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Zhen-Qing Chen

Publications and source records attributed to Zhen-Qing Chen.

At least 73 records · Page 4Linked to original sources
arXiv2017

Skeleton decomposition and law of large numbers for supercritical superprocesses

The goal of this paper has two-folds. First, we establish skeleton and spine decompositions for superprocesses whose underlying processes are general symmetric Hunt processes. Second, we use these decompositions to obtain weak and strong law of large numbers for supercritical superprocesses where the spatial motion is a symmetric Hunt process on a locally compact metric space $E$ and the branching mechanism takes the form $$ψ_β(x,λ)=-β(x)λ+α(x)λ^{2}+\int_{(0,{\infty})}(e^{-λy}-1+λy)π(x,dy)$$ with $β\in\mathcal{B}_{b}(E)$, $α\in \mathcal{B}^{+}_{b}(E)$ and $π$ being a kernel from $E$ to $(0,{\infty})$ satisfying $\sup_{x\in E}\int_{(0,{\infty})} (y\wedge y^{2}) π(x,dy)<{\infty}$. The limit theorems are established under the assumption that an associated Schrödinger operator has a spectral gap. Our results cover many interesting examples of superprocesses, including super Ornstein-Uhlenbeck process and super stable-like process. The strong law of large numbers for supercritical superprocesses are obtained under the assumption that the strong law of large numbers for an associated supercritical branching Markov process holds along a discrete sequence of times, extending an earlier result of Eckhoff, Kyprianou and Winkel [Ann. Probab.,43(5), 2594--2659, 2015] for superdiffusions to a large class of superprocesses. The key for such a result is due to the skeleton decomposition of superprocess, which represents a superprocess as an immigration process along a supercritical branching Markov process.

math.PR↗
arXiv2017

$L\log L$ criterion for a class of multitype superdiffusions with non-local branching mechanisms

In this paper, we provide a pathwise spine decomposition for multitype superdiffusions with non-local branching mechanisms under a martingale change of measure. As an application of this decomposition, we obtain a necessary and sufficient condition (called the $L\log L$ criterion) for the limit of the fundamental martingale to be non-degenerate. This result complements the related results for superprocesses with purely local branching mechanisms and in Kyprianou and Palau (2016) for super Markov chains.

math.PR↗
arXiv2017

Systems of interacting diffusions with partial annihilation through membranes

We introduce an interacting particle system in which two families of reflected diffusions interact in a singular manner near a deterministic interface $I$. This system can be used to model the transport of positive and negative charges in a solar cell or the population dynamics of two segregated species under competition. A related interacting random walk model with discrete state spaces has recently been introduced and studied in Chen and Fan (2014). In this paper, we establish the functional law of large numbers for this new system, thereby extending the hydrodynamic limit in Chen and Fan (2014) to reflected diffusions in domains with mixed-type boundary conditions, which include absorption (harvest of electric charges). We employ a new and direct approach that avoids going through the delicate BBGKY hierarchy.

math.PR↗
arXiv2017

Superdiffusions with large mass creation --- construction and growth estimates

Superdiffusions corresponding to differential operators of the form $\LL u+βu-αu^{2}$ with large mass creation term $β$ are studied. Our construction for superdiffusions with large mass creations works for the branching mechanism $βu-αu^{1+γ},\ 0<γ<1,$ as well. Let $D\subseteq\mathbb{R}^{d}$ be a domain in $\R^d$. When $β$ is large, the generalized principal eigenvalue $λ_c$ of $L+β$ in $D$ is typically infinite. Let $\{T_{t},t\ge0\}$ denote the Schrödinger semigroup of $L+β$ in $D$ with zero Dirichlet boundary condition. Under the mild assumption that there exists an $0<h\in C^{2}(D)$ so that $T_{t}h$ is finite-valued for all $t\ge 0$, we show that there is a unique $\mathcal{M}_{loc}(D)$-valued Markov process that satisfies a log-Laplace equation in terms of the minimal nonnegative solution to a semilinear initial value problem. Although for super-Brownian motion (SBM) this assumption requires $β$ be less than quadratic, the quadratic case will be treated as well. When $λ_c = \infty$, the usual machinery, including martingale methods and PDE as well as other similar techniques cease to work effectively, both for the construction and for the investigation of the large time behavior of the superdiffusions. In this paper, we develop the following two new techniques in the study of local/global growth of mass and for the spread of the superdiffusions: \begin{itemize} \item a generalization of the Fleischmann-Swart `Poissonization-coupling,' linking superprocesses with branching diffusions; \item the introduction of a new concept: the `{\it $p$-generalized principal eigenvalue.}' \end{itemize} The precise growth rate for the total population of SBM with $α(x)=β(x)=1+|x|^p$ for $p\in[0,2]$ is given in this paper.

math.PR↗
arXiv2017

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.

math.PR↗
arXiv2017

Markov processes with darning and their approximations

In this paper, we study darning of general symmetric Markov processes by shorting some parts of the state space into singletons. A natural way to construct such processes is via Dirichlet forms restricted to the function space whose members take constant values on these collapsing parts. They include as a special case Brownian motion with darning, which has been studied in details in \cite{Chen, CF, CFR}. When the initial processes have discontinuous sample paths, the processes constructed in this paper are the genuine extensions of those studied in Chen and Fukushima \cite{CF}. We further show that, up to a time change, these Markov processes with darning can be approximated in the finite dimensional sense by introducing additional large intensity jumps among these compact sets to be collapsed into singletons to the original Markov processes. For diffusion processes, it is also possible to get, up to a time change, diffusions with darning by increasing the conductance on these compact sets to infinity. To accomplish these, we extend the semigroup characterization of Mosco convergence to closed symmetric forms whose domain of definition may not be dense in the $L^2$-space. The latter is of independent interest and potentially useful to study convergence of Markov processes having different state spaces. Indeed, we show in Section 5 of this paper that Brownian motion in a plane with a very thin flag pole can be approximated by Brownian motion in the plane with a vertical cylinder whose horizontal motion on the cylinder is a circular Brownian motion moving at fast speed.

math.PR↗
arXiv2016

Heat kernels for non-symmetric diffusion operators with jumps

For $d\geq 2$, we establish the existence and uniqueness of heat kernels for a large class of time-dependent second order diffusion operator with jumps, which is the sum of time-dependent of a second order elliptic differential operators non-divergence form and a non-local $α$-stable-type operator with bounded time-dependent coefficient. Moreover, we obtain sharp two-sided estimates, gradient estimate and fractional derivative estimate for the heat kernels under some mild conditions. Our approach is mainly analytic but also uses some probabilistic techniques.

math.AP↗
arXiv2016

Reflections at infinity of time changed RBMs on a domain with Liouville branches

Let $Z$ be the transient reflecting Brownian motion on the closure of an unbounded domain $D\subset {\mathbb R}^d$ with $N$ number of Liouville branches. We consider a diffusion $X$ on $\overline D$ having finite lifetime obtained from $Z$ by a time change. We show that $X$ admits only a finite number of possible symmetric conservative diffusion extensions $Y$ beyond its lifetime characterized by possible partitions of the collection of $N$ ends and we identify the family of the extended Dirichlet spaces of all $Y$ (which are independent of time change used) as subspaces of the space ${\rm BL}(D)$ spanned by the extended Sobolev space $H_e^1(D)$ and the approaching probabilities of $Z$ to the ends of Liouville branches.

math.PR↗
arXiv2016

Perturbation by non-local operators

Suppose that $d\ge 1$ and $0<β<α<2$. We establish the existence and uniqueness of the fundamental solution $q^b(t, x, y)$ to a class of (possibly nonsymmetric) non-local operators $L^b=Δ^{α/2}+S^b$, where $$ S^bf(x):=A(d, -β) \int_{R^d} ( f(x+z)-f(x)- \nabla f(x) \cdot z 1_{\{|z|\leq 1\}} ) \frac{b(x, z)}{|z|^{d+β}}dz $$ and $b(x, z)$ is a bounded measurable function on $R^d\times R^d$ with $b(x, z)=b(x, -z)$ for $x, z\in R^d$. Here $A(d, -β)$ is a normalizing constant so that $S^b=Δ^{β/2}$ when $b(x, z)\equiv 1$. We show that if $b(x, z) \geq -\frac{{\cal A}(d, -α)}{A(d, -β)}\, |z|^{β-α}$, then $q^b(t, x, y)$ is a strictly positive continuous function and it uniquely determines a conservative Feller process $X^b$, which has strong Feller property. The Feller process $X^b$ is the unique solution to the martingale problem of $(L^b, {\cal S} (R^d))$, where ${\cal S}(R^d)$ denotes the space of tempered functions on $R^d$. Furthermore, sharp two-sided estimates on $q^b(t, x, y)$ are derived. In stark contrast with the gradient perturbations, these estimates exhibit different behaviors for different types of $b(x, z)$. The model considered in this paper contains the following as a special case. Let $Y$ and $Z$ be (rotationally) symmetric $α$-stable process and symmetric $β$-stable processes on $R^d$, respectively, that are independent to each other. Solution to stochastic differential equations $dX_t=dY_t + c(X_{t-})dZ_t$ has infinitesimal generator $L^b$ with $b(x, z)=| c(x)|^β$.

math.PR↗
arXiv2016

$L^p$-maximal hypoelliptic regularity of nonlocal kinetic Fokker-Planck operators

For $p\in(1,\infty)$, let $u(t,x,v)$ and $f(t,x,v)$ be in $L^p(\mathbb{R} \times \mathbb{R}^d \times \mathbb{R}^d)$ and satisfy the following nonlocal kinetic Fokker-Plank equation on $\mathbb{R}^{1+2d}$ in the weak sense: $$ \partial_t u+v\cdot\nabla_x u=Δ^{α/{2}}_v u+f, $$ where $α\in(0,2)$ and $Δ^{α/{2}}_v$ is the usual fractional Laplacian applied to $v$-variable. We show that there is a constant $C=C(p,α,d)>0$ such that for any $f(t, x, v)\in L^p(\mathbb{R} \times \mathbb{R}^d \times \mathbb{R}^d)=L^p(\mathbb{R}^{1+2d})$, $$ \|Δ_x^{α/{(2(1+α))}}u\|_p+\|Δ_v^{α/{2}}u\|_p\leq C\|f\|_p, $$ where $\|\cdot\|_p$ is the usual $L^p$-norm in $L^p(\mathbb{R}^{1+2d}; d z)$. In fact, in this paper the above inequality is established for a large class of time-dependent non-local kinetic Fokker-Plank equations on $\mathbb{R}^{1+2d}$, with $U_t v$ and $\mathscr{L}^{ν_t}_{σ_t}$ in place of $v\cdot \nabla_x$ and $Δ^{α/2}_v$. See Theorem 3.3 for details.

math.AP↗
arXiv2016

Stochastic Komatu-Loewner evolutions and SLEs

Let $D={\mathbb H}\setminus \bigcup_{j=1}^N C_j$ be a standard slit domain, where ${\mathbb H}$ is the upper half plane and $C_j,1\le j\le N,$ are mutually disjoint horizontal line segments in ${\mathbb H}$. A stochastic Komatu-Loewner evolution denoted by ${\rm SKLE}_{α,b}$ has been introduced in \cite{CF} as a family $\{F_t\}$ of random growing hulls with $F_t\subset D$ driven by a diffusion process $ξ(t)$ on $\partial {\mathbb H}$ that is determined by certain continuous homogeneous functions $α$ and $b$ defined on the space ${\cal S}$ of all labelled standard slit domains. We aim at identifying the distribution of a suitably reparametrized ${\rm SKLE}_{α,b}$ with that of the Loewner evolution on ${\mathbb H}$ driven by the path of a certain continuous semimartingale and thereby relating the former to the distribution of ${\rm SLE}_{α^2}$ when $α$ is a constant. We then prove that, when $α$ is a constant, ${\rm SKLE}_{α,b}$ up to some random hitting time and modulo a time change has the same distribution as ${\rm SLE}_{α^2}$ under a suitable Girsanov transformation. We further show that a reparametrized ${\rm SKLE}_{\sqrt{6},-b_{\rm BMD}}$ has the same distribution as ${\rm SLE}_6$, where $b_{\rm BMD}$ is the BMD-domain constant indicating the discrepancy of $D$ from ${\mathbb H}$ relative to Brownian motion with darning (BMD in abbreviation). A key ingredient of the proof is a hitting time analysis for the absorbing Brownian motion on ${\mathbb H}.$ We also revisit and examine the locality property of ${\rm SLE}_6$ in several canonical domains. Finally K-L equations and SKLEs for other canonical multiply connected planar domains than the standard slit one are recalled and examined.

math.PR↗
arXiv2016

Stochastic Komatu-Loewner evolutions and BMD domain constant

Let $D={\mathbb H} \setminus \cup_{k=1}^N C_k$ be a standard slit domain, where ${\mathbb H}$ is the upper half plane and $C_k$, $1\leq k\leq N$, are mutually disjoint horizontal line segments in $H$. Given a Jordan arc $γ\subset D$ starting at $\partial H$, let $g_t$ be the unique conformal map from $D\setminusγ[0,t]$ onto a standard slit domain $D_t={\mathbb H} \setminus \cup_{k=1}^N C_k(t)$ satisfying the hydrodynamic normalization at infinity. It has been established recently that $g_t$ satisfies an ODE called a Komatu-Loewner equation in terms of the complex Poisson kernel of the Brownian motion with darning (BMD) for $D_t$. We randomize the Jordan arc $γ$ according to a system of probability measures on the family of equivalence classes of Jordan arcs that enjoy a domain Markov property and a certain conformal invariance property. We show that the induced process $(ξ(t), {\bf s}(t))$ satisfies a Markov type stochastic differential equation, where $ξ(t)$ is a motion on $\partial {\mathbb H}$ and ${\bf} s(t)$ represents the motion of the endpoints of the slits $\{C_k(t),\; 1\le k\le N \}.$ Conversely, given such functions $α$ and $b$ with local Lipschitz continuity, the corresponding SDE admits a unique solution $(ξ(t), {\bf s}(t))$. The latter produces random conformal maps $g_t(z)$ via the Komatu-Loewner equation. The resulting family of random growing hulls $\{F_t\}$ from the conformal mappings is called ${\rm SKLE}_{α,b}.$ We show that it enjoys a certain scaling property and a domain Markov property. Among other things, we further prove that ${\rm SKLE}_{α,-b_{\rm BMD}}$ for a constant $α>0$ has a locality property if and only if $α= \sqrt{6}$, where $b_{\rm BMD}$ is a BMD-domain constant that describes the discrepancy of a standard slit domain from ${\mathbb H}$ relative to BMD.

math.PR↗
arXiv2016

Brownian Motion on Spaces with Varying Dimension

In this paper we introduce and study Brownian motion on state spaces with varying dimension. Starting with a concrete case of such state spaces that models a big square with a flag pole, we construct a Brownian motion on it and study how heat propagates on such a space. We derive sharp two-sided global estimates on its transition density functions (also called heat kernel). These two-sided estimates are of Gaussian type, but the measure on the underlying state space does not satisfy volume doubling property. Parabolic Harnack inequality fails for such a process. Nevertheless, we show Holder regularity holds for its parabolic functions. We also derive the Green function estimates for this process on bounded smooth domains. Brownian motion on some other state spaces with varying dimension are also constructed and studied in this paper.

math.PR↗
arXiv2016

Uniqueness of stable-like processes

In this work we consider the following $α$-stable-like operator (a class of pseudo-differential operator) $$ {\mathscr L} f(x):=\int_{\mathbb R^d}[f(x+σ_x y)-f(x)-1_{α\in[1,2)}1_{|y|\leq 1}σ_x y\cdot\nabla f(x)]ν_x(d y), $$ where the Lévy measure $ν_x(d y)$ is comparable with a non-degenerate $α$-stable-type Lévy measure (possibly singular), and $σ_x$ is a bounded and nondegenerate matrix-valued function. Under Hölder assumption on $x\mapstoν_x(d y)$ and uniformly continuity assumption on $x\mapstoσ_x$, 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 $σ$ belongs to the first order Sobolev space $\mathbb W^{1,p}(\mathbb R^d)$ provided $p>d(1+α\vee 1)$ and $ν_x=ν$ is a non-degenerate $α$-stable-type Lévy measure.

math.PR↗
arXiv2016

Boundary Harnack principle and gradient estimates for fractional Laplacian perturbed by non-local operators

Suppose $d\ge 2$ and $0<β<α<2$. We consider the non-local operator $\mathcal{L}^{b}=Δ^{α/2}+\mathcal{S}^{b}$, where $$\mathcal{S}^{b}f(x):=\lim_{\varepsilon\to 0}\mathcal{A}(d,-β)\int_{|z|>\varepsilon}\left(f(x+z)-f(x)\right)\frac{b(x,z)}{|z|^{d+β}}\,dy.$$ Here $b(x,z)$ is a bounded measurable function on $\mathbb{R}^{d}\times\mathbb{R}^{d}$ that is symmetric in $z$, and $\mathcal{A}(d,-β)$ is a normalizing constant so that when $b(x, z)\equiv 1$, $\mathcal{S}^{b}$ becomes the fractional Laplacian $Δ^{β/2}:=-(-Δ)^{β/2}$. In other words, $$\mathcal{L}^{b}f(x):=\lim_{\varepsilon\to 0}\mathcal{A}(d,-β)\int_{|z|>\varepsilon}\left(f(x+z)-f(x)\right) j^b(x, z)\,dz,$$ where $j^b(x, z):= \mathcal{A}(d,-α) |z|^{-(d+α)}+ \mathcal{A}(d,-β) b(x, z)|z|^{-(d+β)}$. It is recently established in Chen and Wang [arXiv:1312.7594 [math.PR]] that, when $j^b(x, z)\geq 0$ on $\mathbb{R}^d\times \mathbb{R}^d$, there is a conservative Feller process $X^{b}$ having $\mathcal{L}^b$ as its infinitesimal generator. In this paper we establish, under certain conditions on $b$, a uniform boundary Harnack principle for harmonic functions of $X^b$ (or equivalently, of $\mathcal{L}^b$) in any $κ$-fat open set. We further establish uniform gradient estimates for non-negative harmonic functions of $X^{b}$ in open sets.

math.PR↗
arXiv2016

Global Dirichlet Heat Kernel Estimates for Symmetric Lévy Processes in Half-space

In this paper, we derive explicit sharp two-sided estimates for the Dirichlet heat kernels of a large class of symmetric (but not necessarily rotationally symmetric) Lévy processes on half spaces for all $t>0$. These Lévy processes may or may not have Gaussian component. When Lévy density is comparable to a decreasing function with damping exponent $β$,our estimate is explicit in terms of the distance to the boundary, the Lévy exponent and the damping exponent $β$ of Lévy density.

math.PR↗