# Perturbation of an $\alpha$-stable type stochastic process by a pseudo-
gradient

Mykola Boiko, Mykhailo Osypchuk

(Vasyl Stefanyk Precarpathian National University,
57, Shevchenko str, 76018 Ivano-Frankivsk, Ukraine)

###### Abstract

We consider a Markov process defined by some pseudo-differential operator of
an order $1<\alpha<2$ as the process generator. Using a pseudo-gradient
operator, that is, the operator defined by the symbol
$i\lambda|\lambda|^{\beta-1}$ with some $0<\beta<1$, the perturbation of the
Markov process under consideration by the pseudo-gradient with a multiplier,
which is integrable at some great enough power, is constructed. Such
perturbation defines a family of evolution operators, properties of which are
investigated. A corresponding Cauchy problem is considered. 00footnotetext:
Keywords: $\alpha$-stable process, perturbation, pseudo-gradient, pseudo-
processMSC2010: 60G52

##  1 Introduction

Let $d$ be some fixed positive integer number. By $\mathbb{R}^{d}$ we denote
the real $d$-dimensional Euclidean space. As usual, we denote by
$(\cdot,\cdot)$ the inner product and by $|\cdot|$ the norm in
$\mathbb{R}^{d}$ (we use the last notation for denoting the absolute value of
a real number and the module of a complex number).

Let us consider a family of pseudo-differential operators $(A(t,x))_{t\geq
0,x\in\mathbb{R}^{d}}$ defined by the symbols
$(a(t,x,\lambda))_{\lambda\in\mathbb{R}^{d}}$ for every $t\geq 0$,
$x\in\mathbb{R}^{d}$. That is,

$A(t,x)f(x)=\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}a(t,x,\lambda)F(\lambda)e^{i(\lambda,x)}\,\mathrm{d}\lambda,\quad
t\leq 0,\ x\in\mathbb{R}^{d},$

where $F$ is the Fourier transform of the function $f$:
$F(\lambda)=\int_{\mathbb{R}^{d}}f(x)e^{-i(\lambda,x)}\,\mathrm{d}x$,
$\lambda\in\mathbb{R}^{d}$. Note that the function $a$ might be complex
valued.

We assume the conditions from [5, Ch. 4] about the function $a$ and the
operator $A$, that is, (we denote by $Lf(r,\cdot)(x)$ the result of the
operator $L$ used on $f(r,x)$ as the function of $x$)

  1. $A_{1}$)

The function $a$ is homogeneous of a degree $\alpha\in(1,2)$ with respect to
the variable $\lambda$ and $\operatorname{Re}a(t,x,\lambda)\geq a_{0}>0$ for
all $t\geq 0$, $x,\lambda\in\mathbb{R}^{d}$, $|\lambda|=1$.

  2. $A_{2}$)

The function $a$ has $N\geq 2d+3$ (it is some natural number) continuous
derivatives in $\lambda$ for all $\lambda\neq 0$, and

$|\partial^{\varkappa}a(t,x,\cdot)(\lambda)|\leq
C_{N}|\lambda|^{\alpha-|\varkappa|},$
$|\partial^{\varkappa}[a(t,x,\cdot)-a(s,y,\cdot)](\lambda)|\leq
C_{N}\left(|x-y|^{\gamma}+|t-s|^{\gamma/\alpha}\right)|\lambda|^{\alpha-|\varkappa|}$

for all multi-indexes $\varkappa$ with $|\varkappa|\leq N$,
$x,y,\lambda\in\mathbb{R}^{d}$, $\lambda\neq 0$, $s,t\in[0,+\infty)$. Here
$\gamma\in(0,1)$ is some constant.

  3. $A_{3}$)

In the representing

$A(t,x)f(x)=\int_{\mathbb{R}^{d}}\Omega\left(t,x,\frac{h}{|h|}\right)\frac{f(x)-2f(x-h)+f(x-2h)}{|h|^{d+\alpha}}\,\mathrm{d}h,$

the function $\Omega(t,x,\cdot)$ is even and non-negative.

######  Remark 1.

Assumptions $A_{1}$, $A_{2}$, $A_{3}$ coincide with assumptions ($A_{41}$),
($A_{42}$), ($A_{44}$) from [5], respectively (see pages 266 and 294 there).

Theorem 4.3 from [5] (see also Theorem 4.1 there) states that there exists a
bounded non-terminating strict Markov process $(x(t))_{t\geq 0}$ without
second kind discontinuities and the fundamental solution $(g(s,x,t,y))_{0\leq
s<t,x,y\in\mathbb{R}^{d}}$ to the equation

$\frac{\partial}{\partial t}u(t,x)+A(t,x)u(t,\cdot)(x)=0,\quad t>0,\
x\in\mathbb{R}^{d}$

is its transition probability density. The function $g$ can be constructed by
the parametrix method (see [5, Sec. 4.1.4]).

If the function $a$ is defined by the equality

$a(t,x,\lambda)=c|\lambda|^{\alpha},\quad t\geq 0,\ x\in\mathbb{R}^{d},\
\lambda\in\mathbb{R}^{d},$

where $c>0$ is some constant, the corresponding Markov process is an isotropic
$\alpha$-stable process. The function $g$ can be presented by the equality

$g(s,x,t,y)=\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}\exp\\{i(x-y,\lambda)-c(t-s)|\lambda|^{\alpha}\\}\mathrm{d}\lambda$

in this case. The operator $A$ (it does not depend on $t$ and $x$) is the
generator of this process. This is the simplest example of the processes
considered here. Therefore we name the process $(x(t))_{t\geq 0}$ by the
$\alpha$-stable type process.

Choosing some $0<\beta\leq 1$, let us denote the $\beta$-order ‘‘pseudo-
gradient’’ by $\nabla_{\beta}$, i.e., it is the operator which is defined by
the symbol $(i\lambda|\lambda|^{\beta-1})_{\lambda\in\mathbb{R}^{d}}$. Note
that $\nabla_{\beta}$ is a vector-valued operator. In the case of $\beta=1$,
it is the ordinary gradient. We restrict ourselves to case $\beta\neq 1$ in
this paper.

Our goal is to construct a perturbation of the considered process using the
operator $\left(b,\nabla_{\beta}\right)$, where $b$ is some measurable
$\mathbb{R}^{d}$-valued function. Under the perturbation of the Markov process
$(x(t))_{t\geq 0}$ we understand the construction of a two parametric family
of operators $(\mathds{T}_{st})_{0\leq s<t}$, defined on the set
$C_{b}(\mathbb{R}^{d})$ of real-valued continuous bounded functions defined on
$\mathbb{R}^{d}$, such that for every $\varphi\in C_{b}(\mathbb{R}^{d})$ and
$t>0$, the function $u(s,x,t)=\mathds{T}_{st}\varphi(x)$ satisfies in some
sense the following Cauchy problem

$\frac{\partial}{\partial s}u(s,x,t)+L(s,x)u(s,\cdot,t)(x)=0,\quad 0\leq s<t,\
x\in\mathbb{R}^{d},$ $\lim_{s\uparrow t}u(s,x,t)=\varphi(x),\quad
x\in\mathbb{R}^{d},$

where $L(s,x)=A(s,x)+(b(s,x),\nabla_{\beta})$.

The problem of perturbation of Markov processes was and remains in the focus
of attention of many researchers. The diffusion case, that is, if $\alpha=2$,
was considered by M. I. Portenko (see, [13, 14] and the references there). The
perturbation operators there is of the form $(b,\nabla)$, where the function
$b$ belongs to some $L_{p}$-space of functions or is some generalized function
of the delta function type. In [18], the parabolic equation of the type
$\nabla(A\nabla u)+B\nabla u-u_{t}=0$ was considered. The case $\alpha<2$ was
considered in the works of K. Bogdan and T. Jakubowski [2, 6], who studied
such a perturbation with the function $b$ from a Kato class. S. I. Podolynny,
M. I. Portenko [12] and M. I. Portenko [15, 16, 17] investigated this
perturbation with the function $b$ from $L_{p}$. In [7], $\alpha$-stable
process was perturbed by the gradient operator with multiplier $b$ satisfying
a certain integral space-time condition. Y. Maekawa and H. Miura were
concerned with non-local parabolic equations in the presence of a divergence
free drift term in [9]. J.-U. Loebus and M. I. Portenko [8] perturbed the
infinitesimal operator of a one-dimensional symmetric $\alpha$-stable process
using the operator $(q\delta_{0},\partial_{\alpha-1})$, where
$\partial_{\alpha-1}$ is some pseudo-differential operator of the order
$\alpha-1$. Perturbation of $\alpha$-stable process by an operator of
fractional Laplacian was considered in [4]. The results of $\alpha\in(1,2)$
and perturbation operators of the type $(b,\nabla_{\alpha-1})$ can be found in
[10, 11]. We studied the case of an $\alpha$-stable process and perturbation
operators $(b,\nabla_{\beta})$ with a $\mathbb{R}^{d}$-valued time independent
function $b$ from $L_{p}(\mathbb{R}^{d})$ and $0<\beta<\alpha$ in [3].

This article is structured as follows. The next section is devoted to some
auxiliary facts. The perturbation equation is solved in Section 3. In Section
4, we study some properties of the corresponding two-parameter evolutionary
family of operators. The last section is devoted to constructing a solution to
a corresponding Cauchy problem.

We will not use different notations for constants, if it is not important, but
we will use the notations $C$. If we need to emphasize the dependence of the
constant $C$ on parameters $\pi$, we will write $C_{\pi}$.

##  2 Auxiliaries

First of all let us note that if $0<\beta<1$ the operator $\nabla_{\beta}$ can
be represented in the following integral form:

$\nabla_{\beta}f(x)=n_{\beta}\int_{\mathbb{R}^{d}}(f(x+y)-f(x))\frac{y}{|y|^{d+\beta+1}}\,\mathrm{d}y,$
(1)

where
$n_{\beta}=-2^{-1}\pi^{-(d-1)/2}\Gamma\left(-\beta/2\right)\Gamma\left((d+\beta+1)/2\right)/\Gamma(-\beta)$
(here one have to use the equality $\Gamma(1+x)=x\Gamma(x)$ to expand the
Euler gamma function $\Gamma$ to negative non-integer arguments).
Representation (1) is true if a function $f$ is at least Lipschitz continuous
and bounded. To obtain the normalizing factor $n_{\beta}$, it is sufficient to
apply the operator $\nabla_{\beta}$ to the function
$f_{\lambda}(x)=e^{i(\lambda,x)}$, $x\in\mathbb{R}^{d}$ for any
$\lambda\in\mathbb{R}^{d}$. Moreover this is the way to prove (1).

Next, we will need some auxiliary statements. The following lemma proved in
[5] (see Lemma 1.11 there). It will be used frequently in this article.

######  Lemma 1.

The inequality ($0\leq s<t$, $x,y\in\mathbb{R}^{d}$, remind that $1<\alpha<2$)

$\displaystyle\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}\frac{(\tau-s)^{\lambda/\alpha}}{((\tau-s)^{1/\alpha}+|z-x|)^{d+l}}\frac{(t-\tau)^{\varkappa/\alpha}}{((t-\tau)^{1/\alpha}+|y-z|)^{d+k}}\,\mathrm{d}z\leq$
$\displaystyle
C\left[B\left(1+\frac{\varkappa-k}{\alpha},1+\frac{\lambda}{\alpha}\right)\frac{(t-s)^{1+(\varkappa+\lambda-k)/\alpha}}{((t-s)^{1/\alpha}+|y-x|)^{d+l}}\right.+$
$\displaystyle\left.B\left(1+\frac{\varkappa}{\alpha},1+\frac{\lambda-l}{\alpha}\right)\frac{(t-s)^{1+(\varkappa+\lambda-l)/\alpha}}{((t-s)^{1/\alpha}+|y-x|)^{d+k}}\right],$
(2)

holds with some constant $C>0$ that depends only on $d$, $\alpha$, $k$ and $l$
for all $\varkappa$, $\lambda$, $k$, $l$, satisfying the inequalities:
$0<k<\alpha+\varkappa$, $0<l<\alpha+\lambda$. Here $B(\cdot,\cdot)$ is the
Euler beta function.

The statement of the following lemma can be obtained from the results of [5,
Ch. 4]).

######  Lemma 2.

If the assumptions $A_{1}$ — $A_{3}$ hold the transition probability density
of the process $(x(t))_{t\geq 0}$ have the following properties ($0\leq
s<t\leq T$, $x,y\in\mathbb{R}^{d}$): the function $g$ is continuous
differentiable with respect to $x\in\mathbb{R}^{d}$ and

$|\partial^{k}g(s,\cdot,t,y)(x)|\leq
N_{k,T}\frac{(t-s)^{1-(\gamma+k)/\alpha}}{((t-s)^{1/\alpha}+|y-x|)^{d+\alpha-\gamma}},\quad
k=0,1,$ (3)

where $\partial^{k}$ means some derivative of the order $k$;

$\displaystyle|\nabla_{\beta}g(s,\cdot,t,y)(x)|\leq
N_{\beta,T}\left(\frac{1}{((t-s)^{1/\alpha}+|y-x|)^{d+\beta}}\right.+$
$\displaystyle\left.\frac{(t-s)^{1-\beta/\alpha}}{((t-s)^{1/\alpha}+|y-x|)^{d+\alpha-\gamma}}\right).$
(4)

Here, positive constants $N_{k,T}$ and $N_{\beta,T}$ can be depended on $T$.

###### Proof.

Let us note that the function $g$ can be constructed by the parametrix method
(see [5, Th. 4.1]). That is, it can be presented by the following equality

$g(s,x,t,y)=g_{0}(s,x,t,y)+h(s,x,t,y),$

where

$g_{0}(s,x,t,y)=\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}\exp\\{i(x-y,\lambda)-a(t,y,\lambda)(t-s)\\}\mathrm{d}\lambda,$
$h(s,x,t,y)=\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}g_{0}(s,x,\tau,z)\Phi(\tau,z,t,y)\mathrm{d}z,$
(5)

and the function $\Phi$ is determined by some integral equation.

Theorem 4.1 cited above states that the following inequalities hold

$|h(s,x,t,y)|\leq
C\left(\frac{(t-s)^{1+\gamma/\alpha}}{((t-s)^{1/\alpha}+|y-x|)^{d+\alpha}}+\frac{t-s}{((t-s)^{1/\alpha}+|y-x|)^{d+\alpha-\gamma}}\right),$
$|\Phi(s,x,t,y)|\leq C\frac{1}{((t-s)^{1/\alpha}+|y-x|)^{d+\alpha-\gamma}}.$
(6)

The function $g_{0}$ satisfies the inequality (see [5, eq. (4.1.25)])

$|\partial^{k}g_{0}(s,\cdot,t,y)(x)|\leq
N_{k}\frac{t-s}{((t-s)^{1/\alpha}+|y-x|)^{d+\alpha+k}}$ (7)

in which $\partial^{k}$ means some derivative of the integer order $0\leq
k\leq N-2d-1$ (the constant $N$ is defined in assumption $A_{2}$), $N_{k}>0$
are some constants. Note that $k$ can be at least 0, 1, or 2.

Using (2) we can obtain the following inequality valid for $k=0$ or $1$

$\displaystyle\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}\left|\partial^{k}g_{0}(s,\cdot,\tau,z)(x)\Phi(\tau,z,t,y)\right|\mathrm{d}z\leq$
$\displaystyle
C_{k}\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}\frac{\tau-s}{((\tau-s)^{1/\alpha}+|z-x|)^{d+\alpha+k}}\frac{1}{((t-\tau)^{1/\alpha}+|y-z|)^{d+\alpha-\gamma}}\,\mathrm{d}z\leq$
$\displaystyle
C_{k}\left(\frac{(t-s)^{1-k/\alpha}}{((t-s)^{1/\alpha}+|y-x|)^{d+\alpha+k}}+\frac{(t-s)^{1+\gamma/\alpha}}{((t-s)^{1/\alpha}+|y-x|)^{d+\alpha-\gamma}}\right)\leq$
$\displaystyle
C_{k}\frac{(t-s)^{1-k/\alpha}}{((t-s)^{1/\alpha}+|y-x|)^{d+\alpha-\gamma}},$

where $C_{k}>0$ are some constants. Therefore (use (7))

$\displaystyle|\partial^{k}g(s,\cdot,t,y)(x)|\leq\max(N_{k},C_{k})\frac{(t-s)^{1-k/\alpha}}{((t-s)^{1-(k+\gamma)/\alpha}+|y-x|)^{d+\alpha+-\gamma}}\times$
$\displaystyle\left[\frac{(t-s)^{(k+\gamma)/\alpha}}{((t-s)^{1/\alpha}+|y-x|)^{k+\gamma}}+(t-s)^{\gamma/\alpha}\right]\leq$
$\displaystyle
C_{k,T}\frac{(t-s)^{1-(\gamma+k)/\alpha}}{((t-s)^{1/\alpha}+|y-x|)^{d+\alpha-\gamma}},$

for all $0\leq s<t\leq T$, $x,y\in\mathbb{R}^{d}$ and every $T>0$. Here the
constants $C_{k,T}>0$ depend on $T$.

For proving inequality (4), we use representation (1) of the operator
$\nabla_{\beta}$ (remind that $0<\beta<1$). So, since $a(t,x,\cdot)$ is a
homogeneous function and

$\displaystyle\nabla_{\beta}g_{0}(s,\cdot,t,y)(x)=$
$\displaystyle\frac{i}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}\lambda|\lambda|^{\beta-1}\exp\\{i(x-y,\lambda)-a(t,y,(t-s)^{1/\alpha}\lambda)\\}\mathrm{d}\lambda=$
$\displaystyle\frac{i}{(2\pi)^{d}(t-s)^{(d+\beta)/\alpha}}\int_{\mathbb{R}^{d}}\lambda|\lambda|^{\beta-1}\exp\left\\{i\left(\frac{x-y}{(t-s)^{1/\alpha}},\lambda\right)-a(t,y,\lambda)\right\\}\mathrm{d}\lambda,$

the result of [5, Lemma 4.2] leads us to the inequality

$|\nabla_{\beta}g_{0}(s,\cdot,t,y)(x)|\leq
C_{\beta}\frac{1}{((t-s)^{1/\alpha}+|y-x|)^{d+\beta}}$ (8)

valid for all $0\leq s<t$, $x,y\in\mathbb{R}^{d}$ with some positive constant
$C_{\beta}$. Inequalities (7) allow us to state that the integral

$\int_{\mathbb{R}^{d}}\frac{u}{|u|^{d+\beta+1}}\mathrm{d}u\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}(g_{0}(s,x+u,\tau,z)-g_{0}(s,x,\tau,z))\Phi(\tau,z,t,y)\,\mathrm{d}z$

is absolutely convergent. Indeed, for
$(\tau,z,u)\in(s,t)\times\mathbb{R}^{d}\times B_{\varepsilon}(0)$ and fixed
$0\leq s<t$, $x,y\in\mathbb{R}^{d}$ ($\varepsilon>0$ is small enough constant
and by $B_{\varepsilon}(0)$ we denote the ball of the radius $\varepsilon$ and
the center placed in the point $0$), we have

$\displaystyle
R=\left|\frac{u}{|u|^{d+\beta+1}}(g_{0}(s,x+u,\tau,z)-g_{0}(s,x,\tau,z))\Phi(\tau,z,t,y)\right|\leq$
$\displaystyle\frac{1}{|u|^{d+\beta-1}}\frac{N_{1}(\tau-s)}{((\tau-s)^{1/\alpha}+|z-x-\theta
u|)^{d+\alpha+1}}\frac{C}{((t-\tau)^{1/\alpha}+|y-z|)^{d+\alpha-\gamma}},$

where $\theta=\theta(\tau,z)\in(0,1)$; if
$(\tau,z,u)\in(s,t)\times\mathbb{R}^{d}\times B_{\varepsilon}(0)^{C}$, we have
the inequality

$\displaystyle
R\leq\frac{N_{0}}{|u|^{d+\beta}}\left(\frac{\tau-s}{((\tau-s)^{1/\alpha}+|z-x-u|)^{d+\alpha}}+\frac{\tau-s}{((\tau-s)^{1/\alpha}+|z-x|)^{d+\alpha}}\right)\times$
$\displaystyle\frac{C}{((t-\tau)^{1/\alpha}+|y-z|)^{d+\alpha-\gamma}}.$

The right-hand sides of these inequalities are integrable as the function of
$(\tau,z,u)$ on the corresponding domains for every fixed $0\leq s<t$ and
$x,y\in\mathbb{R}^{d}$. Using the Fubini’s theorem we can write down the
equality

$\nabla_{\beta}h(s,\cdot,t,y)(x)=\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}\nabla_{\beta}g_{0}(s,\cdot,\tau,z)(x)\Phi(\tau,z,t,y)\mathrm{d}z,$

valid for all $0\leq s<t$, $x,y\in\mathbb{R}^{d}$.

Now, use inequalities (6), (8) and Lemma 1. The result is as follows

$\displaystyle|\nabla_{\beta}h(s,\cdot,t,y)(x)|\leq
C\left(\frac{(t-s)^{\gamma/\alpha}}{((t-s)^{1/\alpha}+|y-x|)^{d+\beta}}\right.$
$\displaystyle+\left.\frac{(t-s)^{1-\beta/\alpha}}{((t-s)^{1/\alpha}+|y-x|)^{d+\alpha-\gamma}}\right),$

where $C$ is some positive constant. Combining this inequality with (8) we
obtain (4), where it is taken into account that $0\leq s<t\leq T$. The lemma
is proved. ∎

##  3 Perturbation equation solving

Let us consider the perturbation equation ($0\leq s<t$,
$x,y\in\mathbb{R}^{d}$)

$G(s,x,t,y)=g(s,x,t,y)+\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}g(s,x,\tau,z)(b(\tau,z),\nabla_{\beta}G(\tau,\cdot,t,y)(z))\,\mathrm{d}z.$
(9)

Using the formal application of the operator $\nabla_{\beta}$ to both sides of
equation (9) with respect to the variable $x$ we can search for its solution
in the form

$G(s,x,t,y)=g(s,x,t,y)+\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}g(s,x,\tau,z)(b(\tau,z),v(\tau,z,t,y))\,\mathrm{d}z,$
(10)

where the function $(v(s,x,t,y))_{0\leq s<t,x,y\in\mathbb{R}^{d}}$ is a
solution to the following equation

$v(s,x,t,y)=v_{0}(s,x,t,y)+\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}v_{0}(s,x,\tau,z)(b(\tau,z),v(\tau,z,t,y))\,\mathrm{d}z,$
(11)

in which

$v_{0}(s,x,t,y)=\nabla_{\beta}g(s,\cdot,t,y)(x).$ (12)

######  Theorem 1.

Let assumptions $A_{1}$ — $A_{3}$ hold and the $\mathbb{R}^{d}$-valued
function $(b(t,x))_{t>0,x\in\mathbb{R}^{d}}$ belongs to
$L_{p}([0,T]\times\mathbb{R}^{d})$ for every $T>0$ with some
$p>\frac{d+\alpha}{\alpha-1}$, including $p=+\infty$. Then the following
statements are true: there exists a unique solution to equation (11) in the
class of functions satisfying the estimate

$|v(s,x,t,y)|\leq
C_{T}\left(\frac{1}{((t-s)^{1/\alpha}+|x-y|)^{d+\beta}}+\frac{(t-s)^{1-\beta/\alpha}}{((t-s)^{1/\alpha}+|x-y|)^{d+\alpha-\gamma}}\right);$
(13)

the function (10) is a solution to equation (9) and satisfies the estimate

$\displaystyle|G(s,x,t,y)|\leq
C_{T}\left(\frac{(t-s)^{\beta/\alpha}}{((t-s)^{1/\alpha}+|x-y|)^{d+\beta}}+\right.$
$\displaystyle\left.\frac{(t-s)^{1-\gamma/\alpha}}{((t-s)^{1/\alpha}+|x-y|)^{d+\alpha-\gamma}}\right);$
(14)

for all $0\leq s<t\leq T$, $x,y\in\mathbb{R}^{d}$ and every $T>0$, where a
positive constant $C_{T}$ can be depended on $T$.

Before proving this theorem, let us prove the following auxiliary statement.

######  Lemma 3.

If the assumptions of Theorem 1 hold and a $\mathbb{R}^{d}$-valued function
$(v(s,x,t,y))_{0\leq s<t,x,y\in\mathbb{R}^{d}}$ satisfies inequality (13) then
the equality

$\displaystyle\nabla_{\beta}\left(\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}g(s,\cdot,\tau,z)(b(\tau,z),v(\tau,z,t,y))\,\mathrm{d}z\right)(x)=$
$\displaystyle\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}\nabla_{\beta}g(s,\cdot,\tau,z)(x)(b(\tau,z),v(\tau,z,t,y))\,\mathrm{d}z$
(15)

is fulfilled.

###### Proof.

Note that the integral

$\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}\nabla_{\beta}g(s,\cdot,\tau,z)(x)(b(\tau,z),v(\tau,z,t,y))\,\mathrm{d}z$

converges (it is sufficiently to use inequalities (4), (13) and Lemma 1).
Indeed, for fixed $0\leq s<t\leq T$, $x,y\in\mathbb{R}^{d}$, we can write the
following111Here and further, we use the obvious inequality $(f+g)^{r}\leq
2^{r}(f^{r}+g^{r})$, valid for all $r\geq 0$ and non-negative functions $f$
and $g$, without any references.

$\displaystyle\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}|\nabla_{\beta}g(s,\cdot,\tau,z)(x)||b(\tau,z)||v(\tau,z,t,y)|\,\mathrm{d}z\leq$
$\displaystyle\|b\|_{p}^{T}C_{T}N_{\beta,T}\times$
$\displaystyle\left(\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}\left(\frac{1}{((\tau-s)^{1/\alpha}+|x-z|)^{d+\beta}}+\frac{(\tau-s)^{1-\beta/\alpha}}{((\tau-s)^{1/\alpha}+|x-z|)^{d+\alpha-\gamma}}\right)^{q}\times\right.$
$\displaystyle\left.\left(\frac{1}{((t-\tau)^{1/\alpha}+|z-y|)^{d+\beta}}+\frac{(t-\tau)^{1-\beta/\alpha}}{((t-\tau)^{1/\alpha}+|z-y|)^{d+\alpha-\gamma}}\right)^{q}\mathrm{d}z\right)^{1/q}\leq$
$\displaystyle C_{T}\left(\frac{(t-s)^{1/q-d/(\alpha
p)-\beta/\alpha}}{((t-s)^{1/\alpha}+|x-y|)^{d+\beta}}+\frac{(t-s)^{1/q-d/(\alpha
p)+(\gamma-\beta)/\alpha}}{((t-s)^{1/\alpha}+|x-y|)^{d+\beta}}+\right.$
$\displaystyle\left.\frac{(t-s)^{1+1/q-d/(\alpha
p)-2\beta/\alpha}}{((t-s)^{1/\alpha}+|x-y|)^{d+\alpha-\gamma}}+\frac{(t-s)^{1+1/q-d/(\alpha
p)-(2\beta-\gamma)/\alpha}}{((t-s)^{1/\alpha}+|x-y|)^{d+\alpha-\gamma}}\right)\leq$
$\displaystyle C_{T}(t-s)^{1/q-d(1/p+1)/\alpha-2\beta/\alpha}.$

To prove the existence of left-hand side of (15) we use representation (1) of
the operator $\nabla_{\beta}$. For fixed $0\leq s<t\leq T$ and
$y\in\mathbb{R}^{d}$ the function

$\left(\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}g(s,x,\tau,z)(b(\tau,z),v(\tau,z,t,y))\,\mathrm{d}z\right)_{x\in\mathbb{R}^{d}}$

is bounded and Lipschitz continuous. Indeed, it is sufficient to prove a
boundedness of the integral

$\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}\nabla_{k}g(s,\cdot,\tau,z)(x)(b(\tau,z),v(\tau,z,t,y))\,\mathrm{d}z,$

as the function of $x\in\mathbb{R}^{d}$ for fixed $0\leq s<t\leq T$ and
$y\in\mathbb{R}^{d}$ in the cases of $\nabla_{0}=I$ (the identical operator)
and $\nabla_{1}=\nabla$ (the gradient).

Using inequalities (3), (13) and Lemma 1, for all $x\in\mathbb{R}^{d}$, we can
write down ($k=0,1$)

$\displaystyle\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}|\nabla_{k}g(s,\cdot,\tau,z)(x)||b(\tau,z)||v(\tau,z,t,y)|\,\mathrm{d}z\leq$
$\displaystyle\|b\|_{p}^{T}C_{T}N_{k,T}\left(\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}\frac{(\tau-s)^{(1-(\gamma+k)/\alpha)q}}{((\tau-s)^{1/\alpha}+|x-z|)^{(d+\alpha-\gamma)q}}\times\right.$
$\displaystyle\left.\left(\frac{1}{((t-\tau)^{1/\alpha}+|z-y|)^{d+\beta}}+\frac{(t-\tau)^{1-\beta/\alpha}}{((t-\tau)^{1/\alpha}+|z-y|)^{d+\alpha-\gamma}}\right)^{q}\mathrm{d}z\right)^{1/q}\leq$
$\displaystyle C_{T}\left(\frac{(t-s)^{1+1/q-d/(\alpha
p)-(\gamma+\beta+k)/\alpha}}{((t-s)^{1/\alpha}+|x-y|)^{d+\alpha-\gamma}}+\frac{(t-s)^{1/q-d/(\alpha
p)-k/\alpha}}{((t-s)^{1/\alpha}+|x-y|)^{d+\beta}}+\right.$
$\displaystyle\left.\frac{(t-s)^{1+1/q-d/(\alpha
p)-(\beta+k)/\alpha}}{((t-s)^{1/\alpha}+|x-y|)^{d+\alpha-\gamma}}\right)\leq
C_{T}(t-s)^{1/q-d(1/p+1)/\alpha-(\beta+k)/\alpha},$

which is bounded for fixed $0\leq s<t\leq T$.

Together with the operator $\nabla_{\beta}$, we consider a family of operators
$\\{\nabla_{\beta}^{(\varepsilon)}:\varepsilon>0\\}$ acting on a function
$(f(x))_{x\in\mathbb{R}^{d}}$ according to the following rule

$\nabla_{\beta}^{(\varepsilon)}\varphi(x)=n_{\beta}\int_{\varepsilon\leq|y|\leq\frac{1}{\varepsilon}}(f(x+y)-f(x))\frac{y}{|y|^{d+\beta+1}}\,\mathrm{d}y.$

It is clear that the following relation $\lim_{\varepsilon\to
0+}\nabla_{\beta}^{(\varepsilon)}f(x)=\nabla_{\beta}f(x)$,
$x\in\mathbb{R}^{d}$ holds for every bounded and Lipschitz continuous function
$f$.

Using inequalities (3), (13) we can state, that the following inequality

$\displaystyle\left|\frac{u}{|u|^{d+\beta+1}}(g(s,x+u,\tau,z)-g(s,x,\tau,z))(b(\tau,z),v(\tau,z,t,y))\right|\leq$
$\displaystyle\frac{{N_{0,T}}C_{T}|b(\tau,z)|}{|u|^{d+\beta}}\times$
$\displaystyle\left(\frac{(\tau-s)^{1-\gamma/\alpha}}{((\tau-s)^{1/\alpha}+|z-x-u|)^{d+\alpha-\gamma}}+\frac{(\tau-s)^{1-\gamma/\alpha}}{((\tau-s)^{1/\alpha}+|z-x|)^{d+\alpha-\gamma}}\right)\times$
$\displaystyle\left(\frac{1}{((t-\tau)^{1/\alpha}+|y-z|)^{d+\beta}}+\frac{(t-\tau)^{1-\beta/\alpha}}{((t-\tau)^{1/\alpha}+|y-z|)^{d+\alpha-\gamma}}\right)$

is true for every $T>0$ and all $0\leq s<\tau<t\leq T$,
$x,y,z,u\in\mathbb{R}^{d}$.

Using Hölder’s inequality, we can write down the following

$\displaystyle\int_{s}^{t}\hskip-5.0pt\mathrm{d}\tau\hskip-5.0pt\int_{\mathbb{R}^{d}}\left|(g(s,x+u,\tau,z)-g(s,x,\tau,z))(b(\tau,z),v(\tau,z,t,y))\right|\,\mathrm{d}z\leq$
$\displaystyle\leq 4C_{T}{N_{0,T}}\|b\|_{p}^{T}$
$\displaystyle\left[\int_{s}^{t}\hskip-5.0pt\mathrm{d}\tau\hskip-5.0pt\int_{\mathbb{R}^{d}}\frac{(\tau-s)^{(1-\gamma/\alpha)q}}{((\tau-s)^{1/\alpha}+|z-x-u|)^{(d+\alpha-\gamma)q}}\frac{1}{((t-\tau)^{1/\alpha}+|y-z|)^{(d+\beta)q}}\,\mathrm{d}z+\right.$
$\displaystyle\left.\int_{s}^{t}\hskip-5.0pt\mathrm{d}\tau\hskip-5.0pt\int_{\mathbb{R}^{d}}\frac{(\tau-s)^{(1-\gamma/\alpha)q}}{((\tau-s)^{1/\alpha}+|z-x|)^{(d+\alpha-\gamma)q}}\frac{1}{((t-\tau)^{1/\alpha}+|y-z|)^{(d+\beta)q}}\,\mathrm{d}z+\right.$
$\displaystyle\left.\int_{s}^{t}\hskip-5.0pt\mathrm{d}\tau\hskip-5.0pt\int_{\mathbb{R}^{d}}\frac{(\tau-s)^{(1-\gamma/\alpha)q}}{((\tau-s)^{1/\alpha}+|z-x-u|)^{(d+\alpha-\gamma)q}}\frac{(t-\tau)^{(1-\beta/\alpha)q}}{((t-\tau)^{1/\alpha}+|y-z|)^{(d+\alpha-\gamma)q}}\,\mathrm{d}z+\right.$
$\displaystyle\left.\int_{s}^{t}\hskip-5.0pt\mathrm{d}\tau\hskip-5.0pt\int_{\mathbb{R}^{d}}\frac{(\tau-s)^{(1-\gamma/\alpha)q}}{((\tau-s)^{1/\alpha}+|z-x|)^{(d+\alpha-\gamma)q}}\frac{(t-\tau)^{(1-\beta/\alpha)q}}{((t-\tau)^{1/\alpha}+|y-z|)^{(d+\alpha-\gamma)q}}\,\mathrm{d}z\right]^{1/q},$

which holds for all $0\leq s<t\leq T$, $x,y,u\in\mathbb{R}^{d}$. Here the
number $q\geq 1$ is defined by the relation $\frac{1}{p}+\frac{1}{q}=1$ (if
$p=+\infty$ then $q=1$) and the norm of the function $b$ in the space
$L_{p}([0,T]\times\mathbb{R}^{d})$ is denoted by $\|b\|_{p}^{T}$. It is easy
to see, that the right hand side integrals satisfy the conditions of Lemma 1.
Therefore these integrals are finite for all fixed $0\leq s<t\leq T$,
$x,y,u\in\mathbb{R}^{d}$. As a consequence, the function

$\left((g(s,x+u,\tau,z)-g(s,x,\tau,z))(b(\tau,z),v(\tau,z,t,y))\frac{u}{|u|^{d+\beta+1}}\right)_{\tau\in(s,t),u,z\in\mathbb{R}^{d}}$

is integrable with respect to
$(u,\tau,z)\in\\{\varepsilon\leq|u|\leq\frac{1}{\varepsilon}\\}\times(s,t)\times\mathbb{R}^{d}$
with small enough $\varepsilon>0$ for every fixed $0\leq s<t\leq T$,
$x,y\in\mathbb{R}^{d}$. Thus, using Fubini’s theorem, we obtain the following
equality

$\displaystyle\nabla_{\beta}^{(\varepsilon)}\left(\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}g(s,\cdot,\tau,z)(b(\tau,z),v(\tau,z,t,y))\,\mathrm{d}z\right)(x)=$
$\displaystyle\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}\nabla_{\beta}^{(\varepsilon)}g(s,\cdot,\tau,z)(x)(b(\tau,z),v(\tau,z,t,y))\,\mathrm{d}z.$
(16)

hold for each $\varepsilon>0$.

So, it is enough to pass to the limit as $\varepsilon\to 0+$ in equality (16)
to complete the proof of the lemma. ∎

###### Proof of Theorem 1.

Let us solve equation (11) for all fixed $0\leq s<t\leq T$,
$x,y\in\mathbb{R}^{d}$, using the method of successive approximations. Namely,
consider the sequence of functions $(v_{k}(s,x,t,y))_{0\leq s<t\leq
T,x,y\in\mathbb{R}^{d}}$, $k=0,1,2,\dots$, given by the recurrence relation

$v_{k+1}(s,x,t,y)=\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}v_{0}(s,x,\tau,z)(b(\tau,z),v_{k}(\tau,z,t,y))\,\mathrm{d}z,$

where $v_{0}$ is defined by (12).

Using inequality (4) one can obtain the relation ($0\leq s<t\leq T$,
$x,y\in\mathbb{R}^{d}$)

$\displaystyle|v_{k+1}(s,x,t,y)|\leq
2N_{\beta,T}\|b\|_{p}^{T}\left(\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}\frac{|v_{k}(\tau,z,t,y)|^{q}}{((\tau-s)^{1/\alpha}+|x-z|)^{(d+\beta)q}}\,\mathrm{d}z+\right.$
$\displaystyle\left.\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}\frac{|v_{k}(\tau,z,t,y)|^{q}(\tau-s)^{(1-\beta/\alpha)q}}{((\tau-s)^{1/\alpha}+|x-z|)^{(d+\alpha-\gamma)q}}\,\mathrm{d}z\right)^{1/q}.$
(17)

Iterating relation (3) by using inequality (2) we come to the estimate

$\displaystyle|v_{k}(s,x,t,y)|\leq R_{k,T}\times$
$\displaystyle\left(\frac{(t-s)^{k\theta}}{((t-s)^{1/\alpha}+|x-y|)^{d+\beta}}+\frac{(t-s)^{k\theta+1-\beta/\alpha}}{((t-s)^{1/\alpha}+|x-y|)^{d+\alpha-\gamma}}\right),$
(18)

where $\theta=1-((d+\alpha)/p+\beta)/\alpha$ (note that
$\theta>(1-\beta)/\alpha$) and the sequence $\\{R_{k,T}:k=0,1,2,\dots\\}$
satisfies the relations $R_{0}=N_{\beta,T}$ and

$\displaystyle
R_{k+1,T}=R_{k,T}8^{1+1/q}N_{\beta,T}\|b\|_{p}^{T}C^{1/q}(1+T^{\gamma/\alpha})\times$
$\displaystyle\max\left(B\left((k+1)\theta q,1\right)^{1/q},B\left(1+k\theta
q,\theta q\right)^{1/q}\right).$ (19)

for all $k=1,2,\dots$. Here $C$ is the maximum of a finite number of constants
taking from inequality (2). Indeed, when $k=0$, inequality (3) coincides with
inequality (4). If (3) is correct for some $k\in\mathbb{N}$ we obtain

$\displaystyle|v_{k+1}(s,x,t,y)|\leq 8N_{\beta,T}\|b\|_{p}^{T}R_{k,T}\times$
$\displaystyle\left[\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}\frac{(\tau-s)^{(1-\beta/\alpha)q}}{((\tau-s)^{1/\alpha}+|x-z|)^{(d+\alpha-\gamma)q}}\frac{(t-\tau)^{k\theta
q}}{((t-\tau)^{1/\alpha}+|z-y|)^{(d+\beta)q}}\,\mathrm{d}z+\right.$
$\displaystyle\left.\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}\frac{(\tau-s)^{(1-\beta/\alpha)q}}{((\tau-s)^{1/\alpha}+|x-z|)^{(d+\alpha-\gamma)q}}\frac{(t-\tau)^{k\theta
q+(1-\beta/\alpha)q}}{((t-\tau)^{1/\alpha}+|z-y|)^{(d+\alpha-\gamma)q}}\,\mathrm{d}z+\right.$
$\displaystyle\left.\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}\frac{1}{((\tau-s)^{1/\alpha}+|x-z|)^{(d+\beta)q}}\frac{(t-\tau)^{k\theta
q+(1-\beta/\alpha)q}}{((t-\tau)^{1/\alpha}+|z-y|)^{(d+\alpha-\gamma)q}}\,\mathrm{d}z+\right.$
$\displaystyle\left.\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}\frac{1}{((\tau-s)^{1/\alpha}+|x-z|)^{(d+\beta)q}}\frac{(t-\tau)^{k\theta
q}}{((t-\tau)^{1/\alpha}+|z-y|)^{(d+\beta)q}}\,\mathrm{d}z\right]^{1/q}.$

Using inequality (2), we can write

$\displaystyle|v_{k+1}(s,x,t,y)|\leq
8N_{\beta,T}\|b\|_{p}^{T}R_{k,T}C^{1/q}\times$
$\displaystyle\left[B\left((k+1)\theta
q,1+\left(1-\frac{\beta}{\alpha}\right)q\right)\frac{(t-s)^{(k+1)\theta
q+(1-\beta/\alpha)q}}{((t-s)^{1/\alpha}+|x-y|)^{(d+\alpha-\gamma)q}}+\right.$
$\displaystyle\left.B\left(1+k\theta q,\theta
q+\frac{\gamma}{\alpha}q\right)\frac{(t-s)^{(k+1)\theta
q+q\gamma/\alpha}}{((t-s)^{1/\alpha}+|x-y|)^{(d+\beta)q}}+\right.$
$\displaystyle\left.B\left((k+1)\theta
q+\frac{\gamma}{\alpha}q,1+\left(1-\frac{\beta}{\alpha}\right)q\right)\frac{(t-s)^{(k+1)\theta
q+(1-\beta/\alpha+\gamma/\alpha)q}}{((t-s)^{1/\alpha}+|x-y|)^{(d+\alpha-\gamma)q}}+\right.$
$\displaystyle\left.B\left(1+k\theta
q+\left(1-\frac{\beta}{\alpha}\right),\theta
q+\frac{\gamma}{\alpha}q\right)\frac{(t-s)^{(k+1)\theta
q+(1-\beta/\alpha+\gamma/\alpha)q}}{((t-s)^{1/\alpha}+|x-y|)^{(d+\alpha-\gamma)q}}+\right.$
$\displaystyle\left.B\left((k+1)\theta
q+\frac{\gamma}{\alpha}q,1\right)\frac{(t-s)^{(k+1)\theta
q+q\gamma/\alpha}}{((t-s)^{1/\alpha}+|x-y|)^{(d+\beta)q}}+\right.$
$\displaystyle\left.B\left(1+k\theta
q+\left(1-\frac{\beta}{\alpha}\right)q,\theta q\right)\frac{(t-s)^{(k+1)\theta
q+(1-\beta/\alpha)q}}{((t-s)^{1/\alpha}+|x-y|)^{(d+\alpha-\gamma)q}}+\right.$
$\displaystyle\left.B\left((k+1)\theta q,1\right)\frac{(t-s)^{(k+1)\theta
q}}{((t-s)^{1/\alpha}+|x-y|)^{(d+\beta)q}}+\right.$
$\displaystyle\left.B\left(1+k\theta q,\theta q\right)\frac{(t-s)^{(k+1)\theta
q}}{((t-s)^{1/\alpha}+|x-y|)^{(d+\beta)q}}\right]^{1/q}.$

Since the function $(B(x,y))_{x>0,y>0}$ decreases with respect to each of its
arguments, we have the following:

$\displaystyle|v_{k+1}(s,x,t,y)|\leq
8N_{\beta,T}\|b\|_{p}^{T}R_{k,T}C^{1/q}\times$
$\displaystyle\max\left(B\left((k+1)\theta q,1\right)^{1/q},B\left(1+k\theta
q,\theta q\right)^{1/q}\right)8^{1/q}(1+T^{\gamma/\alpha})\times$
$\displaystyle\left(\frac{(t-s)^{(k+1)\theta}}{((t-s)^{1/\alpha}+|x-y|)^{d+\beta}}+\frac{(t-s)^{(k+1)\theta+1-\beta/\alpha}}{((t-s)^{1/\alpha}+|x-y|)^{d+\alpha-\gamma}}\right)$

that is, (3) holds for $k+1$. Therefore, it is correct for all $k=0,1,2,\dots$
and relation (3) is true.

Since $\lim_{k\to\infty}B(1+k\theta q,\theta q)=\lim_{k\to\infty}B((k+1)\theta
q,1)=0$ the series $\sum_{k=0}^{\infty}v_{k}(s,x,t,y)$ converges uniformly
with respect to $x,y\in\mathbb{R}^{d}$ and locally uniformly with respect to
$0\leq s<t$. Let $(v(s,x,t,y))_{0\leq s<t,x,y\in\mathbb{R}^{d}}$ be the sum of
this series. The function $v$ is a solution to equation (11) and the following
estimate

$|v(s,x,t,y)|\leq
C_{T}\left(\frac{1}{((t-s)^{1/\alpha}+|x-y|)^{d+\beta}}+\frac{(t-s)^{1-\beta/\alpha}}{((t-s)^{1/\alpha}+|x-y|)^{d+\alpha-\gamma}}\right),$

holds for all $0\leq s<t\leq T$, $x,y\in\mathbb{R}^{d}$ and every $T>0$ with
some constant $C_{T}>0$ depended on $T$.

The uniqueness of this solution follows from the fact that the difference
${v^{*}}$ of every two such solutions satisfies the equation

${v^{*}}(s,x,t,y)=\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}v_{0}(s,x,\tau,z)(b(\tau,z),{v^{*}}(\tau,z,t,y))\,\mathrm{d}z.$

Therefore, since ${v^{*}}$ satisfies estimate (13), using inequality (2), one
can obtain the following ($0\leq s<t\leq T$, $x,y\in\mathbb{R}^{d}$)

$\displaystyle|{v^{*}}(s,x,t,y)|\leq R_{k,T}\times$
$\displaystyle\left(\frac{(t-s)^{k\theta}}{((t-s)^{1/\alpha}+|x-y|)^{d+\beta}}+\frac{(t-s)^{k\theta+1-\beta/\alpha}}{((t-s)^{1/\alpha}+|x-y|)^{d+\alpha-\gamma}}\right),$

for all $k\in\mathbb{N}$, $T>0$, where $R_{k,T}$ is defined by (3). Since
$\lim_{k\to\infty}R_{k,T}T^{k\theta}=0$ for all $T>0$, this means that
${v^{*}}(s,x,t,y)\equiv 0$.

Let us define the function $G$ by equality (10) with the function $v$ just
constructed. Using (2), (3) and Hölder’s inequality we can write the following
chain of inequalities

$\displaystyle|G(s,x,t,y)|\leq
N_{0,T}\frac{(t-s)^{1-\gamma/\alpha}}{((t-s)^{1/\alpha}+|x-y|)^{d+\alpha-\gamma}}+$
$\displaystyle 2^{1+1/q}N_{0,T}C_{T}\|b\|_{p}^{T}\times$
$\displaystyle\left[\left(\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}\frac{(\tau-s)^{(1-\beta/\alpha)q}}{((\tau-s)^{1/\alpha}+|x-z|)^{(d+\alpha-\gamma)q}}\frac{1}{((t-\tau)^{1/\alpha}+|z-y|)^{(d+\beta)q}}\,\mathrm{d}z\right)^{1/q}+\right.$
$\displaystyle\left.\left(\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}\frac{(\tau-s)^{(1-\beta/\alpha)q}}{((\tau-s)^{1/\alpha}+|x-z|)^{(d+\alpha-\gamma)q}}\frac{(t-\tau)^{(1-\beta/\alpha)q}}{((t-\tau)^{1/\alpha}+|z-y|)^{(d+\beta)q}}\,\mathrm{d}z\right)^{1/q}\right]\leq$
$\displaystyle{C}_{T}\left[\frac{(t-s)^{\beta/\alpha}}{((t-s)^{1/\alpha}+|x-y|)^{d+\beta}}+\frac{(t-s)^{1-\gamma/\alpha}}{((t-s)^{1/\alpha}+|x-y|)^{d+\alpha-\gamma}}\right],$

where ${C}_{T}$ (in the last expression) is some positive constant, which
might be depended on $T$.

Let us prove that $\nabla_{\beta}G(s,\cdot,t,y)(x)\equiv v(s,x,t,y)$. Using
the Lemma 3 statement, we obtain the following equality

$\displaystyle\nabla_{\beta}G(s,\cdot,t,y)(x)=$ $\displaystyle
v_{0}(s,x,t,y)+\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}v_{0}(s,x,\tau,z)(b(\tau,z),v(\tau,z,t,y))\,\mathrm{d}z=$
$\displaystyle v(s,x,t,y),$

which is true for all $0\leq s<t$, $x,y\in\mathbb{R}^{d}$.

Consequently the function $G$ is a solution to equation (9) and it satisfies
estimate (1). The theorem is proved. ∎

##  4 Evolution family of operators

Let us define the two-parameter family of operators $\\{\mathds{T}_{st}:0\leq
s<t\\}$ defined in the space of continuous bounded functions
${C}_{b}(\mathbb{R}^{d})$ by the equality

$\mathds{T}_{st}\varphi(x)=\int_{\mathbb{R}^{d}}G(s,x,t,y)\varphi(y)\,\mathrm{d}y,\quad\varphi\in{C}_{b}(\mathbb{R}^{d}),\
x\in\mathbb{R}^{d}.$ (20)

Similarly to the proof of Theorem 1 one can prove the following statement.

######  Lemma 4.

The function
$w(s,x,t,\varphi)=\int_{\mathbb{R}^{d}}v(s,x,t,y)\varphi(y)\,\mathrm{d}y$,
$0\leq s<t$, $x\in\mathbb{R}^{d}$ and $\varphi\in{C}_{b}(\mathbb{R}^{d})$ ($v$
is defined in Theorem 1), is a unique (in the class of functions, which
satisfy the inequality $|w(s,x,t,\varphi)|\leq C_{T}(t-s)^{-\beta/\alpha}$)
solution to the equation

$w(s,x,t,\varphi)=w_{0}(s,x,t,\varphi)+\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}v_{0}(s,x,\tau,z)(b(\tau,z),w(\tau,z,t,\varphi))\,\mathrm{d}z,$
(21)

for all $0\leq s<t\leq T$, $x\in\mathbb{R}^{d}$ and every $T>0$. Here $v_{0}$
is defined by (12) and
$w_{0}(s,x,t,\varphi)=\int_{\mathbb{R}^{d}}v_{0}(s,x,t,y)\varphi(y)\,\mathrm{d}y$.

###### Proof.

Equation (21) is obtained from equation (11) by multiplying it with the
functions $\varphi$ and using the Fubini’s theorem. The justification of the
usage of the Fubini’s theorem bases on estimates (4), (13) (see also (12)).
Indeed,

$\displaystyle\int_{\mathbb{R}^{d}}\,\mathrm{d}y\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}|v_{0}(s,x,\tau,z)||b(\tau,z)||v(\tau,z,t,y)||\varphi(y)|\,\mathrm{d}z\leq$
$\displaystyle 4\|\varphi\|\|b\|_{p}^{T}N_{\beta,T}C_{T}\times$
$\displaystyle\int_{\mathbb{R}^{d}}\,\mathrm{d}y\left(\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}\frac{1}{((\tau-s)^{1/\alpha}+|z-x|)^{(d+\beta)q}}\times\right.$
$\displaystyle\left.\frac{1}{((t-\tau)^{1/\alpha}+|y-z|)^{(d+\beta)q}}\,\mathrm{d}z+\right.$
$\displaystyle\left.\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}\frac{1}{((\tau-s)^{1/\alpha}+|z-x|)^{(d+\beta)q}}\frac{(t-\tau)^{(1-\beta/\alpha)q}}{((t-\tau)^{1/\alpha}+|y-z|)^{(d+\alpha-\gamma)q}}\,\mathrm{d}z+\right.$
$\displaystyle\left.\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}\frac{(\tau-s)^{(1-\beta/\alpha)q}}{((\tau-s)^{1/\alpha}+|z-x|)^{(d+\alpha-\gamma)q}}\frac{1}{((t-\tau)^{1/\alpha}+|y-z|)^{q(d+\beta)}}\,\mathrm{d}z+\right.$
$\displaystyle\left.\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}\frac{(\tau-s)^{(1-\beta/\alpha)q}}{((\tau-s)^{1/\alpha}+|z-x|)^{(d+\alpha-\gamma)q}}\times\right.$
$\displaystyle\left.\frac{(t-\tau)^{(1-\beta/\alpha)q}}{((t-\tau)^{1/\alpha}+|y-z|)^{(d+\alpha-\gamma)q}}\,\mathrm{d}z\right)^{1/q}\leq$
$\displaystyle{C}_{T}\int_{\mathbb{R}^{d}}\left(\frac{1}{((t-s)^{1/\alpha}+|x-y|)^{d+\beta}}+\frac{(t-s)^{1-\beta/\alpha}}{((t-s)^{1/\alpha}+|x-y|)^{d+\alpha-\gamma}}\right)\,\mathrm{d}y\leq$
$\displaystyle{C}_{T}(t-s)^{-\beta/\alpha}$

for each $T>0$ and every $0\leq s<t\leq T$, $x\in\mathbb{R}^{d}$ with some
constants ${C}_{T}>0$. Here we used the well-known formula222Below we will use
this formula without any references.
$\int_{\mathbb{R}^{d}}(a+|x|)^{-d-\varkappa}\,\mathrm{d}x=a^{-\varkappa}B(d,\varkappa)\frac{2\pi^{d/2}}{\Gamma(d/2)}$,
which is valid for all $a>0$ and $\varkappa>0$. ∎

The next theorem contains the properties of the family of operators (20).

######  Theorem 2.

Let the Theorem 1 assumptions hold. Then the following statements are true:

  * •

the operators $\mathds{T}_{st}$, $0\leq s<t$ are linear and bounded on
$C_{b}(\mathbb{R}^{d})$;

  * •

if $\varphi(x)\equiv 1$ then $\mathds{T}_{st}\varphi(x)\equiv 1$;

  * •

the family of operators $\\{\mathds{T}_{st}:0\leq s<t\\}$ has an evolution
property, that is, $\mathds{T}_{s\tau}\mathds{T}_{\tau t}=\mathds{T}_{st}$ for
all $0\leq s<\tau<t$;

  * •

$\operatorname*{\mathit{w}-lim}_{s\uparrow t}\mathds{T}_{st}=I$, where $I$ is
the identical operator, i.e., $\lim_{s\uparrow
t}\mathds{T}_{st}\varphi(x)=\varphi(x)$, $x\in\mathbb{R}^{d}$ for all
$\varphi\in C_{b}(\mathbb{R}^{d})$.

###### Proof.

The linearity of operator $\mathds{T}_{st}$ is evident. Let us prove its
boundedness. If $\varphi\in C_{b}(\mathbb{R}^{d})$ then using inequality (1)
we can write ($\|\varphi\|=\max_{x\in\mathbb{R}^{d}}|\varphi(x)|$)

$\displaystyle|\mathds{T}_{st}\varphi(x)|\leq\|\varphi\|\int_{\mathbb{R}^{d}}|G(s,x,t,y)|\,\mathrm{d}y\leq$
$\displaystyle
C_{T}\|\varphi\|\int_{\mathbb{R}^{d}}\left(\frac{(t-s)^{\beta/\alpha}}{((t-s)^{1/\alpha}+|x-y|)^{d+\beta}}+\frac{(t-s)^{1-\gamma/\alpha}}{((t-s)^{1/\alpha}+|x-y|)^{d+\alpha-\gamma}}\right)\,\mathrm{d}y\leq$
$\displaystyle{C}_{T}\|\varphi\|$

for all $0\leq s<t\leq T$ and each $T>0$. Therefore the operators
$\mathds{T}_{st}$ are bounded.

Next, if $\varphi(x)\equiv 1$ then

$\displaystyle\mathds{T}_{st}\varphi(x)=\int_{\mathbb{R}^{d}}g(s,x,t,y)\,\mathrm{d}y+$
$\displaystyle\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}g(s,x,\tau,z)(b(\tau,z),\int_{\mathbb{R}^{d}}v(\tau,z,t,y)\,\mathrm{d}y)\,\mathrm{d}z.$

The function $w(s,x,t,1)=\int_{\mathbb{R}^{d}}v(\tau,z,t,y)\,\mathrm{d}y$,
$0\leq s<t$, $x\in\mathbb{R}^{d}$ is a solution to equation (21) with
$w_{0}(s,x,t,1)\equiv 0$. So, $w(s,x,t,1)\equiv 0$ and
$\mathds{T}_{st}\varphi(x)=\int_{\mathbb{R}^{d}}g(s,x,t,y)\,\mathrm{d}y\equiv
1$.

Although the evolution property can be proved in a standard way (see, for
example [10, 11, 13, 15, 16]), we will provide it here. For this, let us
choose arbitrary $0\leq s<u<t$, $\varphi\in C_{b}(\mathbb{R}^{d})$,
$x\in\mathbb{R}^{d}$ and consider

$\displaystyle\mathds{T}_{st}\varphi(x)=\mathds{T}_{st}^{0}\varphi(x)+\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}g(s,x,\tau,z)(b(\tau,z),w(\tau,z,t,\varphi))\,\mathrm{d}z=$
$\displaystyle\mathds{T}_{su}^{0}(\mathds{T}_{ut}^{0}\varphi)(x)+\int_{s}^{u}\mathrm{d}\tau\int_{\mathbb{R}^{d}}g(s,x,\tau,z)(b(\tau,z),w(\tau,z,t,\varphi))\,\mathrm{d}z+$
$\displaystyle\int_{\mathbb{R}^{d}}g(s,x,u,y)\,\mathrm{d}y\int_{u}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}g(u,y,\tau,z)(b(\tau,z),w(\tau,z,t,\varphi))\,\mathrm{d}z=$
$\displaystyle\mathds{T}_{su}^{0}(\mathds{T}_{ut}\varphi)(x)+\int_{s}^{u}\mathrm{d}\tau\int_{\mathbb{R}^{d}}g(s,x,\tau,z)(b(\tau,z),w(\tau,z,t,\varphi))\,\mathrm{d}z,$

where

$\mathds{T}_{st}^{0}\varphi(x)=\int_{\mathbb{R}^{d}}g(s,x,t,y)\varphi(y)\,\mathrm{d}y.$
(22)

Using (21), one can obtain (by changing of integration order)

$w(s,x,t,\varphi)=\int_{\mathbb{R}^{d}}\varphi(y)\,\mathrm{d}y\int_{\mathbb{R}^{d}}\nabla_{\beta}g(s,\cdot,u,z)(x)g(u,z,t,y)\,\mathrm{d}z+$
$\displaystyle\int_{s}^{u}\mathrm{d}\tau\int_{\mathbb{R}^{d}}v_{0}(s,x,\tau,z)(b(\tau,z),w(\tau,z,t,\varphi))\,\mathrm{d}z+$
$\displaystyle\int_{u}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}v_{0}(s,x,u,y)g(u,y,\tau,z)\,\mathrm{d}y(b(\tau,z),w(\tau,z,t,\varphi))\,\mathrm{d}z=$
$\displaystyle
w_{0}(s,x,u,\mathds{T}_{ut}\varphi)+\int_{s}^{u}\mathrm{d}\tau\int_{\mathbb{R}^{d}}v_{0}(s,x,\tau,z)(b(\tau,z),w(\tau,z,t,\varphi))\,\mathrm{d}z.$

Here we used the well-known relation (the Chapman-Kolmogorov equation)

$g(s,x,t,y)=\int_{\mathbb{R}^{d}}g(s,x,u,z)g(u,z,t,y)\,\mathrm{d}z,$

which is true for all $0\leq s<u<t$, $x,y\in\mathbb{R}^{d}$.

Since equation (21) has a unique solution, we state that

$w(s,x,t,\varphi)=w(s,x,u,\mathds{T}_{ut}\varphi)$

for all $\varphi\in C_{b}(\mathbb{R}^{d})$, $x\in\mathbb{R}^{d}$, $0\leq
s<u<t$. Therefore
$\mathds{T}_{st}\varphi(x)=\mathds{T}_{su}(\mathds{T}_{ut}\varphi)(x)$ and the
evolution property is proved.

The last statement of this theorem follows from the following two facts:
first, using (22) we have the equality
$\operatorname*{\mathit{w}-lim}_{s\uparrow t}\mathds{T}_{st}^{0}=I$, and
second, (note that $1\leq q<\frac{d+\alpha}{d+1}$)

$\displaystyle\left|\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}g(s,x,\tau,z)(b(\tau,z),w(\tau,z,t,\varphi))\,\mathrm{d}z\right|\leq$
$\displaystyle\|b\|_{p}^{T}N_{0,T}C_{T}\left(\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}\frac{(\tau-s)^{(1-\gamma/\alpha)q}}{((\tau-s)^{1/\alpha}+|x-z|)^{(d+\alpha-\gamma)q}}(t-\tau)^{-q\beta/\alpha}\,\mathrm{d}z\right)^{1/q}\leq$
$\displaystyle{C}_{T}\left(\int_{s}^{t}(\tau-s)^{-(q-1)d/\alpha}(t-\tau)^{-q\beta/\alpha}\,\mathrm{d}\tau\right)^{1/q}=$
$\displaystyle{C}_{T}(t-s)^{1-q\beta/\alpha-(q-1)d/\alpha}\to 0,\mbox{ as
}s\uparrow t.$

This completes the proof of the theorem. ∎

######  Remark 2.

We cannot state that the operators $\mathds{T}_{st}$ keep the cone of non-
negative functions. We have no proof that the function
$\mathds{T}_{st}\varphi(x)$ can have negative values if $\varphi(x)\geq 0$,
$x\in\mathbb{R}^{d}$. But the example of $\alpha$-stable process and
$b(t,x)\equiv b\in\mathbb{R}^{d}$ confirms this fact. Exactly analogous to how
it is done in [1] for the case $\beta=\alpha-1$, we can obtain the following
equality

$G(s,x,t,y)=\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}\exp\\{i(x-y-(t-s)b|\lambda|^{\beta-1},\lambda)-c(t-s)|\lambda|^{\alpha}\\}\mathrm{d}\lambda,$

in our case. The function $G$ here has negative values as the Fourier
transform of a non-positive definite function.

Thus, the evolution operators family $\\{\mathds{T}_{st}:0\leq s<t\\}$ does
not define any of the Markov processes but only a pseudo-process possessing a
‘‘Markov property’’.

##  5 Cauchy problem

In this section we fix some $T>0$ and prove that the function
$u(s,x,t)=\mathds{T}_{st}\varphi(x)$, $0\leq s<t\leq T$, $x\in\mathbb{R}^{d}$
is a solution (in some sense) to the following Cauchy problem:

$\displaystyle\frac{\partial}{\partial
s}u(s,x,t)+L(s,x)u(s,\cdot,t)(x)=0,\quad 0\leq s<t,\ x\in\mathbb{R}^{d}$ (23)
$\displaystyle\lim_{s\uparrow t}u(s,x,t)=\varphi(x),\quad x\in\mathbb{R}^{d},$
(24)

for every $t\in(0,T]$, where $L(s,x)=A(s,x)+(b(s,x),\nabla_{\beta})$ and
$\varphi\in C_{b}(\mathbb{R}^{d})$.

Equality (24) is proved in Theorem 2 (see the last statement). Equality (23)
will be proved in some generalized sense. Namely, let a sequence
$\\{b_{n}:n\in\mathbb{N}\\}\subset L_{p}([0,T]\times\mathbb{R}^{d})$ be such
that $L_{p}\mbox{-}\lim_{n\to\infty}b_{n}=b$ and
$u_{n}(s,x,t)=\mathds{T}_{st}^{(n)}\varphi(x)$ satisfied the Cauchy problem
(23), (24) with the function $b_{n}$ instead of $b$. The operators
$\mathds{T}_{st}^{(n)}$ are constructed using Theorems 1, 2 and functions
$b_{n}$ instead of $b$. If the limit $u(s,x,t)=\lim_{n\to\infty}u_{n}(s,x,t)$,
$0\leq s<t\leq T$, $x\in\mathbb{R}^{d}$ exists, we will call the function $u$
by the generalized solution to the Cauchy problem (23), (24). The following
auxiliary statement will be useful in constructing this generalized solution.

######  Lemma 5.

Let functions $(\tilde{b}(s,x))_{s\geq 0,x\in\mathbb{R}^{d}}$ and
$(\hat{b}(s,x))_{s\geq 0,x\in\mathbb{R}^{d}}$ satisfy the assumptions of
Theorem 1 with the same $p$. Then the corresponding functions $\tilde{G}$ and
$\hat{G}$ satisfy the following inequality

$\displaystyle|\tilde{G}(s,x,t,y)-\hat{G}(s,x,t,y)|\leq
C_{T}\|\tilde{b}-\hat{b}\|_{p}^{T}(1+\|\tilde{b}\|_{p}^{T}+\|\hat{b}\|_{p}^{T})\times$
$\displaystyle\frac{1}{((t-s)^{1/\alpha}+|x-y|)^{d+\beta-\gamma}},$ (25)

for all $0\leq s<t\leq T$, $x,y\in\mathbb{R}^{d}$ and each $T>0$ with some
positive constant $C_{T}$.

###### Proof.

Using (10), we can obtain the following

$\hat{G}(s,x,t,y)-\tilde{G}(s,x,t,y)=\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}g(s,x,\tau,z)W(\tau,z,t,y))\,\mathrm{d}z$
(26)

for all $0\leq s<t$, $x,y\in\mathbb{R}^{d}$, where
$W(s,x,t,y)=(\hat{b}(s,x),\hat{v}(s,x,t,y))-(\tilde{b}(s,x),\tilde{v}(s,x,t,y))$,
in which $\hat{v}$ and $\tilde{v}$ are solutions to equations obtained from
(11) replacing the function $b$ by the functions $\hat{b}$ and $\tilde{b}$,
respectively. Relation (11) leads us to the equation

$\displaystyle
W(s,x,t,y)=W_{0}(s,x,t,y)+\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}(v_{0}(s,x,\tau,z),\hat{b}(s,x))W(\tau,z,t,y))\,\mathrm{d}z+$
$\displaystyle\int_{\mathbb{R}^{d}}W_{0}(s,x,\tau,z)(\tilde{b}(\tau,z),\tilde{v}(\tau,z,t,y))\,\mathrm{d}z,$

where $W_{0}(s,x,t,y)=(\hat{b}(s,x)-\tilde{b}(s,x),v_{0}(s,x,t,y))$.
Considering (4), we obtain the following inequality

$\displaystyle|W_{0}(s,x,t,y)\leq|\hat{b}(s,x)-\tilde{b}(s,x)|N_{\beta,T}\left(\frac{1}{((t-s)^{1/\alpha}+|y-x|)^{d+\beta}}\right.+$
$\displaystyle\left.\frac{(t-s)^{1-\beta/\alpha}}{((t-s)^{1/\alpha}+|y-x|)^{d+\alpha-\gamma}}\right),$

valid for all $0\leq s<t\leq T$, $x,y\in\mathbb{R}^{d}$ and each $T>0$.
Moreover

$\displaystyle\left|\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}W_{0}(s,x,\tau,z)(\tilde{b}(\tau,z),\tilde{v}(\tau,z,t,y))\,\mathrm{d}z\right|\leq$
$\displaystyle
N_{\beta,T}|\hat{b}(s,x)-\tilde{b}(s,x)|C_{T}\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}|\tilde{b}(\tau,z)|\times$
$\displaystyle\left(\frac{1}{((\tau-s)^{1/\alpha}+|z-x|)^{d+\beta}}+\frac{(\tau-s)^{1-\beta/\alpha}}{((\tau-s)^{1/\alpha}+|z-x|)^{d+\alpha-\gamma}}\right)\times$
$\displaystyle\left(\frac{1}{((t-\tau)^{1/\alpha}+|y-z|)^{d+\beta}}+\frac{(t-\tau)^{1-\beta/\alpha}}{((t-\tau)^{1/\alpha}+|y-z|)^{d+\alpha-\gamma}}\right)\,\mathrm{d}z=$
$\displaystyle C_{T}|\hat{b}(s,x)-\tilde{b}(s,x)|\sum_{k=1}^{4}I_{k},$

where

$I_{1}=\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}|\tilde{b}(\tau,z)|\frac{1}{((\tau-s)^{1/\alpha}+|z-x|)^{d+\beta}}\frac{1}{((t-\tau)^{1/\alpha}+|y-z|)^{d+\beta}}\,\mathrm{d}z,$
$\displaystyle
I_{2}=\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}|\tilde{b}(\tau,z)|\frac{1}{((\tau-s)^{1/\alpha}+|z-x|)^{d+\beta}}\times$
$\displaystyle\frac{(t-\tau)^{1-\beta/\alpha}}{((t-\tau)^{1/\alpha}+|y-z|)^{d+\alpha-\gamma}}\,\mathrm{d}z,$
$\displaystyle
I_{3}=\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}|\tilde{b}(\tau,z)|\frac{(\tau-s)^{1-\beta/\alpha}}{((\tau-s)^{1/\alpha}+|z-x|)^{d+\alpha-\gamma}}\times$
$\displaystyle\frac{1}{((t-\tau)^{1/\alpha}+|y-z|)^{d+\beta}}\,\mathrm{d}z,$
$\displaystyle
I_{4}=\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}|\tilde{b}(\tau,z)|\frac{(\tau-s)^{1-\beta/\alpha}}{((\tau-s)^{1/\alpha}+|z-x|)^{d+\alpha-\gamma}}\times$
$\displaystyle\frac{(t-\tau)^{1-\beta/\alpha}}{((t-\tau)^{1/\alpha}+|y-z|)^{d+\alpha-\gamma}}\,\mathrm{d}z.$

Since $\tilde{b}\in L_{p}([0,T]\times\mathbb{R}^{d})$ for each $T>0$, using
inequality (2) we obtain the following estimates

$I_{1}\leq\|\tilde{b}\|_{p}^{T}(2CB(1,\theta
q))^{1/q}\frac{(t-s)^{\theta}}{((t-s)^{1/\alpha}+|y-x|)^{d+\beta}},$
$\displaystyle I_{2}\leq\|\tilde{b}\|_{p}^{T}\left(2C\max\left(B\left(1,\theta
q+q\frac{\gamma}{\alpha}\right),B\left(1+\left(1-\frac{\beta}{\alpha}\right)q,\theta
q\right)\right)\right)^{1/q}\times$
$\displaystyle\left(\frac{(t-s)^{\theta+\gamma/\alpha}}{((t-s)^{1/\alpha}+|y-x|)^{d+\beta}}+\frac{(t-s)^{\theta+1-\beta/\alpha}}{((t-s)^{1/\alpha}+|y-x|)^{d+\alpha-\gamma}}\right),$
$\displaystyle I_{3}\leq\|\tilde{b}\|_{p}^{T}\left(2C\max\left(B\left(1,\theta
q+q\frac{\gamma}{\alpha}\right),B\left(1+\left(1-\frac{\beta}{\alpha}\right)q,\theta
q\right)\right)\right)^{1/q}\times$
$\displaystyle\left(\frac{(t-s)^{\theta+\gamma/\alpha}}{((t-s)^{1/\alpha}+|y-x|)^{d+\beta}}+\frac{(t-s)^{\theta+1-\beta/\alpha}}{((t-s)^{1/\alpha}+|y-x|)^{d+\alpha-\gamma}}\right),$
$I_{4}\leq\|\tilde{b}\|_{p}^{T}\left(2CB\left(1+\left(1-\frac{\beta}{\alpha}\right)q,\theta
q+q\frac{\gamma}{\alpha}\right)\right)^{1/q}\frac{(t-s)^{\theta+1-\beta/\alpha+\gamma/\alpha}}{((t-s)^{1/\alpha}+|y-x|)^{d+\alpha-\gamma}},$

where, as it was above, $\theta=1-((d+\alpha)/p+\beta)/\alpha$, $q=p/(p-1)$
and $C$ is maximum of positive constants derived from inequality (2) in
considered four cases. Therefore

$\displaystyle\left|\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}W_{0}(s,x,\tau,z)(\tilde{b}(\tau,z),\tilde{v}(\tau,z,t,y))\,\mathrm{d}z\right|\leq
C_{T}\|\tilde{b}\|_{p}^{T}|\hat{b}(s,x)-\tilde{b}(s,x)|\times$
$\displaystyle\left(\frac{1}{((t-s)^{1/\alpha}+|y-x|)^{d+\beta}}+\frac{(t-s)^{1-\beta/\alpha}}{((t-s)^{1/\alpha}+|y-x|)^{d+\alpha-\gamma}}\right).$

Let us denote

$W_{0}^{*}(s,x,t,y)=W_{0}(s,x,t,y)+\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}W_{0}(s,x,\tau,z)(\tilde{b}(\tau,z),\tilde{v}(\tau,z,t,y))\,\mathrm{d}z.$

Then we can write down the following

$W(s,x,t,y)=W_{0}^{*}(s,x,t,y)+\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}(v_{0}(s,x,\tau,z),\hat{b}(s,x))W(\tau,z,t,y))\,\mathrm{d}z.$
(27)

This equation can be solved by the method of successive approximations, i. e.,
a solution of it can be found in the form
$W(s,x,t,y)=\sum_{k=0}^{\infty}W_{k}^{*}(s,x,t,y)$. The terms of this series
satisfy the following relation

$W_{k}^{*}(s,x,t,y)=\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}(v_{0}(s,x,\tau,z),\hat{b}(s,x))W_{k-1}^{*}(\tau,z,t,y))\,\mathrm{d}z\quad
k=1,2,\dots.$

To justification this, note that

$\displaystyle|W_{0}^{*}(s,x,t,y)|\leq
C_{T}(1+\|\tilde{b}\|_{p}^{T})|\hat{b}(s,x)-\tilde{b}(s,x)|\times$
$\displaystyle\left(\frac{1}{((t-s)^{1/\alpha}+|y-x|)^{d+\beta}}+\frac{(t-s)^{1-\beta/\alpha}}{((t-s)^{1/\alpha}+|y-x|)^{d+\alpha-\gamma}}\right).$

Moreover, by the method mathematical induction one can prove the following
estimate

$\displaystyle|W_{k}^{*}(s,x,t,y)|\leq
R_{k}|\hat{b}(s,x)|\|\hat{b}-\tilde{b}\|_{p}^{T}\times$
$\displaystyle\left(\frac{(t-s)^{k\theta}}{((t-s)^{1/\alpha}+|y-x|)^{d+\beta}}+\frac{(t-s)^{k\theta+1-\beta/\alpha}}{((t-s)^{1/\alpha}+|y-x|)^{d+\alpha-\gamma}}\right).$

valid for all $k\geq 1$, where $R_{1}=C_{T}(1+\|\tilde{b}\|_{p}^{T})$ and
$C_{T}$ is some positive constant depended on $T$, and, for $k\geq 2$

$\displaystyle
R_{k}=R_{k-1}C_{T}(1+T^{\gamma/\alpha})(1+\|\tilde{b}\|_{p}^{T})\|\hat{b}\|_{p}^{T}\times$
$\displaystyle(8C\max(B(1,k\theta q),B(\theta q,(k-1)\theta q+1)))^{1/q}.$

Therefore the series $\sum_{k=0}^{\infty}W_{k}^{*}(s,x,t,y)$ converges
uniformly with respect to $x,y\in\mathbb{R}^{d}$ and locally uniformly with
respect to $0\leq s<t$. So, its sum $W$ is a solution to equation (27). In
addition we obtain the following estimate:

$\displaystyle|W(s,x,t,y)|\leq{C}_{T}\left((1+\|\tilde{b}\|_{p}^{T})|\hat{b}(s,x)-\tilde{b}(s,x)|+|\hat{b}(s,x)|\|\hat{b}-\tilde{b}\|_{p}^{T}\right)\times$
$\displaystyle\left(\frac{1}{((t-s)^{1/\alpha}+|y-x|)^{d+\beta}}+\frac{(t-s)^{1-\beta/\alpha}}{((t-s)^{1/\alpha}+|y-x|)^{d+\alpha-\gamma}}\right),$
(28)

valid for all $0\leq s<t\leq T$, $x,y\in\mathbb{R}^{d}$ and each $T>0$, where
${C}_{T}$ is some positive constant depended on $T$.

Using (26),(28) and (2) with the Hölder inequality, we obtain (5). ∎

######  Lemma 6.

Let the function $w$ be defined in Lemma 4. For every $0\leq s<t\leq T$,
$x,y\in\mathbb{R}^{d}$, the following inequality

$|w(s,x,t,\varphi)-w(s,y,t,\varphi)|\leq C_{T}|x-y|(t-s)^{-(\beta+1)/\alpha},$

holds with some positive constant $C_{T}$ depended on $T$.

###### Proof.

Using (21) we can write down the relation ($0\leq s<t$,
$x,y\in\mathbb{R}^{d}$)

$\displaystyle
w(s,x,t,\varphi)-w(s,y,t,\varphi)=w_{0}(s,x,t,\varphi)-w_{0}(s,y,t,\varphi)+$
$\displaystyle\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}(v_{0}(s,x,\tau,z)-v_{0}(s,y,\tau,z))(b(\tau,z),w(\tau,z,t,\varphi))\,\mathrm{d}z.$
(29)

Let us remark that (see (5))

$v_{0}(s,x,t,y)=\nabla_{\beta}g_{0}(s,\cdot,t,y)(x)+\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}\nabla_{\beta}g_{0}(s,\cdot,\tau,z)(x)\Phi(\tau,z,t,y)\,\mathrm{d}z.$

Moreover, for all $0\leq s<t$, $x,y,z\in\mathbb{R}^{d}$,

$\displaystyle\nabla_{\beta}g_{0}(s,\cdot,t,z)(x)-\nabla_{\beta}g_{0}(s,\cdot,t,z)(y)=$
$\displaystyle\frac{i}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}\left(e^{i(x-z,\lambda)}-e^{i(y-z,\lambda)}\right)\lambda|\lambda|^{\beta-1}\exp\left\\{-a(t,z,\lambda)(t-s)\right\\}\,\mathrm{d}\lambda=$
$\displaystyle-\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}e^{i(\eta-z,\lambda)}(\lambda,x-y)\lambda|\lambda|^{\beta-1}\exp\left\\{-a(t,z,\lambda)(t-s)\right\\}\,\mathrm{d}\lambda=$
$\displaystyle|x-y|D_{\beta+1}g_{0}(s,\cdot,t,z)(\eta),$

where $\eta=\theta x+(1-\theta)y$ with some $\theta\in(0,1)$ and the operator
$D_{\beta+1}$ is defined by the symbol
$\left(\left(\lambda,\frac{x-y}{|x-y|}\right)\lambda|\lambda|^{\beta-1}\right)_{\lambda\in\mathbb{R}^{d}}$.
This symbol satisfies the assumptions of [5, Lemma 4.2], which leads us to the
estimate

$|D_{\beta+1}g_{0}(s,\cdot,t,z)(\eta)|\leq\frac{C}{((t-s)^{1/\alpha}+|z-\eta|)^{d+\beta+1}}.$

Thus, using (2), we obtain the inequality

$\displaystyle|v_{0}(s,x,t,z)-v_{0}(s,y,t,z)|\leq\frac{C|x-y|}{((t-s)^{1/\alpha}+|z-\eta|)^{d+\beta+1}}+$
$\displaystyle\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}\frac{C|x-y|}{((\tau-s)^{1/\alpha}+|u-\eta|)^{d+\beta+1}}\frac{C}{((t-\tau)^{1/\alpha}+|z-u|)^{d+\alpha-\gamma}}\,\mathrm{d}u\leq$
$\displaystyle|x-y|C_{T}\left(\frac{1}{((t-s)^{1/\alpha}+|z-\eta|)^{d+\beta+1}}+\frac{(t-s)^{1-(\beta+1)/\alpha}}{((t-s)^{1/\alpha}+|z-\eta|)^{d+\alpha-\gamma}}\right),$

valid for all $0\leq s<t\leq T$, $x,y,z\in\mathbb{R}^{d}$. And, as
consequence, the following inequalities are true for all $0\leq s<t\leq T$,
$x,y\in\mathbb{R}^{d}$ and $\varphi\in C_{b}(\mathbb{R}^{d})$:

$|v_{0}(s,x,t,\varphi)-v_{0}(s,y,t,\varphi)|\leq
C_{T}|x-y|\|\varphi\|(t-s)^{-(\beta+1)/\alpha};$
$\displaystyle\left|\int_{s}^{t}\mathrm{d}\tau\int_{\mathbb{R}^{d}}(v_{0}(s,x,\tau,z)-v_{0}(s,y,\tau,z))(b(\tau,z),w(\tau,z,t,\varphi))\,\mathrm{d}z\right|\leq$
$\displaystyle C_{T}|x-y|\|b\|_{p}^{T}\left(\int_{s}^{t}(t-\tau)^{-\beta
q/\alpha}\mathrm{d}\tau\int_{\mathbb{R}^{d}}\left(\frac{1}{((t-s)^{1/\alpha}+|z-\eta|)^{d+\beta+1}}+\right.\right.$
$\displaystyle\left.\left.\frac{(t-s)^{1-(\beta+1)/\alpha}}{((t-s)^{1/\alpha}+|z-\eta|)^{d+\alpha-\gamma}}\right)^{q}\,\mathrm{d}z\right)^{1/q}\leq$
$\displaystyle{C}_{T}|x-y|(t-s)^{1-(1+d/\alpha)/p-(2\beta+1)/\alpha},$

where $q=p/(p-1)$.

Therefore, using (5) and the fact that
$1-(1+d/\alpha)/p-\beta/\alpha>(1-\beta)/\alpha$, we obtain the lemma
statement. ∎

In the next theorem, we construct a generalized solution to the Cauchy problem
formulated at the beginning of the section.

######  Theorem 3.

Let the assumptions of Theorem 1 hold and the function $G$ is constructed
there. Then the function

$u(s,x,t)=\int_{\mathbb{R}^{d}}G(s,x,t,y)\varphi(y)\,\mathrm{d}y,\quad 0\leq
s<t\leq T,\ x\in\mathbb{R}^{d}$

is a generalized solution to the Cauchy problem (23), (24) for each
$\varphi\in C_{b}(\mathbb{R}^{d})$.

###### Proof.

Let us consider a sequence $\left\\{b_{n}:n\in\mathbb{N}\right\\}$ of
$\mathbb{R}^{d}$-valued functions, which are infinity continuous
differentiable, have compact supports and belong to
$L_{p}([0,T]\times\mathbb{R}^{d})$, where $p$ is defined in Theorem 1. Assume
that $L_{p}\mbox{-}\lim_{n\to\infty}b_{n}=b$, where $b$ is the function from
Theorem 1.

We denote by $v_{n}$, $w_{n}$ and $G_{n}$ the objects that are defined as $v$,
$w$ and $G$, respectively, using $b_{n}$ instead of $b$.

Lemma 5 allows us to state that the sequence of corresponding functions
$G_{n}$ constructed in Theorem 1 using the functions $b_{n}$ instead of $b$
converges to the function $G$ uniformly with respect to $y\in\mathbb{R}^{d}$
for each fixed $0\leq s<t\leq T$ and $x\in\mathbb{R}^{d}$.

Let us consider the function

$f_{n}(s,x,t)=\int_{\mathbb{R}^{d}}(b_{n}(s,x),\nabla_{\beta}G_{n}(s,\cdot,t,y)(x))\varphi(y)\,\mathrm{d}y,\quad
0\leq s<t\leq T,\ x\in\mathbb{R}^{d}.$

Remind that $\nabla_{\beta}G_{n}(s,\cdot,t,y)(x)=v_{n}(s,x,t,y)$. Moreover,

$\displaystyle|f_{n}(s,x,t)-f_{n}(s,y,t)|\leq|b_{n}(s,x)||w_{n}(s,x,t,\varphi)-w_{n}(s,y,t,\varphi)|+$
$\displaystyle|b_{n}(s,x)-b_{n}(s,y)||w_{n}(s,y,t,\varphi)|$

Since the function $b_{n}$ is Lipschitz continuous in $x$, uniformly to $s$
and has a compact support, then, as follows from Lemma 6, the function
$f_{n}(s,x,t)$ is Lipschitz continuous in $x$, uniformly to $s\in[0,t)$ for
every fixed $t>0$.

Therefore (see [5, Th. 4.1.]) the function
$u_{n}(s,x,t)=\int_{\mathbb{R}^{d}}G_{n}(s,x,t,y)\varphi(y)\,\mathrm{d}y$ is a
solution to the Cauchy problem (23), (24) for every $t\in(0,T]$.

∎

## References

  * [1] Bigun, G.S., Osypchuk, M.M.: The explicit form of the fundamental solution of one pseudo-differential equation with constant coefficients. Precarpathian bulletin of the Shevchenko scientific society (Number) (1(29)), 123–131 (2015). (In Ukrainian)
  * [2] Bogdan, K., Jakubowski, T.: Estimates of heat kernel of fractional laplacian perturbed by gradient operators. Commun. Math. Phys. 271, 179–198 (2007). doi:10.1007/s00220-006-0178-y
  * [3] Boyko, M.V., Osypchuk, M.M.: Perturbation of a rotationally invariant $\alpha$-stable stochastic process by a pseudo-gradient operator. Precarpathian bulletin of the Shevchenko scientific society (Number) (16(60)), 20–32 (2021). doi:10.31471/2304-7399-2021-16(60)-20-32. (In Ukrainian)
  * [4] Chen, Z.-Q., Wang, J.-M.: Perturbation by non-local operators. Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 54(2), 606–639 (2018). doi:10.1214/16-AIHP816
  * [5] Eidelman, S.D., Ivasyshen, S.D., Kochubei, A.N.: Analytic Methods in the Theory of Differential and Pseudo-Differential Equations of Parabolic Type. Birkhäuser, Basel (2004). doi:10.1007/978-3-0348-7844-9
  * [6] Jakubowski, T.: Fundamental solution of the fractional diffusion equation with a singular drift. J. Math. Sci. (N.Y.) 218(2) (2016). doi:10.1007/s10958-016-3016-6
  * [7] Jakubowski, T., Szczypkowski, K.: Time-dependent gradient perturbations of fractional laplacian. J. Evol. Equ. 10(2), 319–339 (2010). doi:10.1007/s00028-009-0051-5
  * [8] Loebus, J.-U., Portenko, M.I.: On a class of perturbations of a stable process. Theor. Probability and Math. Statist. 52, 102–111 (1995). (In Ukrainian)
  * [9] Maekawa, Y., Miura, H.: On fundamental solutions for non-local parabolic equations with divergence free drift. Advances in Mathematics 247, 123–191 (2013). doi:10.1016/j.aim.2013.07.011
  * [10] Osypchuk, M.M.: On some perturbations of a stable process and solutions to the Cauchy problem for a class of pseudo-differential equations. Carpathian Math. Publ. 7(1), 101–107 (2015). doi:10.15330/cmp.7.1.101-107
  * [11] Osypchuk, M.M.: On some perturbations of a symmetric stable process and the corresponding Cauchy problems. Theory Stoch. Process. 21(37)(1), 64–72 (2016)
  * [12] Podolynny, S.I., Portenko, N.I.: On multidimentional stable processes with locally unbounded drift. Random Oper. Stoch. Equ. 3(2), 113–124 (1995). doi:10.1515/rose.1995.3.2.113
  * [13] Portenko, M.I.: Diffusion processes in media with membranes. In: Proceedings of the Institute of Mathematics of the Ukrainian National Academy of Sciences (1995). (In Ukrainian)
  * [14] Portenko, N.I.: Generalized Diffusion Processes. American Mathematical Society, Providence, Rhode Island (1990)
  * [15] Portenko, N.I.: Some perturbations of drift-type for symmetric stable processes. Random Oper. Stoch. Equ. 2(3), 211–224 (1994). doi:10.1515/rose.1994.2.3.211
  * [16] Portenko, N.I.: On some perturbations of symmetric stable processes. In: Probability Theory and Mathematical Statistics. Proceedings of the Seventh Japan-Russia Symposium, Tokyo, Japan, July 26-30, 1995, pp. 414–422. World Scientific, Singapore (1996)
  * [17] Portenko, N.I.: One class of transformations of a symmetric stable process. Theory Stoch. Process. 3(19)(3-4), 373–387 (1997)
  * [18] Zhang, Q.: Gaussian bounds for the fundamental solutions of $\nabla(a\nabla u)+b\nabla u-u_{t}=0$. Manuscripta Mathematica 93(1), 381–390 (2007). doi:10.1007/BF02677479