Log-Harnack Inequalities for Markov Semigroups Generated by Non-Local Gruschin Type Operators

TL;DR

This paper establishes log-Harnack inequalities for Markov semigroups generated by non-local Gruschin type operators using coupling and regularization techniques, with concrete examples provided.

Contribution

It introduces a novel approach combining coupling and regularization to derive log-Harnack inequalities for a new class of non-local operators.

Findings

Log-Harnack inequalities are proven for the class of operators.

The method applies to various concrete examples.

The approach advances understanding of non-local Markov processes.

Abstract

Based on coupling in two steps and the regularization approximations of the underlying subordinators, we establish log-Harnack inequalities for Markov semigroups generated by a class of non-local Gruschin type operators. Some concrete examples are also presented.

Equations

$$L(x^{(1)},x^{(2)})=\frac{1}{2}\left[\frac{{\partial }^2}{\partial (x^{(1)}{)}^2}+|x^{(1)}{|}^{2l}\frac{{\partial }^2}{\partial (x^{(2)}{)}^2}\right].$$

$$L_{{\varphi }_1,{\varphi }_2}(x^{(1)},x^{(2)})=-\frac{1}{2}\left[{\varphi }_1\left(-\frac{{\partial }^2}{\partial (x^{(1)}{)}^2}\right)+|x^{(1)}{|}^{2l}{\varphi }_2\left(-\frac{{\partial }^2}{\partial (x^{(2)}{)}^2}\right)\right].$$

$${\varphi }_i(u)={\vartheta }_{i}u+{\int }_{(0,\infty )}\left(1-{\mathrm{e}}^{-ux}\right){\nu }_i(\mathrm{d}x),u>0,$$

$$\left\{\begin{array}[]{l}\mathrm{d}X_{t}^{(1)}=\mathrm{d}W_{S_{1}(t)}^{(1)},\\ \mathrm{d}X_{t}^{(2)}=|X_{t}^{(1)}|^{l}\,\mathrm{d}W_{S_{2}(t)}^{(2)},\end{array}\right.$$

$$\text{mathds}E{\mathrm{e}}^{-u{S}_i(t)}={\mathrm{e}}^{-t{\varphi }_i(u)},t\ge 0,u>0.$$

$$\left\{\begin{array}[]{l}\mathrm{d}X_{t}^{(1)}=\sigma_{t}\,\mathrm{d}W_{S_{1}(t)}^{(1)},\\ \mathrm{d}X_{t}^{(2)}=b(t,X_{t}^{(2)})\,\mathrm{d}t+|X_{t}^{(1)}|^{l}\,\mathrm{d}W_{S_{2}(t)}^{(2)},\end{array}\right.$$

$$\big{\langle}b(t,x^{(2)})-b(t,y^{(2)}),x^{(2)}-y^{(2)}\big{\rangle}\leq k(t)|x^{(2)}-y^{(2)}|^{2},\quad x^{(2)},y^{(2)}\in\mathds{R}^{d},\,t\geq 0.$$

$$P_{t}f(x):=\mathds{E}f\big{(}X_{t}(x)\big{)},\quad t\geq 0,\,f\in\mathscr{B}_{b}(\mathds{R}^{m+d}),\,x\in\mathds{R}^{m+d}.$$

$$K(s,t):={\int }_s^{t}k(r)\mathrm{d}r,0\le s\le t,$$

$$P_{2T}\log f(y)\le \log P_{2T}f(x)+\frac{|x^{(1)}-y^{(1)}{|}^2}{2}\text{mathds}E{\left({\int }_0^T{\lambda }_r^{-2}\mathrm{d}{S}_1(r)\right)}^{-1}$$

$$\displaystyle\quad+C\mathrm{e}^{2K(0,T)}\Bigg{\{}|x^{(2)}-y^{(2)}|^{2}\mathds{E}\left(\int_{0}^{T}\lambda_{r}^{-2}\,\mathrm{d}S_{1}(r)\right)^{-l}\cdot\mathds{E}\left(\int_{T}^{2T}\mathrm{e}^{-2K(T,s)}\,\mathrm{d}S_{2}(s)\right)^{-1}$$

$$+\left(\left[|x^{(1)}{|}^{2(l-1{)}^{+}}+|y^{(1)}{|}^{2(l-1{)}^{+}}\right]\text{mathds}E{\left({\int }_0^T{\lambda }_r^{-2}\mathrm{d}{S}_1(r)\right)}^{-l}+\text{mathds}E{\left({\int }_0^T{\lambda }_r^{-2}\mathrm{d}{S}_1(r)\right)}^{-(l\wedge 1)}\right)$$

$$\displaystyle\qquad\quad\times|x^{(1)}-y^{(1)}|^{2(l\wedge 1)}\mathds{E}\frac{\int_{0}^{T}\mathrm{e}^{-2K(0,s)}\,\mathrm{d}S_{2}(s)}{\int_{T}^{2T}\mathrm{e}^{-2K(T,s)}\,\mathrm{d}S_{2}(s)}\Bigg{\}}.$$

$$P_{2T}\log f(y)\le \log P_{2T}f(x)+\frac{|x^{(1)}-y^{(1)}{|}^2}{2}\text{mathds}E{S}_1(T{)}^{-1}$$

$$\displaystyle\;+C\mathds{E}S_{2}(T)^{-1}\cdot\bigg{\{}|x^{(2)}-y^{(2)}|^{2}\mathds{E}S_{1}(T)^{-l}$$

$$\displaystyle\;\quad+\left(\left[|x^{(1)}|^{2(l-1)^{+}}+|y^{(1)}|^{2(l-1)^{+}}\right]\mathds{E}S_{1}(T)^{-l}+\mathds{E}S_{1}(T)^{-(l\wedge 1)}\right)|x^{(1)}-y^{(1)}|^{2(l\wedge 1)}\mathds{E}S_{2}(T)\bigg{\}}.$$

$$P_{2T}\log f(y)$$

$$\displaystyle\qquad+\left(\left[|x^{(1)}|^{2(l-1)^{+}}+|y^{(1)}|^{2(l-1)^{+}}\right]T^{-l/\alpha}+T^{-(l\wedge 1)/\alpha}\right)|x^{(1)}-y^{(1)}|^{2(l\wedge 1)}\bigg{\}}.$$

$$\displaystyle P_{2T}\log f(y)\leq\log P_{2T}f(x)+C\bigg{\{}T^{-1/\alpha}|x^{(1)}-y^{(1)}|^{2}+T^{-l/\alpha}\left(T^{-1/\beta}\vee T^{-1}\right)|x^{(2)}-y^{(2)}|^{2}$$

$$\displaystyle\qquad\,+\left(T^{1-1/\beta}\vee 1\right)\left(\left[|x^{(1)}|^{2(l-1)^{+}}+|y^{(1)}|^{2(l-1)^{+}}\right]T^{-l/\alpha}+T^{-(l\wedge 1)/\alpha}\right)|x^{(1)}-y^{(1)}|^{2(l\wedge 1)}\bigg{\}}.$$

$$\left\{\begin{array}[]{l}\mathrm{d}X_{t}^{(1),\ell_{1}}=\sigma_{t}\,\mathrm{d}W_{\ell_{1}(t)}^{(1)},\\ \mathrm{d}X_{t}^{(2),\ell_{1},\ell_{2}}=b(t,X_{t}^{(2),\ell_{1},\ell_{2}})\,\mathrm{d}t+|X_{t}^{(1),\ell_{1}}|^{l}\,\mathrm{d}W_{\ell_{2}(t)}^{(2)}.\end{array}\right.$$

$$P_{t}^{\ell_{1},\ell_{2}}f(x)=\mathds{E}f\big{(}X_{t}^{\ell_{1},\ell_{2}}(x)\big{)},\quad t\geq 0,\,f\in\mathscr{B}_{b}(\mathds{R}^{m+d}),\,x\in\mathds{R}^{m+d},$$

$$P_{2T}^{{\ell }_1,{\ell }_2}\log f(y)\le \log P_{2T}^{{\ell }_1,{\ell }_2}f(x)+\frac{|x^{(1)}-y^{(1)}{|}^2}{2}{\left({\int }_0^T{\lambda }_r^{-2}\mathrm{d}{\ell }_1(r)\right)}^{-1}$$

$$\displaystyle\qquad\quad+\frac{C\mathrm{e}^{2K(0,T)}}{\int_{T}^{2T}\mathrm{e}^{-2K(T,s)}\,\mathrm{d}\ell_{2}(s)}\Bigg{\{}|x^{(2)}-y^{(2)}|^{2}\left(\int_{0}^{T}\lambda_{r}^{-2}\,\mathrm{d}\ell_{1}(r)\right)^{-l}$$

$$+\left(\left[|x^{(1)}{|}^{2(l-1{)}^{+}}+|y^{(1)}{|}^{2(l-1{)}^{+}}\right]{\left({\int }_0^T{\lambda }_r^{-2}\mathrm{d}{\ell }_1(r)\right)}^{-l}+{\left({\int }_0^T{\lambda }_r^{-2}\mathrm{d}{\ell }_1(r)\right)}^{-(l\wedge 1)}\right)$$

$$\displaystyle\quad\quad\qquad\quad\times|x^{(1)}-y^{(1)}|^{2(l\wedge 1)}\int_{0}^{T}\mathrm{e}^{-2K(0,s)}\,\mathrm{d}\ell_{2}(s)\Bigg{\}}.$$

$${\ell }_i^{{\epsilon }_i}(t):=\frac{1}{{\epsilon }_i}{\int }_t^{t+{\epsilon }_i}{\ell }_i(s)\mathrm{d}s+{\epsilon }_{i}t={\int }_0^1{\ell }_i({\epsilon }_{i}s+t)\mathrm{d}s+{\epsilon }_{i}t,i\in \{1,2\},t\ge 0.$$

$${\ell }_i^{{\epsilon }_i}(t)\downarrow {\ell }_i(t)\text{as εi↓0}$$

$$\left\{\begin{array}[]{l}\mathrm{d}X_{t}^{(1),\ell^{\varepsilon_{1}}_{1}}=\sigma_{t}\,\mathrm{d}W_{\ell_{1}^{\varepsilon_{1}}(t)-\ell_{1}^{\varepsilon_{1}}(0)}^{(1)},\\ \mathrm{d}X_{t}^{(2),\ell_{1}^{\varepsilon_{1}},\ell_{2}^{\varepsilon_{2}}}=b(t,X_{t}^{(2),\ell_{1}^{\varepsilon_{1}},\ell_{2}^{\varepsilon_{2}}})\,\mathrm{d}t+|X_{t}^{(1),\ell_{1}^{\varepsilon_{1}}}|^{l}\,\mathrm{d}W_{\ell_{2}^{\varepsilon_{2}}(t)-\ell_{2}^{\varepsilon_{2}}(0)}^{(2)}.\end{array}\right.$$

$$P_{t}^{\ell_{1}^{\varepsilon_{1}},\ell_{2}^{\varepsilon_{2}}}f(x)=\mathds{E}f\big{(}X_{t}^{\ell_{1}^{\varepsilon_{1}},\ell_{2}^{\varepsilon_{2}}}(x)\big{)},\quad t\geq 0,\,f\in\mathscr{B}_{b}(\mathds{R}^{m+d}),\,x\in\mathds{R}^{m+d}.$$

$$P_{2T}^{{\ell }_1^{{\epsilon }_1},{\ell }_2^{{\epsilon }_2}}\log f(y)\le \log P_{2T}^{{\ell }_1^{{\epsilon }_1},{\ell }_2^{{\epsilon }_2}}f(x)+\frac{|x^{(1)}-y^{(1)}{|}^2}{2}{\left({\int }_0^T{\lambda }_r^{-2}\mathrm{d}{\ell }_1^{{\epsilon }_1}(r)\right)}^{-1}$$

$$\displaystyle\quad\quad+\frac{C\mathrm{e}^{2K(0,T)}}{\int_{T}^{2T}\mathrm{e}^{-2K(T,s)}\,\mathrm{d}\ell^{\varepsilon_{2}}_{2}(s)}\Bigg{\{}|x^{(2)}-y^{(2)}|^{2}\left(\int_{0}^{T}\lambda_{r}^{-2}\,\mathrm{d}\ell^{\varepsilon_{1}}_{1}(r)\right)^{-l}$$

$$+\left(\left[|x^{(1)}{|}^{2(l-1{)}^{+}}+|y^{(1)}{|}^{2(l-1{)}^{+}}\right]{\left({\int }_0^T{\lambda }_r^{-2}\mathrm{d}{\ell }_1^{{\epsilon }_1}(r)\right)}^{-l}+{\left({\int }_0^T{\lambda }_r^{-2}\mathrm{d}{\ell }_1^{{\epsilon }_1}(r)\right)}^{-(l\wedge 1)}\right)$$

$$\displaystyle\quad\quad\qquad\times|x^{(1)}-y^{(1)}|^{2(l\wedge 1)}\int_{0}^{T}\mathrm{e}^{-2K(0,s)}\,\mathrm{d}\ell^{\varepsilon_{2}}_{2}(s)\Bigg{\}}.$$

$$\left\{\begin{array}[]{l}\mathrm{d}Y_{t}^{(1)}=\sigma_{t}\,\mathrm{d}W_{\ell_{1}^{\varepsilon_{1}}(t)-\ell_{1}^{\varepsilon_{1}}(0)}^{(1)}+\xi^{(1)}_{t}\frac{X_{t}^{(1),\ell^{\varepsilon_{1}}_{1}}(x)-Y_{t}^{(1)}}{\big{|}X_{t}^{(1),\ell^{\varepsilon_{1}}_{1}}(x)-Y_{t}^{(1)}\big{|}}\mathds{1}_{[0,\tau^{(1)})}(t)\,\mathrm{d}\ell_{1}^{\varepsilon_{1}}(t),\\ Y_{0}^{(1)}=y^{(1)},\end{array}\right.$$

$$\xi^{(1)}_{t}:=\frac{|x^{(1)}-y^{(1)}|}{\int_{0}^{T}\lambda_{r}^{-2}\,\mathrm{d}\ell^{\varepsilon_{1}}_{1}(r)}\lambda_{t}^{-2}\quad\text{and}\quad\tau^{(1)}:=\inf\big{\{}t\geq 0\,:\,X_{t}^{(1),\ell^{\varepsilon_{1}}_{1}}(x)=Y_{t}^{(1)}\big{\}}.$$

$$\text{mathds}{R}^m\times \text{mathds}{R}^m\ni (z,z^{\prime })\mapsto \text{mathds}{1}_{\{z\ne z^{\prime }\}}\frac{z-z^{\prime }}{|z-z^{\prime }|}\in \text{mathds}{R}^m$$

Full text

Financial support through the National Natural Science Foundation of China (11401442) (for Chang-Song Deng) and the National Natural Science Foundation of China (11626237) (for Shao-Qin Zhang) is gratefully acknowledged.

1. Introduction

The classical Gruschin semigroup on $\mathds{R}^{2}$ with order $l>0$ is generated by the differential operator

[TABLE]

The derivative formula of Bismut-Elworthy-Li’s type (cf. [1, 7]) and log-Harnack inequality, first introduced in [9], have been investigated for the associated diffusion processes in [13] and [15], respectively. As a natural extension, let us consider the following non-local Gruschin type operator

[TABLE]

Here, each $\phi_{i}:(0,\infty)\rightarrow(0,\infty)$ is a Bernstein function with $\lim_{u\downarrow 0}\phi_{i}(u)=0$, i.e. $\phi_{i}$ is given by (see e.g. [10, Theorem 3.2])

[TABLE]

where $\vartheta_{i}\geq 0$ is the drift parameter, and $\nu_{i}$ is a Lévy measure, that is, a Radon measure on $(0,\infty)$ such that $\int_{(0,\infty)}(1\wedge x)\,\nu_{i}(\mathrm{d}x)<\infty$. The Markov process with jumps generated by the non-local operator (1.1) can be constructed by solving the degenerate SDE driven by subordinate Brownian motions

[TABLE]

where $W_{t}^{(1)}$, $W_{t}^{(2)}$, $S_{1}(t)$, and $S_{2}(t)$ are independent processes such that each $W_{t}^{(i)}$ is a standard $1$-dimensional Brownian motion, and $S_{i}(t)$ is a subordinator (i.e. a non-decreasing Lévy process on $[0,\infty)$) determined by its Laplace transform which is of the form

[TABLE]

Due to the importance both in theory and in applications, recently, there has been considerable interest in the study of discontinuous Markov processes. The central aim of this paper is to establish the log-Harnack inequality for Markov semigroups generated by the non-local Gruschin type operator (1.1). The log-Harnack inequality can be regarded as a weaker version of F.-Y. Wang’s dimension-free Harnack inequality with power initialed in [11], and has been thoroughly investigated, especially for diffusion processes; the basic argument was a coupling by change of measure, see [12] and reference therein for recent developments on Harnack type inequalities for various models. Since it is usually very difficult to construct successful couplings for non-linear SDEs driven by pure jump noises, the methods from diffusions cannot be directly applied and we need some technique from the study for jump-diffusion processes. In this article, our tool is based on the coupling approach in [15] and the regularization approximations of time-changes used in [17, 14, 16, 3, 5].

The log-Harnack inequality has become an efficient tool in stochastic analysis, and it can be used to study the strong Feller property, heat kernel estimates, transportation-cost inequalities, and many more; we refer to the monograph by F.-Y. Wang [12, Subsection 1.4.1] for an in-depth explanation of its applications.

For generality, we consider the following SDE for $X_{t}=(X_{t}^{(1)},X_{t}^{(2)})$ on $\mathds{R}^{m+d}=\mathds{R}^{m}\times\mathds{R}^{d}$ ($m,d\in\mathds{N}$):

[TABLE]

where $\sigma:[0,\infty)\rightarrow\mathds{R}^{m}\otimes\mathds{R}^{m}$ is measurable and locally bounded, $b:[0,\infty)\times\mathds{R}^{d}\rightarrow\mathds{R}^{d}$ is measurable, locally bounded in the time variable $t\geq 0$ and continuous in the space variable $x^{(2)}\in\mathds{R}^{d}$, and $W_{t}:=(W_{t}^{(1)},W_{t}^{(2)})$, $S_{1}(t)$, and $S_{2}(t)$ are independent processes on a probability space $(\Omega,\mathscr{A},\mathds{P})$ such that

1. (i)

$W_{t}$ is a standard Brownian motion on $\mathds{R}^{m+d}$; 2. (ii)

Each $S_{i}(t)$ is a subordinator with characteristic exponent (Bernstein function) $\phi_{i}$ given by (1.2).

We will assume the following conditions on $\sigma$ and $b$:

1. (H1)

For every $t\geq 0$, $\sigma_{t}$ is invertible and there exists a non-decreasing function $\lambda:[0,\infty)\rightarrow(0,\infty)$ such that $\|\sigma_{t}^{-1}\|\leq\lambda_{t}$ for all $t\geq 0$. 2. (H2)

There exists a locally bounded measurable function $k:[0,\infty)\rightarrow\mathds{R}$ such that

[TABLE]

It is easy to see that once $X_{t}^{(1)}$ is fixed, then (H2) implies the existence, uniqueness and non-explosion of the solution to the second equation in (1.3). For $x=(x^{(1)},x^{(2)})\in\mathds{R}^{m}\times\mathds{R}^{d}$, denote by $X_{t}(x)=(X_{t}^{(1)}(x),X_{t}^{(2)}(x))$ the solution to (1.3) with $X_{0}=x$. We aim to establish log-Harnack inequalities for the associated Markov semigroup $P_{t}$ on $\mathscr{B}_{b}(\mathds{R}^{m+d})$:

[TABLE]

In order to state our main result, we need the following notation:

[TABLE]

where $k$ is the function appearing in (H2).

Theorem 1.1**.**

Let $l\in(0,m/2)$ and assume that (H1) and (H2) hold. There is some constant $C=C(m,d,l)>0$ such that for any $T>0$, $x=(x^{(1)},x^{(2)}),y=(y^{(1)},y^{(2)})\in\mathds{R}^{m+d}$, and $f\in\mathscr{B}_{b}(\mathds{R}^{d})$ with $f\geq 1$,

[TABLE]

The following result is a direct consequence of Theorem 1.1.

Corollary 1.2**.**

Let $\sigma=I_{m\times m}$, $b=0$, and $l\in(0,m/2)$. There is some constant $C=C(m,d,l)>0$ such that for any $T>0$, $x=(x^{(1)},x^{(2)}),y=(y^{(1)},y^{(2)})\in\mathds{R}^{m+d}$, and $f\in\mathscr{B}_{b}(\mathds{R}^{d})$ with $f\geq 1$,

[TABLE]

Now we apply our result to some concrete examples of subordinators.

Example 1.3**.**

Let $\sigma=I_{m\times m}$, $b=0$, and $l\in(0,m/2)$. Assume that $S_{1}$ is an $\alpha$-stable subordinator, which has no drift and its Lévy measure is given by $c_{1}x^{-1-\alpha}\mathds{1}_{\{x>0\}}\mathrm{d}x$, and $S_{2}$ is a truncated $\beta$-stable subordinator, which has no drift and its Lévy measure is given by $c_{2}x^{-1-\beta}\mathds{1}_{\{0<x<1\}}\mathrm{d}x$, where $\alpha,\beta\in(0,1)$ and $c_{1},c_{2}>0$ are constants. Then there is some constant $C=C(m,d,l,\alpha,\beta,c_{1},c_{2})>0$ such that for any $T>0$, $x=(x^{(1)},x^{(2)}),y=(y^{(1)},y^{(2)})\in\mathds{R}^{m+d}$, and $f\in\mathscr{B}_{b}(\mathds{R}^{d})$ with $f\geq 1$,

[TABLE]

Example 1.4**.**

Let $\sigma=I_{m\times m}$, $b=0$, and $l\in(0,m/2)$. Assume that $S_{1}$ is an $\alpha$-stable subordinator, which has no drift and its Lévy measure is given by $c_{1}x^{-1-\alpha}\mathds{1}_{\{x>0\}}\,\mathrm{d}x$, and $S_{2}$ is a relativistic $\beta$-stable subordinator, which has no drift and its Lévy measure is given by $c_{2}\mathrm{e}^{-\rho^{\beta}x}x^{-1-\beta}\mathds{1}_{\{x>0\}}\,\mathrm{d}x$, where $\alpha,\beta\in(0,1)$ and $c_{1},c_{2},\rho>0$ are constants. Then there is some constant $C=C(m,d,l,\alpha,\beta,c_{1},c_{2},\rho)>0$ such that for any $T>0$, $x=(x^{(1)},x^{(2)}),y=(y^{(1)},y^{(2)})\in\mathds{R}^{m+d}$, and $f\in\mathscr{B}_{b}(\mathds{R}^{d})$ with $f\geq 1$,

[TABLE]

The remaining part of this paper is organized as follows. By using coupling in two steps and an approximation argument, we establish in Section 2 the log-Harnack inequalities for SDEs driven by non-random time-changed Brownian motions. Section 3 is devoted to the proofs of Theorem 1.1 and Examples 1.3 and 1.4.

2. Log-Harnack inequalities under

deterministic time-changes

For each $i\in\{1,2\}$, let $\ell_{i}:[0,\infty)\rightarrow[0,\infty)$ be a non-decreasing and càdlàg function with $\ell_{i}(0)=0$. By (H2), the following SDE for $X_{t}^{\ell_{1},\ell_{2}}=(X_{t}^{(1),\ell_{1}},X_{t}^{(2),\ell_{1},\ell_{2}})$ has a unique non-explosive solution:

[TABLE]

Let

[TABLE]

where $X_{t}^{\ell_{1},\ell_{2}}(x)=(X_{t}^{(1),\ell_{1}}(x),X_{t}^{(2),\ell_{1},\ell_{2}}(x))$ is the solution to (2.1) with $X_{0}^{\ell_{1},\ell_{2}}=x\in\mathds{R}^{m+d}$.

Proposition 2.1**.**

Let $l\in(0,m/2)$ and assume that (H1) and (H2) hold. There is some constant $C=C(m,d,l)>0$ such that for any $T>0$, $x=(x^{(1)},x^{(2)}),y=(y^{(1)},y^{(2)})\in\mathds{R}^{m+d}$, and $f\in\mathscr{B}_{b}(\mathds{R}^{d})$ with $f\geq 1$

[TABLE]

Following the line of [17, 14, 16, 3, 5], for $\varepsilon_{1},\varepsilon_{2}\in(0,1)$, we define

[TABLE]

It is clear that $\ell_{i}^{\varepsilon_{i}}$ is absolutely continuous and strictly increasing with

[TABLE]

for all $t\geq 0$. For each $i\in\{1,2\}$, denote by $\gamma_{i}^{\varepsilon_{i}}:[\ell_{i}^{\varepsilon_{i}}(0),\infty)\rightarrow[0,\infty)$ the inverse function of $\ell_{i}^{\varepsilon_{i}}$. By definition, $\ell_{i}^{\varepsilon_{i}}(\gamma_{i}^{\varepsilon_{i}}(t))=t$ for $t\geq\ell_{i}^{\varepsilon_{i}}(0)$, $\gamma_{i}^{\varepsilon_{i}}(\ell_{i}^{\varepsilon_{i}}(t))=t$ for $t\geq 0$, and $t\mapsto\gamma_{i}^{\varepsilon_{i}}(t)$ is absolutely continuous and strictly increasing.

Consider the approximation equation for $X_{t}^{\ell_{1}^{\varepsilon_{1}},\ell_{2}^{\varepsilon_{2}}}=(X_{t}^{(1),\ell_{1}^{\varepsilon_{1}}},X_{t}^{(2),\ell_{1}^{\varepsilon_{1}},\ell_{2}^{\varepsilon_{2}}})$

[TABLE]

Denote by $X_{t}^{\ell_{1}^{\varepsilon_{1}},\ell_{2}^{\varepsilon_{2}}}(x)=(X_{t}^{(1),\ell_{1}^{\varepsilon_{1}}}(x),X_{t}^{(2),\ell_{1}^{\varepsilon_{1}},\ell_{2}^{\varepsilon_{2}}}(x))$ the unique non-explosive (strong) solution to (2.3) with $X_{0}^{\ell_{1}^{\varepsilon_{1}},\ell_{2}^{\varepsilon_{2}}}=x\in\mathds{R}^{m+d}$, and let

[TABLE]

Note that (2.3) is indeed driven by Brownian motions and thus, as in [15], the method of coupling in two steps and Girsanov transformation can be used to establish the log-Harnack inequality for $P_{t}^{\ell_{1}^{\varepsilon_{1}},\ell_{2}^{\varepsilon_{2}}}$.

Observe that the regular conditional probability $\mathds{P}(\cdot|\mathscr{F}^{(1)})$ given $\mathscr{F}^{(1)}$ exists, where $\mathscr{F}^{(1)}$ is the $\sigma$-algebra generated by $\{W^{(1)}_{t}\,:\,t\geq 0\}$. Let $\mathscr{F}^{(2)}_{t}$ and $\mathscr{F}_{t}$ be the $\sigma$-algebras generated by $\{W^{(2)}_{s}\,:\,0\leq s\leq t\}$ and $\{W_{s}\,:\,0\leq s\leq t\}$, respectively. For any probability measure $\tilde{\mathds{P}}$, we denote by $\mathds{E}_{\tilde{\mathds{P}}}$ the expectation w.r.t. $\tilde{\mathds{P}}$. If $\tilde{\mathds{P}}=\mathds{P}$, we simply denote the expectation by $\mathds{E}$ as usual.

Lemma 2.2**.**

Fix $\varepsilon_{1},\varepsilon_{2}\in(0,1]$, let $l\in(0,m/2)$, and assume that (H1) and (H2) hold. There is some constant $C=C(m,d,l)>0$ such that for any $T>0$, $x=(x^{(1)},x^{(2)}),y=(y^{(1)},y^{(2)})\in\mathds{R}^{m+d}$, and $f\in\mathscr{B}_{b}(\mathds{R}^{d})$ with $f\geq 1$

[TABLE]

Proof.

We divide the proof into four steps.

Step 1: Fix $T>0$, $x=(x^{(1)},x^{(2)}),y=(y^{(1)},y^{(2)})\in\mathds{R}^{m+d}$ and let $Y_{t}^{(1)}$ solve the equation

[TABLE]

where

[TABLE]

Since

[TABLE]

is locally Lipschitz continuous off the diagonal, the coupling $(X_{t}^{(1),\ell^{\varepsilon_{1}}_{1}}(x),Y_{t}^{(1)})$ is well-defined and unique for $t<\tau^{(1)}$. If $\tau^{(1)}<\infty$, we set $Y_{t}^{(1)}=X_{t}^{(1),\ell^{\varepsilon_{1}}_{1}}(x)$ for $t\in[\tau^{(1)},\infty)$. In this way, we can construct a unique solution $(Y_{t}^{(1)})_{t\geq 0}$ to (2.4). By the differential formula

[TABLE]

we have for $t<\tau^{(1)}$ that

[TABLE]

Then it must be $\tau^{(1)}\leq T$. Indeed, if $\tau^{(1)}(\omega)>T$ for some $\omega\in\Omega$, we can take $t=T$ in the above equality to get

[TABLE]

which is absurd. Let

[TABLE]

where

[TABLE]

for $s\geq 0$. By (H1), we have for any $s\in[0,T]$

[TABLE]

which implies that the compensator of the martingale $M_{t}$ satisfies

[TABLE]

This, together with Novikov’s criterion, yields that $\mathds{E}R^{(1)}_{t}=1$, where

[TABLE]

According to Girsanov’s theorem, for any $t\geq 0$, $(\widetilde{W}_{s}^{(1)})_{s\geq 0}$ is an $m$-dimensional Brownian motion under the new probability measure $R^{(1)}_{t}\mathds{P}$. Thus, for all $t\geq 0$,

[TABLE]

Step 2: Consider the following SDE

[TABLE]

where

[TABLE]

Since $Y_{t}^{(1)}$ is now fixed, the equation (2.8) has a unique solution for $t<\tau^{(2)}$. Let $Y_{t}^{(2)}=X_{t}^{(2),\ell^{\varepsilon_{2}}_{2}}(x)$ for $t\in[\tau^{(2)},\infty)$. Thus, $Y_{t}^{(2)}$ solves (2.8) for all $t\geq 0$. Noting that $X_{t}^{(1),\ell_{1}^{\varepsilon_{1}}}(x)=Y_{t}^{(1)}$ for $t\geq T$, it follows from (2.5) and (H2) that for $t\in[T,\tau^{(2)})$

[TABLE]

Similarly as in the first part of the proof, it is easy to see that this implies $\tau^{(2)}\leq 2T$. For $t\geq 0$ and $n\in\mathds{N}$, let

[TABLE]

By Girsanov’s theorem, under the weighted probability measure $R^{(2)}_{\ell_{2}^{\varepsilon_{2}}(2T)-\ell_{2}^{\varepsilon_{2}}(0)}(n)\mathds{P}(\cdot|\mathscr{F}^{(1)})$, the process

[TABLE]

is a standard $d$-dimensional Brownian motion. Then we have for all $n\in\mathds{N}$ and $t\geq 0$

[TABLE]

where in the last equality we have used the fact that, for $t\geq 0$, $R^{(2)}_{t}(n)$ is an $\mathscr{F}_{t}^{(2)}$-martingale under $\mathds{P}(\cdot|\mathscr{F}^{(1)})$ and $R^{(2)}_{\ell_{2}^{\varepsilon_{2}}(T)-\ell_{2}^{\varepsilon_{2}}(0)}(n)=1$. By Itô’s formula, (H2) and the inequality

[TABLE]

we get that for $t\in[0,T]$

[TABLE]

Since it follows from (2.6) that \big{|}X_{t}^{(1),\ell_{1}^{\varepsilon_{1}}}(x)-Y_{t}^{(1)}\big{|}\leq|x^{(1)}-y^{(1)}|, this implies that for $t\in[0,T]$

[TABLE]

Now we know that for all $n\in\mathds{N}$ and $t\geq 0$

[TABLE]

Step 3: For $t\geq 0$ and $n\in\mathds{N}$, let $R_{t}(n)=R^{(1)}_{t}\cdot R^{(2)}_{t}(n)$ and $R_{t}=R^{(1)}_{t}\cdot R^{(2)}_{t}$. Since for any $t\geq 0$ and $s\geq 0$, the distribution of $Y_{s}^{(1)}$ under $R^{(1)}_{t}\mathds{P}$ coincides with that of $X_{s}^{(1),\ell_{1}^{\varepsilon_{1}}}(x)$ under $\mathds{P}$, it follows from (2.7) and (2.9) that

[TABLE]

where

[TABLE]

and

[TABLE]

It follows from the elementary inequality

[TABLE]

that

[TABLE]

where

[TABLE]

According to Lemma 4.1 in the appendix, for any $\theta\in(0,m/2)$, there exists $c=c(m,\theta)>0$ such that

[TABLE]

holds for all $s>0$ and $x\in\mathds{R}^{m+d}$. This yields that for some $c_{1}=c_{1}(m,l)>0$

[TABLE]

Since $l<m/2$, we can pick $p=p(l)>1$ such that $pl<m/2$. It follows from the Hölder inequality and (2.12) that

[TABLE]

for some $c_{2}=c_{2}(m,p,l)>0$. Moreover, we have

[TABLE]

for some $c_{3}=c_{3}(m,p,l)>0$. Thus,

[TABLE]

Combining the above estimates, we get that for some positive constant $C=C(m,d,l)$

[TABLE]

This, together with (2.10), gives that for all $n\in\mathds{N}$ and $t\geq 0$

[TABLE]

It is not hard to verify that this implies that $(R_{t})_{t\geq 0}$ is an $\mathscr{F}_{t}$-martingale under $\mathds{P}$, and thus $\mathds{E}R=1$, where

[TABLE]

Since for any $t\geq 0$, $R_{t}(n)\rightarrow R_{t}$ as $n\rightarrow\infty$, we can let $n\rightarrow\infty$ in (2.13) and use Fatou’s lemma to know that (2.13) holds with $R_{t}(n)$ replaced by $R$.

Step 4: By the Jensen inequality, we have for any random variable $F\geq 1$,

[TABLE]

hence

[TABLE]

Let

[TABLE]

Then

[TABLE]

where

[TABLE]

Clearly, we can rewrite $R_{t}$ as

[TABLE]

It follows from Girsanov’s theorem that under the weighted probability measure $R\mathds{P}$, $\widetilde{W}_{t}$ is a standard Brownian motion on $\mathds{R}^{m+d}$. Noting that $Y_{t}=(Y_{t}^{(1)},Y_{t}^{(2)})$ solves the SDE

[TABLE]

we conclude that the distribution of $Y_{2T}$ under $R\mathds{P}$ coincides with that of $X_{2T}^{\ell_{1}^{\varepsilon_{1}},\ell_{2}^{\varepsilon_{2}}}(y)$ under $\mathds{P}$. Therefore, we obtain from $Y_{2T}=X_{2T}^{\ell_{1}^{\varepsilon_{1}},\ell_{2}^{\varepsilon_{2}}}(x)$ and (2.14) that for any $f\in\mathscr{B}_{b}(\mathds{R}^{m+d})$ with $f\geq 1$

[TABLE]

Inserting the estimate (2.13) with $R_{t}(n)$ replaced by $R$ into this inequality, we complete the proof of the log-Harnack inequality. ∎

To prove Proposition 2.1 by using Lemma 2.2, we need some preparations. First, the following Burkholder-Davis-Gundy type inequality is essentially due to [8, Lemma 2.3]. For the reader’s convenience, we include a simple proof.

Lemma 2.3**.**

Let $\varrho:[0,\infty)\rightarrow\mathds{R}$ be a non-decreasing càdlàg function with $\varrho(0)=0$. For any $p>0$, there exists a constant $C_{p}>0$ depending only on $p$ such that for any $t>0$

[TABLE]

Proof.

Fix $t>0$. Let $g:[0,t]\rightarrow\mathds{R}$ be a bounded measurable function. By [18, Lemma 4.2], one has

[TABLE]

where $\varrho^{-1}(u):=\inf\{s\geq 0\,:\,\varrho(s)>u\}$. A similar argument shows that

[TABLE]

Then by the Burkholder-Davis-Gundy inequality, we get

[TABLE]

The following two assumptions will be used:

1. (A1)

$\sigma$ is piecewise constant, i.e. there exists a sequence $\{t_{n}\}_{n\geq 0}$ with $t_{0}=0$ and $t_{n}\uparrow\infty$ such that

[TABLE] 2. (A2)

$b_{t}:\mathds{R}^{d}\rightarrow\mathds{R}^{d}$ is, uniformly for $t$ in compact intervals, global Lipschitz, i.e. for any $t>0$, there is some $C_{t}>0$ such that

[TABLE]

Lemma 2.4**.**

Assume (A1). Then for any $l>0$, $x\in\mathds{R}^{m+d}$ and $t>0$,

[TABLE]

[TABLE]

Proof.

Fix $l>0$, $x\in\mathds{R}^{m+d}$ and $t>0$. It is not hard to obtain from (A1) that

[TABLE]

which, together with (2.2), implies (2.15).

By the isometry property of stochastic integrals, we have

[TABLE]

According to [17, Lemma 2.3 (i)],

[TABLE]

Using (2.11), we find that

[TABLE]

This, together with Lemma 2.3, yields

[TABLE]

Then it follows from the dominated convergence theorem and (2.18) that

[TABLE]

Next, we obtain from (2.11) and Lemma 2.3 that for any $q\geq 0$

[TABLE]

and

[TABLE]

This means that for any $q\geq 0$

[TABLE]

Using the elementary inequality

[TABLE]

we obtain

[TABLE]

which, together with the dominated convergence theorem and (2.15), yields

[TABLE]

Combining this with (2.17) and (2.19), we get (2.16). ∎

Lemma 2.5**.**

Assume (A1) and (A2). Then for any $l>0$, $x\in\mathds{R}^{m+d}$ and $t>0$,

[TABLE]

Proof.

Fix $l>0$, $x\in\mathds{R}^{m+d}$ and $t>0$. It follows easily from (A2) that

[TABLE]

Since by (A2) $z\mapsto\sup_{s\in[0,t]}|b_{s}(z)|$ grows at most linearly, it is not hard to verify that

[TABLE]

Letting first $\varepsilon_{2}\downarrow 0$ and then $\varepsilon_{1}\downarrow 0$ in (2.21), and using Fatou’s lemma and (2.16), we get

[TABLE]

This, together with Gronwall’s inequality, yields the claim. ∎

Lemma 2.6**.**

Assume (A1) and (A2), and let $h:\mathds{R}^{m+d}\rightarrow\mathds{R}$ be a bounded and uniformly continuous function. Then for any $t>0$ and $x\in\mathds{R}^{m+d}$

[TABLE]

Proof.

Fix $t>0$ and $x\in\mathds{R}^{m+d}$. It follows from (2.15) and (2.20) that

[TABLE]

Since $h$ is uniformly continuous, for any $\epsilon>0$, there exists $\delta=\delta(\epsilon)>0$ such that

[TABLE]

Therefore, we obtain from Chebyshev’s inequality that

[TABLE]

Letting first $\varepsilon_{2}\downarrow 0$ and then $\varepsilon_{1}\downarrow 0$, and using (2.22), we get

[TABLE]

which implies the claim since $\epsilon>0$ is arbitrary. ∎

Now we are ready to prove Proposition 2.1.

Proof of Proposition 2.1.

Fix $T>0$. By a standard approximation argument, we may and do assume that $f$ is bounded and uniformly continuous with $f\geq 1$.

Step 1: Assume (A1) and (A2). Since $\ell_{i}$ is of bounded variation, it is not hard to verify from (2.2) that

[TABLE]

and

[TABLE]

Letting first $\varepsilon_{2}\downarrow 0$ and then $\varepsilon_{1}\downarrow 0$ in Lemma 2.2, and using Lemma 2.6, we get the desired log-Harnack inequality.

Step 2: Assume (A2). Clearly, we can pick a sequence of $\mathds{R}^{m}\otimes\mathds{R}^{m}$-valued functions $\{\sigma^{(n)}\,:\,n\in\mathds{N}\}$ on $[0,\infty)$ such that each $\sigma^{(n)}$ is piecewise constant, $\|(\sigma^{(n)}_{t})^{-1}\|\leq\lambda_{t}$ for all $n\in\mathds{N}$ and $t\in[0,2T]$, and $\sigma^{(n)}\rightarrow\sigma$ in $L^{2}([0,2T];\,\mathrm{d}\ell_{1})$ as $n\rightarrow\infty$. Let $X_{t}^{\ell_{1},\ell_{2},n}(x)=(X_{t}^{(1),\ell_{1},n}(x),X_{t}^{(2),\ell_{1},\ell_{2},n}(x))$ solve (2.1) with $\sigma$ replaced by $\sigma^{(n)}$ and $X_{0}^{\ell_{1},\ell_{2},n}(x)=x\in\mathds{R}^{m+d}$, and denote by $P_{t}^{\ell_{1},\ell_{2},n}$ the associated Markov semigroup. By Step 1, the statement of Proposition 2.1 holds with $P_{2T}^{\ell_{1},\ell_{2}}$ replaced by $P_{2T}^{\ell_{1},\ell_{2},n}$. We have

[TABLE]

It holds from (A2) that

[TABLE]

which implies

[TABLE]

Similarly as in the proof of Lemma 2.4, we can deduce from (2.23) and Lemma 2.3 that

[TABLE]

Letting $n\rightarrow\infty$ in (2.24) and using Fatou’s Lemma, we obtain

[TABLE]

which, together with Gronwall’s inequality, gives

[TABLE]

Then we conclude that for any $x\in\mathds{R}^{m+d}$, $X_{2T}^{\ell_{1},\ell_{2},n}(x)\rightarrow X_{2T}^{\ell_{1},\ell_{2}}(x)$ in $L^{2}(\mathds{P})$, and hence (up to a subsequence)

[TABLE]

Letting $n\rightarrow\infty$, the desired inequality in Proposition 2.1 holds.

Step 3: For the general case, we shall make use of the approximation argument in [14, part (c) of proof of Theorem 2.1]. Let

[TABLE]

By (H2), it is easy to see that the mapping $\operatorname{id}-n^{-1}\tilde{b}_{t}:\mathds{R}^{d}\rightarrow\mathds{R}^{d}$ is injective for any $n\in\mathds{N}$ and $t\geq 0$. Let

[TABLE]

Then we find that for any $n\in\mathds{N}$ and $t\geq 0$, $b_{t}^{(n)}$ is, uniformly for $t$ in compact intervals, globally Lipschitzian, see [2]. Let $\bar{X}_{t}^{\ell_{1},\ell_{2},n}(x)=(\bar{X}_{t}^{(1),\ell_{1},n}(x),\bar{X}_{t}^{(2),\ell_{1},\ell_{2},n}(x))$ solve (2.1) with $b$ replaced by $b^{(n)}$ and $\bar{X}_{0}^{\ell_{1},\ell_{2},n}(x)=x\in\mathds{R}^{m+d}$, and denote by $\bar{P}_{t}^{\ell_{1},\ell_{2},n}$ the associated Markov semigroup. According to the second part of the proof, the statement of Proposition 2.1 holds with $P_{2T}^{\ell_{1},\ell_{2}}$ replaced by $\bar{P}_{2T}^{\ell_{1},\ell_{2},n}$. As in [14, part (c) of proof of Theorem 2.1], one has $\bar{X}_{t}^{\ell_{1},\ell_{2},n}(x)\rightarrow X_{t}^{\ell_{1},\ell_{2}}(x)$ a.s. and therefore, it remains to let $n\rightarrow\infty$ to finish the proof. ∎

3. Proofs of Theorem 1.1

and Examples 1.3 and 1.4

Proof of Theorem 1.1.

Noting that

[TABLE]

we get the desired log-Harnack inequality by using Proposition 2.1 and the Jensen inequality. ∎

Proof of Example 1.3.

By the self-similar property of $\alpha$-stable subordinators, one has

[TABLE]

On the other hand, it is clear that

[TABLE]

Since $\frac{\mathrm{e}-1}{\mathrm{e}}(1\wedge z)\leq 1-\mathrm{e}^{-z}\leq 1\wedge z$ for all $z\geq 0$, we get that for all $u>0$

[TABLE]

Here, $f\asymp g$ means that $c^{-1}f(u)\leq g(u)\leq cf(u)$ for some constant $c\geq 1$ and all $u$. This, together with [4, Theorem 3.8 (a)] (or [6, Theorem 2.1 b)]), yields that for some constant $C_{\beta,c_{2}}>0$

[TABLE]

Combining the above estimates with Corollary 1.2, we finish the proof. ∎

Proof of Example 1.4.

Since $S_{2}$ has finite second moments, (3.2) holds true. Since the characteristic exponent of $S_{2}$ is $\phi_{2}(u)=c_{2}\beta^{-1}\Gamma(1-\beta)[(u+\rho^{1/\beta})^{\beta}-\rho]$, we obtain from [6, Theorem 2.1 b)] that

[TABLE]

for some constant $C_{\beta,c_{2},\rho}>0$. Inserting this bound, (3.1) and (3.2) into Corollary 1.2, the desired estimate follows. ∎

4. Appendix

The following elementary result should be known, but we could not find a reference and so we include a simple proof for the sake of completeness.

Lemma 4.1**.**

Let $\xi$ be an $m$-dimensional Gaussian random variable with mean zero and covariance matrix $\sigma I_{m\times m}$, where $\sigma>0$. Then for any $\theta\in(0,m/2)$, there exists $C=C(m,\theta)>0$ depending only on $m$ and $\theta$ such that

[TABLE]

Proof.

First,

[TABLE]

where $c(m,\theta):=\pi^{-m/2}2^{-\theta}\int_{\mathds{R}^{m}}|y|^{-2\theta}\mathrm{e}^{-|y|^{2}}\,\mathrm{d}y$ is a positive constant since $2\theta<m$. For $\mu\in\mathds{R}^{m}\setminus\{0\}$,

[TABLE]

where

[TABLE]

and

[TABLE]

If $|x-\mu|\geq|\mu|/2$, one has

[TABLE]

and then

[TABLE]

On the other hand, if $|x-\mu|<|\mu|/2$, it follows that

[TABLE]

This implies

[TABLE]

Therefore, we obtain from (4.1), (4.2) and (4.3) that, for any $\mu\in\mathds{R}^{m}\setminus\{0\}$,

[TABLE]

which completes the proof. ∎

Acknowledgement**.**

The authors would like to thank an anonymous referee for useful suggestions.