# Remarks on Stochastic Systems I:
Markov properties, local and global uniqueness, and limits of stochastic
equations

Seiichiro Kusuoka
∗Department of Mathematics, Graduate School of Science, Kyoto University,
Kitashirakawa-Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan
e-mail address<EMAIL_ADDRESS>
###### Abstract

In the present paper, we give some examples of stochastic differential
equations which have delicateness in the Markov and strong Markov properties,
the uniqueness locally in time and globally in time, and initial conditions.
Moreover, we show that such stochastic differential equations appear in the
limits of stochastic differential equations which have the existence and
pathwise uniqueness of solutions. These examples are constructed in motivation
to singular stochastic partial differential equations. We also give some
examples of shifted equations whose sum of the solutions depends on the
choices of the decomposition of the initial condition of the original
equation.

AMS Classification Numbers: 60J60, 60H17, 60H10, 58J65

Key words: stochastic differential equations, singular stochastic partial
differential equations, Markov property, uniqueness of solutions

##  1 Introduction

After stochastic differential equations (SDEs) were introduced by Kiyoshi Itô,
SDEs had been studied very much as the mainstream in the probability theory.
The relations to other fields of mathematics also had been discovered, and the
framework of SDEs are almost fixed now except some new types of SDEs (see [13,
14, 12] for an extension of the framework of SDEs, and [19] for a review of
the history from the extension of SDEs to singular SPDEs). As partial
differential equations (PDEs) including some noises, stochastic partial
differential equations (SPDEs) also have been studied intensively, not only in
view of the motivations of mathematics, but also of the motivations in
physics. We remark that SPDEs can be regarded as the infinite-dimensional
version of SDEs.

Recently singular SPDEs are studied intensively. The key methods for the study
are regularity structures (see [20, 21]) and paracontrolled calculus (see
[18]). The methods enable us to treat SPDEs which require renomalization to be
solved in suitable sense. Such equations appear when we consider stochastic
quantization equations of quantum field theories (see the introductions of [2,
3] and [17] for the history of stochastic quantizations). Because of explicit
applications, now singular SPDEs are studied by many mathematicians and
mathematical physicists from both the mathematical and the physical points of
view.

However, the method of singular SPDEs is very different from the standard
theory of SDEs. Indeed, to solve singular SPDEs we first consider the
linearized SPDE associated to the original singular SPDE. The linearized SPDE
is often of the infinite-dimensional Ornstein-Uhlenbeck type, and hence the
solution is often the Ornstein-Uhlenbeck process. Second, we consider a
transform of the singular SPDE to another SPDE which the difference between
the solutions to the original singular SPDE and the linearized SPDE is
expected to satisfy. We call the equation which the difference of the
solutions satisfies a shifted equation. For the shifted equation to be well-
posed we often need renormalization, which cancels the diverging terms coming
from the nonlinearity in a suitable sense. Then, we give a solution to the
original singular SPDE in a suitable sense by the sum of the solutions to the
shifted equation and the linearized equation. This method is originally
introduced in [10]. Here, we remark that the definition of the solution to
singular SPDEs are different from those to usual SPDEs. There is another way
to understand the solution to singular SPDEs. Before taking the difference
between the solutions to the original singular SPDE and the linearized SPDE,
by regularizing the noise (or the nonlinear term) of singular SPDEs we can
define the solution to the regularized singular SPDE in usual sense of
solutions to SPDEs. The solutions to the shifted equations of the regularized
singular SPDE are obtained, and we can show the convergence of the solutions
as removing regularization if the shifted equation is solved without
regularization. Then, the sum of the solutions to the shifted equations and
the linearized SPDE is also convergent in a suitable sense. Hence, the
solution to the singular SPDE can be regarded as the limit of the solutions to
the regularized equations. Here, we remark that the limits are known to be
independent of regularizations in suitable classes (see [20] and [18]). See
Section 2.1 for more details of singular SPDEs.

The points that we have to be careful in the methods of singular SPDEs are the
following.

  * •

The solutions to singular SPDEs are defined by limits of solutions to
approximate equations, or defined by a projection of solutions to shifted
equations.

  * •

There exist exceptional sets of initial points of original equations and
shifted equations.

These do not appear in the standard theory of SDEs, and seem to make delicate
problems differently from normal SDEs. See Section 2 for the details of the
methods of singular SPDEs and the standard theory of SDEs.

From these points of view, in the present paper we consider examples under the
framework of SDEs which make phenomena different from the cases of
sufficiently nice SDEs. Precisely, we focus on the importance of the
uniqueness in all initial conditions and the delicateness of taking limits of
strong Markov processes.

Now we summarize the results in the present paper as the following theorem.

######  Theorem 1.1.

  1. (i)

There exists a sequence of SDEs which have the pathwise uniqueness, such that
their solutions have the strong Markov property and converge to a stochastic
process which does not have the strong Markov property. (See Theorem 3.5.)

  2. (ii)

There exists a SDE which has the uniqueness locally in time for every initial
point except one point, whose any solution is extendable globally in time.
However, the SDE does not have the uniqueness globally in time. (See Theorem
4.1.)

  3. (iii)

There exist sequences of SDEs which have the pathwise uniqueness for every
initial point, such that the solutions to the both sequences of SDEs converges
to solutions to a SDE which does not have the uniqueness of solutions.
Moreover, the limits of the solutions are different from each other. (See
Theorem 4.3.)

  4. (iv)

There exists a path-dependent SDE which has the uniqueness locally in time for
every initial point except one point, whose any solution is extendable
globally in time. However, the SDE does not have the uniqueness globally in
time. (See Theorem 4.4.)

  5. (v)

There exists a sequence of SDEs which have the pathwise uniqueness for every
initial point, such that the solutions to the both sequences of SDEs converges
to solutions to a SDE which does not have the uniqueness of solutions.
Moreover, the limit of the sequence of SDEs can be regarded a shifted
equation. However, the limit of the solutions depends on the decomposition of
initial conditions for the shifted equation. (See Theorem 5.7.)

In view of the examples in Theorem 1.1 we see the difference of the frameworks
between singular SPDEs and the standard theory of SDEs. Moreover, arguments in
some papers in the field of singular SPDEs seem loose. We dare not to refer
them here in the present paper. It should be emphasized that the results in
the present paper do not conflict with known results in singular SPDEs, but
suggest careful arguments to apply known facts in conventional stochastic
analysis. Here, we also remark that in the case of singular SPDEs on the
Euclidean spaces, we need to take limits not only for regularization, but also
for infinite volumes (see [26] for the $\Phi^{4}$-stochastic quantization on
${\mathbb{R}}^{2}$ and [3, 16, 17] for that on ${\mathbb{R}}^{3}$). Hence,
also with respect to the limits for the infinite volumes, we have to pay
attentions in view of such examples in Theorem 1.1.

On the other hand, some papers are written very carefully. In the first
memorable paper [20] of the theory of regularity structures, the uniqueness of
the solution is discussed in the framework of the regularity structures (see
[20, Theorem 7.8]), while in applications to singular SPDEs the author did not
mention the uniqueness of the solution, but mentioned only the independence of
regularizations in a suitable class is mentioned (see [20, Section 1.5]). Also
in the first memorable paper [18] of paracontrolled calculus, the dependence
of inputs and the sense of the unique solutions are clarified (see [18,
Theorem 3.3, Theorem 4.1, Corollary 5.9]). By the way, the continuities in the
initial conditions in suitable senses are obtained in both theories (see [20]
and [18]), and from the continuities we may prove that many singular SPDEs do
not make problems in Theorem 1.1.

Finally we remark that in the cases of the $\Phi^{4}_{2}$-stochastic
quantization and the $\exp(\Phi)_{2}$-stochastic quantization, the stochastic
quantization equations are explicitly written by means of Wick products. This
means that the singular SPDEs are well-defined without taking limits.
Moreover, we are able to construct the associated Dirichlet forms, which
generate the strong Markov processes. See [1, 27] for the case of the
$\Phi^{4}_{2}$-model and [23, 22] for the case of the $\exp(\Phi)_{2}$-model.
In view of these facts, the author in the present paper guesses that singular
SPDEs which have sufficiently low singularity of noises do not make delicate
problems. It also should be remarked that even these cases exceptional sets
appear, because Wick products are defined almost everywhere with respect to
the free field measure, and the Markov processes constructed by Dirichlet
forms are defined only for quasi-every initial points with respect to the
reference measure.

The construction of the present paper is as follows. In Section 2 we recall
the method of singular SPDEs via the $\Phi^{4}$-stochastic quantization model,
and the standard theory of SDEs. Precisely, in Section 2.1 we give a short
review of the methods of singular SPDEs to clarify the techniques to construct
solutions to singular SPDEs. And in Section 2.2 we recall the standard theory
of SDEs to clarify the differences between them. In Section 3 we see that for
the strong Markov property, the transition probability should be defined at
every initial points in the state space. Precisely, we give an example of a
stochastic process that the process is Markov, but not strong Markov, because
of a problematic point in the state space. Moreover, we show that such a
situation occurs even in the limit of the strong Markov processes which are
the solutions to pathwise-unique SDEs. This implies that we need to be careful
in the strong Markov property of the limit processes. In Section 4 we give an
example of a SDE so that the solutions have the uniqueness locally in time for
every initial points except one point, any solution is extendable to one
globally in time, but the SDE does not have the uniqueness globally in time.
Moreover, we also give examples that such a situation can occur in the case of
a limit of SDEs which have the pathwise uniqueness for all initial points, and
also in the case of a path-dependent SDE with the pathwise uniqueness locally
in time for all initial points. In Section 5 we see a delicateness for the
implication from the uniqueness of shifted equations to the uniqueness of the
original equation. In particular, we give the dependence of the choice of
initial conditions of shifted equations on the solutions to the original
equation.

Acknowledgements. The author is grateful to Emeritus Professors Yozo Tamura
and Tokuzo Shiga for giving me many knowledges and techniques while the author
was a student. Indeed, the proofs in the present paper are based on the
knowledges and the techniques given by them many years ago. This work was
partially supported by JSPS KAKENHI Grant Numbers 21H00988.

Notation. Denote $a\wedge b:=\min\\{a,b\\}$ and $a\vee b:=\max\\{a,b\\}$ for
$a,b\in{\mathbb{R}}$. We define the topology of the space
$C([0,\infty))=C([0,\infty);{\mathbb{R}})$ by the uniform convergence on each
compact set. We denote by $P_{x}$ the law of some Markov processes with
respect to initial point $x$, and by $E_{x}$ the expectation with respect to
$P_{x}$. Let $C_{0}({\mathbb{R}})$ be the total set of continuous functions on
${\mathbb{R}}$ which have compact supports.

##  2 Preliminary

###  2.1 The technique in singular SPDEs

Here, we see the motivation of the present paper by a short review of the
technique in singular SPDEs. As an explicit example, consider the stochastic
quantization equation of the $\Phi^{4}$-quantum field model

(2.1)
$\partial_{t}\Phi(t,x)=(\triangle-1)\Phi(t,x)-\Phi(t,x)^{3}+\dot{W}(t,x),\quad(t,x)\in[0,\infty)\times{\mathbb{T}}^{d}$

where $\dot{W}(t,x)$ is the time-space white noise (often regarded as weak
derivative in time $t$ of the cylindrical Brownian motion $W(t,x)$ on
$L^{2}({\mathbb{T}}^{d})$). In the case that the dimension $d=1$, the
singularity of $\dot{W}$ is not high and (2.1) is solved by usual methods in
SPDEs. However, when $d\geq 2$, the singularity of $\dot{W}$ is too high to
solve (2.1) without renormalization. The problem is on the nonlinear term
$\Phi(t,x)^{3}$, which is not well-defined as it is, if $\Phi$ is not a
function, but a distribution. Indeed, (2.1) is ill-defined in the sense that
the expected regularity of $\Phi$ is too low, precisely $\Phi$ is expected to
be a distribution, and $\Phi^{3}$ is not well-defined as it is.

In the case that $d=2$, (2.1) is modified by renormalization as

(2.2)
$\partial_{t}\Phi(t,x)=(\triangle-1)\Phi(t,x)-:\Phi(t,x)^{3}:+\dot{W}(t,x),\quad(t,x)\in[0,\infty)\times{\mathbb{T}}^{d}$

where $:\phi^{n}:$ is the wick product with respect to the free field measure
for $n\in{\mathbb{N}}$. Here, we note that the map $\phi\mapsto:\phi^{n}:$ is
well-defined for almost every $\phi$ with respect to the free field measure. A
direct approach by methods of SPDEs to (2.2) was first introduced in [10]. Now
we see the strategy of the approach, because it is closely related to the
motivation of the present paper. Consider the stationary solution to the
linearized equation of (2.2):

(2.3)
$\partial_{t}Z(t,x)=(\triangle-1)Z(t,x)+\dot{W}(t,x),\quad(t,x)\in[0,\infty)\times{\mathbb{T}}^{d},$

which is given by

$Z(t):=\int_{-\infty}^{t}e^{(t-s)(\triangle-1)}dW(s),\quad t\in[0,\infty),$

where $(W(t);t\in(-\infty,\infty))$ is an extension in the time parameter $t$
of the cylindrical Brownian motion $(W(t);t\in[0,\infty))$ which satisfies
$\partial_{t}W(t)=\dot{W}(t,\cdot)$. If $\Phi$ and $Z$ are solutions to (2.2)
and (2.3) in a suitable sense, respectively, then $Y:=\Phi-Z$ satisfies

(2.4)
$\begin{array}[]{rl}\partial_{t}Y(t,x)&=(\triangle-1)Y(t,x)-Y(t,x)^{3}-3Z(t,x)Y(t,x)^{2}\\\
&\hskip
56.9055pt-3:Z(t,x)^{2}:Y(t,x)-:Z(t,x)^{3}:,\quad(t,x)\in[0,\infty)\times{\mathbb{T}}^{2}.\end{array}$

We often call (2.4) a shifted equation. Since the coefficients $Z(t,x)$,
$:Z(t,x)^{2}:$ and $:Z(t,x)^{3}:$ of this equations are explicit and have
better regularities than the original noise $\dot{W}(t,x)$, (2.4) is solved in
the sense of mild solutions. Hence, by letting $\Phi:=Y+Z$ we have a solution
to (2.2) (recall that $Z$ is an explicit stochastic process). This is the
strategy of the argument in [10]. Here, it should be remarked that $\Phi:=Y+Z$
is the definition of the solution to (2.2) (see [10, Definition 4.1]), and the
renormalized equation (2.2) is not solved directly. Moreover, there is
ambiguity on the initial condition $\Phi(0,x)=Y(0,x)+Z(0,x)$, because $Z$ is a
stationary solution to (2.3) and has exceptional sets with respect to the
probability measure. Furthermore, for given initial condition $\Phi(0,x)$
there exist infinitely many choices to decompose it into $Y(0,x)$ and $Z(0,x)$
by $\Phi(0,x)=Y(0,x)+Z(0,x)$, and the different decomposition can give
different solutions $\Phi$. These facts make difficulties to discuss the
uniqueness of solutions. The uniqueness is closely related to the strong
Markov property of solutions (see Section 2.2 for a brief review of the
standard theory of SDEs, and Section 3 for examples of non-strong Markov
solutions). The motivation of the present paper is to study the differences
between the solutions obtained by this approach and by the standard theory of
SDEs. Precisely, we give some explicit examples which make different
situations from the standard theory of SDEs.

It also should be remarked that the Dirichlet form associated to (2.1) had
been constructed in [1] before [10] appeared. The advantage of the method of
the Dirichlet form is that we obtain the unique Hunt process associated to the
bilinear form, which has the strong Markov property. The solution to (2.2) is
also constructed by [1]. However, there also exists exceptional sets, which
are polar sets of the Hunt process, and a similar situation occurs to the
approach by [10]. See [15] for the theory of the Dirichlet forms.

Now we turn to the case that $d=3$. In this case, the singularity of the noise
$\dot{W}(t,x)$ is higher than the case that $d=2$, and solving (2.1) via
renormalization had been an open problem for many years. However, recently it
became solvable in the sense of singular SPDEs by the regularity structure
introduced in [20] and the paracontrolled calculus in [18]. Here, we remark
that the regularity structure and the paracontrolled calculus are different
theories, but share the key idea that similarly to [10] we decompose the
formal solution $\Phi$ by grades of regularities (singularities), and
transform the original SPDE (2.1) into solvable one. Indeed, the method in
[10] can be regarded as the most simple case of singular SPDEs. In the
regularity structure the pair of the decomposed terms of $\Phi$ is regarded as
an element the model space, the solution $\Phi$ to the renormalized version of
(2.1) is constructed by the projection of the solution in the model space to
the regularity structure associated to (2.1). See [21] for a survey of the
approach by the regularity structure to the $\Phi^{4}_{3}$-stochastic
quantization equation, and [8] for the regularity structure approach to other
singular SPDEs.

In the paracontrolled calculus, the formal solution $\Phi$ is decomposed by
the paracontrolled ansatz, which is also a decomposition by the regularities
(singularities). For (2.1) the paracontrolled ansatz is given by

(2.5)
$\Phi=Z+I(Z^{3})-3\int_{-\infty}^{t}e^{(t-s)(\triangle-1)}\left[(\Psi_{1}+\Psi_{2}-I(:Z^{3}:))\mbox{\scriptsize$<$⃝}:Z^{2}:\right](s)ds+\Psi_{2}$

where $I(:Z^{3}:)(t)=\int_{-\infty}^{t}e^{(t-s)(\triangle-1)}:Z^{3}(s):ds$ and
\scriptsize$<$⃝ is a notation of the paraproduct. By solving nonlinear partial
differential equation of $(\Phi_{1},\Phi_{2})$:

(2.6)
$\left\\{\begin{array}[]{rl}\partial_{t}\Psi_{1}(t)&=(\triangle-1)\Psi_{1}(t)-3\left[\Psi_{1}+\Psi_{2}-I(:Z^{3}:))\mbox{\scriptsize$<$⃝}:Z^{2}:\right](t)\\\
\partial_{t}\Psi_{2}(t)&=(\triangle-1)\Psi_{2}(t)+F(\Psi_{1},\Psi_{2},Z,:Z^{2}:,I(:Z^{3}:))(t),\end{array}\right.$

which is obtained by a transformation of (2.1) together with renormalization,
we can define a solution $\Phi$ by (2.5) to the renormalized version of (2.1).
The term $F(\Psi_{1},\Psi_{2},Z,:Z^{2}:,I(:Z^{3}:))$ is too complicated to
write down explicitly, because it includes paraproducts, resonance, and their
commutators. But here, we remark that
$F(\Psi_{1},\Psi_{2},Z,:Z^{2}:,I(:Z^{3}:))(t)$ depends on the trajectory of
$Z$ before the present time $t$, i.e. (2.6) is a partial differential equation
of the path-dependent type. Indeed, $F$ depends on $I(:Z^{3}:)$. Hence, even
if (2.6) has the uniqueness of the solution $(\Phi_{1},\Phi_{2})$ for all
initial conditions in a suitable state space, it is not trivial for
$(\Phi_{1},\Phi_{2})$ to have the Markov and strong Markov properties. For the
details of this argument, see [9], [25] or [2]. The definition of the solution
$\Phi$ given by (2.5) is same as in the case that $d=2$ (see above or [10,
Definition 4.1] for the precise definition). The renormalized equation of
(2.1) for $d=3$ is written formally by

(2.7)
$\partial_{t}\Phi(t,x)=(\triangle-1)\Phi(t,x)-(\Phi(t,x)^{3}-\infty\cdot\Phi(t,x))+\dot{W}(t,x),\quad(t,x)\in[0,\infty)\times{\mathbb{T}}^{3}.$

We remark that the term $(\Phi(t,x)^{3}-\infty\cdot\Phi(t,x))$ cannot be
written by Wick products, because we need further renormailzation than Wick
products. This means that we do not have explicit representation of the term,
and it is jutisfied by limits, i.e. approximations by regularization. It is
also remarked that the renormalization constants are written explicitly, once
we fix an approximation. For the renormalization constants to be independent
of $t$, we need a restriction on the initial condition $Z_{0}$ of $Z$, and
hence there is anbiguity of the initial condition of $\Phi$ similar to the
case that $d=2$. We enphasize here that the target (2.7) is a formal equation,
and the solution to (2.7) is defined by the solution to a path-dependent
equation (2.6) via approximations.

Now shortly, we mention a few references of the approach by the paracontrolled
calculus to the $\Phi^{4}_{3}$-stochastic quantization equation. The approach
of the paracontrolled calculus to the $\Phi^{4}_{3}$-stochastic quantization
equation was first introduced in [9]. The solution obtained in [9] is locally
in time, and unique in the sense of the paracontrolled calculus. In [25] the
extendability globally in time of the solutions to (2.5) was obtained for
sufficiently regular initial conditions of $(\Psi_{1},\Psi_{2})$. Here, we
remark that the global exisntence and the local uniqueness of the solution
imply neither the global uniquness nor the strong Markov property of
solutions, as we will see in the following sections. On the other hand, in [2]
the solution (in the sense of the paracontrolled calculus) globally in time
for almost every initial point with respect to the $\Phi^{4}_{3}$-measure was
obtained by stationary approximation. This approach by stationary
approximation enables us to constuct a time-global solution for almost every
initial points with respect to the $\Phi^{4}_{3}$-measure which is constructed
by the approach itself.

We remark that the regularity structure and the paracontrolled calculus are
explained above independently, but the precise relation between the regularity
structure and the paracontrolled calculus has been studied in [4, 5].

What we would like to emphasize here is that the methods in singular SPDEs are
very different from the standard theory of SDEs. To clarify the difference we
recall the standard theory of SDEs in Section 2.2.

###  2.2 Markov processes and SDEs

In this section we recall the standard theories of Markov processes and SDEs.

First, we recall the key theorem to construct the strong Makov processes form
Markov semigroups. For a locally compact Hausdorff space $S$ with the second
axiom of countability, let $C_{\infty}(S)$ be the total sets of continuous
functions $f$ such that $f$ is extended to a continuous function $\tilde{f}$
on $S\cup\\{\Delta\\}$ which is the one-point compactification of $S$, and
that $\tilde{f}(\Delta)=0$.

######  Theorem 2.1 (cf. [7]).

Let $\\{T_{t}\\}$ be a nonnegative contractive and semigroup on
$C_{\infty}(S)$. Assume that $\\{T_{t}\\}$ satisfies the Feller property, i.e.
$\\{T_{t}\\}$ is a strong continuous semigroup on $C_{\infty}(S)$. Then, there
exists a Hunt process which has $\\{T_{t}\\}$ as its transition semigroup.

See [15, Section A.2] for the definition of Hunt processes. We remark that
Hunt processes have the strong Markov property (see [15, Theorem A.2.1]). The
Feller property, in particular $\lim_{x\rightarrow\Delta}(T_{t}f)(x)=0$ for
$f\in C_{\infty}(S)$, implies that, the approaching of the associated
stochastic process to $\Delta$ is regarded as explosion. In Section 3 we will
see the importance of the Feller property, in particular of the assumption
that the semigroup is defined on $C_{\infty}(S)$.

For the strong Markov property of solutions to (time-homogenious) SDEs, the
following is known.

If a SDE has uniquness in law for all initial values
in the state space, then the solutions satisfy the strong Markov property.

For the precise statement and the proof, see e.g. [24, Theorem 5.1, Chapter
IV] and [29, Theorem 6.2.2]. For notations, let $X$ be the solution to a SDE,
$P_{x}$ the law of $X$ with respect to initial point $x$, $E_{x}$ the
expectation with respect to $P_{x}$, $\\{{\mathcal{F}}_{t}\\}_{t}$ the
filtration for solutions, and let $B$ be the Brownian motion of the driving
noise in the SDE. The key points in the proof are the following facts.

  1. (s1)

For any finite stiopping time $\tau$, $t\mapsto B_{t+\tau}-B_{\tau}$ is again
a Brownian motion with the filtration $\\{{\mathcal{F}}_{t+\tau}\\}_{t}$.

  2. (s2)

The uniqueness for all initial value $x\in S$ implies the connection of
solutions i.e.

$P_{x}(X_{\cdot\wedge\tau}\in A_{1},X_{\cdot+\tau}\in
A_{2})=E_{x}[{\mathbb{I}}_{A_{1}}(X_{\cdot\wedge\tau})P_{X_{\tau}}(A_{2})]$

for any $A_{1},A_{2}\in{\mathcal{B}}(C([0,\infty);{\mathbb{R}}^{n}))$ and a
finite stopping time $\tau$.

The elementary fact (s1) is necessary. Unless (s1), the behaviour of $X$ after
$\tau$ can be different from the law of $X$ starting at $X_{\tau}$, because of
the difference of the law of the driving noises. On the other hand, (s1) is
hardly paid attention, because by the martingale representation theorem (see
e.g. [24, Section 7, Chapter II]) we can transform SDEs driven by semi-
Martingales into those driven by Brownian motions and drift terms. Thanks to
(s1) and the uniqueness of the solution, we have (s2). If the uniqueness does
not hold for an initial point $x_{0}\in S$ and $\\{x_{0}\\}$ is not a polar
set, then by letting $\tau$ be the hitting time at $x_{0}$ we can choose two
different $P_{X_{\tau}}(A_{2})$ for a
$A_{2}\in{\mathcal{B}}(C([0,\infty);{\mathbb{R}}^{n}))$. Here, we remark that
in the argument of singular SPDEs the driving noise is the Ornstein-Uhlenbeck
process rather than a white noise, and does not satisfy the property like
(s1). We also remark that for the strong Markov property the uniquness for all
initial points in the state space is important. Precisely, the acceptable
exceptional sets are only polar sets.

In the theory of SDEs, if there exist some problematic points for initial
conditions, we often remove the points from the state space and regard them as
boundaries. Now recall that the background of the present paper is on singular
SPDEs, and that as in Section 2.1 we do not regard such points as boundaries.
Another way to treat SDEs on a state space $S$ which is not an Euclidean
space, is regarding them as SDEs on a manifold $S$. The solution to SDEs on
manifolds are defined by connections of paths via consistency of the local
solutions which are solved on local charts. The consistency comes from the
strong Markov property of the solutions, which is guaranteed from the
uniqueness of the property on local charts. As mentioned above, to obtain the
strong Markov property we need the uniqueness of the solution first. See [24,
Chapter V] and [11] for the details of SDEs on manifolds.

On the other hand, once some points are removed from the Euclidean spaces, the
space after removed is not complete. So, we have to regard the exit time
$\zeta$ from $S$ as the explosion time (life time). This is very reasonable in
view of the facts above. Indeed, regarding the exit time $\zeta$ from $S$ as
the explosion time corresponds to fixing a boundary condition on $\partial S$,
which is the killing boundary condition. Moreover, once we put the killing
boundary condition on $\partial S$, the processes starting at the problematic
initial points in $\partial S$ are uniquely determined as the processes
staying at the points. From these arguments, it is reasonable to regard
hitting at problematic points as explosion of the processes. It is also
remarked again that the Feller property means explosion at $\partial S$ for
the processes as mentioned above.

In Section 3 we give some explicit examples of the stochastic processes which
do not have the strong Markov property, but have the Markov property. In
Section 4 we see explicit examples of SDEs such that the existence and
uniqueness hold locally in time except one point, but the global uniqueness
does not hold. These examples are constructed by providing a problematic point
for initial conditions. This fact implies the importance for stochastic
processes to have the uniqueness in all initial points in the state space.
Moreover, in the same sections, we see that these phenomena can happen for the
limits of SDE which have the existence and uniqueness of the solutions. This
implies that the delicateness is hidden by the limiting procedure.

As mentioned in Sections 1 and 2.1, the motivation of the present paper is not
on SDEs, but on SPDEs. However, since SPDEs can be regarded as infinite-
dimensional versions of SDEs, the examples are sufficient to see the
delicateness of the arguments in singular SPDEs.

##  3 Markov processes without strong Markov properties

In this section, we see examples of the Markov processes without strong Markov
properties.

######  Proposition 3.1.

Let $S:=[-\pi,\pi]$ and $Y=(Y_{t})$ be the Brownian motion on $S$ with
reflecting boundary condition at $-\pi$ and $\pi$. Let $S^{\circ}:=(-\pi,\pi)$
and define a family of operators $\\{T_{t}^{\circ};t\in[0,\infty)\\}$ on
$C_{b}(S^{\circ})$ by

$T_{t}^{\circ}f(x):=E_{x}[f(Y_{t});Y_{t}\in S^{\circ}],\quad x\in S^{\circ},\
f\in C_{b}(S^{\circ}).$

Then, $\\{T_{t}^{\circ}\\}$ is a nonnegative, contractive and conservative
semigroup on $C_{b}(S^{\circ})$. Moreover, for $f\in C_{\infty}(S^{\circ})$ it
holds that

$\lim_{t\downarrow 0}\|T_{t}^{\circ}f-f\|_{\infty}=0.$

###### Proof.

For each $t\in[0,\infty)$ and $x\in S^{\circ}$, $P_{x}(Y_{t}\in S\setminus
S^{\circ})=0$. From this and the fact that $Y_{t}$ is a continuous Markov
process on $S$, we have the assertion. ∎

######  Remark 3.2.

It is easy to see that $T_{t}^{\circ}$ in Proposition 3.1 does not generate a
continuous Markov process on $S^{\circ}$. Indeed, the candidate of the
continuous Markov process is $Y$ in Proposition 3.1, and the state space of
$Y$ cannot be $S^{\circ}$, but $S$. The reason why Theorem 2.1 is not
applicable to $T_{t}^{\circ}$ is that

$\lim_{x\rightarrow\pm\pi}T_{t}^{\circ}f(x)=E_{\pm\pi}[f(Y_{t})]\quad(\mbox{double-
sign in same order}),$

and the right-hand side is not $0$ for some $f\in C_{\infty}(S^{\circ})$.

######  Proposition 3.3.

Define $Y_{t}$ as in Proposition 3.1 and let $Z_{t}:=(\cos Y_{t},\sin Y_{t})$
for $t\in[0,\infty)$. Then, $Z$ is a continuous stochastic process on the unit
circle $S^{1}$ in ${\mathbb{R}}^{2}$. Let

$p_{t}^{Z}(x,dz):=P_{y}((\cos Y_{t},\sin Y_{t})\in dz),\quad x\in
S^{1}\setminus(-1,0)$

where $y\in(-\pi,\pi)$ such that $x=(\cos y,\sin y)$. Then, it holds that

(3.1) $\displaystyle E_{x}[g(Z_{s})f(Z_{t+s})]$
$\displaystyle=\int_{S^{1}\setminus(-1,0)}g(w)\left(\int_{S^{1}\setminus(-1,0)}f(z)p_{t}^{Z}(w,dz)\right)p_{s}^{Z}(x,dw),$
$\displaystyle\hskip 56.9055ptf,g\in C(S^{1}),\ s,t\in[0,\infty),\ x\in
S^{1}\setminus(-1,0).$

In particular, there exists a modification $\widetilde{Z}$ of $Z$ (i.e.
$P(\widetilde{Z}_{t}=Z_{t})=1$ for all $t\in[0,\infty)$) so that
$\widetilde{Z}_{t}\in S^{1}\setminus(-1,0)$ for all $t\in[0,\infty)$, and all
such modifications $\widetilde{Z}$ is a Markov process on
$S^{1}\setminus(-1,0)$. However, neither $Z$ nor $\widetilde{Z}$ satisfy the
strong Markov property.

###### Proof.

Since $P_{y}(Y_{s}=\pm\pi)=0$ for all $y\in(-\pi,\pi)$ and $t\in[0,\infty)$,
it holds that

$E_{y}[f((\cos Y_{t},\sin
Y_{t}))]=\int_{S^{1}\setminus(-1,0)}f(z)p_{t}^{Z}(x,dz),\quad f\in
L^{\infty}(S^{1}),$

where $y\in(-\pi,\pi)$ and $x=(\cos y,\sin y)\in S^{1}\setminus(-1,0)$. Hence,
by the Markov property of $Y$, we have for $f,g\in C(S^{1})$,
$s,t\in[0,\infty)$ and $x\in S^{1}\setminus(-1,0)$

$\displaystyle E_{x}[g(Z_{s})f(Z_{t+s})]$ $\displaystyle=E_{y}[g((\cos
Y_{s},\sin Y_{s}))f(\cos Y_{t+s},\sin Y_{t+s})]$
$\displaystyle=E_{y}\left[g((\cos Y_{s},\sin Y_{s}))E_{(\cos Y_{s},\sin
Y_{s})}[f(\cos Y_{t},\sin Y_{t})]\right]$
$\displaystyle=\int_{S^{1}\setminus(-1,0)}g(w)\left(\int_{S^{1}\setminus(-1,0)}f(z)p_{t}^{Z}(w,dz)\right)p_{s}^{Z}(x,dw).$

Thus, (3.1) is obtained.

By considering the hitting time $\tau$ of $Z$ at $(-1,0)$, it is easy to prove
that $Z$ does not satisfy the strong Markov property. For $\widetilde{Z}$,
consider the sequence of the hitting times $\tau_{n}$ of $\widetilde{Z}$ at
$\\{z\in S^{1}\setminus\\{(-1,0)\\};{\rm dist}(z,(-1,0))<1/n\\}$ and the limit
$\tau:=\lim_{n\rightarrow\infty}\tau_{n}$. Then, similarly to the case of $Z$,
we can prove that $\widetilde{Z}$ does not satisfy the strong Markov property.
∎

######  Remark 3.4.

  * •

Under the setting in Proposition 3.3 $Y$ is a strong Markov process on
$[-\pi,\pi]$, while $Z$ is a stochastic process on $S^{1}$ but not strong
Markov. Intuitively these facts seem natural, because $[-\pi,\pi]\ni y\mapsto
x\in S^{1}$ is not injective.

  * •

For $Z$ to be a strong Markov process we have to give a suitable boundary
condition at $(-1,0)$, which is the reflecting boundary condition. The
reflecting boundary condition decompose the point $(-1,0)$ into the two
points: the limits along $S^{1}$ from the upper-half and lower-half planes.

  * •

The construction of a strong Markov process from $\widetilde{Z}$ is more
serious, because $\widetilde{Z}$ cannot be an almost-sure right-continuous
process on $S^{1}\setminus(-1,0)$. To obtain a right-continuous modification
of $\widetilde{Z}$ we have to extend the state space $S^{1}\setminus(-1,0)$.
When we choose $S^{1}$ as the extension of $S^{1}\setminus(-1,0)$, the right-
continuous modification of $\widetilde{Z}$ is $Z$ and we need the procedure
above to construct a strong Markov process. The best choice of the extended
space of $S^{1}\setminus(-1,0)$ is a space $\bar{S}$ so that the limits along
$S^{1}$ from the upper-half and lower-half planes are decoupled. Then, by
choosing a right-continuous modification of $\widetilde{Z}$ on $\bar{S}$ we
obtain a strong Markov process.

From Propositions 3.1 and 3.3 and Remark 3.4 we see that for the construction
of the strong Markov processes, the state space should be chosen suitably so
that the process would be right-continuous and the transition probability
should be determined for all initial point in the state space. On the other
hand, in view of Theorem 2.1 if we provide the one-point compactification of
the state space and regard reaching of the stochastic process to the extra
point as explosion of the process, we obtain a strong Markov process. In the
case of Proposition 3.3, this procedure corresponds to giving the absorbing
boundary condition (Dirichlet boundary condition) at $(-1,0)$ for $Z$ or
$\widetilde{Z}$.

In the setting of Proposition 3.3, $Z$ is a stochastic process on $S^{1}$ such
that $Z$ behaves like the Brownian motion on $S^{1}\setminus(-1,0)$ and
reflected at $(-1,0)$. It is also possible to construct a stochastic process
$Z^{+}$ so that $Z^{+}$ behaves like $Z$ on $S^{1}\setminus(-1,0)$, but
$Z^{+}$ passes trough from the lower-half plane to the upper-half plane and is
reflected when $Z$ comes from the upper-half plane. The precise procedure of
the construction is as follows. Let $Y^{+,x}$ be the Brownian motion on
$[-\pi,\pi]$ starting at $x$ with reflecting boundary condition at $\pi$ and
trapped at $-\pi$, and let $\\{\widetilde{Y}^{+,i};i\in{\mathbb{N}}\\}$ be
independent copies of $Y^{+,0}$. Denote $\tau_{-\pi}^{0}$ and
$\tau_{-\pi}^{i}$ be the first hitting times of $Y^{+,x}$ and
$\widetilde{Y}^{+,i}$ at $-\pi$, respectively. Define the stochastic process
$Z^{+}$ by

$Z_{t}^{+,x}:=\left\\{\begin{array}[]{l}(\cos Y_{t}^{+,x},\sin
Y_{t}^{+,x}),\quad 0\leq t<\tau_{-\pi}^{0},\\\
\left(\cos\widetilde{Y}_{t-\tau_{-\pi}^{0}+\tau_{-\pi}^{1}+\cdots+\tau_{-\pi}^{k-1}}^{+,k},\sin\widetilde{Y}_{t-\tau_{-\pi}^{0}+\tau_{-\pi}^{1}+\cdots+\tau_{-\pi}^{k-1}}^{+,k}\right),\\\
\qquad\tau_{-\pi}^{0}+\tau_{-\pi}^{1}+\cdots+\tau_{-\pi}^{k-1}\leq
t<\tau_{-\pi}^{0}+\tau_{-\pi}^{1}+\cdots+\tau_{-\pi}^{k},\
k=0,1,2,\dots\end{array}\right.$

We remark that $(Z_{t}^{+,x};t\in[0,\infty))$ is well-defined as a continuous
stochastic process, because
$\lim_{k\rightarrow\infty}(\tau_{-\pi}^{0}+\tau_{-\pi}^{1}+\cdots+\tau_{-\pi}^{k})=\infty$
almost surely and

$\displaystyle\lim_{t\rightarrow\tau_{-\pi}^{0}+\tau_{-\pi}^{1}+\cdots\tau_{-\pi}^{k}}\left(\cos\widetilde{Y}_{t-\tau_{-\pi}^{0}+\tau_{-\pi}^{1}+\cdots+\tau_{-\pi}^{k-1}}^{+,k},\sin\widetilde{Y}_{t-\tau_{-\pi}^{0}+\tau_{-\pi}^{1}+\cdots+\tau_{-\pi}^{k-1}}^{+,k}\right)$
$\displaystyle\quad=(-1,0)=(\cos\widetilde{Y}_{0}^{+,k+1},\sin\widetilde{Y}_{0}^{+,k+1}),\quad
k=0,1,2,\dots.$

Then, it is easy to see the stochastic process $(Z_{t}^{+,x};t\in[0,\infty))$
satisfies the properties mentioned above. We remark that
$(Z_{t}^{+,x};t\in[0,\infty))$ satisfies the strong Markov property, because
the behavior of $Z^{+,x}$ after hitting $(-1,0)$ does not depend on where
$Z^{+,x}$ comes from.

Next we give an example of a stochastic process without the strong Markov
property which is constructed by limits of strong Markov processes. The
construction is given by the penalization method, and such a method has been
given in [28].

######  Theorem 3.5.

For $n\in{\mathbb{N}}$, let $\varphi\in C^{1}({\mathbb{R}})$ be a nonnegative
function satisfying

$\displaystyle\varphi(x)=0,\quad x\in(-\infty,-1]\cup[1,\infty),$
$\displaystyle\varphi^{\prime}(x)\geq 0,\quad x\in[-1,0],$
$\displaystyle\varphi^{\prime}(x)\leq 0,\quad x\in[0,1]$
$\displaystyle\int_{[-1,1]}\varphi(x)dx=1.$

Define $\varphi_{n}(x)$ by $\varphi_{n}(x):=n\varphi(nx)$ for
$x\in{\mathbb{R}}$. Consider the solution $X^{n}$ to the SDE

(3.2)
$dX^{n}_{t}=\sigma(X^{n}_{t})dB_{t}+b(X^{n}_{t})dt-\varphi_{n}^{\prime}(X^{n}_{t})dt$

where $\sigma$ and $b$ are Lipschitz continuous functions such that
$\sigma(x)>0$ for all $x\in{\mathbb{R}}$.

Then, when $\\{X^{n}\\}$ is the family of the solutions to (3.2) with tight
initial laws, then the laws of $\\{X^{n}\\}$ in $C([0,\infty);{\mathbb{R}})$
is tight. Moreover, if $X$ is a limit of a subsequence of the laws of $X^{n}$,
$X$ hits $0$ with positive probability, but does not cross the origin almost
surely, and $\\{t\in[0,\infty);X_{t}=0\\}$ is a null set with respect to the
Lebesgue measure almost surely. In particular, $X$ is not a strong Markov
process.

###### Proof.

First we show the tightness of the laws of $X^{n}$. Let $T>0$, $\varepsilon>0$
and choose a nondecreasing smooth function $f_{\varepsilon}$ such that

$\displaystyle f_{\varepsilon}(x)$ $\displaystyle=x,\quad
x\in\left(-\infty,-\frac{\varepsilon}{4}\right]\cup\left[\frac{\varepsilon}{4},\infty\right),$
$\displaystyle f_{\varepsilon}(x)$ $\displaystyle=0,\quad
x\in\left[-\frac{\varepsilon}{8},\frac{\varepsilon}{8}\right].$

The fact that $|x-f_{\varepsilon}(x)|\leq\varepsilon/4$ for $x\in{\mathbb{R}}$
and the Itô formula imply that for $h>0$

$\displaystyle
P\left(\sup_{s,t\in[0,T];|t-s|<h}|X^{n}_{t}-X^{n}_{s}|>\varepsilon\right)$
$\displaystyle\leq
P\left(\sup_{s,t\in[0,T];|t-s|<h}|f_{\varepsilon}(X^{n}_{t})-f_{\varepsilon}(X^{n}_{s})|>\frac{\varepsilon}{2}\right)$
$\displaystyle\quad+P\left(\sup_{s,t\in[0,T];|t-s|<h}\left|[X^{n}_{t}-f_{\varepsilon}(X^{n}_{t})]-[X^{n}_{s}-f_{\varepsilon}(X^{n}_{s})]\right|>\frac{\varepsilon}{2}\right)$
$\displaystyle\leq
P\left(\sup_{s,t\in[0,T];|t-s|<h}\left|\int_{s}^{t}f_{\varepsilon}^{\prime}(X^{n}_{u})\sigma(X^{n}_{u})dB_{u}+\int_{s}^{t}f_{\varepsilon}^{\prime}(X^{n}_{u})b(X^{n}_{u})du\right.\right.$
$\displaystyle\quad\hskip
85.35826pt\left.\left.-\int_{s}^{t}f_{\varepsilon}^{\prime}(X^{n}_{u})\varphi_{n}^{\prime}(X^{n}_{u})du+\frac{1}{2}\int_{s}^{t}f_{\varepsilon}^{\prime\prime}(X^{n}_{u})\sigma(X^{n}_{u})^{2}du\right|>\frac{\varepsilon}{2}\right).$

Note that $f_{\varepsilon}^{\prime}(x)\varphi_{n}^{\prime}(x)=0$ for
$x\in{\mathbb{R}}$ and $n>8/\varepsilon$. This inequality and the Chebyshev
inequality imply that for $n>8/\varepsilon$

$\displaystyle
P\left(\sup_{s,t\in[0,T];|t-s|<h}|X^{n}_{t}-X^{n}_{s}|>\varepsilon\right)$
$\displaystyle\leq
P\left(\sup_{s,t\in[0,T];|t-s|<h}\left|\int_{s}^{t}f_{\varepsilon}^{\prime}(X^{n}_{u})\sigma(X^{n}_{u})dB_{u}\right|>\frac{\varepsilon}{4}\right)$
$\displaystyle\quad+P\left(\sup_{s,t\in[0,T];|t-s|<h}\left|\int_{s}^{t}f_{\varepsilon}^{\prime}(X^{n}_{u})b(X^{n}_{u})du+\frac{1}{2}\int_{s}^{t}f_{\varepsilon}^{\prime\prime}(X^{n}_{u})\sigma(X^{n}_{u})^{2}du\right|>\frac{\varepsilon}{4}\right)$
$\displaystyle\leq
P\left(\sup_{s,t\in[0,T];|t-s|<h}\left|\int_{s}^{t}f_{\varepsilon}^{\prime}(X^{n}_{u})\sigma(X^{n}_{u})dB_{u}\right|>\frac{\varepsilon}{4}\right)$
$\displaystyle\quad+\frac{16}{\varepsilon^{2}}E\left[\sup_{s,t\in[0,T];|t-s|<h}\left|\int_{s}^{t}\left(f_{\varepsilon}^{\prime}(X^{n}_{u})b(X^{n}_{u})+\frac{1}{2}f_{\varepsilon}^{\prime\prime}(X^{n}_{u})\sigma(X^{n}_{u})^{2}\right)du\right|^{2}\right]$
$\displaystyle\leq
P\left(\sup_{s,t\in[0,T];|t-s|<h}\left|\int_{s}^{t}f_{\varepsilon}^{\prime}(X^{n}_{u})\sigma(X^{n}_{u})dB_{u}\right|>\frac{\varepsilon}{4}\right)+\frac{16}{\varepsilon^{2}}(\|f_{\varepsilon}^{\prime\prime}\|_{\infty}^{2}\|\sigma\|_{\infty}^{4}h+\|f_{\varepsilon}^{\prime}\|_{\infty}^{2}\|b\|_{\infty}^{2}h^{2}).$

On the other hand, the Burkholder-Davis-Gundy inequality implies

$\displaystyle
E\left[\left|\int_{s}^{t}f_{\varepsilon}^{\prime}(X^{n}_{u})\sigma(X^{n}_{u})dB_{u}\right|^{4}\right]$
$\displaystyle\leq
CE\left[\left(\int_{s}^{t}|f_{\varepsilon}^{\prime}(X^{n}_{u})|^{2}\sigma(X^{n}_{u})^{2}du\right)^{2}\right]$
$\displaystyle\leq
C\|f_{\varepsilon}^{\prime}\|_{\infty}^{4}\|\sigma\|_{\infty}^{4}(t-s)^{2}.$

As in the proof of [24, Theorem 4.3 of Chapter I], this estimate yields

$\lim_{h\downarrow
0}\sup_{n}P\left(\sup_{s,t\in[0,T];|t-s|<h}\left|\int_{s}^{t}f_{\varepsilon}^{\prime}(X^{n}_{u})\sigma(X^{n}_{u})dB_{u}\right|>\frac{\varepsilon}{4}\right)=0.$

Thus, we obtain

$\lim_{h\downarrow
0}\sup_{n}P\left(\sup_{s,t\in[0,T];|t-s|<h}|X^{n}_{t}-X^{n}_{s}|>\varepsilon\right)=0.$

This convergence and the tightness of the initial laws of $X^{n}$ yield the
tightness of the laws of $X^{n}$ in $C([0,T];{\mathbb{R}})$ (see [24, Theorem
4.2 of Chapter I]).

Next we show that $X$ hits $0$ with positive probability, but does not cross
the origin almost surely. Let

$L:=\frac{1}{2}\sigma(x)^{2}\frac{d^{2}}{dx^{2}}+b(x)\frac{d}{dx}$

and let $l,r\in{\mathbb{R}}$ such that $l<r$. Define a function $s_{n}$ on
${\mathbb{R}}$ by

$s_{n}(x):=\int_{l}^{x}\exp\left(-\int_{l}^{y}\frac{2(b(z)-\varphi_{n}^{\prime}(z))}{\sigma(z)^{2}}dz\right)dy.$

Then, we have

(3.3) $\left(L-\varphi_{n}(x)\frac{d}{dx}\right)s_{n}(x)=0.$

Let $X^{n,x}_{t}$ be the solution starting at $x\in(-1,1)$, and let $\tau_{c}$
be the first hitting time of $X^{n,x}_{t}$ at $c$. The general theory of one-
dimensional diffusion processes and (3.3) imply that for $l<x<r$

$P(X^{n,x}_{\tau_{l}\wedge\tau_{r}}=r)=\frac{s_{n}(x)-s_{n}(l)}{s_{n}(r)-s_{n}(l)}=\frac{s_{n}(x)}{s_{n}(r)}.$

Note that $\varphi(0)>0$. In the case that $l<0<x<r$, an explicit calculation
implies that

$\displaystyle s_{n}(x)$
$\displaystyle\geq\int_{l}^{0}\exp\left(-\int_{l}^{y}\frac{2(b(z)-\varphi_{n}^{\prime}(z))}{\sigma(z)^{2}}dz\right)dy$
$\displaystyle\geq\int_{l}^{0}\exp\left(\left(\inf_{z\in[-(1/n),0]}\frac{1}{\sigma(z)^{2}}\right)\int_{l}^{y}\varphi_{n}^{\prime}(z)dz-\int_{l}^{y}\frac{2b(z)}{\sigma(z)^{2}}dz\right)dy$
$\displaystyle=\int_{l}^{0}\exp\left(\left(\inf_{z\in[-(1/n),0]}\frac{1}{\sigma(z)^{2}}\right)\left(\varphi_{n}(y)-\varphi_{n}(l)\right)-\int_{l}^{y}\frac{2b(z)}{\sigma(z)^{2}}dz\right)dy$
$\displaystyle=\int_{l/n}^{0}\exp\left(\left(\inf_{z\in[-(1/n),0]}\frac{1}{\sigma(z)^{2}}\right)\cdot
n\left(\varphi(\tilde{y})-\varphi(nl)\right)-\int_{l}^{\tilde{y}/n}\frac{2b(z)}{\sigma(z)^{2}}dz\right)d\tilde{y}$
$\displaystyle\rightarrow\infty\qquad\mbox{as}\ n\rightarrow\infty.$

Similarly, in the case that $l<x<0<r$,

$\displaystyle s_{n}(x)$
$\displaystyle\leq\int_{l}^{x}\exp\left(\left(\sup_{z\in[-(1/n),0]}\frac{1}{\sigma(z)^{2}}\right)\int_{l}^{y}\varphi_{n}^{\prime}(z)dz-\int_{l}^{y}\frac{2b(z)}{\sigma(z)^{2}}dz\right)dy$
$\displaystyle\rightarrow\int_{l}^{x}\exp\left(-\int_{l}^{y}\frac{2b(z)}{\sigma(z)^{2}}dz\right)dy<\infty\qquad\mbox{as}\
n\rightarrow\infty.$

These yield

(3.4) $\lim_{n\rightarrow\infty}P(X^{n,x}_{\tau_{l}\wedge\tau_{r}}=r)=0,\quad
l<x<0<r.$

Moreover, the fact that
$P(X^{n,x}_{\tau_{l}\wedge\tau_{r}}=l)+P(X^{n,x}_{\tau_{l}\wedge\tau_{r}}=r)=1$
for all $n$ implies that

(3.5) $\lim_{n\rightarrow\infty}P(X^{n,x}_{\tau_{l}\wedge\tau_{r}}=l)=1,\quad
l<x<0<r.$

Similarly, for $l<x<r<0$ we also have

$\lim_{n\rightarrow\infty}P(X^{n,x}_{\tau_{l}\wedge\tau_{r}}=r)=\lim_{n\rightarrow\infty}\frac{s_{n}(x)}{s_{n}(r)}=\frac{\int_{l}^{x}\exp\left(-\int_{l}^{y}\frac{2b(z)}{\sigma(z)^{2}}dz\right)dy}{\int_{l}^{r}\exp\left(-\int_{l}^{y}\frac{2b(z)}{\sigma(z)^{2}}dz\right)dy}.$

This implies that for $l<x<0$ there exists a constant $c_{l,x}\in(0,1)$ such
that

(3.6) $\lim_{r\uparrow
0}\lim_{n\rightarrow\infty}P(X^{n,x}_{\tau_{l}\wedge\tau_{r}}=r)=c_{l,x}.$

From (3.4), (3.5) and (3.6) we see that the limit process $X$ of $X^{n}$ in
law does not cross the origin from $\\{x<0\\}$ to $\\{x>0\\}$, and $X$
starting at $x<0$ hits $0$ with positive probability. The proofs that $X$ does
not cross the origin from $\\{x>0\\}$ to $\\{x<0\\}$ and $X$ starting at $x<0$
hits $0$ with positive probability, are similar to the argument above. So we
omit them.

Finally, we see that the limit process $X$ is not trapped at $0$. For the
proof it is sufficient to show that

(3.7) $\lim_{\varepsilon\downarrow
0}E\left[\int_{0}^{T}{\mathbb{I}}_{[-\varepsilon,\varepsilon]}(X_{t})dt\right]=0$

for $T>0$ under the condition
$\sup_{n\in{\mathbb{N}}}E[|X_{0}^{n}|^{2}]<\infty$. Fix $\varepsilon\in(0,1)$
and $T>0$. Choose a sequence $\\{g_{m}\\}\subset C_{0}({\mathbb{R}})$ so that
${\mathbb{I}}_{[-\varepsilon,\varepsilon]}\leq g_{m}\leq{\mathbb{I}}_{[-1,1]}$
for $m\in{\mathbb{N}}$ and
$g_{m}(x)\downarrow{\mathbb{I}}_{[-\varepsilon,\varepsilon]}(x)$ as
$m\in{\mathbb{N}}$ for each $x\in{\mathbb{R}}$, and define $G_{m}\in
C^{2}({\mathbb{R}})$ by

$G_{m}(x):=\int_{0}^{x}\left(\int_{0}^{y}g_{m}(z)dz\right)dy.$

Then, the Itô formula implies

$\displaystyle E[G_{m}(X^{n}_{T})]-E[G_{m}(X^{n}_{0})]$
$\displaystyle=\frac{1}{2}E\left[\int_{0}^{T}g_{m}(X^{n}_{t})\sigma(X^{n}_{t})^{2}dt\right]+E\left[\int_{0}^{T}G_{m}^{\prime}(X^{n}_{t})b(X^{n}_{t})dt\right]-E\left[\int_{0}^{T}G_{m}^{\prime}(X^{n}_{t})\varphi_{n}^{\prime}(X^{n}_{t})dt\right].$

Noting that the definition of $G_{m}$ yields that

$G_{m}^{\prime}(x)\varphi_{n}^{\prime}(x)\leq 0,\quad x\in{\mathbb{R}},$

we have

$E\left[\int_{0}^{T}g_{m}(X^{n}_{t})\sigma(X^{n}_{t})^{2}dt\right]\leq
2E[G_{m}(X^{n}_{T})]-2E[G_{m}(X^{n}_{0})]-2E\left[\int_{0}^{T}G_{m}^{\prime}(X^{n}_{t})b(X^{n}_{t})dt\right].$

By the convergence of $X^{n}$ to $X$ in law on $C([0,\infty))$, it follows
that

$E\left[\int_{0}^{T}g_{m}(X_{t})\sigma(X_{t})^{2}dt\right]\leq
2E[G_{m}(X_{T})]-2E[G_{m}(X_{0})]-2E\left[\int_{0}^{T}G_{m}^{\prime}(X_{t})b(X_{t})dt\right].$

Here, we remark that $|G_{m}(x)|\leq|x|$ for $x\in{\mathbb{R}}$ and that
$\sup_{t\in[0,T]}E[|X_{t}^{n}|^{2}]<\infty$ follows from application of the
Itô formula to $f(X^{n}_{t})$ for some $f\in C^{2}({\mathbb{R}})$ such that

$\left\\{\begin{array}[]{l}x^{2}\leq f(x)\leq 2(1+x^{2}),\quad
x\in{\mathbb{R}},\\\ f(x)=1,\quad x\in[-1,1],\end{array}\right.$

and a standard argument (c.f. e.g. [24, Theorem 2.4, Chapter IV]). Since

$\displaystyle\lim_{m\rightarrow\infty}G_{m}^{\prime}(x)$
$\displaystyle=\int_{0}^{x}{\mathbb{I}}_{[-\varepsilon,\varepsilon]}(z)dz=x{\mathbb{I}}_{(-\varepsilon,\varepsilon)}(x)+\varepsilon{\mathbb{I}}_{[\varepsilon,\infty)}(x)-\varepsilon{\mathbb{I}}_{(-\infty,-\varepsilon]}(x),$
$\displaystyle\lim_{m\rightarrow\infty}G_{m}(x)$
$\displaystyle=\int_{0}^{x}\left(\int_{0}^{y}{\mathbb{I}}_{[-\varepsilon,\varepsilon]}(z)dz\right)dy=\frac{x^{2}}{2}{\mathbb{I}}_{(-\varepsilon,\varepsilon)}(x)+\varepsilon\left(|x|-\frac{\varepsilon}{2}\right){\mathbb{I}}_{(-\infty,-\varepsilon]\cup[\varepsilon,\infty)}(x),$

by the dominate convergence theorem we have

$\displaystyle
E\left[\int_{0}^{T}{\mathbb{I}}_{[-\varepsilon,\varepsilon]}(X_{t})\sigma(X_{t})^{2}dt\right]$
$\displaystyle\leq
2E\left[\frac{X_{T}^{2}}{2}{\mathbb{I}}_{(-\varepsilon,\varepsilon)}(X_{T})+\varepsilon|X_{T}|\right]+2E\left[\frac{X_{0}^{2}}{2}{\mathbb{I}}_{(-\varepsilon,\varepsilon)}(X_{0})+\varepsilon|X_{0}|\right]$
$\displaystyle\quad+2\varepsilon E\left[\int_{0}^{T}|b(X_{t})|dt\right].$

Therefore, by noting that

$E\left[\int_{0}^{T}{\mathbb{I}}_{[-\varepsilon,\varepsilon]}(X_{t})dt\right]\leq\left(\inf_{x\in[-\varepsilon,\varepsilon]}\frac{1}{\sigma(x)^{2}}\right)E\left[\int_{0}^{T}{\mathbb{I}}_{[-\varepsilon,\varepsilon]}(X_{t})\sigma(X_{t})^{2}dt\right].$

and applying the dominate convergence theorem again for $\varepsilon\downarrow
0$, we obtain (3.7). ∎

Theorem 3.5 implies that in general the limit in law of strong Markov
processes is not a strong Markov process. The origin $0$ should be regarded as
a reflecting boundary for the limit $X$ in Theorem 3.5, and a situation
similar to Proposition 3.3 occurs. As we have seen from Proposition 3.3 and
the discussion after that, for the strong Markov property it is important for
stochastic processes to be determined for all initial points in the state
space. Additionally to this fact, from Theorem 3.5 it is seen that whether the
stochastic process determined for all initial points in the state space or not
is blinded by taking approximations.

######  Remark 3.6.

In [12] “virtual solutions” of SDEs with distributional drifts are considered.
However, Assumption 5 in [12], because the derivative $\delta_{0}^{\prime}$ of
the Dirac delta function on ${\mathbb{R}}$ does not belong to the Sobolev
space $H_{p}^{-\beta}$ for $\beta<(p-1)/p$.

##  4 Non-unique global solutions

In this section, we introduce some examples of SDEs that have the uniqueness
locally in time and global extendability of solutions, but does not have
uniqueness globally in time.

######  Theorem 4.1.

Consider the SDE

(4.1)
$\left\\{\begin{array}[]{rl}dX_{t}&={\mathbb{I}}_{{\mathbb{R}}\setminus\\{0\\}}(X_{t})dB_{t}\\\
X_{0}&=\xi\end{array}\right.$

where $B=(B_{t};t\in[0,\infty))$ is a one-dimensional Brownian motion and
$\xi$ is a random variable independent of $B$. If $P(\xi=0)=0$ and
$E[|\xi|^{2}]<\infty$, then the followings hold.

  1. (i)

(4.1) has the unique local-in-time solution $X=(X_{t};t\in[0,\tau_{0}(X)])$ up
to the first hitting time $\tau_{0}(X)$ at $0$, which is positive almost
surely.

  2. (ii)

Any solution $X=(X_{t};t\in[0,T])$ has the estimate

$E\left[\sup_{t\in[0,T]}|X_{t}|^{2}\right]<\infty$

for any nonrandom $T\in[0,\infty)$. In particular, any solution $X$ is
extended to a global-in-time solution $X=(X_{t};t\in[0,\infty))$.

  3. (iii)

For any nonrandom $T\in[0,\infty)$ the two different processes
$X_{t}=\xi+B_{t}$ and
$\widetilde{X}_{t}=\xi+B_{t\wedge\tau_{0}(\widetilde{X})}$ are solutions to
(4.1). In particular, (4.1) does not have the uniqueness of solutions on the
time interval $[0,T]$.

###### Proof.

The first assertion (i) is follows from the facts that the solution $X$ to
(4.1) behaves as $B_{t}$ up to the hitting time $\tau_{0}(X)$ and that
$P(\xi=0)=0$ implies $\tau_{0}(X)>0$ almost surely. To prove (ii), let
$X=(X_{t};t\in[0,\infty))$ be a solution to (4.1). Since the Burkholder-Davis-
Gundy inequality implies

$\displaystyle E\left[\sup_{t\in[0,T]}|X_{t}|^{2}\right]$ $\displaystyle\leq
2E\left[\left|\xi\right|^{2}\right]+2E\left[\sup_{t\in[0,T]}\left|\int_{0}^{t}{\mathbb{I}}_{{\mathbb{R}}\setminus\\{0\\}}(X_{s})dB_{s}\right|^{2}\right]$
$\displaystyle\leq
2E\left[\left|\xi\right|^{2}\right]+CE\left[\int_{0}^{T}{\mathbb{I}}_{{\mathbb{R}}\setminus\\{0\\}}(X_{s})ds\right]$
$\displaystyle\leq 2E\left[\left|\xi\right|^{2}\right]+CT.$

where $C$ is an absolute constant, (ii) follows. Since it is easy to see that
the two continuous stochastic processes

$X_{t}:=\xi+B_{t}\quad\mbox{and}\quad\widetilde{X}_{t}:=\xi+B_{t\wedge\tau_{0}(\widetilde{X})}$

are solutions to (4.1), (iii) follows. ∎

######  Remark 4.2.

  * •

The first assertion of Theorem 4.1(ii) implies that any solution does not
explode in finite time. Hence, any solution $(X_{t};t\in[0,\tau))$ to (4.1)
with a stopping time $\tau$ can be extended to a continuous stochastic process
$(\bar{X}_{t};t\in[0,\tau])$. Therefore, we have a global-in-time solution $Z$
by

$Z_{t}=\left\\{\begin{array}[]{cl}\bar{X}_{t},&t\in[0,\tau],\\\
Y_{t-\tau},&t\in(\tau,\infty),\end{array}\right.$

where $(Y_{t};t\in[0,\infty))$ is another solution with initial condition
$Y_{0}=\bar{X}_{\tau}$ whose driving noise is
$\widetilde{B}_{t}:=B_{t+\tau}-B_{\tau}$. Here, note that the existence of
solutions for all initial points follows from (iii), in particular such $Y$
exists.

  * •

The SDE (4.1) is also treated in [6], and infinitely many solutions are
constructed in [6] in view of sticky points. In particular, there exist
infinitely many strong Markov processes such that they satisfy (4.1) and the
Lebesgue measures of their spending time at $0$ equals to $0$.

The point of Theorem 4.1 is that the SDE (4.1) does not have the uniqueness of
the solution starting at $0$, and hence solutions of the SDE (4.1) have
branches at $0$. The trick to make the local-in-time unique and globally
extendable, but not unique globally in time, is that the initial low do not
have a positive mass at the branching points almost surely, i.e.
$P(X_{0}=0)=0$ in the case of (4.1). We usually do not assume $P(X_{0}=0)=0$,
and the assumption seems eccentric. However, two different solutions to (4.1)
in Theorem 4.1 are obtained by the unique limits of the solutions to different
approximations of the SDE, as follows.

######  Theorem 4.3.

Let $\xi\in{\mathbb{R}}$.

  1. (i)

Let $\varepsilon>0$ and consider the SDE

(4.2)
$\left\\{\begin{array}[]{rl}dX_{t}^{\varepsilon}&\displaystyle=\left[\left(\frac{1}{\varepsilon}\sqrt{|X_{t}^{\varepsilon}|}\right)\wedge
1\right]dB_{t}\\\ X_{0}^{\varepsilon}&=\xi\end{array}\right.$

where $B$ is a one-dimensional Brownian motion. Then, for each $\varepsilon>0$
the solution $X^{\varepsilon}=(X_{t};t\in[0,\infty))$ to (4.2) exists
pathwise-uniquely, and the limit $X$ of $X^{\varepsilon}$ in the topology
$L^{2}(\Omega;C([0,\infty)))$ as $\varepsilon\downarrow 0$ exists and is given
by $X_{t}=\xi+B_{t\wedge\tau_{0}(X)}$ where $\tau_{0}(X)$ is the first hitting
time of $X$ at $0$. In particular, the limit $X$ is a solution to (4.1).

  2. (ii)

Let $\varepsilon>0$ and consider the SDE

(4.3)
$\left\\{\begin{array}[]{rl}dY_{t}^{\varepsilon}&\displaystyle={\mathbb{I}}_{{\mathbb{R}}\setminus\\{0\\}}(Y_{t}^{\varepsilon})dB_{t}+\varepsilon
d\widetilde{B}_{t}\\\ Y_{0}^{\varepsilon}&=\xi\end{array}\right.$

where $B$ and $\widetilde{B}$ are two independent one-dimensional Brownian
motions. Then, for each $\varepsilon>0$ the solution
$Y^{\varepsilon}=(Y_{t};t\in[0,\infty))$ to (4.3) exists pathwise-uniquely,
and the limit $Y$ of $Y^{\varepsilon}$ in the topology
$L^{2}(\Omega;C([0,\infty)))$ as $\varepsilon\downarrow 0$ exists and is given
by $Y_{t}=\xi+B_{t}$. In particular, the limit $Y$ is a solution to (4.1).

###### Proof.

First we prove (i). Since the function
$x\mapsto\left(\frac{1}{\varepsilon}\sqrt{|x|}\right)\wedge 1$ is
$(1/2)$-Hölder continuous, by the Yamada-Watanabe theorem (cf. Theorem 1 in
[30] or Theorem 3.2 of Chapter IV in [24]) we have the existence and pathwise
uniqueness of the solution $X^{\varepsilon}$ to (4.2) for any
$X_{0}^{\varepsilon}\in{\mathbb{R}}$. In particular, $X^{\varepsilon}$
satisfies the strong Markov property (cf. Section 5 of Chapter IV in [24]).
These and the fact that $X_{t}^{\varepsilon}=0$ for $t\in[0,\infty)$ is the
solution to (4.2) with the initial condition $X_{0}^{\varepsilon}=0$, imply
that

$X_{t}^{\varepsilon}=X^{\varepsilon}_{t\wedge\tau_{0}(X^{\varepsilon})},\quad
t\in[0,\infty)$

almost surely. Let $X_{t}=\xi+B_{t\wedge\tau_{0}(X)}$. Then, for any $T>0$, by
the Burkholder-Davis-Gundy inequality we have

$\displaystyle
E\left[\sup_{t\in[0,\tau_{0}(X^{\varepsilon})\wedge\tau_{0}(X)\wedge
T]}\left|X_{t}^{\varepsilon}-X_{t}\right|^{2}\right]$
$\displaystyle=E\left[\sup_{t\in[0,T]}\left|\int_{0}^{\tau_{0}(X^{\varepsilon})\wedge\tau_{0}(X)\wedge
T}\left\\{\left[\left(\frac{1}{\varepsilon}\sqrt{|X_{s}^{\varepsilon}|}\right)\wedge
1\right]-1\right\\}dB_{s}\right|^{2}\right]$ $\displaystyle\leq
CE\left[\int_{0}^{\tau_{0}(X^{\varepsilon})\wedge\tau_{0}(X)\wedge
T}\left\\{\left[\left(\frac{1}{\varepsilon}\sqrt{|X_{s}^{\varepsilon}|}\right)\wedge
1\right]-1\right\\}^{2}ds\right].$

Hence, the dominate convergence theorem yields

(4.4) $\lim_{\varepsilon\downarrow
0}E\left[\sup_{t\in[0,\tau_{0}(X^{\varepsilon})\wedge\tau_{0}(X)\wedge
T]}\left|X_{t}^{\varepsilon}-X_{t}\right|^{2}\right]=0,\quad T>0.$

From (4.4) it is easy to see that

$\lim_{\varepsilon\downarrow 0}P\left(\tau_{0}(X^{\varepsilon})\wedge
T\neq\tau_{0}(X)\wedge T\right)=0,\quad T>0.$

From this fact, (4.4) and the fact that $X_{t}^{\varepsilon}=X_{t}=0$ for
$t\geq\tau_{0}(X^{\varepsilon})\vee\tau_{0}(X)$ almost surely, by the dominate
convergence theorem we obtain the convergence of $X^{\varepsilon}$ to $X$ in
the topology $L^{2}(\Omega;C([0,\infty)))$. The part “In particular” has been
obtained in Theorem 4.1(iii).

Next we prove (ii). Let $Y^{\varepsilon}$ be a solution to (4.3). Note that
similarly to the proof of Theorem 4.1, we have

(4.5) $E\left[\sup_{t\in[0,T]}|Y_{t}^{\varepsilon}|^{2}\right]\leq
C(|\xi|^{2}+T(1+\varepsilon^{2}))$

for any $T>0$ and an absolute constant $C$. For any $\delta>0$ take
$g_{\delta}\in C_{0}({\mathbb{R}})$ such that
${\mathbb{I}}_{[-\delta,\delta]}\leq
g_{\delta}\leq{\mathbb{I}}_{[-2\delta,2\delta]}$ and let

(4.6)
$G_{\delta}(x):=\int_{-2\delta}^{x}\left(\int_{-2\delta}^{y}g_{\delta}(z)dz\right)dy,\quad
x\in{\mathbb{R}}.$

Then, the Itô formula implies

$\displaystyle E[G_{\delta}(Y_{t}^{\varepsilon})]-E[G_{\delta}(\xi)]$
$\displaystyle=\frac{1}{2}\int_{0}^{t}E\left[g_{\delta}(Y_{s}^{\varepsilon}){\mathbb{I}}_{{\mathbb{R}}\setminus\\{0\\}}(Y_{s}^{\varepsilon})\right]ds+\frac{\varepsilon^{2}}{2}\int_{0}^{t}E\left[g_{\delta}(Y_{s}^{\varepsilon})\right]ds$
$\displaystyle\geq\frac{\varepsilon^{2}}{2}\int_{0}^{t}E\left[{\mathbb{I}}_{[-\delta,\delta]}(Y_{s}^{\varepsilon})\right]ds.$

Since $|G_{\delta}(x)|\leq 4\delta(|x|+2\delta)$, in view of (4.5) we have

$\int_{0}^{t}E\left[{\mathbb{I}}_{[-\delta,\delta]}(Y_{s}^{\varepsilon})\right]ds\leq
C\delta\varepsilon^{-2}(\delta+|\xi|^{2}+T(1+\varepsilon^{2})).$

for $t\in[0,T]$, where $C$ is an absolute constant. Hence, by letting
$\delta\downarrow 0$ we have

$E\left[\int_{0}^{t}{\mathbb{I}}_{\\{0\\}}(Y_{s}^{\varepsilon})ds\right]=0$

for $t\in[0,\infty)$ and $\varepsilon>0$. This equality and the Itô isometry
imply that

$\int_{0}^{t}{\mathbb{I}}_{{\mathbb{R}}\setminus\\{0\\}}(Y_{s}^{\varepsilon})dB_{s}=B_{t}$

for any $t\in[0,\infty)$ almost surely. Thus, we obtain that
$Y_{t}^{\varepsilon}$ satisfies

(4.7) $Y_{t}^{\varepsilon}=\xi+B_{t}+\varepsilon\widetilde{B}_{t},\quad
t\in[0,\infty)$

almost surely. Now it is easy to see that
$Y^{\varepsilon}=(Y_{t};t\in[0,\infty))$ given by (4.7) is the pathwise-unique
solution to (4.3), and $Y^{\varepsilon}$ converges to $\xi+B$ in the topology
$L^{2}(\Omega;C([0,\infty)))$. The part “In particular” has been obtained in
Theorem 4.1(iii). ∎

In Theorem 4.3, we constructed two different solutions to (4.1) by limits of
the strong Markov processes which are the unique solutions starting at any
given initial point to approximate SDEs. On the other hand, by considering
path-dependent SDEs, even without approximations, we can construct a SDE which
has the local-in-time pathwise-unique solutions starting at any initial point
and all the solutions are extendable to globally-in-time solutions, but the
solutions globally in time are not unique.

######  Theorem 4.4.

Consider the path-dependent SDE

(4.8)
$\left\\{\begin{array}[]{rl}dX_{t}&=\left({\mathbb{I}}_{{\mathbb{R}}\setminus\\{0\\}}(X_{t})+{\mathbb{I}}_{[0,1)}\left(\int_{0}^{t}|X_{s}|ds\right)\right)dB_{t}\\\
X_{0}&=\xi,\end{array}\right.$

where $B$ is a one-dimensional Brownian motion and $\xi$ is a random variable
independent of $B$.

  1. (i)

(4.8) has the pathwise-unique local-in-time solution
$X=(X_{t};t\in[0,\tilde{\tau}_{0}(X)])$ up to the stopping time
$\tilde{\tau}_{0}(X)$ defined by

$\tilde{\tau}_{0}(X):=\inf\left\\{t>0;\int_{0}^{t}|X_{s}|ds\geq 1\ \mbox{and}\
X_{t}=0\right\\},$

which is positive almost surely.

  2. (ii)

When $E[|\xi|^{2}]<\infty$, any solution $X$ to (4.8) has the estimate

$E\left[\sup_{t\in[0,T]}|X_{t}|^{2}\right]<\infty$

for any nonrandom $T\in[0,\infty)$. In particular, any solution $X$ is
extended to a global-in-time solution $X=(X_{t};t\in[0,\infty))$.

  3. (iii)

For any nonrandom $T\in[0,\infty)$, (4.8) does not have the uniqueness of
solutions on the time interval $[0,T]$.

###### Proof.

The proof of Theorem 4.4 is similar to that of Theorem 4.3. For the proof of
(i) we prove only the case that $\xi$ is nonrandom, because the general case
follows from this case by consider it under the regular conditional
probability measure given by the $\sigma$-field generated by $\xi$. Let

$\bar{\tau}_{1}(X):=\inf\left\\{t>0;\int_{0}^{t}|X_{s}|ds\geq 1\right\\},$

Then, for any solution $X$ to (4.8) satisfies

$X_{t}=\xi+\int_{0}^{t}\left({\mathbb{I}}_{{\mathbb{R}}\setminus\\{0\\}}(X_{s})+1\right)dB_{s},\quad
t\in[0,\bar{\tau}_{1}(X)].$

For $\delta>0$, define $g_{\delta}\in C_{0}({\mathbb{R}})$ and $G_{\delta}\in
C^{2}({\mathbb{R}})$ as in the proof of Theorem 4.3 (see (4.6)). By the Itô
formula we have

$\displaystyle E[G_{\delta}(X_{t\wedge\bar{\tau}_{1}(X)})]-E[G_{\delta}(\xi)]$
$\displaystyle=\frac{1}{2}E\left[\int_{0}^{t\wedge\bar{\tau}_{1}(X)}g_{\delta}(X_{s})\left({\mathbb{I}}_{{\mathbb{R}}\setminus\\{0\\}}(X_{s})+1\right)^{2}ds\right]$
$\displaystyle\geq\frac{1}{2}E\left[\int_{0}^{t\wedge\bar{\tau}_{1}(X)}{\mathbb{I}}_{[-\delta,\delta]}(X_{s})ds\right].$

Similarly to the proof of Theorem 4.3(ii), we have

$E\left[\int_{0}^{t\wedge\bar{\tau}_{1}(X)}{\mathbb{I}}_{[-\delta,\delta]}(X_{s})ds\right]\leq
C\delta\varepsilon^{-2}(\delta+|\xi|^{2}+t(1+\varepsilon^{2})).$

for $t\in[0,\infty)$, where $C$ is an absolute constant. Hence, by letting
$\delta\downarrow 0$ and $t\uparrow\infty$ we have

$E\left[\int_{0}^{\bar{\tau}_{1}(X)}{\mathbb{I}}_{\\{0\\}}(X_{s})ds\right]=0.$

This equality and the Itô isometry imply that

(4.9) $X_{t}=\xi+2B_{t},\quad t\in[0,\bar{\tau}_{1}(X)]$

almost surely. This yields the uniqueness of the solution on the time interval
$[0,\bar{\tau}_{1}(X)]$. From (4.9) we also have that
$0<\bar{\tau}_{1}(X)<\infty$ almost surely. It is easy to see that
$(X_{t};t\in[0,\bar{\tau}_{1}(X)])$ given by (4.9) satisfies (4.8) on
$[0,\overline{\tau}_{1}(X)]$. Let $Y_{t}:=X_{t+\bar{\tau}_{1}(X)}$
($t\in[0,\infty)$). Then, $Y_{t}$ satisfies (4.1) with replacement of $\xi$ by
$X_{\overline{\tau}_{1}(X)}$. Note that $X_{\overline{\tau}_{1}(X)}\neq 0$
almost surely, because of (4.9). Hence, by Theorem 4.1 $Y_{t}$ is uniquely
determined for $t\in[0,\tau_{0}(Y)]$ where

$\tau_{0}(Y):=\inf\\{t>0;Y_{t}=0\\}.$

Since $\tilde{\tau}_{0}(X)=\bar{\tau}_{1}(X)+\tau_{0}(Y)$, we obtain (i).

Similarly to the proof of Theorem 4.1(ii), we have (ii) from the boundedness
of the coefficient of (4.8).

Let $T>0$. Since (4.9) implies that $P(\bar{\tau}_{1}(X)<T)>0$, we obtain
(iii) from Theorem 4.1(iii) and the fact that any stochastic process $X$ which
satisfies

$X_{t}=\left\\{\begin{array}[]{cl}\xi+2B_{t},&t\in[0,\bar{\tau}_{1}(X)]\\\
Y_{t-\bar{\tau}_{1}(X)},&t\in[\bar{\tau}_{1}(X),\infty)\end{array}\right.$

where $Y_{t}$ satisfies (4.1) with replacement of $\xi$ by
$X_{\overline{\tau}_{1}(X)}$, is a solution to (4.8). ∎

The example appeared in Theorem 4.4 is of a path-dependent SDE. However, since
the dependence of the path is very simple, we can transform it to a
multidimensional SDE which is not path-dependent, as follows.

######  Corollary 4.5.

Consider the two-dimensional SDE

(4.10) $\left\\{\begin{array}[]{rl}d\left(\begin{array}[]{c}X_{t}\\\
Y_{t}\end{array}\right)&=\left(\begin{array}[]{c}\left({\mathbb{I}}_{{\mathbb{R}}\setminus\\{0\\}}(X_{t})+{\mathbb{I}}_{[0,1)}\left(Y_{t}\right)\right)dB_{t}\\\
|X_{t}|dt\end{array}\right)\\\ \left(\begin{array}[]{c}X_{0}\\\
Y_{0}\end{array}\right)&=\left(\begin{array}[]{c}\xi\\\
0\end{array}\right),\end{array}\right.$

where $B$ is a one-dimensional Brownian motion and $\xi$ is a random variable
independent of $B$.

###### Proof.

For any solution $X$ to (4.8), by letting

$Y_{t}:=\int_{0}^{t}|X_{s}|ds,$

we have a solution $(X,Y)$ to (4.10). On the other hand, any solution $(X,Y)$
to (4.10) satisfies

$Y_{t}=\int_{0}^{t}|X_{s}|ds.$

Hence, $X$ is a solution to (4.8). Clearly, the relation between the solutions
to (4.8) and (4.10) is a one-to-one correspondence. Therefore, Corollary 4.5
follows from Theorem 4.4. ∎

######  Remark 4.6.

In this section we consider a one-dimensional SDE, which has branches of
solutions at one point in the state space. We chose this SDE, because it is
the most simple example without the uniqueness of solutions. Indeed, we have
given different solutions explicitly. We remark that the same phenomena occur
also for other SDEs which have branches of solutions at non-polar sets.

##  5 Shifted equations

In this section we consider relations between solutions to shifted equations
and to original SDEs. Shifted equations appear in approaches in singular
SPDEs, in particular in the paracontrolled calculus (see [18]).

First we consider general SDEs and their shifted equations. Consider a SDE

(5.1)
$\left\\{\begin{array}[]{rl}dX_{t}&=\sigma_{X}(t,X_{t})dB_{t}+b_{X}(t,X_{t})dt\\\
X_{0}&=x.\end{array}\right.$

Let $k\in{\mathbb{N}}$ and consider a system of SDEs

(5.2)
$\left\\{\begin{array}[]{rl}dY_{t}^{i}&=\sigma_{Y^{i}}(t,Y_{t}^{1},Y_{t}^{2},\dots,Y_{t}^{k})dB_{t}+b_{Y^{i}}(t,Y_{t}^{1},Y_{t}^{2},\dots,Y_{t}^{k})dt\\\
Y_{0}^{i}&=y^{i}.\end{array}\right.$

for $i=1,2,\dots,k$. Here, the dimensions of equations are not mentioned,
because any dimension is allowed in next (trivial) proposition. Let ${\rm
Sol}_{X}$ and ${\rm Sol}_{Y}$ be the total sets of solutions $X$ and
$(Y^{1},Y^{2},\dots,Y^{k})$ to (5.1) and (5.2), respectively. Note that any
sense of the solutions is allowed for the definitions of ${\rm Sol}_{X}$ and
${\rm Sol}_{Y}$.

Trivially, the following proposition holds.

######  Proposition 5.1.

  1. (i)

If ${\rm Sol}_{Y}\neq\emptyset$ and there exists a map $f_{Y\rightarrow
X}:{\rm Sol}_{Y}\rightarrow{\rm Sol}_{X}$, then the existence of the solution
to (5.2) implies the existence of the solution to (5.1) (i.e. ${\rm
Sol}_{X}\neq\emptyset$).

  2. (ii)

If there exists a injective map $f_{X\rightarrow Y}:{\rm
Sol}_{X}\rightarrow{\rm Sol}_{Y}$, then the uniqueness of the solution to
(5.2) (i.e. $\\#{\rm Sol}_{Y}=1$) implies the uniqueness of the solution to
(5.1) (i.e. $\\#{\rm Sol}_{X}=1$).

######  Remark 5.2.

Since the solutions $X$ and $(Y^{1},Y^{2},\dots,Y^{k})$ to (5.1) and (5.2),
respectively, have no interaction with each other, the existence (resp.
uniqueness) of the solution to (5.1) and of the solution to (5.2) is
equivalent to the existence (resp. uniqueness) of the coupled equation of
(5.1) and (5.2).

In view of the paracontrolled calculus, we consider a sufficient conditions
that for a given solution $(Y^{1},Y^{2},\dots,Y^{k})\in
C([0,\infty);{\mathbb{R}}^{d})^{\otimes k}$ to (5.2), the stochastic process
$X\in C([0,\infty);{\mathbb{R}}^{d})$ defined by

(5.3) $X=Y^{1}+Y^{2}+\cdots+Y^{k}$

satisfies (5.1). A sufficent condition is that

(5.4)
$\left\\{\begin{array}[]{rl}\displaystyle\sigma_{X}(\eta^{0},\eta^{1}+\eta^{2}\dots+\eta^{k})&\displaystyle=\sum_{i=1}^{k}\sigma_{Y^{i}}(\eta^{0},\eta^{1},\dots,\eta^{k}),\quad(\eta^{0},\eta^{1},\dots,\eta^{k})\in{\rm
Supp}_{Y},\\\ \displaystyle
b_{X}(\eta^{0},\eta^{1}+\eta^{2}\dots+\eta^{k})&\displaystyle=\sum_{i=1}^{k}b_{Y^{i}}(\eta^{0},\eta^{1},\dots,\eta^{k}),\quad(\eta^{0},\eta^{1},\dots,\eta^{k})\in{\rm
Supp}_{Y},\\\ \displaystyle
x&\displaystyle=\sum_{i=1}^{k}y^{i},\end{array}\right.$

where ${\rm Supp}_{Y}$ is the support of the map
$(t,\omega)\mapsto(t,Y_{t}^{1}(\omega),Y_{t}^{2}(\omega),\dots,Y_{t}^{k}(\omega))$,
i.e. ${\rm
Supp}_{Y}:=\\{(t,Y_{t}^{1}(\omega),Y_{t}^{2}(\omega),\dots,Y_{t}^{k}(\omega));t\in[0,T],\omega\in\Omega\\}$.
Here, we note that (5.4) can be relaxed for $X$ to satisfy (5.1) by allowing
exceptional sets of $(t,\omega)$ with respect to $dt\otimes P(d\omega)$.

Next, in view of Proposition 5.1, we consider a sufficient condition for the
existence of an injective map $f_{X\rightarrow Y}:{\rm Sol}_{X}\rightarrow{\rm
Sol}_{Y}$ which satisfies (5.3) with
$(Y^{1},Y^{2},\dots,Y^{k})=f_{X\rightarrow Y}(X)$ for $X\in{\rm Sol}_{X}$. In
the case that $Y^{1}$ is given by $X$, and $Y^{i+1}$ is uniquely determined by
$X,Y^{1},Y^{2},\dots,Y^{i}$ for $i=1,2,\dots,k-1$ inductively, it is easy to
construct $f_{X\rightarrow Y}$. For example, the following proposition holds.

######  Proposition 5.3.

Let $k\in{\mathbb{N}}$ such that $k\geq 2$, let $X$ be a solution to (5.1),
and let $y^{i}$ be given for $i=1,2,\dots,k-1$. Consider coupled SDEs with a
solution $(Y^{1},Y^{2},\dots,Y^{k-1})$:

(5.5)
$\left\\{\begin{array}[]{rl}dY_{t}^{i}&=\tilde{\sigma}_{Y^{i}}(t,Y_{t}^{1},Y_{t}^{2},\dots,Y_{t}^{i},X_{t})dB_{t}+\tilde{b}_{Y^{i}}(t,Y_{t}^{1},Y_{t}^{2},\dots,Y_{t}^{i},X_{t})dt\\\
Y_{0}^{i}&=y^{i}\end{array}\right.$

for $i=1,2,\dots,k-1$. Assume that (5.5) is solved pathwise-uniquely and
recursively for $i=1,2,\dots,k-1$. Then,
$Y_{t}^{k}:=X_{t}-(Y_{t}^{1}+Y_{t}^{2}+\dots+Y_{t}^{k-1})$ satisfies

(5.6)
$\left\\{\begin{array}[]{rl}dY_{t}^{k}&=\tilde{\sigma}_{Y^{k}}(t,Y_{t}^{1},Y_{t}^{2},\dots,Y_{t}^{k},X_{t})dB_{t}+\tilde{b}_{Y^{k}}(t,Y_{t}^{1},Y_{t}^{2},\dots,Y_{t}^{k},X_{t})dt\\\
Y_{0}^{k}&=y^{k}:=x-(y^{1}+y^{2}+\cdots y^{k-1})\end{array}\right.$

where

$\displaystyle\tilde{\sigma}_{Y^{k}}(t,\eta^{1},\eta^{2},\dots,\eta^{k},\xi)$
$\displaystyle:=\sigma_{X}(t,\xi)-\sum_{i=1}^{k-1}\tilde{\sigma}_{Y^{i}}(t,\eta^{1},\eta^{2},\dots,\eta^{i},\xi),$
$\displaystyle\tilde{b}_{Y^{k}}(t,\eta^{1},\eta^{2},\dots,\eta^{k},\xi)$
$\displaystyle:=b_{X}(t,\xi)-\sum_{i=1}^{k-1}\tilde{b}_{Y^{i}}(t,\eta^{1},\eta^{2},\dots,\eta^{i},\xi)$

for
$(t,\eta^{1},\eta^{2},\dots,\eta^{k},\xi)\in[0,T]\times({\mathbb{R}}^{d})^{\otimes(k+1)}$.

###### Proof.

This is trivial, because of the definitions of $\tilde{\sigma}_{Y^{k}}$ and
$\tilde{b}_{Y^{k}}$. ∎

######  Remark 5.4.

When we consider the implication of the (time-inhomogeneous) strong Markov
property of $X$ form that of $(Y^{1},Y^{2},\dots,Y^{k})$, we need the
uniqueness in law of $(Y^{1},Y^{2},\dots,Y^{k})$ for all
$(s,y^{1},y^{2},\dots,y^{k})\in{\rm Supp}_{Y}$ (See Section 2.2 or [24,
Theorem 5.1, Chapter IV]). Here, we emphasize that for the condition “for all
$(s,y^{1},y^{2},\dots,y^{k})\in{\rm Supp}_{Y}$” is necessary, because the
solution to (5.2) has to be determined uniquely for initial points
$y_{i}=Y^{i}_{\tau}$ for $i=1,2,\dots,k$ and any stopping time $\tau$.

The discussion above is almost trivial, but is the background of the method of
singular SPDEs. Indeed, we can regard the strategy of the method, as follows.

  * (Step 1)

We regularize the target singular SPDE. Once we regularize the target
equation, we have a solution $X^{(n)}$ to the regularized equation.

  * (Step 2)

Applying an analogue of Proposition 5.3 to the SPDE and we construct the
solution $(Y^{(n),1},Y^{(n),2},\dots,Y^{(n),k})$ to shifted equations. Note
that $X^{(n)}=Y^{(n),1}+Y^{(n),2}+\cdots+Y^{(n),k}$.

  * (Step 3)

Showing the existence of the limit of $(Y^{(n),1},Y^{(n),2},\dots,Y^{(n),k})$
as $n\rightarrow\infty$ (removing the regularization), we have the limit $X$
of $X^{(n)}$ as $n\rightarrow\infty$.

  * (Step 4)

We define the solution to the target singular SPDE by $X$.

See Section 2.1 for the detail of the method in singular SPDEs. Here, it
should be remark that for Proposition 5.3 the existence of the solution $X$ to
the original equation is necessary.

Now we see the delicateness of the discussion of the solutions to original
equations from the shifted equations by explicit examples.

######  Proposition 5.5.

Consider SDEs for $X$ on ${\mathbb{R}}$ and for $(Y^{1},Y^{2})$ on
${\mathbb{R}}^{2}$, as follows.

(5.9)
$\displaystyle\left\\{\begin{array}[]{rl}dX_{t}&={\mathbb{I}}_{{\mathbb{R}}\setminus\\{0\\}}(X_{t})dB_{t}\\\
X_{0}&=x,\end{array}\right.$ (5.13)
$\displaystyle\left\\{\begin{array}[]{rl}dY^{1}_{t}&={\mathbb{I}}_{|y^{1}|>1}{\mathbb{I}}_{[0,\tau_{0}(Y^{1}+Y^{2}))}(t)dB_{t}\\\
dY^{2}_{t}&={\mathbb{I}}_{|y^{1}|\leq 1}dB_{t}\\\
(Y^{1}_{0},Y^{2}_{0})&=(y^{1},y^{2}).\end{array}\right.$

Then, we have the following.

  1. (i)

For all $(y^{1},y^{2})\in{\mathbb{R}}^{2}$, the solution $(Y^{1},Y^{2})$ to
(5.13) exists uniquely.

  2. (ii)

Let $x\in{\mathbb{R}}$ and $(y^{1},y^{2})\in{\mathbb{R}}^{2}$. If
$x=y^{1}+y^{2}$, then $X:=Y^{1}+Y^{2}$ is a solution to (5.9).

  3. (iii)

Let $x\in{\mathbb{R}}$. Let $(Y_{1},Y_{2})$ be the solution to (5.13) with
initial condition $(y_{1},y_{2})=(0,x)$, and let
$(\widetilde{Y}_{1},\widetilde{Y}_{2})$ be the solution to (5.13) with initial
condition $(y_{1},y_{2})=(2,x-2)$. Then, $Y_{1}+Y_{2}$ and
$\widetilde{Y}_{1}+\widetilde{Y}_{2}$ are different solutions to (5.9) with
the same initial condition $X_{0}=x$.

###### Proof.

We have (i) by solving the SDEs of $(Y^{1},Y^{2})$ as

(5.14) $\left(\begin{array}[]{c}Y_{t}^{1}\\\
Y_{t}^{2}\end{array}\right)=\left\\{\begin{array}[]{c}\left(\begin{array}[]{c}y^{1}+B_{t\wedge\tau_{0}(Y^{1}+Y^{2})}\\\
y^{2}\end{array}\right),\quad\mbox{if}\ |y^{1}|>1,\\\\[11.38109pt]
\left(\begin{array}[]{c}y^{1}\\\ y^{2}+B_{t}\end{array}\right),\quad\mbox{if}\
|y^{1}|\leq 1.\end{array}\right.$

Let $x:=y^{1}+y^{2}$ and $X:=Y^{1}+Y^{2}$. From (5.14) it holds that

(5.15)
$Y_{t}^{1}+Y_{t}^{2}=\left\\{\begin{array}[]{c}(y^{1}+B_{t\wedge\tau_{0}(Y^{1}+Y^{2})})+y^{2}=x+B_{t\wedge\tau_{0}(X)},\quad\mbox{if}\
|y^{1}|>1,\\\\[11.38109pt] y^{1}+(y^{2}+B_{t})=x+B_{t},\quad\mbox{if}\
|y^{1}|\leq 1.\end{array}\right.$

Hence, we have (ii) by Theorem 4.1. From (5.15), (iii) follows. ∎

######  Remark 5.6.

Proposition 5.5 is prepared to see the delicateness of the initial conditions
of shifted equations. From Theorem 4.1 we see that the uniqueness of solutions
to (5.9) does not hold, and that two different solutions to (5.9) are chosen
by $|y^{1}|>1$ or $|y^{1}|\leq 1$.

Similarly to Theorem 4.3, the SDEs concerned in Proposition 5.5 can be written
as a limits of uniquely-solved SDEs, but depending on the initial condition,
as follows.

######  Theorem 5.7.

Let $\varepsilon>0$ and consider SDEs for
$(Y^{1,\varepsilon},Y^{2,\varepsilon})$ on ${\mathbb{R}}^{2}$, as follows.

(5.19)
$\displaystyle\left\\{\begin{array}[]{rl}dY^{1,\varepsilon}_{t}&\displaystyle={\mathbb{I}}_{(1,\infty)}(|y^{1}|)\left[\left(\frac{1}{\varepsilon}\sqrt{|Y^{1,\varepsilon}_{t}+Y^{2,\varepsilon}_{t}|}\right)\wedge
1\right]dB_{t}\\\
dY^{2,\varepsilon}_{t}&\displaystyle={\mathbb{I}}_{[0,1]}(|y^{1}|){\mathbb{I}}_{{\mathbb{R}}\setminus\\{0\\}}(Y^{1,\varepsilon}_{t}+Y^{2,\varepsilon}_{t})dB_{t}+\varepsilon
d\widetilde{B}_{t}\\\ (Y^{1}_{0},Y^{2}_{0})&=(y^{1},y^{2}).\end{array}\right.$

Then, we have the following.

  1. (i)

For all $(y^{1},y^{2})\in{\mathbb{R}}^{2}$, the solution
$(Y^{1,\varepsilon},Y^{2,\varepsilon})$ to (5.19) exists uniquely, and
converges in the topology of $L^{2}(\Omega;C([0,\infty)))$ as
$\varepsilon\downarrow 0$.

  2. (ii)

Let $x\in{\mathbb{R}}$, $(y^{1},y^{2})\in{\mathbb{R}}^{2}$ and let
$(Y^{1},Y^{2})$ be the limit in the topology of $L^{2}(\Omega;C([0,\infty)))$
as $\varepsilon\downarrow 0$ of the unique solutions
$(Y^{1,\varepsilon},Y^{2,\varepsilon})$ to (5.19) obtained in (i). If
$x=y^{1}+y^{2}$, then $X:=Y^{1}+Y^{2}$ is a solution to (5.9) in Proposition
5.5.

  3. (iii)

Let $x\in{\mathbb{R}}$. For each $\varepsilon>0$, let
$(Y^{1,\varepsilon},Y^{2,\varepsilon})$ be the unique solution to (5.19) with
initial condition $(y_{1},y_{2})=(0,x)$, and let
$(\widetilde{Y}^{1,\varepsilon},\widetilde{Y}^{2,\varepsilon})$ be the unique
solution to (5.19) with initial condition $(y^{1},y^{2})=(2,x-2)$. Denote the
limits $(Y^{1},Y^{2})$ and $(\widetilde{Y}^{1},\widetilde{Y}^{2})$ in the
topology of $L^{2}(\Omega;C([0,\infty)))$ as $\varepsilon\downarrow 0$ of
$(Y^{1,\varepsilon},Y^{2,\varepsilon})$ and
$(\widetilde{Y}^{1,\varepsilon},\widetilde{Y}^{2,\varepsilon})$, respectively.
Then, $Y^{1}+Y^{2}$ and $\widetilde{Y}^{1}+\widetilde{Y}^{2}$ are different
solutions to (5.9) with the same initial condition $X_{0}=x$.

###### Proof.

Note that

$\left\\{\begin{array}[]{l}Y^{2,\varepsilon}_{t}=y^{2},\quad\mbox{if}\
|y^{1}|>1,\\\ Y^{1,\varepsilon}_{t}=y^{1},\quad\mbox{if}\ |y^{1}|\leq
1.\end{array}\right.$

From Theorem 4.3, (i) follows. Since

(5.20)
$X=(Y^{1}+y^{2}){\mathbb{I}}_{(1,\infty)}(|y^{1}|)+(y^{1}+Y^{2}){\mathbb{I}}_{[0,1]}(|y^{1}|),$

(ii) also follows from Theorem 4.3.

Now we prove (iii). In view of (5.20) and Theorems 4.3, by the SDEs which
$Y^{1,\varepsilon}+Y^{2,\varepsilon}$ and
$\widetilde{Y}^{1,\varepsilon}+\widetilde{Y}^{2,\varepsilon}$ satisfy, we have

$Y_{t}^{1}+Y_{t}^{2}=x+B_{t},\quad\widetilde{Y}_{t}^{1}+\widetilde{Y}_{t}^{2}=x+B_{t\wedge\tau_{0}(\widetilde{Y}^{1}+\widetilde{Y}^{2})}.$

Now it is easy to see that $Y^{1}+Y^{2}$ and
$\widetilde{Y}^{1}+\widetilde{Y}^{2}$ are different solutions to (5.9) (see
Theorem 4.1(iii)). ∎

######  Remark 5.8.

Note that the SDEs in Proposition (5.5) does not satisfy (5.4). On the other
hand, the limit of (5.19) in Theorem 5.7 is a shifted equation of (5.9),
because

$\displaystyle{\mathbb{I}}_{(1,\infty)}(|y^{1}|)\left[\left(\frac{1}{\varepsilon}\sqrt{|Y^{1,\varepsilon}_{t}+Y^{2,\varepsilon}_{t}|}\right)\wedge
1\right]dB_{t}+{\mathbb{I}}_{[0,1]}(|y^{1}|){\mathbb{I}}_{{\mathbb{R}}\setminus\\{0\\}}(Y^{1,\varepsilon}_{t}+Y^{2,\varepsilon}_{t})dB_{t}+\varepsilon
d\widetilde{B}_{t}$
$\displaystyle\mathop{\longrightarrow}^{\varepsilon\downarrow
0}{\mathbb{I}}_{(1,\infty)}(|y^{1}|){\mathbb{I}}_{{\mathbb{R}}\setminus\\{0\\}}(Y^{1}_{t}+Y^{2}_{t})dB_{t}+{\mathbb{I}}_{[0,1]}(|y^{1}|){\mathbb{I}}_{{\mathbb{R}}\setminus\\{0\\}}(Y^{1}_{t}+Y^{2}_{t})dB_{t}$
$\displaystyle={\mathbb{I}}_{{\mathbb{R}}\setminus\\{0\\}}(Y^{1}_{t}+Y^{2}_{t})dB_{t}.$

######  Remark 5.9.

The examples in Proposition (5.5) and Theorem 5.7 are SDEs whose coefficients
depending on the initial condition $y^{1}$. But, the dependence of the initial
condition can be removed easily. For example, we can replace (5.13) by

$\left\\{\begin{array}[]{rl}dY^{1}_{t}&={\mathbb{I}}_{|Y^{3}_{t}|>1}{\mathbb{I}}_{[0,\tau_{0}(Y^{1}+Y^{2}))}(t)dB_{t}\\\
dY^{2}_{t}&={\mathbb{I}}_{|Y^{3}_{t}|\leq 1}dB_{t}\\\ dY^{3}_{t}&=0\\\
(Y^{1}_{0},Y^{2}_{0},Y^{3}_{0})&=(y^{1},y^{2},y^{3}).\end{array}\right.$

As mentioned in Section 2.1, the solution $X$ to a singular SPDE is defined by
$X:=Y_{1}+Y_{2}+\cdots+Y_{k}$ with a solution $(Y_{1},Y_{2},\dots,Y_{k})$ to
the shifted equation. Note that we often choose a specific initial condition
for the shifted equation. Indeed, in [2, Section 4] and [3, Section 4], for
the shifted equation of the $\Phi^{4}_{3}$-stochastic quantization equations
we choose the initial conditions so that $X_{0}^{N,(2),<}=0$ and
$X_{0}^{M,N,(2),<}=0$, respectively. In [23, Section 3.3] and [22, Section
4.3], to construct the solution to the $\exp(\Phi)_{2}$-stochastic
quantization equation, we choose the initial condition $0$ for the shifted
equation. It should be recalled that in many cases, in order for the
renormalization constants to be independent of the time parameter, the initial
condition of the solution to the linearized equation, which appear as a
component of shifted equations (see Section 2.1), is chosen from suitable
distributions except a null set with respect to the free field measure. We
often choose the initial conditions like these, because of technical
difficulties. In view of Proposition 5.5 and Theorem 5.7, we keep in mind that
the solutions of singular SPDEs are some specific ones for original singular
SPDEs, even if the solutions to shifted equations are unique for each initial
conditions.

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