r}=(w^{r,i})_{i}$ denotes the solution of
$\mathcal{G}^{\mathscr{V}}y^{r,i}=0$ with $y^{r,i}_{r}=V^{i}_{r}$. The right-
hand side vanishes by the uniform Lipschitz bound from [KP23, Corollary 4.14].
Thus, together (5.35) follows. Finally, we can of course rename
$(Y,W,\mathbb{W})$ as $(X,Z,\mathbb{Z}^{V})$. ∎

The following theorem generalizes Itô’s formula for rough weak solutions $X$.

######  Theorem 5.11.

Let $\mathscr{V}\in\mathcal{X}^{\beta,\gamma}$ for
$\beta\in(\frac{2-2\alpha}{3},0)$ and let $(X,L,\mathbb{Z}^{V})$ be a weak
solution started in $x\in\mathbb{R}^{d}$. Let $u\in D_{T}(V)$ be such that
$v:=(\partial_{t}-\mathfrak{L}^{\alpha}_{\nu})u$ is paracontrolled by $V$ in
the sense that $v=v^{\sharp}+v^{\prime}\varolessthan V$ with $v^{\sharp}\in
C_{T}\mathscr{C}^{(2-\gamma)\alpha+2\beta-1}$ and $v^{\prime}\in
C_{T}\mathscr{C}^{\alpha+\beta-1}_{\mathbb{R}^{d}}$ and such that $u\in
C^{1}([0,T],\mathscr{C}^{\beta})$ (i.e. continuously differentiable in time
with values in $\mathscr{C}^{\beta}$). Then, if $\alpha\in(1,2)$, the
following Itô-formula holds true:

$\displaystyle u(t,X_{t})$
$\displaystyle=u(0,x)+\int_{0}^{t}(\partial_{s}-\mathfrak{L}^{\alpha}_{\nu})u(s,X_{s})ds+\int_{0}^{t}\nabla
u(s,X_{s})\cdot d(Z,\mathbb{Z}^{V})_{s}$
$\displaystyle\qquad+\int_{0}^{t}\int_{\mathbb{R}^{d}\setminus\\{0\\}}[u(s,X_{s-}+y)-u(s,X_{s-})]\hat{\pi}(ds,dy),$

where $Z=X-x-L$ and $\hat{\pi}$ denotes the compensated Poisson random measure
of $L$. If $\alpha=2$ and $L=B$ for a Brownian motion $B$, the martingale is
replaced by $\int_{0}^{t}\nabla u(s,X_{s})\cdot dB_{s}$. In the formula above,
$\int_{0}^{t}(\partial_{s}-\mathfrak{L}^{\alpha}_{\nu})u(s,X_{s})ds$ is
defined as the limit of
$\big{(}\int_{0}^{t}(\partial_{s}-\mathfrak{L}^{\alpha}_{\nu})u^{n}(s,X_{s})ds\big{)}_{n}$
in $L^{2}(\mathds{P})$ for the smooth mollifications
$(u^{n}=P_{n^{-1}}u)_{n}$.

######  Remark 5.12.

The assumptions on $u$ are satisfied for solutions $u$ of Kolmogorov backward
equations with regular terminal conditions
$u^{T}\in\mathscr{C}^{2\alpha+2\beta-1}$ and right-hand sides $f\in
C_{T}L^{\infty}$.

###### Proof of Theorem 5.11.

We give the proof in the case $\alpha\in(1,2)$, $\alpha=2$ is similar. Since
$u\in C^{1}([0,T],\mathscr{C}^{\beta})$ the mollification satisfies for
$n\in\mathbb{N}$,

$\displaystyle u^{n}_{t}=P_{n^{-1}}u_{t}\in C^{1}([0,T],C^{\infty}_{b})$

In particular, $u^{n}\in C^{1,2}_{b}([0,T]\times\mathbb{R}^{d})$ and applying
[CJMS06, Theorem 3.1] to $u^{n}(t,X_{t})$ yields

$\displaystyle u^{n}(t,X_{t})$
$\displaystyle=u^{n}(0,x)+\int_{0}^{t}(\partial_{s}-\mathfrak{L}^{\alpha}_{\nu})u^{n}(s,X_{s})ds+\int_{0}^{t}\nabla
u^{n}(s,X_{s})\cdot dZ_{s}$
$\displaystyle\quad+\int_{0}^{t}\int_{\mathbb{R}^{d}\setminus\\{0\\}}[u^{n}(s,X_{s-}+y)-u^{n}(s,X_{s-})]\hat{\pi}(ds,dy).$

Due to convergence of the mollifications $u^{n}\to u$ in
$\mathcal{L}^{0,\alpha+\beta}_{T}$ and analogue arguments as for [KP22, Lemma
4.4] (with Burkholder-Davis-Gundy inequality and separating in large and small
jumps), we obtain that the $\hat{\pi}$-martingales converge in
$L^{2}(\mathds{P})$ to the one with $u^{n}$ replaced by $u$.
By Theorem 4.4 it follows that almost surely,

$\displaystyle\int_{0}^{t}\nabla u^{n}(s,X_{s})\cdot dZ_{s}=\int_{0}^{t}\nabla
u^{n}(s,X_{s})\cdot d(Z,\mathbb{Z}^{V})_{s}.$

The stability of the rough stochastic integral from Proposition 5.6 for the
rough stochastic integral yields convergence of the integrals if
$(u^{n},u^{\prime})\to(u,u^{\prime})$ in $D_{T}(V)$. The latter follows from
the convergence of the mollification $u^{n}\to u$ in
$\mathcal{L}_{T}^{0,\alpha+\beta}$ and

$\displaystyle u^{n,\sharp}$ $\displaystyle=u^{n}-u^{\prime}\varolessthan
J^{T}(V)$
$\displaystyle=P_{n^{-1}}u^{\sharp}+\big{(}P_{n^{-1}}[u^{\prime}\varolessthan
J^{T}(V)]-u^{\prime}\varolessthan P_{n^{-1}}J^{T}(V)\big{)}$ $\displaystyle\to
u^{\sharp}$

in $\mathcal{L}_{T}^{0,2(\alpha+\beta)-1}$ due to [KP23, Corollary 3.2] and
the commutator [KP23, Lemma 3.4]. It remains to show that the additive
functional

$\displaystyle\int_{0}^{t}(\partial_{s}-\mathfrak{L}^{\alpha}_{\nu})u^{n}(s,X_{s})ds$

converges in $L^{2}(\mathds{P})$. For $r\in[0,T]$ we have by assumption for
$v_{r}:=(\partial_{r}-\mathfrak{L}^{\alpha}_{\nu})u_{r}$, that

$\displaystyle v_{r}=v_{r}^{\sharp}+v^{\prime}_{r}\varolessthan V_{r},$

for $v_{r}^{\sharp}\in\mathscr{C}^{(2-\gamma)\alpha+2\beta-1}$ and
$v_{r}^{\prime}\in\mathscr{C}^{\alpha+\beta-1}$. Thus $v_{r}$ is an admissible
terminal condition (in the sense of Theorem 2.9) for solving the Kolmogorov
equation on $[0,r]$ with paracontrolled terminal condition $v_{r}$.
By Theorem 5.7, $X$ is a martingale solution and in particular a strong Markov
process. Denote again by $(T_{s,r})_{0\leqslant s\leqslant r\leqslant T}$ its
semigroup. Utilizing the ideas of [Lê20, section 3], we define,

$\displaystyle\Xi_{s,t}:=\int_{s}^{t}T_{s,r}v_{r}(X_{s})dr.$

We let $y_{s}:=T_{s,r}v_{r}$ for $s\in[0,r]$. Then $y$ is a mild solution of
the Kolmogorov equation with terminal condition $v_{r}$.