# Plug-and-Play Algorithm Convergence Analysis From The Standpoint of
Stochastic Differential Equation

Zhongqi Wang, Bingnan Wang, Maosheng Xiang
School of Electronic Electrical and Communication Engineering
University of Chinese Academy of Science
Beijing, China
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

The Plug-and-Play (PnP) algorithm is popular for inverse image problem-
solving. However, this algorithm lacks theoretical analysis of its convergence
with more advanced plug-in denoisers. We demonstrate that discrete PnP
iteration can be described by a continuous stochastic differential equation
(SDE). We can also achieve this transformation through Markov process
formulation of PnP. Then, we can take a higher standpoint of PnP algorithms
from stochastic differential equations, and give a unified framework for the
convergence property of PnP according to the solvability condition of its
corresponding SDE. We reveal that a much weaker condition, bounded denoiser
with Lipschitz continuous measurement function would be enough for its
convergence guarantee, instead of previous Lipschitz continuous denoiser
condition.

##  1 Introduction

The inverse problem emerges in various fields, such as astrophysics [21],
computational imaging [2], and numerical computation [11]. Consider the
inverse problem with a given measurement system $M$,

$y=Mx+\epsilon.$ (1)

We meant to compute the target $x$ according to the acquired signal $y$ and
eliminate perturbation $\epsilon$. The challenge would be the irreversible of
the measurement model $M$. There could be information loss due to the
insufficient measurement and the perturbations, which makes this an
underdetermined problem.

[29] proposed the Plug-and-Play (PnP) methods to solve inverse image problems,
such as image denoising, image inpainting, demosaic, super-resolution, phase
retrieval, microscopy image reconstruction, etc. In image processing, a
general form of inverse problem solving would be:

$\mathop{\min}\limits_{x}f(x,y)+\gamma g(x),$ (2)

where $x$ is the image we would like to reconstruct, $f(x,y)=y-Mx$ is the data
fidelity term and $g(x)$ are the constraints of image properties, usually the
qualification of smoothness . $\gamma$ is a parameter that controls the weight
of $g$. The above formulation can also be interpreted as Maximum A Posteriori
problems. Given the posterior probability

$P(x|y)=\frac{P(y|x)P(x)}{P(y)},$ (3)

we seek to compute the optimal $\hat{x}$ that maximizes the conditional
distribution

$\displaystyle\hat{x}=\mathop{\arg\min}\limits_{x}-\ln P(y|x)-\ln P(x).$ (4)

with $f(x)=-\ln P(y|x)$ being the likelihood function and $g(x)=-\ln P(x)$
being the prior function.

For solving this joint optimization problem, the first step is to split the
variable $x$ by adding an auxiliary variable $v$

$(\hat{x},\hat{v})=\mathop{\arg\min}\limits_{x=v}f(x,y)+\gamma g(v)$ (5)

To handle this constrained optimization problem [3], the augmented lagrangian
would be

$L_{\lambda}(x,v;u)=f(x,y)+\gamma
g(v)+\frac{1}{2\lambda^{2}}\|x-v+u\|_{2}^{2}-\frac{\|u\|_{2}^{2}}{2\lambda^{2}},$
(6)

where $\lambda>0$ is the augmented Lagrangian parameter.

There are plenty of variable splitting algorithms, like ADMM (Alternative
Direction Method of Multipliers) [3], HQS (Half-Quadratic Splitting) [33],
FIST (Fast Iterative Soft Thresholding) [14], etc. Consider the ADMM
algorithm, which consists of iteration steps:

$\displaystyle\hat{x}$
$\displaystyle\leftarrow\mathop{\arg\min}\limits_{x\in\mathbb{R}^{N}}L_{\lambda}(x,\hat{v};u)$
(7) $\displaystyle\hat{v}$
$\displaystyle\leftarrow\mathop{\arg\min}\limits_{v\in\mathbb{R}^{N}}L_{\lambda}(\hat{x},v;u)$
(8) $\displaystyle u$ $\displaystyle\leftarrow u+(\hat{x}-\hat{v}),$ (9)

ADMM also gets a global convergence guarantee escorted if the two functions
$f(x)$ and $g(x)$ are both proper, closed, and convex and a saddle point
solution also exists.

Hereafter, we would like to express the ADMM iterations as two operators.

$\displaystyle
F(\tilde{x};\lambda)=\mathop{\arg\min}\limits_{x\in\mathbb{R}^{N}}\\{f(x)+\frac{\|x-\tilde{x}\|_{2}^{2}}{2\lambda^{2}}\\}$
(10) $\displaystyle
G(\tilde{v};\sigma)=\mathop{\arg\min}\limits_{v\in\mathbb{R}^{N}}\\{\frac{\|\tilde{v}-v\|_{2}^{2}}{2\sigma^{2}}+g(v)\\}$
(11)

where the operator $F$ corresponds to the inversion of the measurement model,
the denoising operator $G$ corresponds to the prior model of images and the
parameter $\sigma=\sqrt{\gamma}\lambda$ can be interpreted as the assumed
noise standard deviation. So $G$ can be conducted completely by an off-the-
shelf denoiser [29]. In our opinion, the very progress that the Plug-and-Play
algorithm has made is the separation of these two operators.

With much more advanced denoisers employed, this algorithm obtains state-of-
the-art performance [32, 31] that even comparable with end-to-end deep
learning methods, while preserves the advantage of applying to multiple tasks.
However, we can not acquire a corresponding prior model expression for the
given plug-in denoiser. Therefore, Plug-and-Play algorithms would not inherit
the well-established convergence properties of ADMM algorithms [1].

There are many following theoretical research for its convergence analysis
[26, 6, 24]. Most of them continue to pave along the fixed-point type
methodology. Briefly speaking, they mainly relies on nonexpansive or Lipschitz
continuous denoisers assumptions to construct a Banach contraction operator,
or form a Cauchy convergence sequence equally. However, most denoisers do not
obey this seemingly simple assumption, they would not give a reasonable
theoretical guarantee for the convergence of the practical PnP algorithm.

To bridge the gap between practical plug-in denoiser and theoretical analysis,
we propose to describe PnP iteration by a continuous stochastic differential
equation. This view endows us with a higher standpoint so that we can discuss
PnP convergence property via SDE solvability. The main contributions are as
follows:

  * •

We give the SDE description of the PnP algorithm. Furthermore, we show two
approaches to realize this transformation between PnP and SDE.

  * •

Then, we constructed a unified framework for PnP convergence analysis through
the solvability of the corresponding SDE description of the PnP.

  * •

Accordingly, we propose a much weaker condition, which is more applicable to
current advanced denoiser priors without additional constrains.

##  2 SDE description of PnP

Continuous descriptions of discrete algorithms could facilitate the decoupling
of theoretical analysis and technical implementation. We will see in the next
section that convergence analysis may take advantage of continuous form. Here,
we give a direct transformation between a simplified version of PnP and SDE,
and two other possible pathways to link residual Gaussian denoiser to the pure
stochastic part of Ito-type SDE.

###  2.1 Simplified PnP Problem Formulation

######  Definition 2.1 (PnP iteration).

In a simplified PnP framework without auxillary variable $u=u+(x-v)$, just the
two step iterations:

$\displaystyle{x^{t+1}}$
$\displaystyle=\mathop{\arg\min}\limits_{x\in\mathbb{R}^{N}}\\{f(x)+\frac{\|x-\tilde{x}\|_{2}^{2}}{2\lambda^{2}}\\}=h(v^{t},y)$
(12) $\displaystyle{v^{t+1}}$
$\displaystyle=\mathop{\arg\min}\limits_{x\in\mathbb{R}^{N}}\\{\frac{\|x^{t+1}-x^{t}\|_{2}^{2}}{2\sigma_{t}^{2}}-\log
P(x)\\}=D(x^{t+1},\sigma_{t})$ (13)

Or a more succinct iteration mapping:

$\displaystyle v^{t+1}=D(h(v^{t};y),\sigma_{t})$ (14)

The first step Eq. (12) is a close form solution function $h(\cdot)$ as long
as $f(\cdot)$ is differentiable, this step may operate the inverse mapping of
image restoration likes uperresolution, inpainting, demosic, etc. The second
step Eq. (13) is one step denoising $D(\cdot)$. The variance sequence
$\sigma_{t}$ are settled as hyperparameters beforehand, the adjust strategy
may be linear decay, exponential decay, etc.

Note that the reason why the proximal mapping in the second step Eq. (13) can
be considered as a Gaussian denoising process is that
$\mathop{\arg\min}\limits_{x\in\mathbb{R}^{N}}\\{\frac{\|x^{t+1}-x^{t}\|_{2}^{2}}{2\sigma_{t}^{2}}-\log
P(x)\\}$ resembles to a Maximum A Posteriori(MAP) problem of Gaussian
denoising, with measurement function being
$M(x^{t+1},x^{t},\sigma_{t})=\frac{\|x^{t+1}-x^{t}\|_{2}^{2}}{2\sigma_{t}^{2}}$,
where the expectation is $x^{t}$ and the variance is $\sigma_{t}$. And this
Gaussian denoising analogy firstly appeared in [29], which is the origination
of Plug-and-Play methods. So this is a solid evidance to support the Gaussian
noise assumption.

######  Assumption 2.2 (Gaussian noise in PnP denoising step).

The inversed transition kernel of Plug-and-Play denoising step is assumed to
follow a Gaussian distribution.

$\displaystyle P(x^{t+1}|v^{t+1})\sim\mathcal{N}(v^{t+1},\sigma_{t}).$ (15)

Similarly, [22] also adopted the Gaussian denoising in the score matching
paradigm to interpret RED (Regularization by Denoising) method[24], which is
one of the PnP (Plug-and-Play) variants. Denoising Score Matching gives the
Gaussian noise assumption for a single step as equation (10) in [30]:

$\frac{\partial\log
q_{\sigma}(\tilde{x}|x)}{\partial\tilde{x}}=\frac{1}{\sigma^{2}}(x-\tilde{x})$
(16)

This equation shows that the stein’s score function (i.e. gradient of log
density) equals to the Gaussian denoising.

###  2.2 Main Theorem

######  Theorem 2.3 (SDE description of PnP).

The discrete iteration of PnP algorithm in 2.1 can be described by a
continuous SDE as:

$\displaystyle dv_{t}=b(t,v_{t})dt-\sigma(t,v_{t})dW_{t}.$ (17)

Where $b(x,y)=h(x,y)-x$ is the drift term of the PnP-SDE, the diffusion term
$\sigma(t)$ only depends on time $t$, which is exactly the variance parameter
$\sigma_{t}$ of the Gaussian denoiser $D_{\sigma}(\cdot)$ of PnP.

###### Proof.

The difference of PnP iteration steps in definition 2.1 will be:

$\displaystyle{x^{t+1}}-v^{t}$ $\displaystyle=b(v^{t},y)$ (18)
$\displaystyle{v^{t+1}}-x^{t+1}$ $\displaystyle=D(x^{t+1},\sigma_{t})-x^{t+1}$
(19)

Where the function $b(v^{t},y)=h(v^{t},y)-v^{t}$ for the first step. And in
the second step, $D(x^{t+1},\sigma_{t})-x^{t+1}$ is the one step denoising
operation, and its inverse process follows Gaussian distribution according to
assumption 2.2.

$\displaystyle
x^{t+1}-{v^{t+1}}=x^{t+1}-D(x^{t+1},\sigma_{t})=x^{t+1}-P(v^{t+1}|v^{t+1})\sim\mathcal{N}(0,\sigma_{t}^{2})$
(20)

So we get
${v^{t+1}}-x^{t+1}=D(x^{t+1},\sigma_{t})-x^{t+1}\sim-\mathcal{N}(0,\sigma_{t}^{2})$.
Combining these two we can get
$v^{t+1}-v^{t}=b(v^{t},y)-\mathcal{N}(0,\sigma_{t}^{2})$, which is a discrete
difference equation.

Taking an infinitesimal time interval, this will transform to a continuous
equation that contains a stochastic term. $dV=b(V,y)dt+\sigma^{2}dW_{t}$,
where the $W_{t}$ is a standard Gaussian process (Wiener process). So this
resembles a Ito type Stochastic Differential Equation:

$\displaystyle dX_{t}=b(t,X_{t})dt+\sigma(t,X_{t})dW_{t}$ (21)

where $b_{i}(t,x),\sigma_{ij}(t,x)$ are the drift term and diffusion term in
SDE, respectively. By the definition of SDE, these two function should both be
Borel-measurable, and in this case $b(\cdot)$ take an extra input $y$ and
$\sigma_{ij}(t,x)$ only depends on $t$.

∎

###  2.3 Another Proof Based on Backward Markov Process

###### Proof of Theorem 2.3.

The Markov chain description of PnP iteration is natural because the
trajectory of intermediate variables are random variables and this is a
discrete process.

In a simplified PnP framework without auxillary variable $u=u+(x-v)$, just the
two step iterations:

$\displaystyle{x^{t+1}}$ $\displaystyle=h(v^{t},y)$ (22)
$\displaystyle{v^{t+1}}$ $\displaystyle=D(x^{t+1},\sigma_{t})$ (23)

Combining these two steps, we get the transformation function:

${v^{t+1}}=D(h(v^{t},y),\sigma_{t})$ (24)

And this transformation function decides the kernel of the corresponding
Markov chain $P(v^{t+1}|v^{t})$.

Then, the Markov Chain of can be transformed into a continuous Markov process
description by taking an infinitesimal time interval. And continuous Markov
process can be described by the Fokker-Plank equation, which is an
differential SDE.

$\frac{\partial v}{\partial t}=\nabla_{v}P(v^{t+1}|v^{t})={v^{t+1}}-v^{t},$
(25)

The corresponding intergration SDE is that:

$\displaystyle v_{t}$
$\displaystyle=v_{0}+\int_{0}^{t}\nabla_{v}P(v^{t+1}|v^{t})dt$ (26)
$\displaystyle=v_{0}+\int_{0}^{t}[x^{t+1}-v^{t}]dt+[v^{t+1}-x^{t+1}]dt,$ (27)
$\displaystyle=v_{0}+\int_{0}^{t}b(t,v_{t})dt-\sigma(t)dW_{t},$ (28)

where $dW_{t}$ is the standard Gaussian process (Wiener process),
$b(t,v_{t}),\sigma(t)$ are the drift term and diffusion term in SDE,
respectively. The third equivalence depends on the formulation of drift
function $b(\cdot)$ and the Gaussian denoising assumption 2.2.

∎

##  3 Convergence Analysis

With the SDE description of PnP, we can transform the convergence analysis of
PnP into an equivalent SDE solvability problem. There are two kinds of
solvability of SDE [15, Chapter 5], strong and weak solutions. We form a
unified framework for PnP convergence property by these two solvability,
including the traditional fixed-point or non-expansive ones as strong
solvability and our novel weak convergence conditions according to weak
solvability.

######  Proposition 3.1 (Bridge between PnP convergence and SDE solvability.).

If the SDE description of PnP is solvable, then it achieves to a certain
solution (strong solvability) or obey the probabilistic law of solutions (weak
solvability). Then, the corresponding PnP algorithm converges to that
stationary point (strong convergence) or the stationary distribution (weak
convergence).

However, we will show later on that it is non-trival to apply the SDE
solvability conditions to PnP convergence analysis. In a word, the strong
convergence condition about PnP mapping function $h(\cdot)$ would be different
with the standard Lipschitz condition by a constant multiplier, and the weak
convergence condition about PnP mapping function $h(\cdot)$ would be a relaxed
Lipschitz condition, instead of bounded.

###  3.1 Preliminary: Strong and Weak Solvability of Ito-type SDE

Given a typical Ito-type stochastic differential equation:

$dX_{t}=b(t,X_{t})dt+\sigma(t,X_{t})dW_{t},$ (29)

where $b_{i}(t,x),\sigma_{ij}(t,x)$ are Borel-measurable functions. And
$W_{t}$ is a standard Gaussian process(Wiener process), which is the main
reason why SDE suffers from the problem of solvability that is different to
ODE.

Briefly summarizing, the strong solvability requires a pathwise consistency
(i.e. a determined evolution path and only one terminal point as the
solution), and the strong solvability condition needs
$b_{i}(t,x),\sigma_{ij}(t,x)$ both to be Lipschitz continuous. The weak
solvability of SDE is under probability law, so the path would not consistent.
The weak condition only needs $b_{i}(t,x),\sigma_{ij}(t,x)$ both to be
bounded. For more details we would like to refer to [15, Chapter 5], and we
also exhibit them in the appendix.

###  3.2 Strong Convergence and Lipschitz Condition

First, we will define strong convergence for the PnP algorithm.

######  Definition 3.2 (Strong convergence).

Given PnP iteration

$x^{t+1}=D(h(x^{t};y),\sigma_{t}),$ (30)

and initial distribution $x^{0}\sim\xi$, strong convergence claims that the
intermediate variable sequence $\\{x^{0},x^{1},\cdots,x^{T}\\}$ is Cauchy
sequence.

Then, we derive the conditions for strong convergence of PnP algorithms as:

######  Theorem 3.3 (Strong Convergence Conditions).

Consider a PnP algorithm with iteration:

$\displaystyle x^{t+1}=D(h(x^{t};y),\sigma_{t}).$ (31)

If both the measurement model $h$ and denoiser $D$ satisfies Lipschitz
continuous condition,

$\displaystyle|h(x_{t_{1}};y)-h(x_{t_{2}};y)|$
$\displaystyle\leq(K+1)|x_{t_{1}}-x_{t_{2}}|,$ (32)
$\displaystyle|D(x_{t_{1}},\sigma_{t_{1}})-D(x_{t_{2}},\sigma_{t_{2}})|$
$\displaystyle\leq a|x_{t_{1}}-x_{t_{2}}|,$ (33)

Then, strong convergence holds for this PnP iteration.

###### Proof.

The discrete iteration of PnP algorithm

$x^{t+1}=D(h(x^{t};y),\sigma_{t})$ (34)

can be described by a continuous SDE as stated in 2.3:

$\displaystyle x^{t+1}-v^{t}=b(x^{t},y)+\sigma_{t}^{2}dW_{t}$ (35)

Where $b(x,y)=h(x,y)-x$ is the drift term of the PnP-SDE, the diffusion term
$\sigma_{t}$ only depends on time $t$, which is exactly the variance parameter
$\sigma_{t}$ of the Gaussian denoiser $D(\cdot,\sigma_{t})$ of PnP.

Substituting $h$ by the drift term $b(t,x)$ in SDE gives that

$|b(t,x)-b(t,y)|\leq K|x-y|.$ (36)

With the inverse interpretation of the pure Gaussian process for denoiser $D$,
we can constrain the diffusion term by the variance of this Gaussian process:

$|\sigma(t,x)-\sigma(t,y)|\leq a|x-y|.$ (37)

Then, according to Yamada and Watanabe’s Theorem [15, Proposition 5.2.31], the
corresponding SDE

$dX_{t}=b(t,X_{t})dt+\sigma(t,X_{t})dW_{t}$ (38)

maintains strong uniqueness for its strong solution. So that strong
convergence holds for the given PnP iteration. ∎

Above Lipschitz conditions with an extra scaling condition $a<1$ equal to
those in [26], and this scaling condition can be achieved by restricting PnP
mixing weight $\sigma$ in Eq. (19).

There is a relaxed Lipschitz condition in [6]:

$|D(x_{t_{1}},\sigma_{t_{1}})-x_{t_{1}}|\leq\sigma C$ (39)

where $C$ is a large enough constant. A simple deduction goes from Lipschitz’s
condition would give:

$\displaystyle|D(x_{t_{1}},\sigma_{t_{1}})-D(x_{t_{2}},\sigma_{t_{2}})|$
$\displaystyle\leq a|x_{t_{1}}-x_{t_{2}}|$ (40)
$\displaystyle\Rightarrow\qquad|D(x_{t_{1}},\sigma_{t_{1}})-x_{t_{1}}|$
$\displaystyle\leq\frac{(a-1)}{2}|x_{t_{1}}|$ (41)

So this relaxed Lipschitz condition for the denoiser remains to be a variant
of Lipschitz condition with the choice $x_{t_{2}}=0$ and an upper-bound for
the variable $|x_{t}|\leq\frac{2\sigma C}{a-1}$.

Hence, the strong converngence condition of Lipschitz bounded measurement
function and denoiser in PnP coincides with most previous theoretic analysis
of PnP convergence guarantee. And the deterministic evolution path of SDE
strong solution can be viewed as a direct generalization of the fixed point
analysis of Partial Differential Equations (PDE) described dynamic systems,
there is no doubt that a similar methodology would be adopted within broad
fixed-point-based convergence analysis.

These conditions bring too strong restrains on the denoiser, for example in
[25] the author propose to normalize the spectral of neural network layers to
construct the Lipschitz continuous denoiser. On the contrary, unlimited
denoisers perform quite well experimentally, even in the starting era of PnP-
ADMM algorithm[29, 32]. The gap between theory and application will be bridged
by the weak convergence condition in the following subsection.

###  3.3 Weak Convergence and Bounded Denoiser

First, we show the weak convergence definition according to the weak solution
in SDE [15, Definition 5.3.1]:

######  Definition 3.4 (Weak Convergence).

Weak convergence for PnP algorithm is a combination
$(X,W),(\Omega,\mathscr{F},P)$, where $W$ is an Brownian motion,
$(\Omega,\mathscr{F},P)$ is a probability space, and the distribution of $X$
and $W$ obey a sampling trajectory (a filtration) in $(\Omega,\mathscr{F},P)$.
If this combination was given, the iteration $\\{x^{0},x^{1},\cdots,x^{T}\\}$
would be finite valued and the final distribution $P(x^{T})$ remains the same
under probability law.

Weak convergence of PnP is defined as related to the distribution along the
dynamics. Next, we want to derive the existing condition of weak convergence.

######  Theorem 3.5 (Weak convergence conditions).

Consider the PnP iteration

$x^{t+1}=D(h(x^{t};y),\sigma_{t}),$ (42)

If the measurement model $h$ is relaxed Lipschitz continuous

$|h(x_{t})-x_{t}|\leq\kappa,$ (43)

while denoiser $D$ is only bounded

$\displaystyle|D(x_{t_{1}},\sigma_{t_{1}})|\leq\alpha,$ (44)

Then, weak convergence holds for this PnP iteration.

###### Proof.

The discrete iteration of PnP algorithm

$x^{t+1}=D(h(x^{t};y),\sigma_{t}),$ (45)

can be described by a continuous SDE as stated in Sec. 2.2:

$\displaystyle x^{t+1}-v^{t}=b(x^{t},y)+\sigma_{t}^{2}dW_{t}$ (46)

Where $b(x,y)=h(x,y)-x$ is the drift term of the PnP-SDE, the diffusion term
$\sigma_{t}$ only depends on time $t$, which is exactly the variance parameter
of the Gaussian denoiser $D(\cdot,\sigma_{t})$ of PnP.

Substituting the measurement model by the drift term in SDE via the equation
$h(x_{t})=b(x_{t})+x_{t-1}$ gives that:

$|b(x_{t})|\leq\kappa.$ (47)

The diffusion term is the inversion of the denoising process, so that is
bounded as well:

$\sigma\leq\alpha.$ (48)

Then we adopt Stroock & Varadhan Theorem [27], that bounded drift and
diffusion terms indicate the existence of a weak solution. Then, the
corresponding PnP iteration is weakly converged. ∎

Consequently, the measurement model of PnP should be Lipschitz continuous,
which is not a big deal for most image restoration tasks as stated in [26, 6].
The denoiser only needs to be bounded instead of Lipschitz continuous.

######  Remark 3.6 (Bounded Denoisers).

Deep denoisers with a bounded output layer, like Sigmoid and Tanh function,
would satisfy this weak condition. Even without an explicit bounding tool in
the output module, a well-functioning denoiser would be implicitly bounded,
since the desired images are within a target distribution embedded on the
manifold of all images [16, 18]. Outliers that diverge violently would be
punished so that the nature of denoising encourages bounded conditions.

###  3.4 Relation with Convergence Conditions for Markov Chain

We have viewed the PnP iteration sequence as a Markov chain as above.
Therefore, an inquiry would be, can we analyze the convergence of PnP by means
of convergence condition (detailed balance) of the Markov chain? Of course,
the convergence property of discrete Markov chain is pretty mature, and even
its applications are fruitful. For example, Markov Chain Monte Carlo algorithm
[19] is founded on detailed balance, so that its resulting samples follow that
target distribution.

Some research in statistical physics [10, 23] has already investigated the
relationship between the solution of the Fokker-Planck equation and detailed
balance. Their resulting convergence condition is bounded eigenvalues for the
coefficients of the Fokker-Planck equation, which is consistent with the weak
solution of SDE and weak convergence of PnP as above. We can concrete PnP
convergence in an analog of detailed balance, that the drift term would
contribute either tiny enough or pure Gaussian noise which can be handled
thoroughly by the denoiser.

##  4 Related Works

In the founding work of the PnP algorithm [29], the authors did not discuss
the theoretical convergence properties of the Plug-and-Play framework. They
just mentioned that the convergence guarantee of ADMM algorithms would succeed
in PnP-ADMM. Following theoretical research [26] of the PnP algorithm
presented the sufficient conditions of the denoising operator that strictly
guarantee the convergence of the PnP algorithm. The non-expansive condition of
the denoiser makes the denoising operator a proximal mapping. And then, we can
obtain an explicit expression of the convex prior function according to the
denoiser.

The non-expansive property is too strong to be satisfied by commonly used
denoisers like NLM and BM3D. [6] gave a clear analysis of the fixed-point
convergence of the PnP framework by constructing a Cauchy sequence of
variables during the iteration in a Banach space. This fixed-point convergence
relies on the Lipschitz continuous property of the likelihood function and the
denoiser. However, they achieved this only under diminishing stepsizes. Thus,
non-expansive condition can easily converted into Lipschitz continuous via a
scale factor.

In [25], the author constructed a Lipschitz-bounded deep denoiser to provide
provably convergence property for the PnP method. PnP-SGD [17] uses a
stochastic gradient descent method to solve the PnP problem. Its convergence
depends on the contractive residual condition on the denoiser.

The above theoretical analyses all imposed seemingly strong constraints on the
denoiser. In the meantime, experimental results [32, 31] surprisingly rush
ahead of the corresponding theoretical researches. Apparently, there is a huge
gap between theoretical research and practical experiments. Our novel
interpretation of the PnP algorithm by SDE could possibly bypass the
previously indispensable conditions on denoisers, with only a mild assumption
of bounded outputs of the denoising algorithm embedded.

##  5 Examples

The supportive example is an experimental comparison between PnP algorithms
with Lipschitz continuous denoiser and bounded denoiser. And the
counterexample of unbounded denoiser is presented by deduction.

###  5.1 Lipschitz Continuous Denoiser Or Bounded Denoiser

In the baseline work of Lipschitz continuous deep denoiser within PnP[25], the
authors give the relationship between different $\alpha$ as the raio parameter
and contraction factor. With $\sigma^{2}=\alpha\gamma$ and fixed $\gamma$, the
parameter $\alpha$ is then closely related to the variance parameter $\sigma$.
So we adopted the same experimental setting up as in [25], we use the
relationship between the ratio parameter $\alpha$ and PSNR.

We compare PnP with the vanila DnCNN and with the Real Spectral Normalized[25]
DnCNN as shown in Tab. 1. The previous theoretical analysis in [25] considered
the Lipschitz continuous condition of DnCNN is the key for PnP-CNN to
converge. But in the experimental results below we show that although with
little performance drop, the PnP-CNN converges as well, instead of diverges.

Table 1: PSNR values for convergence of vanila denoiser(PnP-CNN) and Lipschitz
continuous denoiser(PnP-RSN-CNN) with varying noise variance related ratio
parameter $\alpha$.
$\alpha$ | 0.001 | 0.003 | 0.01 | 0.03 | 0.1 | 0.3 | 1.0
---|---|---|---|---|---|---|---
PnP-RSN-CNN | 4.74984 | 5.33115 | 6.75710 | 9.11250 | 12.95924 | 17.1120 | 19.47662
PnP-CNN | 3.37694 | 4.10075 | 5.63357 | 7.86317 | 12.23303 | 16.9121 | 19.39487

We would also present in Fig. 4 the evolution dynamics of above two methods
when $\alpha=1.0$. The resulting PSNR curves of these two methods are close
(even superposition) with each other, and hard to tell between them. We can
see that the main advantage of Lipschitz continuous denoiser is their smoother
trajectories. As a conclusion, with/without Lipschitz continuous denoisers,
the PnP algorithm converges, which indicates that the Lipschitz condition is
not a necessary guarantee for PnP convergence.

Figure 1: PSNR curves during PnP iteration comparing two methods. Orange lines
show the results of PnP with Lipschitz continuous denoisers (PnP-RSN-CNN),
while Indigo lines are with Lipschitz discontinuous denoisers (PnP-CNN).
Lipschitz continuous denoiser ensures a smoother trajectory. In general, both
Lipschitz continuous and discontinuous denoiser converges.

###  5.2 Unbounded denoiser

A counterexample of divergence PnP iteration may employ an unbounded denoiser.

######  Remark 5.1 (PnP Divergence).

PnP iteration $v^{t+1}=D(h(v^{t};y),\sigma_{t})$ with an unbounded denoiser
$D(\cdot)$ would diverge unless with a contractive measurement function
$h(\cdot)$.

Usually, the measurement function would not perform as contractive operator,
thus do no good to the convergence of PnP. Hence, PnP with unbounded denoiser
would diverge in general.

An unbounded denoiser would not perform well in denoising tasks,
instinctively. At least, a well-performed denoiser is bounded by these pixel
values’ range. Even a deep denoiser without the explicit restriction of its
output range would follow an implicit bounded strategy if it was trained on an
elaborate dataset. Since the training dataset is bounded beforehand, extreme
pixel values are doomed to be penalized for the extreme gain of loss.

######  Remark 5.2 (Bounded Denoiser).

With empirical risk minimization training of deep denoiser, the expected risk
would be the approximating error of training, the exceed risk corresponds to
the generalization error.

$\displaystyle\ell_{empirical}=\ell_{expected}+\ell_{exceed}.$ (49)

With over-parameterized deep neural network and sufficient amount of data for
training, the expected risk would approximate zero. For Gaussian distributed
noise, the exceed risk would be constrained between the range of
$[\mu-3\sigma,\mu+3\sigma]$ with parabability of 99.74%. Note that we set the
sequence $\sigma_{t}$ to decay as $t$ grows. So the bounded denoiser within
range $[-256-3\sigma_{T},256+3\sigma_{T}]$ is guaranteed with high
probability, where $T$ is the stopping time of PnP.

##  6 Conclusion

To close the gap between practical PnP with advanced denoiser prior and
theoretical convergence analysis, we considered the continuous counterpart of
the PnP algorithm, and related it to stochastic differential equations.

Then, with the aid of the solvability theory of SDE, we give a unified
framework for convergence analysis of the PnP algorithm. Strong solution
corresponds to Lipschitz’s continuous condition in previous research. With the
weak solution under probability law, we proposed weak convergence for PnP
analysis. Then, we obtained a much weaker convergence condition of only
bounded denoisers and Lipschitz continuous measurement models. Finally, we
give a few counterexamples to further illustrate our convergence theory for
PnP algorithms.

Most current SOTA denoisers are not Lipschitz continuous but bounded
inherently, provided that the denoising task functions as expected. So weak
conditions are more suitable for practical denoisers than previous ones, which
would further potentiate a more confident usage of the PnP algorithm with more
advanced bounded denoisers.

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##  Appendix A Related Works

Other than Plug-and-Play ADMM framework, there are also regularization by
denoiser (RED) and consensus equilibrium (CE) frameworks as the variants of
PnP. We also review them and beyond for the completeness of the related works.

###  A.1 Convergence Analysis of RED

Instead of proximal mapping theory in PnP-ADMM, the regularization by denoiser
[24] framework offered a systematic use of denoisers for regularization in
inverse problems. The regularization term is

$\rho(x)=\frac{1}{2}x^{T}[x-f(x)],$ (50)

where the denoiser takes the candidate’s image as input. Intuitively, the
penalty term is proportional to the inner product between the image and its
denoising residual. With the aid of the RED framework, the authors proposed
the following necessary conditions on the denoiser $f(x)$ to guarantee
convergence:

  * •

Local Homogeneity: $\forall
x\in\mathbb{R}^{n},f((1+\epsilon)x)=(1+\epsilon)f(x)$ for sufficiently small
$\epsilon>0$.

  * •

Differentiability: The denoiser $f(\cdot)$ is differentiable where $\nabla f$
denotes its Jacobian.

  * •

Jacobian Symmetry: $\nabla f(x)^{T}=\nabla f(x),\forall x\in\mathbb{R}^{n}$.

  * •

Strong Passivity: The spectral radius the Jacobian satisfies $\eta(\nabla
f(x))\leq 1$.

There are two highlighting researches about improving RED. [22] interpreted
the denoiser using score-matching theory, and bypassed the constraint of
Jacobian symmetry. [8] introduced the assumption of demi-contractive
denoisers, which is a relaxation compared to the previous convergence analysis
under the fixed-point theory.

###  A.2 Other Convergence Analysis of PnP

Another mainstream framework is consensus equilibrium. In [4], the authors
used the convergence of Mann iteration during consensus equilibria solving to
guarantee the convergence of the corresponding PnP algorithm.

Deep unfolding methods [12] also constitute a particular viewpoint of the PnP
algorithm. [9] centered around the denoising deep neural network and treated
the observation model as a priori. Then, they proposed a Lyapunov function-
based convergence guarantee. This Lyapunov condition, again, assured that the
denoiser should be Lipschitz bounded.

The convergence of other variable splitting algorithms, like ISTA [28, 7], AMP
[5] etc., are also analyzed via fixed-point mathematic tools.

##  Appendix B Experiments

According to previous theoretical convergence analysis[26], the vanila CNN is
expansive and without Lipschitz continuous condition, so vanila PnP-CNN should
diverge. But in fact the vanila PnP-CNN converges in so many previous works
and performs pretty well in downstream application of image restoration. We
added extra experiments to investigate the performance With/Without Lipschitz
continuous condition over different noise variance.

###  B.1 PnP-SDE Implementation

We would call our proposed algorithm PnP-SDE. To keep consistency with the SDE
description, we need to add noise during iteration in order to converge in the
sense of probability law instead of pathwise. Because the denoising process of
a give trained denoiser is deterministic, without stochastic trajectory during
denoising. We should elucidate the denoiser via its variance $\sigma$.

Normally, PnP-ADMM algorithms would use a scale factor to control the dynamics
of PnP iteration [29, 6, 26, 13]. We can correspond the scale factors to the
mixing weight and the Lagrangian multiplier. Intuitively, this progressive
evolution would stratify the processing of different granular features so that
each feature’s granules would be less degraded by more intense noise levels.

However, from the viewpoint of the philosophy of fundamentalism, [25] insists
that we should analyze the PnP convergence property without these scale
factors. To some extent, this research is more like an ablation variant of the
PnP algorithm, which violates the usual setting for PnP. Our formulation of
the SDE description for PnP follows the usual setting, which also intends to
satisfy the diffusion analog of the inverse of denoising.

For the demonstration of counterexample, Lipschitz discontinuous denoiser, we
would adopt the same fundamentalism-style setting that discards the
confounding of scaling factors. More detailed experiments are as follows.

###  B.2 Weak Convergence

Weak convergence with random noise added during PnP iteration is similar to
the implementation in [13]. However, a distinctive difference is that our PnP-
SDE models the noise within each iteration using a precise variance parameter
$\sigma$, while [13] employed bias-free denoiser [20] to achieve a
generalization over different noise levels implicitly. So we implement our
proposed PnP-SDE using DnCNN denoiser [32, 31] with explicit variance $\sigma$
as an extra channel of inputs.

Figure 2: PSNR curves during PnP iteration. Purple lines are under strong
convergence, and they clearly demonstrate pathwise consistency. These lines
with multiple colors are under weak convergence so that they go through
different trajectories. Weak convergence also converges at the end of each
trajectory, and achieve a bonus on performance. Figure 3: SSIM curves during
PnP iteration. The instability at the beginning is due to the independence of
SSIM value and noise level. Purple lines are under strong convergence and are
pathwise unique. These lines with multiple colors are under weak convergence,
with different trajectories. Weak convergence also gains a bonus on
performance. Table 2: PSNR and SSIM results for strong and weak convergence.
The results shows that both of them converge, although with a little
performace gap.
| PSNR | SSIM
---|---|---
Strong | 20.5232 | 0.7207
Weak | 20.6417 | 0.7361

In Fig. 2 and Fig. 3, we can see that PnP converges both under strong and weak
convergence situations. While strong convergence is pathwise unique, weak
convergence is in the sense of probability law, with different trajectories.
Surprisingly, as analyzed before, noises added in weak convergence help the
PnP algorithm to escape from local minima. Therefore, this achieves extra
performance gains, as shown in Tab. 2.

###  B.3 Lipschitz discontinuous denoiser

A simple question would be, is there a counterexample that transgresses
previously proposed conditions on denoisers, but still converges.

Firstly, we can unify previously proposed constraints on the denoisers using
Lipschitz bounded [25]. Other equivalent expressions like non-expansive [26],
contractive property, demi-contractive [8], can be view as the a scaled
version of it.

[25] proposed to constrain the denoiser’s Lipschitz constant by spectral
normalization and 1-Lipschitz non-linearity. Even in their experiments, we can
see that the PnP algorithm without Lipschitz continuous denoisers also
converges to the desired image distribution. As they demonstrated, there is
only a tiny difference between the PSNR of Lipschitz continuous and
discontinuous ones.

Figure 4: Performance curves during PnP iteration under weak convergence.
Orange lines show the results of PnP with Lipschitz continuous denoisers,
while indigo lines are with Lipschitz discontinuous denoisers. Lipschitz
continuous denoiser ensures a slightly smoother trajectory and also shows a
tiny performance gain. In general, both Lipschitz continuous and discontinuous
denoiser converges under weak probability uniqueness.

In fact, practical PnP algorithms up-to-date with deep denoisers have already
given fruitful expositions without divergence problems. Here, we show the
performance curve under weak convergence in Fig. 4. We can see that
Lipschitz’s continuous denoiser makes their corresponding trajectories
smoother. And this is consistent with what we anticipated. A tiny performance
gain is also reasonable for the robustness of the Lipschitz continuous
denoiser. However, this is a bonus, not the main dish. Apparently, both
Lipschitz continuous and discontinuous denoisers converge to the terminal
without oscillation. Note that the setting in [25] discards the scale factor,
which we also adopt here so that we would ignore the disturbance of scaling.

###  B.4 Unbounded denoiser

For the existence of SDE’s weak solution, we need a mild condition of bounded
drift and diffusion term. Therefore, the convergence of PnP-SDE iteration can
be guaranteed by Lipschitz continuous drift term and bounded diffusion term.
This condition is so loose that a counterexample should employ an unbounded
denoiser. Apparently, it would diverge under this circumstance. An unbounded
denoiser would not perform well in denoising tasks, instinctively.

At least, a well-performed denoiser is bounded by these pixel values’ range.
Even a deep denoiser without the explicit restriction of its output range
would follow an implicit bounded strategy if it was trained on an elaborate
dataset. Since the training dataset is bounded beforehand, extreme pixel
values are doomed to be penalized.

So we have to stop after the above deduction because unbounded denoiser exceed
the range of practical implementations.

##  Appendix C SDE Strong Solvability

Given a typical stochastic differential equation

$dX_{t}=b(t,X_{t})dt+\sigma(t,X_{t})dW_{t},$ (51)

where $b_{i}(t,x),\sigma_{ij}(t,x)$ are Borel-measurable functions. The
definition of a strong solution for SDE is as follows:

######  Definition C.1 (Strong solution).

Given a probability space $(\Omega,\mathscr{F},P)$, the fixed Brownian motion
$W$ and initial condition $\xi$, a strong solution of the stochastic
differential equation is a process $X=\\{X_{t};0\leq t<\infty\\}$ with
continuous sample paths and with the following properties:

  * •

X is adapted to the filtration $\\{\mathscr{F}_{t}\\}$,

  * •

$P[X_{0}=\xi]=1$,

  * •

$P[\int_{0}^{t}\\{|b_{i}(s,X_{s})|+\sigma_{ij}^{2}(s,X_{s})\\}ds<\infty]=1$
holds for every $1\leq i\leq d,1\leq j\leq r$ and $0\leq t<\infty$,

  * •

the integral

$X_{t}=X_{0}+\int_{0}^{t}b(s,X_{s})ds+\int_{0}^{t}\sigma(s,X_{s})dW_{s},$ (52)

holds almost surely, for $0\leq t<\infty$.

Next, we would investigate the existence and uniqueness of strong solutions
for a given SDE.

######  Theorem C.2 (Strong Pathwise Uniqueness).

Suppose that the coefficients $b(t,x),\sigma(t,x)$ are locally Lipschitz-
continuous in the space variable; i.e., for every integer $n\leq 1$ there
exists a constant $K_{n}>0$ such that for every $t\geq 0,\|x\|\leq n$ and
$\|y\|\leq n$:

$\|b(t,x)-b(t,y)\|+\|\sigma(t,x)-\sigma(t,y)\|\leq K_{n}\|x-y\|.$ (53)

Then strong uniqueness holds.

######  Theorem C.3 (Strong existence).

Suppose that the coefficients $b(t,x),\sigma(t,x)$ satisfy the global
Lipschitz and linear growth conditions

$\displaystyle\|b(t,x)-b(t,y)\|+\|\sigma(t,x)-\sigma(t,y)\|\leq K\|x-y\|,$
(54) $\displaystyle\|b(t,x)\|^{2}+\|\sigma(t,x)\|^{2}\leq K^{2}(1+\|x\|^{2}).$
(55)

for every $0\leq t<\infty,x\in\mathbb{R}^{d},y\in\mathbb{R}^{d}$, where $K$ is
a positive constant. On some probability space $(\Omega,\mathscr{F},P)$, let
$\xi$ be an $\mathbb{R}^{d}$-valued random vector, independent of the
$r$-dimensional Brownian motion $W=\\{W_{t},\mathbb{F}_{t}^{W},0\leq
t<\infty\\}$, and with finite second moment:

$E\|\xi\|^{2}<\infty.$ (56)

Let $\\{\mathscr{F}_{t}\\}$ be the filtration. Then there exists a continuous,
adapted process $X=\\{X_{t},\mathscr{F}_{t},0\leq t<\infty\\}$ which is a
strong solution of the stochastic differential equation relative to $W$, with
initial condition $\xi$.

Moreover, this process is square-integrable: for every $T>0$, there exists a
constant $C$, depending only on $K$ and $T$, such that

$E\|X_{t}\|^{2}\leq C(1+E\|\xi\|^{2})e^{Ct},0\leq t<T.$ (57)

There is a further proposition on the uniqueness that disjoins the condition
of drift term and diffusion term.

######  Proposition C.4 (Yamada & Watanabe).

Let us suppose that the coefficients of the one-dimensional equation $(d=r=1)$

$dX_{t}=b(t,X_{t})dt+\sigma(t,X_{t})dW_{t}$ (58)

satisfy the conditions

$\displaystyle|b(t,x)-b(t,y)|\leq K|x-y|$ (59)
$\displaystyle|\sigma(t,x)-\sigma(t,y)|\leq a|x-y|$ (60)

for every $0\leq t<\infty$ and $x\in\mathbb{R},y\in\mathbb{R}$, where $K$ is a
positive constant and $a:[0,\infty)\rightarrow[0,\infty)$ is a strictly
increasing function with $a(0)=0$ and

$\int_{(0,\varepsilon)}a^{-2}(u)du=\infty,\forall\varepsilon>0.$ (61)

Then strong uniqueness holds for the stochastic differential equation.

##  Appendix D SDE Weak Solvability

######  Definition D.1 (Weak solution).

A weak solution of a given stochastic differential equation is a triple
$(X,W),(\Omega,\mathscr{F},P),\\{\mathscr{F}_{t}\\}$, where

  * •

$(\Omega,\mathscr{F},P)$ is a probability space, and $\\{\mathscr{F}_{t}\\}$
is a filtration of sub-$\sigma$-fields of $\mathscr{F}$ satisfying the usual
conditions,

  * •

$X=\\{X_{t},\mathscr{F}_{t};0\leq t<\infty\\}$ is a continuous, adapted
$\mathbb{R}^{d}$-valued process, $W=\\{W_{t},\mathscr{F}_{t};0\leq
t<\infty\\}$ is an $r$-dimensional Brownian motion,

  * •

$P[\int_{0}^{t}\\{|b_{i}(s,X_{s})|+\sigma_{ij}^{2}(s,X_{s})\\}ds<\infty]=1$
holds for every $1\leq i\leq d,1\leq j\leq r$ and $0\leq t<\infty$,

  * •

the integral

$X_{t}=X_{0}+\int_{0}^{t}b(s,X_{s})ds+\int_{0}^{t}\sigma(s,X_{s})dW_{s},$ (62)

holds almost surely, for $0\leq t<\infty$.

A weak solution of SDE is related to the distribution along the dynamics. We
would remark here that this triple
$(X,W),(\Omega,\mathscr{F},P),\\{\mathscr{F}_{t}\\}$ have to be considered as
the solution itself, not assumptions.

######  Definition D.2 (Weak Probability Uniqueness).

We say that uniqueness in the sense of probability law holds for a given
stochastic equation if, for any two weak solutions
$(X,W),(\Omega,\mathscr{F},P),\\{\mathscr{F}_{t}\\}$, and
$(\tilde{X},\tilde{W}),(\tilde{\Omega},\tilde{\mathscr{F}},\tilde{P}),\\{\tilde{\mathscr{F}_{t}}\\}$,
with the same initial distribution $\Gamma$, i.e.,

$P[X_{0}\in\Gamma]=\tilde{P}[\tilde{X_{0}}\in\Gamma],\forall\Gamma\in\mathscr{R}(\mathbb{R}^{d}),$
(63)

the two processes $X,\tilde{X}$ have the same law.

######  Proposition D.3 (Weak existence).

Consider the stochastic differential equation

$dX_{t}=b(t,X_{t})dt+dW_{t},0\leq t\leq T,$ (64)

where $T$ is a fixed positive number, $W$ is a $d$-dimensional Brownian
motion, and $b(t,x)$ is a Borel-measurable, $\mathbb{R}^{d}$-valued function
on $[0,T]\times\mathbb{R}^{d}$ which satisfies

$\|b(t,x)\|\leq K(1+\|x\|),0\leq t\leq T,x\in\mathbb{R}^{d}$ (65)

for some positive constant $K$. For any probability measure $\mu$ on
$(\mathbb{R}^{d},\mathscr{R}(\mathbb{R}^{d}))$, the stochastic differential
equation has a weak solution with initial distribution $\mu$.

######  Theorem D.4 (Girsanov Theorem).

Given a probability space $(\Omega,\mathscr{F},P)$ and a $d$-dimensional
Brownian motion
$W=\\{W_{t}=(W_{t}^{(1)},\cdots,W_{t}^{(d)}),\mathscr{F}_{t},0\leq
t<\infty\\}$ defined on it, with $P[W_{0}=0]=1$. Let $\mathscr{F}_{t}$,
$X=\\{X_{t}=(X_{t}^{(1)},\cdots,X_{t}^{(d)}),\mathscr{F}_{t},0\leq
t<\infty\\}$ be a vector of measurable, adapted square-integrable processes.
We set

$Z_{t}(X)\triangleq\exp[\sum_{i=1}^{d}\int_{0}^{t}X_{s}^{(i)}dW_{s}^{(i)}-\frac{1}{2}\int_{0}^{t}\|X_{s}\|^{2}ds].$
(66)

Assume that $Z(X)$ is a martingale. Define a process
$\tilde{W}=\\{\tilde{W}_{t}=(\tilde{W}_{t}^{(1)},\cdots,\tilde{W}_{t}^{(d)}),\mathscr{F}_{t},0\leq
t<\infty\\}$ by

$\tilde{W}_{t}^{(i)}\triangleq W_{t}^{(i)}-\int_{0}^{t}X_{s}^{(}i)ds,1\leq
i\leq d,0\leq t<\infty.$ (67)

For each fixed $T\in[0,\infty)$, the process
$\\{\tilde{W}_{t},\mathscr{F}_{t},0\leq t\leq T\\}$ is a $d$-dimensional
Brownian motion on $(\Omega,\mathscr{F}_{T},\tilde{P}_{T})$.

###### Proof.

We begin with a $d$-dimensional Brownian family
$X=\\{X_{t},\mathscr{F}_{t},0\leq t\leq
T\\},(\omega,\mathscr{F}),\\{P^{x}\\}_{x\in\mathbb{R}^{d}}$. Let

$Z_{t}(X)\triangleq\exp[\sum_{i=1}^{d}\int_{0}^{t}X_{s}^{(i)}dW_{s}^{(i)}-\frac{1}{2}\int_{0}^{t}\|X_{s}\|^{2}ds].$
(68)

be a martingale under each measure $P^{x}$, so the Theorem D.4 implies that,
under $Q^{x}$ given by $(dQ^{x}/dP^{x})=Z_{T}$, the process

$W_{t}\triangleq X_{t}-X_{0}-\int_{0}^{t}b(s,X_{s})ds,0\leq t\leq T$ (69)

is a Brownian with $Q^{\mu}(A)\triangleq\int_{\mathbb{R}^{d}}Q^{x}(A)\mu(dx)$,
the triple $(X,W),(\Omega,\mathscr{F},P),\\{\mathscr{F}_{t}\\}$ is a weak
solution of the SDE. ∎

######  Theorem D.5 (Stroock & Varadhan (1969)).

Consider the stochastic differential equation

$dX_{t}=b(X_{t})dt+\sigma(X_{t})dW_{t}$ (70)

where the coefficients $b_{i},\sigma_{ij}:\mathbb{R}^{d}\rightarrow\mathbb{R}$
are bounded and continuous functions. Corresponding to every initial
distribution $\mu$ on $\mathscr{R}(\mathbb{R}^{d})$ with

$\int\limits_{\mathbb{R}^{d}}\|x\|^{2m}\mu(dx)<\infty,$ (71)

for some $m>1$, there exists a weak solution of the SDE.

###### Proof.

For integers $j\geq 0,n\geq 1$ we consider the dyadic rationals
$t_{j}^{(n)}=j2^{-n}$ and introduce the functions $\psi_{n}(t)=t_{j}^{(n)}$;
$t\in[t_{j}^{(n)},t_{j+1}^{n})$. We define the new coefficients

$\displaystyle b^{(n)}(t,y)\triangleq
b(y(\psi_{n}(t))),\sigma^{(n)}(t,y)\triangleq\sigma(y(\psi(t)));$ (72)
$\displaystyle 0\leq t<\infty,y\in C[0,\infty)^{d},$ (73)

which are progressively measurable functionals. Now let us consider on some
probability space $(\Omega,\mathscr{F},P)$ an $r$-dimensional Brownian motion
$W=\\{W_{t},\mathscr{F}_{t}^{W};0\leq t<\infty\\}$ and an independent random
vector $\xi$ with the given initial distribution $\mu$, and let us construct
the filtration $\\{\mathscr{F}_{t}\\}$. For each $n\geq 1$, we define the
continuous process $X^{(n)}=\\{X_{t}^{(n)},\mathscr{F}_{t};0\leq t<\infty\\}$
by setting $X_{0}^{(n)}=\xi$ and then recursively:

$X_{t}^{(n)}=X_{t_{j}^{(n)}}^{(n)}+b(X_{t_{j}^{(n)}}^{(n)})(t-t_{j}^{(n)})+\sigma(X_{t_{j}^{(n)}}^{(n)})(W_{t}-W_{t_{j}^{(n)}});$
(74)

for $j\geq 0,t_{j}^{(n)}<t\leq t_{j+1}^{(n)}$. Then $X^{(n)}$ solves the
functional stochastic integral equation

$X_{t}^{(n)}=\xi+\int_{0}^{t}b^{(n)}(s,X^{(n)})ds+\int_{0}^{t}\sigma^{(n)}(s,X^{(n)})dW_{s};$
(75)

for $0\leq t<\infty$. Fix $0<T<\infty$. We use the inequality

$\sup\limits_{n\geq 1}E\|X_{t}^{(n)}-X_{s}^{(n)}\|^{2m}\leq
C(1+E\|\xi\|^{2m})(t-s)^{m};$ (76)

where $0\leq s<t\leq T$, $C$ is a constant depending only on $m,T,$ the
dimension $d$, and the bound on $\|b\|^{2}+\|\sigma\|^{2}$. Let
$P^{(n)}\triangleq P(X^{(n)})^{-1};n\geq 1$ be the sequence of probability
measures induced on $(C[0,\infty)^{d},\mathscr{R}(C[0,\infty)^{d}))$ by these
processes; and this sequence is tight. We may then assert by the Prohorov
theorem, relabeling indices if necessary, that the sequence
$\\{P^{(n)}\\}_{n=1}^{\infty}$ converges weakly to a probability measure
$P^{\ast}$ on this canonical space. ∎