Heat kernels for time-dependent non-symmetric stable-like operators

When studying non-symmetric nonlocal operators $${\cal L} f(x) = \int_{{\bf R}^d} \left( f(x+z)-f(x)-\nabla f(x)\cdot z 1_{\{|z|\leq 1\}} \right) \frac{\kappa (x, z)}{|z|^{d+\alpha}} d z ,$$ where $0<\alpha<2$ and $\kappa (x, z)$ is a function on ${\bf R}^d\times {\bf R}^d$ that is bounded between two positive constants, it is customary to assume that $\kappa (x, z)$ is symmetric in $z$. In this paper, we study heat kernel of ${\cal L}$ and derive its two-sided sharp bounds without the symmetric assumption $\kappa(x,z)=\kappa(x,-z)$. In fact, we allow the kernel $\kappa$ to be time-dependent and also derive gradient estimate when $\beta\in(0\vee (1-\alpha),1)$ as well as fractional derivative estimate of order $\theta\in(0,(\alpha+\beta)\wedge 2)$ for the heat kernel, where $\beta$ is the H\"older index of $x\mapsto\kappa(x,z)$. Moreover, when $\alpha\in(1,2)$, the drift perturbation with drift in Kato's class is also considered. As an application, when $\kappa(x,z)=\kappa(z)$ does not depend on $x$, we show the boundedness of nonlocal Riesz's transorfmation: for any $p>2d/(d+2\alpha)$, $$\| {\cal L}^{1/2}f\|_p\asymp \|\Gamma(f)^{1/2}\|_p,$$ where $\Gamma(f):=\frac{1}{2}{\cal L} (f^2)-f {\cal L} f$ is the carr\'e du champ operator associated with ${\cal L}$, and ${\cal L}^{1/2}$ is the square root operator of ${\cal L}$ defined by using Bochner's subordination. Here $\asymp$ means that both sides are comparable up to a constant multiple

Propagation of regularity in LpL^p-spaces for Kolmogorov type hypoelliptic operators

Consider the following Kolmogorov type hypoelliptic operator \mathscr L_t:=\mbox{$\sum_{j=2}^n$}x_j\cdot\nabla_{x_{j-1}}+{\rm Tr} (a_t \cdot\nabla^2_{x_n}), where $n\geq 2$, $x=(x_1,\cdots,x_n)\in(\mathbb R^d)^n =\mathbb R^{nd}$ and $a_t$ is a time-dependent constant symmetric $d\times d$-matrix that is uniformly elliptic and bounded.. Let $\{\mathcal T_{s,t}; t\geq s\}$ be the time-dependent semigroup associated with $\mathscr L_t$; that is, $\partial_s {\mathcal T}_{s, t} f = - {\mathscr L}_s {\mathcal T}_{s, t}f$. For any $p\in(1,\infty)$, we show that there is a constant $C=C(p,n,d)>0$ such that for any $f(t, x)\in L^p(\mathbb R \times \mathbb R^{nd})=L^p(\mathbb R^{1+nd})$ and every $\lambda \geq 0$, $$\left\|\Delta_{x_j}^{{1}/{(1+2(n-j)})}\int^{\infty}_0 e^{-\lambda t} {\mathcal T}_{s, s+t }f(t+s, x)dt\right\|_p\leq C\|f\|_p,\quad j=1,\cdots, n,$$ where $\|\cdot\|_p$ is the usual $L^p$-norm in $L^p(\mathbb R^{1+nd}; d s\times d x)$. To show this type of estimates, we first study the propagation of regularity in $L^2$-space from variable $x_n$ to $x_1$ for the solution of the transport equation $\partial_t u+\sum_{j=2}^nx_j\cdot\nabla_{x_{j-1}} u=f$.Comment: 25 page

Uniqueness of stable-like processes

In this work we consider the following $\alpha$-stable-like operator (a class of pseudo-differential operator) $${\mathscr L} f(x):=\int_{\mathbb R^d}[f(x+\sigma_x y)-f(x)-1_{\alpha\in[1,2)}1_{|y|\leq 1}\sigma_x y\cdot\nabla f(x)]\nu_x(d y),$$ where the L\'evy measure $\nu_x(d y)$ is comparable with a non-degenerate $\alpha$-stable-type L\'evy measure (possibly singular), and $\sigma_x$ is a bounded and nondegenerate matrix-valued function. Under H\"older assumption on $x\mapsto\nu_x(d y)$ and uniformly continuity assumption on $x\mapsto\sigma_x$, we show the well-posedness of martingale problem associated with the operator $\mathscr L$. Moreover, we also obtain the existence-uniqueness of strong solutions for the associated SDE when $\sigma$ belongs to the first order Sobolev space $\mathbb W^{1,p}(\mathbb R^d)$ provided $p>d(1+\alpha\vee 1)$ and $\nu_x=\nu$ is a non-degenerate $\alpha$-stable-type L\'evy measure.Comment: 3

Heat Kernels for Non-symmetric Non-local Operators

We survey the recent progress in the study of heat kernels for a class of non-symmetric non-local operators. We focus on the existence and sharp two-sided estimates of the heat kernels and their connection to jump diffusions.Comment: Survey article. To appear as a chapter in "Recent Developments in the Nonlocal Theory" by De Gruyte

Stochastic flows for L\'evy processes with H\"{o}lder drifts

In this paper we study the following stochastic differential equation (SDE) in ${\mathbb R}^d$: $$\mathrm{d} X_t= \mathrm{d} Z_t + b(t, X_t)\mathrm{d} t, \quad X_0=x,$$ where $Z$ is a L\'evy process. We show that for a large class of L\'evy processes ${Z}$ and H\"older continuous drift $b$, the SDE above has a unique strong solution for every starting point $x\in{\mathbb R}^d$. Moreover, these strong solutions form a $C^1$-stochastic flow. As a consequence, we show that, when ${Z}$ is an $\alpha$-stable-type L\'evy process with $\alpha\in (0, 2)$ and $b$ is bounded and $\beta$-H\"older continuous with $\beta\in (1- {\alpha}/{2},1)$, the SDE above has a unique strong solution. When $\alpha \in (0, 1)$, this in particular solves an open problem from Priola \cite{Pr1}. Moreover, we obtain a Bismut type derivative formula for $\nabla {\mathbb E}_x f(X_t)$ when ${Z}$ is a subordinate Brownian motion. To study the SDE above, we first study the following nonlocal parabolic equation with H\"older continuous $b$ and $f$: $$\partial_t u+{\mathscr L} u+b\cdot \nabla u+f=0,\quad u(1, \cdot )=0,$$ where $\mathscr L$ is the generator of the L\'evy process ${Z}$.Comment: 22page

Well-posedness of supercritical SDE driven by L\'evy processes with irregular drifts

In this paper, we study the following time-dependent stochastic differential equation (SDE) in ${\bf R}^d$: $$d X_{t}= \sigma_t(X_{t-}) d Z_t + b_t(X_{t})d t, \quad X_{0}=x\in {\bf R}^d,$$ where $Z$ is a $d$-dimensioanl nondegenerate $\alpha$-stable-like process with $\alpha \in(0,2)$ (including cylindrical case), and uniform in $t\geq 0$, $x\mapsto \sigma_t(x): {\bf R}^d\to {\bf R}^d\otimes {\bf R}^d$ is Lipchitz and uniformly elliptic and $x\mapsto b_t (x)$ is $\beta$-order H\"older continuous with $\beta\in(1-\alpha/2,1)$. Under these assumptions, we show the above SDE has a unique strong solution for every starting point $x \in {\bf R}^d$. When $\sigma_t (x)={\bf I}_{d\times d}$, the $d\times d$ identity matrix, our result in particular gives an affirmative answer to the open problem of Priola (2015)

H\"older estimates for nonlocal-diffusion equations with drifts

We study a class of nonlocal-diffusion equations with drifts, and derive a priori $\Phi$-H\"older estimate for the solutions by using a purely probabilistic argument, where $\Phi$ is an intrinsic scaling function for the equation.Comment: Minor revision. To appear in Communications in Mathematics and Statistic

Heat kernels for time-dependent non-symmetric mixed L\'evy-type operators

In this paper we establish the existence and uniqueness of heat kernels to a large class of time-inhomogenous non-symmetric nonlocal operators with Dini's continuous kernels. Moreover, quantitative estimates including two-sided estimates, gradient estimate and fractional derivative estimate of the heat kernels are obtained

Active modulation of electromagnetically induced transparency analogue in terahertz hybrid metal-graphene metamaterials

Metamaterial analogues of electromagnetically induced transparency (EIT) have been intensively studied and widely employed for slow light and enhanced nonlinear effects. In particular, the active modulation of the EIT analogue and well-controlled group delay in metamaterials have shown great prospects in optical communication networks. Previous studies have focused on the optical control of the EIT analogue by integrating the photoactive materials into the unit cell, however, the response time is limited by the recovery time of the excited carriers in these bulk materials. Graphene has recently emerged as an exceptional optoelectronic material. It shows an ultrafast relaxation time on the order of picosecond and its conductivity can be tuned via manipulating the Fermi energy. Here we integrate a monolayer graphene into metal-based terahertz (THz) metamaterials, and realize a complete modulation in the resonance strength of the EIT analogue at the accessible Fermi energy. The physical mechanism lies in the active tuning the damping rate of the dark mode resonator through the recombination effect of the conductive graphene. Note that the monolayer morphology in our work is easier to fabricate and manipulate than isolated fashion. This work presents a novel modulation strategy of the EIT analogue in the hybrid metamaterials, and pave the way towards designing very compact slow light devices to meet future demand of ultrafast optical signal processing

Tunable light trapping and absorption enhancement with graphene ring arrays

Surface plasmon resonance (SPR) has been intensively studied and widely employed for light trapping and absorption enhancement. In the mid-infrared and terahertz (THz) regime, graphene supports the tunable SPR via manipulating its Fermi energy and enhances light-matter interaction at the selective wavelength. In this work, periodic arrays of graphene rings are proposed to introduce tunable light trapping with good angle polarization tolerance and enhance the absorption in the light-absorbing materials nearby to more than one order. Moreover, the design principle here could be set as a template to achieve multi-band plasmonic absorption enhancement by introducing more graphene concentric rings into each unit cell. This work not only opens up new ways of employing graphene SPR, but also leads to practical applications in high-performance simultaneous multi-color photodetection with high efficiency and tunable spectral selectivity