# A Note on the Practice of Logical Inferentialism: The State-Effect
Interpretation, Definitional Reflection, and Completeness

Alexander V. Gheorghiu  Department of Computer Science
University College London
London WC1E 6BT, UK<EMAIL_ADDRESS>,  Tao Gu  Department of
Computer Science
University College London
London WC1E 6BT, UK<EMAIL_ADDRESS>and  David J. Pym  Institute of
Philosophy, University of London, London WC1E 7HU, UK and
Department of Computer Science and Department of Philosophy, University
College London, London WC1E 6BT, UK<EMAIL_ADDRESS>
The prevailing paradigm of semantics in logic is _denotationalism_ , wherein
the meaning of logical structures is grounded in reference to abstract
algebraic structures, exemplified by _model-theoretic semantics_ (M-tS). An
alternative paradigm is _inferentialism_ (Brandom, 2009), wherein meaning
emerges from rules of inference. It may be viewed in the context of the
_meaning-as-use_ principle (Wittgenstein, 2009), with ‘use’ understood in
logic as ‘inference’.

For instance, consider the statement ‘Luna is a vixen’, which intuitively
means ‘Luna is a fox’ _and_ ‘Luna is female’. In inferentialism, its meaning
is given by the following rules:

$\begin{array}[]{c}\text{Luna is a vixen}\lx@proof@logical@and\text{Luna is a
fox}\text{Luna is female}\qquad\text{Luna is female}\text{Luna is a
vixen}\qquad\text{Luna is a fox}\text{Luna is a vixen}\\\ \end{array}$

These merit comparison with the laws governing conjunction ($\land$) (Szabo,
1969), which justify the ‘and’ above,

$\phi\land\psi\lx@proof@logical@and\phi\psi\qquad\phi\phi\land\psi\qquad\psi\phi\land\psi$

_Proof-theoretic semantics_ (P-tS) provides a mathematical expression of
inferentialism (Schroeder-Heister, 2018). Unlike M-tS, P-tS views _proofs_ as
the basis of meaning (as opposed to _truth_). Below, we propose an analysis of
P-tS that explicates the semantic clauses for various standard logics. In
particular, we justify the P-tS for _intuitionistic propositional logic_ (IPL)
by Sandqvist (2015) (cf. Piecha & Schroeder-Heister (2019)), and its
subsequent developments by Gheorghiu et al. (2023) and Gu et al. (2023).

Heuristically, logical constants are _defined_ by their introduction rules;
for instance, $\land_{\mathsf{{I}}}$ can be understood as _defining_ $\land$.
A central challenge in P-tS is to formalize this heuristic rigorously.
However, merely providing introduction rules for a logic is insufficient for a
complete definition; one requires elimination rules. Notably, the choice of
elimination rules is not arbitrary (Prior, 1961), leading to the study of
_harmony_ (Tennant, 1978b) and _inversion_ (Prawitz, 2006) in P-tS.

Notwithstanding these challenges, the elimination rules in the natural
deduction systems on which much of P-tS has been based — for example, NJ
(Szabo, 1969) — are thought of as being _consequences_ of their corresponding
introduction rules. Indeed, a central approach in P-tS has been to witness and
exploit this relationship (Schroeder-Heister, 2006). One way to characterize
the relationship is through the following idea:

> The introductions represent the ‘state’ required for a logical constant to
> be asserted, and the corresponding eliminations represent the ‘effect’ of
> that constant.

This _state-effect_ duality is useful for explicating the differences between
M-tS and P-tS. Of course, the ‘effect’ of a logical constant must be warranted
by the ‘state’ it signifies; whence, harmony and inversion principles.

Some explanation is required for what we mean by state and effect. They
correspond to two perspectives on logic, the _ontological_ and the
_teleological_.

The ontological perspective views logic as the study of properties of objects
(Hofweber, 2023) and ‘state’ refers to the state of objects. This view is the
foundation for M-tS and the clauses of the logical constants are, therefore,
closely related to their corresponding introduction rules. For example, in the
standard possible-world semantics for IPL by Kripke (1965), the clause for
disjunction closely mirrors the introduction rule $\lor_{\mathsf{1{I}}}$ and
$\lor_{\mathsf{2{I}}}$ (Szabo, 1969),

$\mathfrak{M},w\Vdash\phi\lor\psi\qquad\mbox{iff}\qquad\mbox{$\mathfrak{M},w\Vdash\phi$
or $\mathfrak{M},w\Vdash\psi$}$

The correctness of the elimination rules follows from ‘backward’ reasoning on
states, which are represented by possible worlds (Gheorghiu & Pym, 2023c, a;
Gheorghiu, 2024).

In contrast, we shall call the ‘teleological’ perspective on logic the view
that is concerned with understanding _reasoning_. Tennant (1978a) exemplifies
this view on logic in the ‘consequentialist’ reading of entailment. By
‘effect’ we mean what reasoning is rendered possible by the assertion of the
logical constants. This explains why clauses of the semantics typically mirror
the elimination rules. For example, in the work on IPL by Sandqvist (2015),
the clause for disjunction ($\lor$) closely mirrors the elimination rule
$\lor_{\mathsf{{E}}}$ (Szabo, 1969) restricted to atoms,

$\begin{array}[]{lcl}\Vdash_{\mathscr{B}}\phi\lor\psi&\mbox{iff}&\mbox{for any
$\mathscr{C}\supseteq\mathscr{B}$ and any atom $p$,}\\\ &&\mbox{if
$\phi\Vdash_{\mathscr{C}}p$ and $\psi\Vdash_{\mathscr{C}}p$, then
$\Vdash_{\mathscr{C}}p$}\end{array}$

The details — such as the significance of $\mathscr{B}$ and $\mathscr{C}$ —
are not essential for this discussion. The point is that forcing a clause
informed by a state-based view yields incompleteness; that is, Piecha et al.
(2015) have shown that using

$\Vdash_{\mathscr{B}}\phi\lor\psi\qquad\mbox{iff}\qquad\Vdash_{\mathscr{B}}\phi\qquad\mbox{or}\qquad\Vdash_{\mathscr{B}}\psi$

does not give a semantics for IPL. The correctness of the introduction rules
follows from the transitivity of the judgment relation.

In the practice of logical inferentialism, while we treat introduction rules
as definitions, our central concern is to understand the effects implied by a
state. More broadly, how one transitions from an ontological interpretation to
a teleological one. We argue that _Definitional Reflection_ (DR) (Hallnäs,
1987; Hallnäs & Schroder-Heister, 1990, 1991; Schroeder-Heister, 1993) plays
an essential role:

> “Whatever follows from all the defining conditions of an assertion follows
> from the assertion itself.”

Not only does DR offer a mathematically elegant solution to these issues but
also embodies the intended conceptual shift.

Indeed, DR gives one approach for generating elimination rules from
introduction rules. For example, let $\lor_{\mathsf{1{I}}}$ and
$\lor_{\mathsf{2{I}}}$ be the ‘definition’ of disjunction ($\lor$),

$\phi\lor\psi\phi\qquad\phi\lor\psi\psi$

Hence, the defining conditions of $\phi\lor\psi$ are $\phi$ and $\psi$.
Whence, by DR, whatever follows from both $\phi$ and $\psi$ also follows from
$\phi\lor\psi$. This motivates $\lor_{\mathsf{{E}}}$,

$\chi\lx@proof@logical@and\phi\lor\psi\chi[\phi]\chi[\psi]$

as required. Observe that $\land_{\mathsf{1{E}}}$ and $\land_{\mathsf{2{E}}}$
are together obtained from and equally expressive as the generalized
elimination rule for $\land_{\mathsf{{I}}}$ (Schroeder-Heister, 1984).This is
all very well as a mechanism, but why ought we accept it?

According to Schroeder-Heister (1993), DR follows from reading the
introduction rules as expressing _all_ conditions for the assertion to be
made. Essentially, this embodies a form of _closed-world assumption_ (Reiter,
1981), which in the form _negation-as-failure_ (Clark, 1977) has been studied
more directly in the context of the P-tS of IPL (Gheorghiu & Pym, 2023b). In
this sense, it is the completion (Clark, 1977) of the inference system
afforded by the introduction rules; in the state-effect reading proposed
above, the generalized elimination rules represent the _maximal_ effect of a
logical constant corresponding to its introduction rules.

This justifies DR as a general phenomenon. It remains to show that it
represents the intended conceptional shift from the ontological to the
teleological perspective when doing logical inferentialism.

Goldfarb (2016) has given an alternative sound and complete inferential
semantics of IPL that is based on the ontological view of logic where the
_objects_ are ‘canonical’ proofs in the Dummett-Prawitz sense (Schroeder-
Heister, 2006). In this case, canonical proofs act as states, and the
introduction rules act as _constructors_ , yielding clauses based on
introduction rules. Gheorghiu & Pym (2022) have shown that moving from
constructors to constructiveness — that is, from states to effects — in the
approach by Goldfarb (2016) recovers the semantics by Sandqvist (2015). The
move is conceptionally simple and best illustrated by an example:

> Consider the judgment $\phi\lor\psi\vdash\psi\lor\phi$. We must show that on
> the assumption that there is a valid argument for $\phi\lor\psi$, there is a
> valid argument for $\psi\lor\phi$. Without loss of generality, a valid
> argument for $\phi\lor\psi$ ends by a constructor $\lor_{\mathsf{1{I}}}$ or
> $\lor_{\mathsf{2{I}}}$. Hence, a valid argument exists for either $\phi$ or
> $\psi$. Whence, by the use of a constructor $\lor_{\mathsf{1{I}}}$ or
> $\lor_{\mathsf{2{I}}}$, there is a valid argument for $\psi\lor\phi$, as
> required.

Observe that we have implicitly used DR: we have argued from _all the defining
conditions_ of the assertion $\phi\lor\psi$ that we can conclude
$\psi\lor\phi$.

To summarize, adopting the state-effect interpretation of introduction and
elimination rules aids in delineating the distinctions between P-tS and M-tS.
This approach offers a unified understanding of the practice of logical
inferentialism. Additionally, DR emerges as a central tenet in transitioning
from a state-based semantics to an effect-based semantics of a logical system.
It not only works at a mathematical level, but actually arises naturally as
one moves between the perspectives.

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# A Wong-Zakai theorem for SDEs with singular drift

Chengcheng Ling, Sebastian Riedel and Michael Scheutzow  Chengcheng Ling:
Technische Universität Berlin, Fakultät II, Institut für Mathematik, 10623
Berlin, Germany
Email<EMAIL_ADDRESS>Sebastian Riedel: Leibniz Universität Hannover,
Institute of Analysis, 30167 Hannover, Germany
Email<EMAIL_ADDRESS>Michael Scheutzow: Technische Universität
Berlin, Fakultät II, Institut für Mathematik, 10623 Berlin, Germany
Email<EMAIL_ADDRESS>
###### Abstract.

We study stochastic differential equations (SDEs) with multiplicative
Stratonovich-type noise of the form

$\displaystyle dX_{t}=b(X_{t})\,{\mathord{{\rm
d}}}t+\sigma(X_{t})\circ{\mathord{{\rm d}}}W_{t},\quad
X_{0}=x_{0}\in\mathbb{R}^{d},\quad t\geqslant 0,$ (0.1)

with a possibly singular drift $b\in L^{{p}}(\mathbb{R}^{d})$, $p>d$ and
$p\geqslant 2$, and show that such SDEs can be approximated by random ordinary
differential equations by smoothing the noise and the singular drift at the
same time. We further prove a support theorem for this class of SDEs in a
rather simple way using the Girsanov theorem.

AMS 2020 Mathematics Subject Classification: 60H10, 60F15, 60J60.

Keywords: singular stochastic differential equations; stability with respect
to singular drifts; Wong-Zakai approximation; Zvonkin’s transformation;
support theorem.

##  1\. Introduction

Consider the following stochastic differential equation (abbreviated as SDE)
in ${\mathbb{R}}^{d}$:

$\displaystyle{\mathord{{\rm d}}}X_{t}=b(t,X_{t})\,{\mathord{{\rm
d}}}t+\sigma(t,X_{t})\,{\mathord{{\rm d}}}W_{t},\quad
X_{0}=x_{0}\in\mathbb{R}^{d},\quad t\geqslant 0,$ (1.1)

where $d\geqslant 1$, $b=(b^{(i)})_{1\leqslant i\leqslant
d}:{\mathbb{R}}_{+}\times{\mathbb{R}}^{d}\rightarrow{\mathbb{R}}^{d}$ and
$\sigma=(\sigma_{ij})_{1\leqslant i,j\leqslant
d}:{\mathbb{R}}_{+}\times{\mathbb{R}}^{d}\rightarrow L(\mathbb{R}^{d})$
$(:=d\times d$ real valued matrices$)$ are Borel measurable, and
$(W_{t})_{t\geqslant 0}$ is a standard $d$-dimensional Brownian motion defined
on some filtered probability space
$(\Omega,{\mathscr{F}},({\mathscr{F}}_{t})_{t\geqslant 0},{\mathbb{P}})$.
Using Picard iteration, it is easy to see that the corresponding ordinary
differential equation (abbreviated as ODE)

$\displaystyle{\mathord{{\rm d}}}X_{t}=b(t,X_{t})\,{\mathord{{\rm d}}}t,\quad
X_{0}=x_{0}\in\mathbb{R}^{d},\quad t\geqslant 0,$ (1.2)

has a unique solution provided that $b$ is Lipschitz continuous. With a
similar argument, under suitable growth and regularity assumptions on
$\sigma$, it can be shown that there exists a unique strong solution to the
SDE (1.1), assuming again that $b$ is Lipschitz continuous. On the other hand,
when $b$ fails to be Lipschitz continuous, e.g. only $\alpha$-Hölder with
$\alpha\in(0,1)$, it is well known that the ODE (1.2) is generally ill-posed.
Interestingly, due to a ’regularization by noise’ effect, the SDE (1.1)
defined with a very singular $b$, maybe even discontinuous, might still be
well-posed and possesses a unique solution even in the strong sense. There are
many works on this aspect. In the following, we mention a few of them. A
remarkable result due to Zvonkin [30] shows that if $d=1$, $\sigma$ is
uniformly elliptic, $b$ and $\sigma$ are bounded, and $\sigma$ is continuous
with respect to space and time variables, then the SDE (1.1) admits a unique
strong solution $(X_{t}(x_{0}))_{t\geqslant 0}$ for each
$x_{0}\in{\mathbb{R}}$. The key idea is to use a certain transformation of the
SDE, nowadays called _Zvonkin’s transformation_ in the literature. Zvonkin’s
result was then extended to the multidimensional case by Veretennikov [22]. A
further generalization was obtained by Krylov and Röckner [13] where existence
and uniqueness of a strong solution for the SDE (1.1) was shown when $\sigma$
is identity matrix in ${\mathbb{R}}^{d}$ (i.e. the SDE is driven by additive
noise) and

$\displaystyle b\in
L^{q}_{loc}({\mathbb{R}}_{+};L^{p}_{loc}({\mathbb{R}}^{d}))\quad\text{with}\quad
q,p\in[2,\infty)\quad\text{and}\quad d/p+2/q<1.$ (1.3)

Besides, Fedrizzi and Flandoli [3] introduced a new method to prove existence
and uniqueness of a global strong solution to the SDE (1.1) by using
regularizing properties of solutions to the Kolmogorov equation corresponding
to (1.1), assuming that (1.3) holds globally. For a well-posedness result for
(1.1) driven by multiplicative noise (i.e. $\sigma$ is not a constant matrix
only), we refer to Zhang [27] who applied Zvonkin’s transformation and Von der
Lühe [17] where Fedrizzi and Flandoli’s method in [3] was generalized to the
multiplicative noise case. More recently, Zhang and Zhao [29] obtained a well-
posedness result for the martingale problem induced by the SDE (1.1) when the
drift is a Schwartz distribution using Zvonkin’s transformation. Another way
to investigate singular SDEs is rough paths theory: Harang and Perkowski in
[9], Catellier and Gubinelli in [2], and Galeati and Gubinelli in [4] showed
the pathwise uniqueness for a large class of singular SDEs.

A Wong-Zakai approximation to an SDE is given by a series of random ODEs which
converge to the original equation. For classical SDEs, the approximating
random ODEs can be obtained by just smoothing the noise. The first results for
classical SDEs are contained in the works by Wong and Zakai [23, 24]. In [23,
Theorem 2], it was proved that in dimension one, under the assumptions that
$b\in\mathcal{C}_{b}^{1}([0,T]\times\mathbb{R})$ and
$\sigma\in\mathcal{C}_{b}^{2}([0,T]\times\mathbb{R})$, if
$(w^{n}_{t})_{n\geqslant 1}$ is a continuous and piecewise smooth sequence of
approximations to the Brownian motion $(W_{t})_{t\geqslant 0}$, then the
solutions $X_{t}^{w^{n}}$ to the following equation

${\mathord{{\rm d}}}X_{t}^{w^{n}}=b(t,X_{t}^{w^{n}})\,{\mathord{{\rm
d}}}t+\sigma(t,X_{t}^{w^{n}})\,{\mathord{{\rm d}}}w^{n}_{t},\quad
X_{0}=x_{0}\in\mathbb{R},\quad t\geqslant 0,$

converge to $X_{t}$ almost surely, where $X_{t}$ solves

$\displaystyle{\mathord{{\rm d}}}X_{t}=b(t,X_{t}){\mathord{{\rm
d}}}t+\sigma(t,X_{t})\circ{\mathord{{\rm d}}}W_{t},\quad
X_{0}=x_{0}\in\mathbb{R},\quad t\geqslant 0.$ (1.4)

Here $\sigma(t,X_{t})\circ{\mathord{{\rm
d}}}W_{t}:=\sigma(t,X_{t})\,{\mathord{{\rm
d}}}W_{t}+\frac{1}{2}\sigma(t,X_{t})\frac{\partial\sigma(t,X_{t})}{\partial
x}\,{\mathord{{\rm d}}}t$, i.e. the $\circ$ integral denotes the usual
Stratonovich integral, and
$\frac{1}{2}\sigma(t,X_{t})\frac{\partial\sigma(t,X_{t})}{\partial
x}\,{\mathord{{\rm d}}}t$ is the Itô correction term. The fact that the
limiting SDE has to be understood in Stratonovich sense is not surprising: if
the approximating equations satisfy the usual chain rule, the same should be
true for the limiting equation. In the multidimensional case, one needs to be
careful about how approximating the noise. For instance, P. Lévy showed that
if $(w^{n,(1)},w^{n,(2)})$ denotes the piecewise linear approximation to a
two-dimensional Brownian motion $(W^{(1)},W^{(2)})$, one has

$\displaystyle\frac{1}{2}\int_{0}^{t}w^{n,(1)}_{s}\,{\mathord{{\rm
d}}}w^{n,(2)}_{s}-w^{n,(2)}_{s}\,{\mathord{{\rm
d}}}w^{n,(1)}_{s}\to\frac{1}{2}\int_{0}^{t}W^{(1)}_{s}\circ{\mathord{{\rm
d}}}W^{(2)}_{s}-W^{(2)}_{s}\circ{\mathord{{\rm d}}}W^{(1)}_{s}.$

The integral on the right hand side is called _Lévy area_. From this
convergence, one can deduce that approximating the Brownian motion piecewise
linearly in a multidimensional SDE converges, again, to the Stratonovich
version of the SDE. However, E. J. McShane [15] gave an example of an
approximation to a two-dimensional Brownian motion for which the limit is
_not_ the usual Lévy area, but the area plus another correction term (see
Example 3.8 below). This implies that a multidimensional SDE with this
approximation will not converge to the usual Stratonovich SDE. More results
and surveys on this topic can be found in [11], [10, Chapter 7], [5] and the
references therein. The study of Wong-Zakai approximations was further
extended to stochastic partial differential equations (abbreviated as SPDEs),
too, cf. e.g. [21], [18], and [8]. In [6] and [7], Gyöngy, Shmatkov and Stinga
showed that the convergence rate of the Wong-Zakai approximation for SPDEs is
$\frac{1}{2}$, which is the same as for SDEs.

Wong-Zakai approximations bridge the gap between the ODE and the SDE world.
They are important for many purposes, e.g. in numerics or in subjects
concerning the limiting behavior of SDEs. However, all known results
exclusively treat SDEs with regular coefficients, i.e. $\sigma$ and $b$ were
at least assumed to be Lipschitz continuous. In this article, we are
interested in studying the case when the coefficients in the SDE might not be
continuous at all but only belong to some $L^{p}$-space with $p>d$. Then
immediately we are in the problematic situation that the approximating
equation may not be well-defined for singular coefficients if we simply smooth
out the noise. In fact, the roughness of the noise is essential to render the
equation well-defined, most visible in the above cited works where a pathwise
rough path approach is used. In order to handle this issue, we approximate the
singular drift by a smooth one at the same time. Using a stability result that
we obtain for singular SDEs (see Theorem 3.3 below), we can show that the
solution of the approximated equation converges to the original singular SDE
in the mean square sense. In the following, we loosely formulate our main
result, the precise formulation can be found in Theorem 3.12.

######  Theorem 1.1.

Let $b$ and $\sigma$ be coefficients defining a singular SDE. In particular,
$|b|\in L^{p}(\mathbb{R}^{d})$, $p\in(d,\infty)$ and $p\geqslant 2$. Let
$(W_{t}^{n})_{n\geqslant 1}$ resp. $(b_{n})_{n\geqslant 1}$ be suitable
approximations for a $d$-dimensional Brownian motion $W_{t}$ resp. for the
coefficient $b$. Assume that $W^{n}$ converges sufficiently fast to $W$
compared to the convergence of $b_{n}$ to $b$. Then the solution
$(X_{t})_{t\geqslant 0}$ to the singular equation

$\displaystyle X_{t}=x_{0}+\int_{0}^{t}b(X_{s})\,{\mathord{{\rm
d}}}s+\int_{0}^{t}\sigma(X_{s})\,{\mathord{{\rm
d}}}W_{s}+\Big{(}\sum_{i,j=1}^{d}\sum_{l=1}^{d}c_{ij}\int_{0}^{t}\sigma_{il}\cdot\nabla_{l}\sigma_{jk}(X_{s})\,{\mathord{{\rm
d}}}s\Big{)}_{k=1,\cdots,d}$ (1.5)

is the mean square limit of the solution to the equation

$\displaystyle X_{t}^{n}=x_{0}+\int_{0}^{t}b_{n}(X_{s}^{n}){\mathord{{\rm
d}}}s+\int_{0}^{t}\sigma(X_{s}^{n}){\mathord{{\rm d}}}W_{s}^{n}.$ (1.6)

Above, $(c_{ij})_{1\leqslant i,j\leqslant d}$ are coefficients depending on
the approximating sequence $(W_{t}^{n})_{n\geqslant 1}$.

The above result shows that certain random, regular ODEs (1.6) converge to the
singular SDE (1.5). The condition that the speed of convergence of the noise
approximation has to be faster than for the vector field approximation seems
plausible, but it is open for us whether this condition is really necessary.

A typical application for the Wong-Zakai theorem is the proof of a support
theorem which describes the topological support of the law of an SDE solution
[20, 5, 19, 1]. To our knowledge, there was no support theorem for singular
SDEs in the literature until now. In principle, it is possible to mimic the
proof of the classical support theorem for regular SDEs using Theorem 1.1.
However, it turns out that there is a much simpler proof, using a change of
measure method to eliminate the drift which allows to apply a well-established
support theorem for non-degenerate diffusions [20]. Here is a short form of
our support theorem, cf. Theorem 4.1.

######  Theorem 1.2.

It holds that

$\textup{supp }\mu=\mathcal{C}([0,T],\mathbb{R}^{d}),$

where $\textup{supp }\mu$ is the topological support of the distribution of
the solution $(X_{t})_{t\geqslant 0}$ to the singular SDE

$\displaystyle X_{t}=x_{0}+\int_{0}^{t}b(s,X_{s}){\mathord{{\rm
d}}}s+\int_{0}^{t}\sigma(s,X_{s})\circ{\mathord{{\rm d}}}W_{s}.$

The structure of this paper is as follows: We introduce some notation and
preliminaries in Section 2. In Section 3, we first obtain the stability of the
solution to a singular SDE with multiplicative noise with respect to the
singular drift by using Zvonkin’s transformation. In the second part, we prove
the Wong-Zakai Theorem for singular SDEs. The support theorem is shown in
Section 4. The appendix collects some necessary results in PDE theory which
are used when establishing the stability result in Section 2.

##  2\. Notation and Preliminaries

We first introduce some notation. We write ${\mathbb{R}}^{+}:=[0,\infty)$ and
$\mathbb{N}=\\{1,2,...\\}$.

For $v\in{\mathbb{R}}^{m}$, let $v^{(i)}$, $i=1,\cdots,m$, be the $i$-th
component of $v$ and $|v|$ its $l^{2}$ Euclidean norm. For a matrix
$\sigma\in{\mathbb{R}}^{m\times n}$, we denote its components by
$\sigma_{ij}$, $i=1,\cdots,m$, $j=1,\cdots,n$. We use $\sigma^{*}$ to denote
its transpose and $\|\sigma\|$ is its Euclidean norm in ${\mathbb{R}}^{m\times
n}$, i.e. the root of the sum of the squares of all entries.

For $p$, $q\in[1,\infty)$ and $0\leqslant S<T\leqslant\infty$, let
$\mathbb{L}_{p}^{q}(S,T)$ denote the space of all real Borel measurable
functions on $[S,T)\times\mathbb{R}^{d}$ with the norm

$\|f\|_{\mathbb{L}_{p}^{q}(S,T)}:=\bigg{(}\int_{S}^{T}\big{(}\int_{\mathbb{R}^{d}}|f(t,x)|^{p}{\mathord{{\rm
d}}}x\big{)}^{q/p}\,{\mathord{{\rm d}}}t\bigg{)}^{1/q}<+\infty.$

For simplicity, we write

$\mathbb{L}_{p}^{q}=\mathbb{L}_{p}^{q}(\mathbb{R}_{+}),\quad\mathbb{L}_{p}^{q}(T)=\mathbb{L}_{p}^{q}(0,T),\quad\mathbb{L}_{p}^{q,loc}=L^{q}_{loc}(\mathbb{R}_{+},L^{p}(\mathbb{R}^{d})).$

Let $\mathcal{C}([0,\infty),\mathbb{R}^{d})$ denote the space of all
continuous $\mathbb{R}^{d}$-valued functions defined on $[0,\infty)$. By
$\mathcal{C}_{b}^{n}(\mathbb{R}^{d})$ we mean the set of all bounded $n$ times
continuously differentiable functions on $\mathbb{R}^{d}$ with bounded
derivatives up to order $n$. $\mathcal{C}(\mathbb{R}^{d})$ collects all
continuous functions defined on $\mathbb{R}^{d}$. For $f\in
L_{loc}^{1}(\mathbb{R}^{d})$ we define $\nabla_{j}f(x):=\frac{\partial
f}{\partial x_{j}}(x)$ and $\nabla f:=(\nabla_{i}f)_{1\leqslant i\leqslant d}$
denotes the gradient of $f$,
$\partial_{ij}^{2}f(x):=\frac{\partial^{2}f}{\partial x^{(i)}\partial
x^{(j)}}$. Here the derivatives are meant in the sense of distributions. Below
when we write $\nabla f$ for a function depending on
$(t,x)\in\mathbb{R}^{d+1}$, we always mean $\nabla_{x}f(t,x)$, i.e. the
derivative with respect to $x\in\mathbb{R}^{d}$, and $\partial_{t}f$ denotes
the derivative with respect to $t\in\mathbb{R}_{+}$. For any
$\alpha\in{\mathbb{R}}$ and ${p}\in[1,\infty)$, let
$H_{p}^{\alpha}(\mathbb{R}^{d}):=(1-\Delta)^{-\alpha/2}\big{(}L^{p}({\mathbb{R}}^{d})\big{)}$
be the usual Bessel potential space with norm
$\|f\|_{H_{p}^{\alpha}(\mathbb{R}^{d})}:=\|({\mathbb{I}}-\Delta)^{\alpha/2}f\|_{L^{p}(\mathbb{R}^{d})},$
where $({\mathbb{I}}-\Delta)^{\alpha/2}f$ is defined through Fourier’s
transform. For $\alpha\in{\mathbb{R}}$, let
${\mathbb{H}}_{\alpha,{p}}^{q}(T):=L^{q}\big{(}[0,T];H^{\alpha}_{p}({\mathbb{R}}^{d})\big{)},$
and the space ${\mathscr{H}}_{\alpha,{p}}^{q}(T)$ consists of the functions
$u=u(t)$ on $[0,T]$ with values in the space of distributions on
${\mathbb{R}}^{d}$ such that $u\in{\mathbb{H}}_{\alpha,{p}}^{q}(T)$ and
$\partial_{t}u\in{\mathbb{L}}^{q}_{p}(T)$.

In the following, $\int_{0}^{t}\sigma(s,X_{s})\circ{\mathord{{\rm d}}}W_{s}$
is understood as Stratonovich integral, i.e.

$\int_{0}^{t}\sigma(s,X_{s})\circ{\mathord{{\rm
d}}}W_{s}:=\int_{0}^{t}\sigma(s,X_{s})\,{\mathord{{\rm
d}}}W_{s}+\frac{1}{2}\int_{0}^{t}\sigma(s,X_{s})\frac{\partial\sigma(s,X_{s})}{\partial
x}\,{\mathord{{\rm d}}}s,$

where $\int_{0}^{t}\sigma(s,X_{s})\,{\mathord{{\rm d}}}W_{s}$ is the Itô
integral whenever it makes sense.

##  3\. Wong-Zakai Approximation for SDEs with singular drift driven by
multiplicative noise

The proof of our main result consists of two steps. First, we prove a
stability result for singular SDEs that shows that singular SDEs can be
approximated by “classical” SDEs. Then, we formulate a quantitative version of
the Wong-Zakai theorem. Both results together yield our main theorem. We start
with step 1.

###  3.1. Stability of SDEs with singular drift driven by multiplicative noise

We present a stability result under the following condition:

######  Assumption 3.1.

For $\frac{d}{p}+\frac{2}{q}<1$, $p,q\in[2,\infty)$,

  * ($H_{b}$)

$b:{\mathbb{R}}^{+}\times{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ is measurable
and $|b|\in{\mathbb{L}}_{p}^{q,loc}$.

  * ($H_{\sigma}$)

$\sigma:{\mathbb{R}}^{+}\times{\mathbb{R}}^{d}\to{\mathbb{R}}^{d\times d}$ is
measurable, $|\nabla\sigma|\in{\mathbb{L}}_{p}^{q,loc}$, and there exists
$K\geqslant 1$ such that for all $(t,x)\in[0,\infty)\times\mathbb{R}^{d}$,

$K^{-1}|\xi|^{2}\leqslant(\sigma\sigma^{*}(t,x)\xi,\xi)\leqslant
K|\xi|^{2},\quad\forall\xi\in\mathbb{R}^{d}.$

Further, $a:=\sigma\sigma^{*}$ is uniformly continuous in $x\in\mathbb{R}^{d}$
locally uniformly with respect to $t\in[0,\infty)$.

Concerning existence and uniqueness of a global strong solution to (1.1), [25]
shows the following result.

######  Theorem 3.2.

[25, Theorem 1.1] Suppose the conditions in Assumption 3.1 hold. Then, for any
$x\in\mathbb{R}^{d}$, there exists a unique strong solution
$(X_{t}(x))_{t\geqslant 0}$ to the SDE (1.1) such that

$\displaystyle{\mathbb{P}}\left\\{\omega:\int_{0}^{T}|b(r,X_{r}(\omega))|\,{\mathord{{\rm
d}}}r+\int_{0}^{T}|\sigma(r,X_{r}(\omega))|^{2}\,{\mathord{{\rm
d}}}r<\infty,\forall T\in[0,\infty)\right\\}=1,$

and, almost surely,

$\displaystyle X_{t}=x+\int_{0}^{t}b(r,X_{r})\,{\mathord{{\rm
d}}}r+\int_{0}^{t}\sigma(r,X_{r})\,{\mathord{{\rm d}}}W_{r},\quad\forall
t\in[0,\infty).$

Next, we formulate the stability result.

######  Theorem 3.3.

Assume that $b^{1},b^{2}$ satisfy ($H_{b^{1}}$), ($H_{b^{2}}$) respectively,
and $\sigma$ satisfies ($H_{\sigma}$). Let $(X_{t}^{1}(x_{1}))_{t\geqslant 0}$
and $(X_{t}^{2}(x_{2}))_{t\geqslant 0}$ be the unique strong solutions to the
equations

$\displaystyle dX_{t}^{1}=b^{1}(t,X_{t}^{1})\,{\mathord{{\rm
d}}}t+\sigma(t,X_{t}^{1})\,{\mathord{{\rm d}}}W_{t},\quad
X_{0}^{1}=x_{1}\in\mathbb{R}^{d},\quad t\geqslant 0,$ (3.1)

and

$\displaystyle dX_{t}^{2}=b^{2}(t,X_{t}^{2})\,{\mathord{{\rm
d}}}t+\sigma(t,X_{t}^{2})\,{\mathord{{\rm d}}}W_{t},\quad
X_{0}^{2}=x_{2}\in\mathbb{R}^{d},\quad t\geqslant 0.$ (3.2)

Then, for each $T>0$, there exist positive constants $C$ and $C^{\prime}$ such
that

$\displaystyle{\mathbb{E}}{\Big{(}}\sup_{s\in[0,T]}|X_{s}^{1}(x_{1})-X_{s}^{2}(x_{2})|^{2}{\Big{)}}\leqslant
C\|b^{1}-b^{2}\|_{{\mathbb{L}}_{p}^{q}(T)}^{2}+C^{\prime}|x_{1}-x_{2}|^{2}$
(3.3)

where $C$ and $C^{\prime}$ may depend on
$T,d,p,q,K,\|\nabla\sigma\|_{{\mathbb{L}}_{p}^{q}(T)},\|b^{1}\|_{{\mathbb{L}}_{p}^{q}(T)}$,
and $\|b^{2}\|_{{\mathbb{L}}_{p}^{q}(T)}$.

###### Proof.

We follow the argument from [28]. We show (3.3) via the following two steps.

_Step 1. SDEs with different diffusion terms and no drift._

Let $\sigma^{1}$ and $\sigma^{2}$ satisfy ($H_{\sigma^{1}}$) and
($H_{\sigma^{2}}$) respectively. Let $(Y_{t}^{1}(y_{1}))_{t\geqslant 0}$ and
$(Y_{t}^{2}(y_{2}))_{t\geqslant 0}$ be the unique strong solutions to

$\displaystyle dY_{t}^{1}=\sigma^{1}(t,Y_{t}^{1})\,{\mathord{{\rm
d}}}W_{t},\quad Y_{0}^{1}=y_{1}\in\mathbb{R}^{d}$ (3.4)

and

$\displaystyle dY_{t}^{2}=\sigma^{2}(t,Y_{t}^{2})\,{\mathord{{\rm
d}}}W_{t},\quad Y_{0}^{2}=y_{2}\in\mathbb{R}^{d}.$ (3.5)

By [28, Theorem 5.3], [27, Lemma 3.2] and the triangle inequality, we get that
there exist
$C_{1}=C_{1}(K,d,p,q,\|\nabla\sigma^{1}\|_{{\mathbb{L}}_{p}^{q}(T)},\|\nabla\sigma^{2}\|_{{\mathbb{L}}_{p}^{q}(T)},T)>0$
and $C_{2}=C_{2}(K,d,p,q,\|\nabla\sigma^{2}\|_{{\mathbb{L}}_{p}^{q}(T)},T)>0$
such that

$\displaystyle{\mathbb{E}}$
$\displaystyle{\Big{(}}\sup_{t\in[0,T]}|Y_{t}^{1}(y_{1})-Y_{t}^{2}(y_{2})|^{2}{\Big{)}}\leqslant
C_{1}\|\sigma^{1}-\sigma^{2}\|_{{\mathbb{L}}_{p}^{q}(T)}^{2}+C_{2}|y_{1}-y_{2}|^{2}.$
(3.6)

_Step 2. SDEs with different drifts and common noise._ Fix $T>0$. For $k=1,2$
and $l=1,\cdots,d$, following from Theorem A.2, let
$u^{k,(l)}\in{\mathscr{H}}_{2,p}^{q}(T)$ be the unique solution to the
equation

$\partial_{t}u^{k,(l)}+\frac{1}{2}\sum_{i,j=1}^{d}a_{ij}\partial_{ij}^{2}u^{k,(l)}+b^{k,(l)}\cdot\nabla
u^{k,(l)}+b^{k,(l)}=0,\quad t\in[0,T],u^{k,(l)}(T,x)=0,l=1,\cdots,d,$

where $(a_{ij})_{1\leqslant i,j\leqslant d}=\sigma\sigma^{*}$. Let
$U_{b^{k}}:=(u^{k,(l)})_{1\leqslant l\leqslant d}$, and define

$\displaystyle\Phi^{k}(t,x):=x+U_{b^{k}}(t,x)\text{ for
}(t,x)\in[0,T]\times\mathbb{R}^{d}.$

Lemma A.3 shows that there exists some $T>0$ such that $\Phi^{k}(t,.)$ is a
diffeomorphism for each $t\in[0,T]$ and that its derivative is jointly
continuous in $(t,x)\in[0,T]\times{\mathbb{R}}^{d}$. Let

$Y^{1}:=\Phi^{1}(t,X_{t}^{1}),\quad\Psi^{1}(t,y)=({\Phi^{1}})^{-1}(t,y).$

By [27, Lemma 4.3], we see that $(X_{t}^{1}(x_{1}))_{t\geqslant 0}$ solves the
SDE (3.1) if and only if $(Y_{t}^{1}(y_{1}))_{t\geqslant 0}$ solves (3.4) with

$\sigma^{1}(t,y):=[\nabla\Phi^{1}\cdot\sigma]\circ({\Psi^{1}}(t,y)),\quad
x_{1}=\Psi^{1}(0,y_{1}),$

and similarly for $(X_{t}^{2}(x_{2}))_{t\geqslant 0}$ and
$(Y_{t}^{2}(y_{2}))_{t\geqslant 0}$, i.e., for $i=1,2$,
$(X_{t}^{i}(x_{i}))_{t\geqslant 0}:=(\Psi^{i}(t,Y_{t}^{i}(y_{i})))_{t\geqslant
0}$ is the solution to (3.1) and (3.2) respectively. Notice that by (A.6) and
Lemma A.3 (2), there exists $N$ such that

$\displaystyle|y_{1}-y_{2}|$
$\displaystyle=|\Phi^{1}(0,x_{1})-\Phi^{2}(0,x_{2})|$
$\displaystyle\leqslant{\frac{3}{2}}|x_{1}-x_{2}|+N\|b^{1}-{b^{2}}\|_{{\mathbb{L}}_{p}^{q}([s_{0},t_{0}])}.$

By (A.6) and Lemma A.3 (1), (2), we have

$\displaystyle\sup_{y\in\mathbb{R}^{d}}|\Psi^{1}(t,y)-\Psi^{2}(t,y)|$
$\displaystyle=\sup_{y\in\mathbb{R}^{d}}|y-\Psi^{2}(t,\cdot)\circ\Phi^{1}(t,y)|$
$\displaystyle\leqslant
2\sup_{y\in\mathbb{R}^{d}}|\Phi^{2}(t,y)-\Phi^{1}(t,y)|$
$\displaystyle\leqslant C_{3}\|b^{1}-b^{2}\|_{{\mathbb{L}}_{p}^{q}(T)},$ (3.7)

where
$C_{3}=C(K,d,p,q,\|b^{1}\|_{{\mathbb{L}}_{p}^{q}([s_{0},t_{0}])},\|{b^{2}}\|_{{\mathbb{L}}_{p}^{q}([s_{0},t_{0}])})$.
Then, combining (3.1) with (3.6) and Lemma A.3 (1), we get

$\displaystyle{\mathbb{E}}$
$\displaystyle\sup_{t\in[0,T]}|X_{t}^{1}(x_{1})-X_{t}^{2}(x_{2})|^{2}$
$\displaystyle={\mathbb{E}}\sup_{t\in[0,T]}|\Psi^{1}(t,Y_{t}^{1}(y_{1}))-\Psi^{2}(t,Y_{t}^{2}(y_{2}))|^{2}$
$\displaystyle\leqslant
2{\mathbb{E}}\sup_{t\in[0,T]}|\Psi^{1}(t,Y_{t}^{1}(y_{1}))-\Psi^{1}(t,Y_{t}^{2}(y_{2}))|^{2}+2{\mathbb{E}}\sup_{t\in[0,T]}|\Psi^{1}(t,Y_{t}^{2}(y_{2}))-\Psi^{2}(t,Y_{t}^{2}(y_{2}))|^{2}$
$\displaystyle\leqslant
2\|\nabla\Psi^{1}\|_{{\mathbb{L}}^{\infty}(T)}^{2}{\mathbb{E}}\sup_{t\in[0,T]}|Y_{t}^{1}(y_{1})-Y_{t}^{2}(y_{2})|^{2}+C_{4}\|b^{1}-b^{2}\|_{{\mathbb{L}}_{p}^{q}(T)}^{2}$
$\displaystyle\leqslant
C_{5}\|b^{1}-b^{2}\|_{{\mathbb{L}}_{p}^{q}(T)}^{2}+C_{6}|y_{1}-y_{2}|^{2}$
$\displaystyle\leqslant
C_{5}\|b^{1}-b^{2}\|_{{\mathbb{L}}_{p}^{q}(T)}^{2}+C_{7}|x_{1}-x_{2}|^{2},$
(3.8)

where all constants appearing are positive. Therefore, we obtain (3.3) for
$T=T_{0}$ sufficiently small. For general $T$, we iterate the argument (as in
[28, Theorem 1.1]) and show that (3.3) holds on the interval
$[\frac{T_{0}}{2},\frac{3T_{0}}{2}]$ etc. By the pathwise uniqueness property,
we can patch up the solution and conclude the proof. ∎

######  Remark 3.4.

In [28, Theorem 1.1 E] and [26, Theorem 3.10], similar stability results were
obtained under slightly stronger assumptions. In [28, Theorem 6.1], the
authors assumed in addition that $\nabla\sigma,b^{i}\in\mathbb{L}^{q}_{p}(T)$
with $p=q$ and $p>d+2$, $i=1,2$, to show well-posedness of the corresponding
Kolmogorov equation and, consequently, the stability result. In [26, Theorem
3.10], the condition $\nabla b^{1}\in\mathbb{L}^{q^{\prime}}_{p^{\prime}}(T)$
for some $p^{\prime},q^{\prime}\in(1,\infty)$ was assumed to obtain the
stability result.

###  3.2. Wong-Zakai approximations

Next, we prove a quantitative version of the classical Wong-Zakai theorem. We
follow the exposition in [10].

Let $(\mathbb{W},{\mathbb{P}},\mathcal{F},(\mathcal{F}_{t})_{t\geqslant 0})$
be a Wiener space, i.e.
$\mathbb{W}:=\\{\omega\in\mathcal{C}([0,\infty),\mathbb{R}^{d}):\omega(0)=0\\}$,
${\mathbb{P}}$ is the Wiener measure defined on the Borel $\sigma$-algebra
$\mathcal{F}$ of $\mathbb{W}$, $W_{t}(\omega):=\omega(t),\,t\geqslant 0$ is a
Wiener process and $(\mathcal{F}_{t})_{t\geqslant 0}$ is the filtration
generated by $(W_{t})_{t\geqslant 0}$. We shall consider the following class
of approximations:

######  Definition 3.5.

[10, VI. Definition 7.1] By an approximation of the Wiener process
$W_{t}(\omega):=\omega(t),\,t\geqslant 0$, we mean a family
$\\{W^{n}\\}_{n\geqslant 0}$ of $d$-dimensional continuous processes defined
on the Wiener space
$(\mathbb{W},{\mathbb{P}},\mathcal{F},(\mathcal{F}_{t})_{t\geqslant 0})$ such
that

  * (1)

for every $\omega\in\mathbb{W},t\mapsto W^{n}_{t}(\omega)$ is continuous and
piecewise continuously differentiable,

  * (2)

$W^{n}_{0}(\omega)$ is $\mathcal{F}_{\frac{1}{n}}$-measurable, and
${\mathbb{E}}W^{n}_{0}=0$,

  * (3)

$W^{n}_{t+\frac{k}{n}}(\omega)=W^{n}_{t}(\theta_{\frac{k}{n}}\omega)+\omega(\frac{k}{n})$,
for every $k=1,2,\cdots,t\geqslant 0$ and $\omega$, where
$\theta_{t},t\geqslant 0$ is the shift operator defined by
$(\theta_{t}\omega)(s)=\omega(t+s)-\omega(t)$ for $\omega\in\mathbb{W}$,

  * (4)

there exists a positive constant $C$ such that

${\mathbb{E}}[|W^{n}_{\frac{1}{n}}|^{6}]\leqslant
C\frac{1}{n^{3}},\quad{\mathbb{E}}{\Big{(}}\int_{0}^{\frac{1}{n}}|\frac{{\mathord{{\rm
d}}}}{{\mathord{{\rm d}}}s}W^{n}_{s}|{\mathord{{\rm
d}}}s{\Big{)}}^{6}\leqslant C\frac{1}{n^{3}}.$

From [10, VI. Section 7] we know that for such a sequence $W^{n}$ we have

$\lim_{n\rightarrow\infty}{\mathbb{E}}\sup_{0\leqslant t\leqslant
T}|W_{t}-W_{t}^{n}|^{2}=0$

for every $T>0$. We further introduce the following notation for $t>0$:

$\displaystyle
S_{ij}(t):=\frac{{\Big{(}}\int_{0}^{t}W_{s}^{i}\circ{\mathord{{\rm
d}}}W^{j}_{s}-W_{s}^{j}\circ{\mathord{{\rm
d}}}W^{i}_{s}{\Big{)}}}{{{2}}t},\quad i,j=1,\cdots,d,\quad,t>0.$ (3.9)

Further,

$\displaystyle
s_{ij}^{n}(t):=s_{ij}(t,n):={\mathbb{E}}S_{ij}(t,n)=\frac{{\mathbb{E}}{\Big{(}}\int_{0}^{t}W^{n,i}_{s}\frac{{\mathord{{\rm
d}}}}{{\mathord{{\rm d}}}s}W^{n,j}_{s}-W^{n,j}_{s}\frac{{\mathord{{\rm
d}}}}{{\mathord{{\rm d}}}s}W^{n,i}_{s}{\mathord{{\rm d}}}s{\Big{)}}}{2t},\quad
i,j=1,\cdots,d,$ (3.10)

and

$\displaystyle
c_{ij}^{n}(t):=c_{ij}(t,n):=\frac{{\mathbb{E}}\Big{[}\int_{0}^{t}\frac{{\mathord{{\rm
d}}}}{{\mathord{{\rm
d}}}s}W^{n,i}_{s}{\Big{(}}W^{n,j}_{t}-W^{n,j}_{s}{\Big{)}}{\mathord{{\rm
d}}}s\Big{]}}{t},\quad i,j=1,\cdots,d.$ (3.11)

Further, $S_{ij}(0):=0$, $s_{ij}^{n}(0):=s_{ij}(0):=0$ and
$c_{ij}^{n}(0):=c_{ij}(0):=0$.

Observe that $(s_{ij}(t,n)):=(s_{ij}(t,n))_{1\leqslant i,j\leqslant d}$ is a
skew-symmetric $d\times d$-matrix for each $t$ and $n$, i.e.
$s_{ij}(t,n)=-s_{ji}(t,n)$.

The process $S_{ij}(t)$ is known as the _Lévy area_ in the literature and
plays a fundamental role in rough paths theory. As explained in [11], for such
a general approximation $\\{W^{n}_{t}\\}$ of $W_{t}$, $t\geqslant 0$, there
are cases in which $(S_{ij}(t,n))$ converges to $(S_{ij}(t))$ (e.g.[14]) and
others for which $(S_{ij}(t,n))$ does not converge to $S_{ij}(t)$ as
$n\rightarrow\infty$ but $S_{ij}(t)+$ ’another correction term’ does [15].
Here we consider a class of approximations of the Wiener process including
both cases by assuming the following condition:

######  Assumption 3.6.

There exists a skew-symmetric $d\times d$-matrix $(s_{ij})_{1\leqslant
i,j\leqslant d}$ and a rate function $f_{n}$ such that

$\displaystyle\big{|}s_{ij}^{n}(\frac{1}{n})-s_{ij}\big{|}\leqslant
f_{n},\quad\lim_{n\rightarrow\infty}f_{n}=0,\quad i,j=1,\ldots,d.$

The following proposition follows from the proof of [10, VI. Lemma 7.1] in a
straightforward manner.

######  Proposition 3.7.

Define

$\displaystyle c_{ij}:=s_{ij}+\frac{1}{2}\delta_{ij},\quad i,j=1,\ldots,d,$
(3.12)

and let $Z:(0,1]\to\mathbb{N}$ be a function such that
$\lim_{\delta\rightarrow 0}Z(\delta)=\infty$. Under Assumption 3.6, there
exists a positive constant $C$, independent of $n$, such that

$\displaystyle\Big{|}c_{ij}\Big{(}\frac{1}{n}Z\big{(}\frac{1}{n}\big{)},n\Big{)}-c_{ij}\Big{|}$
$\displaystyle\leqslant\Big{|}s_{ij}^{n}(\frac{1}{n})-s_{ij}\Big{|}+C\Big{(}Z(\frac{1}{n})^{-1}+Z(\frac{1}{n})^{-1/2}\Big{)}$
$\displaystyle\leqslant f_{n}+2CZ(\frac{1}{n})^{-1/2}.$ (3.13)

In particular,

$\lim_{n\rightarrow\infty}c_{ij}\Big{(}\frac{1}{n}Z\big{(}\frac{1}{n}\big{)},{n}\Big{)}=c_{ij}.$

We provide three classical examples for such approximations. For this purpose,
we introduce the following notation. Denote

$\mathcal{D}:=\\{f:[0,1]\rightarrow\mathbb{R}\text{ continuously
differentiable},\quad f(0)=0\text{ and }f(1)=1.\\}$

######  Example 3.8.

  * (1)

For $f\in\mathcal{D}$, let

$W_{t}^{n}:=W_{\frac{k}{n}}+f(t-\frac{k}{n})(W_{\frac{k+1}{n}}-W_{\frac{k}{n}}),\quad
t\in[\frac{k}{n},\frac{k+1}{n}),\quad k=1,2,\cdots.$

Then according to [10, VI. Example 7.1] such $(W^{n}_{t})_{n\geqslant 1}$
satisfies Definition 3.5, hence it is an approximation of $W_{t}$, $t\geqslant
0$. In this case, we easily observe that

$s_{ij}^{n}(\frac{1}{n})=0,\quad s_{ij}=0,\quad
c_{ij}=\frac{1}{2}\delta_{ij},\quad i,j=1,\cdots,d.$

It implies that Assumption 3.6 is fulfilled by taking $f_{n}=0$ for
$n\in\mathbb{N}_{+}$.

In particular if we take $f(t)=t$, then $(W^{n})_{n\geqslant 1}$ is the
familiar piecewise linear approximation.

  * (2)

Mollification: Let $\rho$ be a non-negative $\mathcal{C}^{\infty}$-function
with support contained in $[0,1]$ and $\int_{0}^{1}\rho(s){\mathord{{\rm
d}}}s=1$. Set

$\rho_{n}(s)=n\rho(ns),\quad n\geqslant 0$

and

$W_{s}^{n}=W_{\cdot}\ast\rho_{n}(s)=\int_{0}^{\infty}W_{r}\rho_{n}(s-r)\,{\mathord{{\rm
d}}}r.$

We can verify that $(W^{n})_{n\geqslant 1}$ is an approximation of a Wiener
process (see, e.g. [10, VI. Example 7.3]). Also in this case

$s_{ij}^{n}(\frac{1}{n})=0,\quad s_{ij}=0,\quad i,j=1,\cdots,d.$

Hence Assumption 3.6 is satisfied by taking $f_{n}=0$ and
$c_{ij}=\frac{1}{2}\delta_{ij}$.

  * (3)

(E.J. McShane[15]) Let $d=2$, $f^{i}\in\mathcal{D}$, $i=1,2$. For $i=1,2$, we
define for $t\in[\frac{k}{n},\frac{k+1}{n}),$ where $n\in\mathbb{N}_{+}$,
$k=0,1,\cdots,n-1,$

$W^{n,i}_{t}=\left\\{\begin{aligned}
&W^{i}_{\frac{k}{n}}+f^{i}(n(t-\frac{k}{n}))(W_{\frac{k+1}{n}}^{i}-W_{\frac{k}{n}}^{i}),\text{
if
}(W_{\frac{k+1}{n}}^{1}-W_{\frac{k}{n}}^{1})(W_{\frac{k+1}{n}}^{2}-W_{\frac{k}{n}}^{2})\geqslant
0,\\\
&W^{i}_{\frac{k}{n}}+f^{3-i}(n(t-\frac{k}{n}))(W_{\frac{k+1}{n}}^{i}-W_{\frac{k}{n}}^{i}),\text{
otherwise. }\\\ \end{aligned}\right.$

$(W^{n})_{n\geqslant 1}$ fulfills all conditions in Definition 3.5 (e.g. [10,
VI. Example 7.2]). In this case, we can verify that

$s_{11}=s_{22}=0,\quad s_{12}=\frac{1}{\pi},\quad s_{21}=-\frac{1}{\pi}.$

In the rest of this section, we only consider time homogeneous coefficients,
i.e. $b:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ and
$\sigma:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d\times d}$. We believe that the
extension to inhomogeneous coefficients is straightforward, but we restrict
ourselves to this case for the sake of simplicity.

To prove our main result, we need a quantitative version of the classical
Wong-Zakai theorem, i.e. an upper bound for the convergence rate. It turns out
that a careful inspection of the proof of [10, VI. Theorem 7.2] even leads to
the known optimal convergence rate. We state this result in the next theorem.

######  Theorem 3.9.

Suppose that for $1\leqslant i,j\leqslant d$,
$\sigma_{ij}\in\mathcal{C}_{b}^{2}(\mathbb{R}^{d})$,
$b^{(i)}\in\mathcal{C}_{b}^{1}(\mathbb{R}^{d})$. Let $(W^{n}_{t})_{n\geqslant
1}$ be a sequence of approximations for $W_{t},t\geqslant 0$ satisfying all
conditions in Definition 3.5. Assume that $(W_{t}^{n})_{n\geqslant
1},t\geqslant 0,$ satisfies Assumption 3.6. Consider the following equations

$\displaystyle X_{t}=x_{0}+\int_{0}^{t}b(X_{s})\,{\mathord{{\rm
d}}}s+\int_{0}^{t}\sigma(X_{s})\,{\mathord{{\rm
d}}}W_{s}+\Big{(}\sum_{i,j=1}^{d}\sum_{l=1}^{d}c_{ij}\int_{0}^{t}\sigma_{il}\cdot\nabla_{l}\sigma_{jk}(X_{s})\,{\mathord{{\rm
d}}}s\Big{)}_{k=1,\cdots,d},$ (3.14)

and

$\displaystyle X_{t}^{n}=x_{0}+\int_{0}^{t}b(X_{s}^{n})\,{\mathord{{\rm
d}}}s+\int_{0}^{t}\sigma(X_{s}^{n})\,{\mathord{{\rm d}}}W_{s}^{n},\quad
t\geqslant 0.$

Then, for each $T>0$ and $\delta\in(0,1)$,

$\displaystyle{\mathbb{E}}{\Big{(}}\sup_{t\in[0,T]}|X_{t}-X_{t}^{n}|^{2}{\Big{)}}\leqslant
C_{1}e^{C_{2}\|b\|_{\mathcal{C}_{b}^{1}(\mathbb{R}^{d})}^{2}}(f^{2}_{n}+(1+\sup_{x\in\mathbb{R}^{d}}|b(x)|^{2})\frac{1}{n^{1-\delta}}),$
(3.15)

where the constants $C_{1},C_{2}$ depend on $\delta$, $\sigma$ and $T$ but not
on $b$ and $n$, and $f_{n}$ is from Assumption 3.6.

###### Proof.

We closely follow the argument in the proof of [10, VI. Theorem 7.2] but we
keep track of the convergence rate of the approximating sequence
$\\{X_{t}^{n}\\}_{n\geqslant 1}$ to $X_{t}$ in the mean square sense on
$t\in[0,T]$.

For $\delta\in(0,1)$, let $Z_{\delta}(s):=\lfloor
s^{-\frac{\delta}{4}}\rfloor$, $s\in(0,1)$. Notice that such functions
$Z_{\delta}$ satisfy

$\displaystyle
Z_{\delta}:(0,1)\rightarrow\mathbb{N},\quad\lim_{n\rightarrow\infty}Z_{\delta}\Big{(}\frac{1}{n}\Big{)}^{4}n^{-1}\leqslant\lim_{n\rightarrow\infty}\frac{1}{n^{1-\delta}}=0,\quad\lim_{s\rightarrow
0}Z_{\delta}(s)=\infty.$ (3.16)

Therefore for such $Z_{\delta}$, by (3.7), we have

$c_{n}:=\Big{|}c_{ij}\Big{(}\frac{1}{n}Z_{\delta}\Big{(}\frac{1}{n}\Big{)},\frac{1}{n}\Big{)}-c_{ij}\Big{|}\leqslant
f_{n}+CZ_{\delta}\Big{(}\frac{1}{n}\Big{)}^{-1/2}.$

Then, as in [10, VI. (7.74)], there exist constants
$C^{\prime}(\sigma),C^{\prime\prime}(\sigma)>0$ independent of $b$ such that

$\displaystyle{\mathbb{E}}$
$\displaystyle{\Big{(}}\sup_{t\in[0,T]}|X_{t}-X_{t}^{n}|^{2}{\Big{)}}$
$\displaystyle\leqslant
C^{\prime}\|b\|_{\mathcal{C}_{b}^{1}(\mathbb{R}^{d})}^{2}\int_{0}^{T}{\mathbb{E}}{\Big{(}}|X_{t}-X_{t}^{n}|^{2}{\Big{)}}{\mathord{{\rm
d}}}t+C^{\prime\prime}[(1+\sup_{x\in\mathbb{R}^{d}}|b(x)|^{2})Z_{\delta}(n^{-1})^{4}n^{-1}+c_{n}^{2}].$

Finally, Gronwall’s inequality together with (3.16) show that (3.15) holds. ∎

######  Remark 3.10.

If $f_{n}=0$ for every $n$, Theorem 3.9 shows that the Wong-Zakai
approximation converges with rate arbitrarily close to $\frac{1}{2}$ uniformly
in $L^{2}(\mathbb{P})$. This is in line with the known optimal (strong)
convergence rate for the Wong-Zakai approximation.

As we already mentioned earlier, to formulate a Wong-Zakai theorem for
singular SDEs, we cannot just simply replace the noise by a smooth
approximation since this will render the equation ill-posed. To solve this
problem, we will approximate the singular coefficient $b$ at the same time. We
will now introduce a class of approximations for this $b$.

######  Definition 3.11 ($\mathcal{A}(b,p,h)$: approximation of $|b|\in
L^{p}(\mathbb{R}^{d})$).

Let $(b_{n})_{n\geqslant 1}$ be a sequence of functions
$b_{n}\colon\mathbb{R}^{d}\to\mathbb{R}^{d}$. For a function
$h\in[0,\infty)\rightarrow[0,\infty)$, we write $(b_{n})_{n\geqslant
1}\in\mathcal{A}(b,p,h)$ if

  * (i)

$b_{n}^{(i)}\in\mathcal{C}_{b}^{1}(\mathbb{R}^{d})$, $i=1,\ldots,d$,

  * (ii)

$\|b-b_{n}\|_{L^{p}(\mathbb{R}^{d})}\rightarrow 0$ as $n\rightarrow\infty$,

  * (iii)

$\|b_{n}\|_{\mathcal{C}_{b}^{1}(\mathbb{R}^{d})}\leqslant
h(n)\|b\|_{L^{p}(\mathbb{R}^{d})}$.

Next, we state our main theorem.

######  Theorem 3.12.

Assume that $b\colon\mathbb{R}^{d}\to\mathbb{R}^{d}$ is measurable and that
$|b|\in L^{p}(\mathbb{R}^{d})$ for some $p\in(d,\infty)$ and $p\geqslant 2$.
Let $\sigma$ satisfy condition $(H_{\sigma})$ in Assumption 3.1 and for
$1\leqslant i,j\leqslant d$,
$\sigma_{ij}\in\mathcal{C}_{b}^{2}(\mathbb{R}^{d})$. Assume further that
$(W^{n}_{t})$ is sequence of approximations for $W_{t},t\geqslant 0$,
satisfying all the conditions in Definition 3.5 and Assumption 3.6. Let
$(X_{t})_{t\geqslant 0}$ be the solution to

$\displaystyle X_{t}=x_{0}+\int_{0}^{t}b(X_{s})\,{\mathord{{\rm
d}}}s+\int_{0}^{t}\sigma(X_{s})\,{\mathord{{\rm
d}}}W_{s}+\Big{(}\sum_{i,j=1}^{d}\sum_{l=1}^{d}c_{ij}\int_{0}^{t}\sigma_{il}\cdot\nabla_{l}\sigma_{jk}(X_{s})\,{\mathord{{\rm
d}}}s\Big{)}_{k=1,\cdots,d}$ (3.17)

with $c_{ij}$ defined as in Assumption 3.6. Let $(b_{n})_{n\geqslant
1}\in\mathcal{A}(b,p,h)$ be a smooth approximation sequence of $b$ such that

$\displaystyle\lim_{n\rightarrow\infty}e^{h(n)^{2}\|b\|_{L^{p}(\mathbb{R}^{d})}^{2}}(f^{2}_{n}+(1+h(n)^{2}\|b\|_{L^{p}(\mathbb{R}^{d})}^{2})\frac{1}{n^{1-\delta}})=0,\quad\text{for
some }\delta\in(0,1),$ (3.18)

where $f_{n}$ is given in Assumption 3.6. Then for every $T>0$ we have

$\displaystyle\lim_{n\rightarrow\infty}{\mathbb{E}}\Big{[}\sup_{0\leqslant
t\leqslant T}|X_{t}-X_{t}^{n}|^{2}\Big{]}=0.$ (3.19)

Here $(X_{t}^{n})_{t\geqslant 0}$ is the solution to the equation

$\displaystyle X_{t}^{n}=x_{0}+\int_{0}^{t}b_{n}(X_{s}^{n})\,{\mathord{{\rm
d}}}s+\int_{0}^{t}\sigma(X_{s}^{n})\,{\mathord{{\rm d}}}W_{s}^{n}.$

###### Proof.

First under Assumption 3.1 we know that
$|((\sum_{i,j=1}^{d}\sum_{l=1}^{d}\sigma_{il}\cdot\nabla_{l}\sigma_{ik})^{(k)})_{k=1,\cdots,d}|\in{L}^{p}(\mathbb{R}^{d})$,
therefore by Theorem 3.2 there exists a unique solution
$(X_{t}(x))_{t\geqslant 0}$ to the SDE (3.17). For
$(b_{m})_{m\in\mathbb{N}}\in\mathcal{A}(b,p,h)$, let $(X_{m,t})_{t\geqslant
0}$ be the solution to

$\displaystyle X_{m,t}=x_{0}+\int_{0}^{t}b_{m}(X_{m,s}){\mathord{{\rm d}}}s$
$\displaystyle+\int_{0}^{t}\sigma(X_{m,s}){\mathord{{\rm d}}}W_{s}$
$\displaystyle+\Big{(}\sum_{i,j=1}^{d}\sum_{l=1}^{d}c_{ij}\int_{0}^{t}\sigma_{il}\cdot\nabla_{l}\sigma_{jk}(X_{m,s}){\mathord{{\rm
d}}}s\Big{)}_{k=1,\cdots,d}.$

By Theorem 3.9, we know that for any $m\geqslant 1$,

$\displaystyle{\mathbb{E}}\Big{[}\sup_{0\leqslant t\leqslant
T}|X_{m,t}-X_{m,t}^{n}|^{2}\Big{]}\leqslant$ $\displaystyle
C^{\prime\prime}e^{C^{\prime}\|b_{m}\|_{\mathcal{C}_{b}^{1}(\mathbb{R}^{d})}^{2}}(f^{2}_{n}+(1+\|b_{m}\|_{\mathcal{C}_{b}^{1}(\mathbb{R}^{d})}^{2})\frac{1}{n^{1-\delta}})$
$\displaystyle\leqslant$ $\displaystyle
C^{\prime\prime}e^{C^{\prime}h(m)^{2}\|b\|_{L^{p}(\mathbb{R}^{d})}^{2}}(f^{2}_{n}+(1+h(m)^{2}\|b\|_{L^{p}(\mathbb{R}^{d})}^{2})\frac{1}{n^{1-\delta}}),$

where $C^{\prime}$ and $C^{\prime\prime}$ are positive constants, independent
of $m$ and $n$, and $(X_{m,t}^{n})_{t\geqslant 0}$ solves

$\displaystyle X_{m,t}^{n}=x_{0}+\int_{0}^{t}b_{m}(X_{m,s}^{n}){\mathord{{\rm
d}}}s+\int_{0}^{t}\sigma(X_{m,s}^{n}){\mathord{{\rm d}}}W_{s}^{n}.$

By choosing $m=n$ and using the condition (3.18) we can obtain that

$\displaystyle\lim_{n\rightarrow\infty}{\mathbb{E}}\Big{[}\sup_{0\leqslant
t\leqslant T}|X_{n,t}-X_{n,t}^{n}|^{2}\Big{]}=0.$

Then by Theorem 3.3,

$\displaystyle{\mathbb{E}}\Big{[}\sup_{0\leqslant t\leqslant
T}|X_{t}-X_{t}^{n}|^{2}\Big{]}$
$\displaystyle\leqslant{\mathbb{E}}\Big{[}\sup_{0\leqslant t\leqslant
T}|X_{t}-X_{n,t}|^{2}\Big{]}+{\mathbb{E}}\Big{[}\sup_{0\leqslant t\leqslant
T}|X_{n,t}-X_{n,t}^{n}|^{2}\Big{]}$ $\displaystyle\leqslant
C^{*}(p,d,T,K,\|\nabla\sigma\|_{L_{p}(\mathbb{R}^{d})},\|b\|_{L_{p}(\mathbb{R}^{d})})\|b-b_{n}\|_{L^{p}(\mathbb{R}^{d})}^{2}$
$\displaystyle\quad\quad+{\mathbb{E}}\Big{[}\sup_{0\leqslant t\leqslant
T}|X_{n,t}-X_{n,t}^{n}|^{2}\Big{]}.$ (3.20)

Therefore (3.19) holds by letting $n\rightarrow\infty$. ∎

######  Remark 3.13.

  1. (1)

Theorem 3.12 states that an approximation to a singular SDE converges to this
SDE provided that the approximation of the noise converges sufficiently fast
compared to the approximation of the singular coefficient $b$, cf. condition
(3.18). Although this condition seems plausible and is true in the “extreme”
case (one can not first let $b_{n}\to b$ and then $W^{n}\to W$), we do not
know at the moment whether such a condition is really necessary. We leave this
point as an open problem.

  2. (2)

It is easy to deduce a convergence rate of the Wong-Zakai approximation.
However, this can arbitrarily slow, depending on the convergence rate of
$b_{n}\to b$. For a smooth $b$, we rediscover the known convergence rate of
(almost) $\frac{1}{2}$ in the classical Wong-Zakai theorem.

  3. (3)

The approximation class $\mathcal{A}(b,p,h)$ turns out to be of own interest
and is discussed in more detail in [12].

In the end we give some examples to show how this theorem works in different
cases.

######  Example 3.14.

  * (1)

If for $i=1,\cdots,d$, $b^{(i)}\in\mathcal{C}_{b}^{1}(\mathbb{R}^{d})$ and
$|b|\in L^{p}(\mathbb{R}^{d})$, $p>d$ and $p\geqslant 4$, then we can simply
take $b_{n}\equiv b$ and choose $h(n)\equiv 1$. This is the special setting of
the standard Wong-Zakai theorem for regular SDEs, e.g. see [10, VI. Section
7].

  * (2)

Let $d=1$ and define

$b(x)=1_{[0,1]}(x),\quad x\in\mathbb{R}.$

Then $|b|\in L^{p}(\mathbb{R})$ for $p\in[1,\infty]$. For
$\chi:[0,\infty)\rightarrow(0,\infty)$ satisfying
$\lim_{n\rightarrow\infty}\chi(n)=\infty$, let

$b_{n}(x)=\left\\{\begin{aligned} 0,&\text{ if
}x\in(-\infty,-\frac{2}{\chi(n)})\cup(1+\frac{2}{\chi(n)},\infty),\\\
\frac{\chi(n)x}{2}+1,&\text{ if }x\in[-\frac{2}{\chi(n)},0),\\\
-\frac{\chi(n)x}{2}+\frac{\chi(n)+2}{2},&\text{ if
}x\in(1,1+\frac{2}{\chi(n)}],\\\ 1,&\text{ if }x\in[0,1].\end{aligned}\right.$

We find that $b_{n}\in\mathcal{C}_{b}^{1}(\mathbb{R})$ almost
everywhere.111Theorem 3.12 assumes $b_{n}$ bounded and differentiable
_everywhere_ , but it is easy to see that $b_{n}$ can be replaced by smooth
versions for the price of adding arbitrary small $\epsilon$ in the following
bounds. This will not affect the overall conclusion. Furthermore,

${\Big{(}}\int_{-\infty}^{+\infty}|b_{n}(x)-b(x)|^{p}{\mathord{{\rm
d}}}x{\Big{)}}^{1/p}=2{\Big{(}}\frac{2}{\chi(n)(p+1)}{\Big{)}}^{1/p},$

and

$\sup_{x\in\mathbb{R}}|\frac{{\mathord{{\rm d}}}b_{n}(x)}{{\mathord{{\rm
d}}}x}|+\sup_{x\in\mathbb{R}}|b_{n}(x)|\leqslant\frac{\chi(n)+2}{2}\|b\|_{L^{p}(\mathbb{R})}=\frac{\chi(n)+2}{2}.$

Now we take

$\chi(n):=\max({2\sqrt{|\log(n^{\alpha})|}}-2,0),$

$\alpha\in(0,1)$, and choose $\alpha$ and $\delta$ such that
$\lim_{n\rightarrow\infty}n^{\alpha}(f_{n}^{2}+\frac{(1+\log
n^{\alpha})}{n^{1-\delta}})=0$, assuming that this is possible. Here $f_{n}$
and $\delta$ keep the same meaning as in Theorem 3.12. For instance, if
$f_{n}=0$, given any $\alpha\in(0,1)$, we can choose $\delta$ such that
$\alpha+\delta<1$. For $h(n):=\frac{\chi(n)+2}{2}$, this implies

$\lim_{n\rightarrow\infty}e^{h(n)^{2}\|b\|_{L^{p}(\mathbb{R}^{d})}^{2}}(f^{2}_{n}+(1+h(n)^{2}\|b\|_{L^{p}(\mathbb{R}^{d})}^{2})\frac{1}{n^{1-\delta}})=0.$

Therefore $(b_{n})_{n\geqslant 1}\in\mathcal{A}(b,p,h)$ and fulfills all
conditions in Theorem 3.12. We can apply this approximation sequence to the
SDE with singular drift $b$ and obtain the Wong-Zakai approximation result.

Actually, we can also take a mollification sequence of $b$ by convoluting with
a smooth function. That is to say, let

$b_{n}(x):=b\ast
g_{n}(x):=\sqrt{\frac{\kappa_{n}}{2\pi}}\int_{-\infty}^{+\infty}b(x-y)e^{\frac{-\kappa_{n}y^{2}}{2}}{\mathord{{\rm
d}}}y,$

where $\kappa_{n}>0$ we will choose later. Notice that such $b_{n}$ is smooth
with bounded derivatives, and there exists a positive constant $C>0$,
independent of $n$, such that

$\sup_{x\in\mathbb{R}}|\frac{{\mathord{{\rm d}}}b_{n}(x)}{{\mathord{{\rm
d}}}x}|+\sup_{x\in\mathbb{R}}|b_{n}(x)|\leqslant C(\kappa_{n}+1).$

By taking

$\kappa_{n}:=\max(\frac{\sqrt{|\log(n^{\alpha})|}}{C}-1,0),$

with $\alpha\in(0,1)$ such that
$\lim_{n\rightarrow\infty}n^{\alpha}(f_{n}^{2}+\frac{1+\log
n^{\alpha}}{n^{1-\delta}})=0$ and $h_{n}:=C(\kappa_{n}+1)$, we see that (3.18)
holds. As before, we know that such approximation can be applied in order to
establish the Wong-Zakai approximation to the singular SDE.

##  4\. Support theorem for Singular SDEs

Let $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geqslant 0},{\mathbb{P}})$ be a
probability spaces and $(W_{t})_{t\geqslant 0}$ a Wiener process in
$\mathbb{R}^{d}$. Consider the SDE

$\displaystyle X_{t}=x_{0}+\int_{0}^{t}b(s,X_{s})\,{\mathord{{\rm
d}}}s+\int_{0}^{t}\sigma(s,X_{s})\circ{\mathord{{\rm d}}}W_{s},\quad
t\in[0,T].$ (4.1)

We are interested in describing the topological support of the distribution of
the solution $(X_{t})_{t\in[0,T]}$ in the space
$\mathcal{C}([0,T],\mathbb{R}^{d})$ equipped with the topology of uniform
convergence when $b$ is a singular drift. The classical strategy to derive a
description of the support uses the Wong-Zakai theorem, see e.g. [10], [5],
[18]. Here, we use a different approach which turns out to be much easier in
our context. Namely, we use Girsanov theory to show that actually the support
of the distribution of the solution to (4.1) is the full canonical space
$\mathcal{C}([0,T],\mathbb{R}^{d})$, which essentially is because of the non
degenerated noise.

######  Theorem 4.1.

Assume that the conditions in Assumption 3.1 are satisfied. Let

$\Omega_{x_{0}}([0,T],\mathbb{R}^{d}):=\\{\omega\in\mathcal{C}([0,T],\mathbb{R}^{d}):w_{0}=x_{0}\\}.$

Then

$\textup{supp }\mu=\Omega_{x_{0}}([0,T],\mathbb{R}^{d}),$

where $\textup{supp }\mu$ is the topological support of the distribution of
the solution $(X_{t}(x_{0}))_{t\in[0,T]}$ to the SDE (4.1) in
$\mathcal{C}([0,T],\mathbb{R}^{d})$.

###### Proof.

First we consider the following equation

$\displaystyle Y_{t}=x_{0}+\int_{0}^{t}\sigma(s,Y_{s})\circ{\mathord{{\rm
d}}}W_{s},\quad t\geqslant 0.$

By Theorem 3.2, there is an unique and strong solution
$(Y_{t}(x_{0}))_{t\geqslant 0}$ to the above SDE.

Let $\textup{supp }\mu_{Y}$ be the topological support of the distribution of
$(Y_{t}(x_{0}))_{t\geqslant 0}$ in $\mathcal{C}([0,T],\mathbb{R}^{d})$. Under
the condition $(H_{\sigma})$, by [20, Theorem 3.1] we know that

$\textup{supp }\mu_{Y}=\Omega_{x_{0}}([0,T],\mathbb{R}^{d}).$

Besides, because of [25, Lemma 4.1 (ii)], for any constant $\kappa>0$, for any
$0\leqslant T<\infty$, we have

$\displaystyle{\mathbb{E}}\exp\bigg{\\{}\kappa\int_{0}^{T}|b(s,Y_{s})|^{2}{\mathord{{\rm
d}}}s\bigg{\\}}\leqslant
C(T,\kappa,d,{p},q,\|b\|_{\mathbb{L}_{p}^{q}(T)})<\infty,$ (4.2)

i.e. Novikov’s condition is fulfilled. As a result, we have

$\displaystyle{\mathbb{E}}\rho_{T}:={\mathbb{E}}\exp\bigg{\\{}{\int_{0}^{T}\big{[}b^{*}\sigma^{-1}\big{]}(s,Y_{s}){\mathord{{\rm
d}}}W_{s}}-\frac{1}{2}\int_{0}^{T}\big{[}b^{*}(\sigma\sigma^{*})^{-1}b\big{]}(s,Y_{s}){\mathord{{\rm
d}}}s\bigg{\\}}=1.$

Then by the Girsanov theorem, for any bounded functional $f$ defined on
$\mathcal{C}([0,T],\mathbb{R}^{d})$,

${\mathbb{E}}[f(X_{\cdot})]={\mathbb{E}}[\rho_{T}f(Y_{\cdot})],\quad
0\leqslant T<\infty.$

This shows that the distribution of the solution $(X_{t})_{t\geqslant 0}$ to
SDE (4.1) in $\mathcal{C}([0,T],\mathbb{R}^{d})$ is absolutely continuous with
respect to the distribution of $(Y_{t})_{t\geqslant 0}$ in
$\mathcal{C}([0,T],\mathbb{R}^{d})$ and vice versa, therefore they share the
same support, which completes our proof. ∎

##  A. Auxiliary results on Kolmogorov backward equations

In this part, we collect the known results on regularity estimates for
corresponding PDEs. [25, Theorem 3.2] and [16, Theorem 2.1] show that the
Kolmogorov backward equation corresponding to the solution to SDE (1.1) can be
solved in $\mathbb{L}_{p}^{q}(T)$ under the following assumption.

######  Assumption A.1.

$(a_{ij})_{1\leqslant i,j\leqslant d}$ is uniformly continuous in
$x\in\mathbb{R}^{d}$ locally uniformly with respect to $t\in\mathbb{R}_{+}$,
and for some $\delta\geqslant 1$ and for
$(t,x)\in[0,\infty)\times\mathbb{R}^{d}$,

$\delta^{-1}|\xi|^{2}\leqslant|a(t,x)\xi|^{2}\leqslant\delta|\xi|^{2},\quad\forall\xi\in\mathbb{R}^{d}.$

######  Theorem A.2.

[16, Theorem 2.1] Let Assumption A.1 be satisfied, and let
$f\in{{\mathbb{L}}}_{p}^{q}(T)$ with $T>0$, $p,q\in(1,\infty)$ and
$d/p+2/q<1$. Assume that $b$ is measurable and that
$|b|\in{{\mathbb{L}}}_{p_{1}}^{q_{1}}(T)$ with $p_{1}\in[p,\infty)$,
$q_{1}\in[q,\infty)$ such that $d/p_{1}+2/q_{1}<1$. Then in
${{\mathscr{H}}}_{2,p}^{q}(T)$, there is a unique solution to the equation

$\displaystyle\partial_{t}u+\frac{1}{2}a_{ij}\partial_{ij}^{2}u+b\cdot\nabla
u+f=0,\quad t\in[0,T],\quad u(T,x)=0,$ (A.1)

which further satisfies the following regularity estimates: for any
$\alpha\in[0,2-\tfrac{2}{q})$, there exists a constant
$C_{T}=C(d,{p},q,\|b\|_{{\mathbb{L}}^{q_{1}}_{p_{1}}(T)},T)$ satisfying
$\lim_{T\to 0}C_{T}=0$ such that

$\displaystyle\|u\|_{\mathbb{H}_{\alpha,p}^{\infty}(T)}\leqslant
C_{T}\|f\|_{\mathbb{L}^{q}_{p}(T)}.$ (A.2)

Furthermore, we have

$\displaystyle\|u\|_{{\mathbb{L}}^{\infty}(T)}\leqslant\hat{C}_{T}\|f\|_{{\mathbb{L}}^{q}_{p}(T)},\quad\text{if}\quad
2/q+d/p<2,$ (A.3)

and

$\displaystyle\|\nabla
u\|_{{\mathbb{L}}^{\infty}(T)}\leqslant\hat{C}_{T}\|f\|_{{\mathbb{L}}^{q}_{p}(T)},\quad\text{if}\quad
2/q+d/p<1,$ (A.4)

where $\hat{C}_{T}>0$ is a constant satisfying $\lim_{T\to 0}\hat{C}_{T}=0$.

For $l=1,\cdots,d$, we consider the unique solution
$u_{b}^{(l)}\in{\mathscr{H}}_{2,p}^{q}(T)$ to the equation

$\partial_{t}u^{(l)}+\frac{1}{2}\sum_{i,j=1}^{d}a_{ij}\partial_{ij}^{2}u^{(l)}+b\cdot\nabla
u^{(l)}+b^{(l)}=0,\quad t\in[0,T],\quad u^{(l)}(T,x)=0,\quad l=1,\cdots,d.$

Let $U_{b}:=(u_{b}^{(l)})_{1\leqslant l\leqslant d}$ and define

$\displaystyle\Phi^{b}(t,x):=x+U_{b}(t,x)\text{ for
}(t,x)\in[0,T]\times\mathbb{R}^{d}.$ (A.5)

The map $\Phi^{b}(t,x)$ is usually called _Zvonkin’s transformation map_ in
the literature (see, e.g. [27], [25], [29], [26]). It was proven to be a
$\mathcal{C}^{1}$-diffeomorphism on $\mathbb{R}^{d}$ for each $t\in[0,T]$ when
$T$ is small. From a similar argument as in [27], [25] and [28], based on the
regularity estimates (A.2), (A.3) and (A.4), we can obtain:

######  Lemma A.3.

Under Assumption 3.1, there exists a positive constant
$\epsilon=\epsilon(K,d,p,q,\lambda)$ such that if
$t_{0}-s_{0}\leqslant\epsilon$ and
$\|b\|_{{\mathbb{L}}_{p}^{q}([s_{0},t_{0}])}\leqslant\lambda$, then for each
$t\in[s_{0},t_{0}]$, $x\rightarrow\Phi^{b}(t,x)$ is a
$\mathcal{C}^{1}$-diffeomorphism with

$\displaystyle\frac{1}{2}|x-y|\leqslant|\Phi^{b}(t,x)-\Phi^{b}(t,y)|\leqslant\frac{3}{2}|x-y|.$
(A.6)

Moreover, let $\delta:=\frac{1}{2}-\frac{d}{2p}-\frac{1}{q}>0$, and define
$\Psi^{b}:=(\Phi^{b})^{-1}$, i.e. the inverse of $\Phi^{b}$. Then. the
following holds:

  * (1)

There exists a universal positive constant $C$ such that
$\|\nabla\Phi^{b}(t,\cdot)\|_{\infty}+\|\nabla{\Psi^{b}}(t,\cdot)\|_{\infty}\leqslant
C$.

  * (2)

Let $b^{\prime}\in{\mathbb{L}}_{p}^{q}(T)$ with
$\|b^{\prime}\|_{{\mathbb{L}}_{p}^{q}([s_{0},t_{0}])}\leqslant\lambda$ and
$\Phi^{b^{\prime}}$ defined as above. Then we have for some $N>0$,

$\displaystyle\|\Phi^{b}-$
$\displaystyle\Phi^{b^{\prime}}\|_{{\mathbb{L}}^{\infty}([s_{0},t_{0}])}+\|\nabla\Phi^{b}-\nabla\Phi^{b^{\prime}}\|_{{\mathbb{L}}_{p}^{q}([s_{0},t_{0}])}\leqslant
N\|b-{b^{\prime}}\|_{{\mathbb{L}}_{p}^{q}([s_{0},t_{0}])}.$

Here $N$ depends on $\lambda,K,p,q,d,\delta$ and the modulus of continuity of
$(a_{ij})_{1\leqslant i,j\leqslant d}$.

  * (3)

Let
$\tilde{\sigma}^{b}(t,y):=[\nabla\Phi^{b}\cdot\sigma]\circ({\Psi^{b}}(t,y))$
and similarly define $\tilde{\sigma}^{b^{\prime}}$ with
$\|b^{\prime}\|_{{\mathbb{L}}_{p}^{q}([s_{0},t_{0}])}\leqslant\lambda$. Then
$\tilde{\sigma}^{b}$ satisfies ($H_{\tilde{\sigma}^{b}}$),
$\tilde{\sigma}^{b^{\prime}}$ satisfies ($H_{\tilde{\sigma}^{b^{\prime}}}$),
and there exists a positive constant $N$ depending on
$\lambda,K,p,q,d,\delta$, and the modulus of continuity of
$(a_{ij})_{1\leqslant i,j\leqslant d}$ such that

$\displaystyle\|\tilde{\sigma}^{b}-\tilde{\sigma}^{b^{\prime}}\|_{{\mathbb{L}}_{p}^{q}([s_{0},t_{0}])}\leqslant
N\|b-{b^{\prime}}\|_{{\mathbb{L}}_{p}^{q}([s_{0},t_{0}])}.$

######  Remark A.4.

Using the well-posedness result in Theorem A.2, the proof of the stated lemma
use now standard techniques, explained in a series of works. We give an
outline of the way on how to get it. First, based on the regularity estimates
from Theorem A.2 (see also [25, Theorem 3.2], by the same argument of
obtaining [28, Theorem 6.1], correspondingly we can also get the same results
for $p,q\in[2,\infty)$ and $\frac{d}{p}+\frac{2}{q}<1$. Then, following the
proof of [28, Lemma 6.2], we can show the above conclusion.

### Acknowledgements

CL, SR and MS acknowledge financial support by the DFG via Research Unit FOR
2402.

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