Searcharxiv⌕ Search

arXiv subjects

Jie-Ming Wang

Publications and source records attributed to Jie-Ming Wang.

6 recordsLinked to original sources
arXiv2026

Boundary Harnack principle for diffusion with jumps on metric measure spaces

A non-scale invariant BHP on any open set is obtained for a large class of discontinuous Hunt processes on metric measure spaces that are in weak duality with another Hunt process under suitable conditions in terms of estimates of exit distributions from open sets, and bounds on Green functions and the comparability of the jump density function. These conditions are easy to verify in concrete cases. Several examples are given to illustrate the main results and the new contributions of this paper. Our result in particular establishes the BHP on any open set for any Hunt process associated with a regular symmetric Dirichlet form on a metric measure space having both the strongly local term and the pure-jump term that admits a two-sided heat kernel estimates of the mixture of (sub-)Gaussian and stable-like form, as well as for a wide class of non-symmetric diffusion processes with jumps on $\mathbb R^d.$

math.PR↗
arXiv2024

Dirichlet heat kernel estimates of subordinate diffusion processes with diffusive components in $C^{1, α}$ open sets

In this paper, we derive explicit sharp two-sided estimates of the Dirichlet heat kernels for a class of symmetric subordinate diffusion processes with diffusive components in $C^{1, α}(α\in (0, 1])$ open sets in $\mathbb R^d$ when the scaling order of the Laplace exponent of purely discontinuous part of the subordinator is between $0$ and $1$ including $1.$ The main result of this paper shows the stability of Dirichlet heat kernel estimates for such processes in $C^{1, α}$ open sets in the sense that the estimates depend on the divergence elliptic operator only via its uniform ellipticity constant and the Dini continuity modulus of the diffusion coefficients. As a corollary, we obtain the sharp two-sided estimates for Green functions of those processes in bounded $C^{1, α}$ open sets.

math.PR↗
arXiv2023

Boundary Harnack principle for non-local operators on metric measure spaces

In this paper, a necessary and sufficient condition is obtained for the scale invariant boundary Harnack inequality (BHP in abbreviation) for a large class of Hunt processes on metric measure spaces that are in weak duality with another Hunt process. We next consider a discontinuous subordinate Brownian motion with Gaussian component $X_t=W_{S_t}$ in ${\bf R}^d$ for which the Lévy density of the subordinator $S$ satisfies some mild comparability condition. We show that the scale invariant BHP holds for the subordinate Brownian motion $X$ in any Lipschitz domain satisfying the interior cone condition with common angle $θ\in (\cos^{-1}(1/\sqrt d), π)$, but fails in any truncated circular cone with angle $θ\leq \cos^{-1}(1/\sqrt d)$, a Lipschitz domain whose Lipschitz constant is larger than or equal to $1/\sqrt{d-1}.$

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↗
arXiv2014

Laplacian perturbed by non-local operators

Suppose that $d\geq 1$ and $0<β<2$. We establish the existence and uniqueness of the fundamental solution $q^b(t, x, y)$ to the operator $\mathcal{L}^b=Δ+S^b$, where $$S^bf(x) := \int_{\mathbb{R}^d} \left( f(x+z) - f(x) - \nabla f(x) \cdot z\mathbb{1}_{\{|z| \leq 1\}} \right) \frac{b(x, z)}{|z|^{d+β}} dz$$ and $b(x, z)$ is a bounded measurable function on $\mathbb{R}^d \times \mathbb{R}^d$ with $b(x, z)=b(x, -z)$ for $x, z\in \mathbb{R}^d$. We show that if for each $x\in\mathbb{R}^d, b(x, z) \geq 0$ for a.e. $z\in\mathbb{R}^d$, 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. Furthermore, sharp two-sided estimates on $q^b(t, x, y)$ are derived.

math.PR↗