[Paper Review] Dirichlet heat kernel estimates for fractional Laplacian under non-local perturbation

Zhen-Qing Chen, Ting Yang|arXiv (Cornell University)|Mar 18, 2015

Nonlinear Partial Differential Equations14 references3 citations

TL;DR

This paper establishes sharp two-sided estimates for the Dirichlet heat kernel of a non-local operator $σ^{b} = \Delta^{\alpha/2} + \mathcal{S}^{b}$, where $\mathcal{S}^{b}$ is a non-local perturbation of the fractional Laplacian, in bounded $C^{1,1}$ domains. The key result is a precise asymptotic estimate for the transition density $p^{b}_{D}(t,x,y)$, showing it behaves like $e^{-\lambda_{1}t}(1\wedge\delta_{D}(x)^{\alpha/2})(1\wedge\delta_{D}(y)^{\alpha/2})$, with $\lambda_{1}$ the first eigenvalue of the generator on $D$, under a mild positivity condition on the perturbation kernel.

ABSTRACT

For $d\ge 2$ and $0\varepsilon\}} (f(x+z)-f(x))\frac{b(x,z)}{|z|^{d+β}}\,dz, $$ and $b(x,z)$ is a bounded measurable function on $\mathbb{R}^{d} imes\mathbb{R}^{d}$ with $b(x,z)=b(x,-z)$ for every $x,z\in\mathbb{R}^{d}$. Here ${\cal A}(d, -β)$ is a normalizing constant so that $\mathcal{S}^b=-(-Δ)^{β/2}$ when $b(x, z)\equiv 1$. It was recently shown in Chen and Wang [arXiv:1312.7594 [math.PR]] that when $b(x, z) \geq -\frac{\mathcal{A}(d, -α)} {\mathcal{A}(d, -β)}\, |z|^{β-α}$, then $\mathcal{L}^b$ admits a unique fundamental solution $p^b(t, x, y)$ which is strictly positive and continuous. The kernel $p^b(t, x, y)$ uniquely determines a conservative Feller process $X^b$, which has strong Feller property. The Feller process $X^b$ is also the unique solution to the martingale problem of $(\mathcal{L}^b, \mathcal{S}(\mathbb{R}^d))$, where $\mathcal{S}(\mathbb{R}^d)$ denotes the space of tempered functions on $\mathbb{R}^d$. In this paper, we are concerned with the subprocess $X^{b,D}$ of $X^{b}$ killed upon leaving a bounded $C^{1,1}$ open set $D\subset \mathbb{R}^d$. We establish explicit sharp two-sided estimates for the transition density function of $X^{b, D}$.

Motivation & Objective

• To derive sharp two-sided estimates for the Dirichlet heat kernel of a non-local operator $\mathcal{L}^{b} = \Delta^{\alpha/2} + \mathcal{S}^{b}$ in bounded $C^{1,1}$ domains.
• To establish the existence and regularity of the transition density $p^{b}_{D}(t,x,y)$ for the subprocess killed upon exiting such domains.
• To obtain explicit bounds on the Green function of $\mathcal{L}^{b}$ in bounded $C^{1,1}$ domains using the heat kernel estimates.
• To analyze the spectral properties of $\mathcal{L}^{b}$ on bounded domains, particularly the first eigenvalue $\lambda_{1}$, and its role in the long-time behavior of the heat kernel.

Proposed method

• The analysis relies on the existence of a strictly positive, jointly continuous fundamental solution $p^{b}(t,x,y)$ for the full generator $\mathcal{L}^{b}$ on $\mathbb{R}^d$, established in prior work.
• The Dirichlet heat kernel $p^{b}_{D}(t,x,y)$ is defined as the transition density of the subprocess $X^{b,D}$ killed upon exiting a bounded $C^{1,1}$ domain $D$.
• Sharp two-sided estimates for $p^{b}_{D}(t,x,y)$ are derived using the Chapman-Kolmogorov equation, domain monotonicity, and the known estimates for the full-space kernel $p^{b}(t,x,y)$.
• The method involves decomposing the transition density over time intervals and applying Hölder’s inequality and the lower/upper bounds from prior estimates.
• The eigenfunction $\phi$ associated with the first eigenvalue $\lambda_{1}$ of $\mathcal{L}^{b,D}$ is used to derive bounds on $\lambda_{1}$ and to relate the long-time decay of the heat kernel to $e^{-\lambda_{1}t}$.
• The final estimate is obtained by combining the boundary behavior $1\wedge\delta_{D}(x)^{\alpha/2}$ with the exponential decay $e^{-\lambda_{1}t}$, using the spectral gap and the positivity of the eigenfunction.

Experimental results

Research questions

• RQ1 What are the sharp two-sided estimates for the Dirichlet heat kernel $p^{b}_{D}(t,x,y)$ of the non-local operator $\mathcal{L}^{b} = \Delta^{\alpha/2} + \mathcal{S}^{b}$ in a bounded $C^{1,1}$ domain $D$?
• RQ2 How does the perturbation $\mathcal{S}^{b}$, defined via a bounded, symmetric kernel $b(x,z)$, affect the transition density and its long-time decay?
• RQ3 What is the precise dependence of the heat kernel on the distance to the boundary $\delta_{D}(x)$ and $\delta_{D}(y)$?
• RQ4 How is the first eigenvalue $\lambda_{1}$ of the generator $\mathcal{L}^{b,D}$ related to the long-time asymptotics of the heat kernel?
• RQ5 Can the Green function of $\mathcal{L}^{b}$ in $D$ be estimated sharply using the heat kernel estimates?

Key findings

• The Dirichlet heat kernel $p^{b}_{D}(t,x,y)$ satisfies the sharp two-sided estimate $p^{b}_{D}(t,x,y) \asymp e^{-\lambda_{1}t}(1\wedge\delta_{D}(x)^{\alpha/2})(1\wedge\delta_{D}(y)^{\alpha/2})$ for all $t>0$ and $x,y\in D$, where $\lambda_{1}$ is the first eigenvalue of $\mathcal{L}^{b,D}$.
• The first eigenvalue $\lambda_{1}$ is positive and uniformly bounded away from zero and infinity depending on $d$, $\alpha$, $\beta$, $D$, $A$, and $M$, ensuring exponential decay in time.
• The lower bound for $p^{b}_{D}(t,x,y)$ holds uniformly for $t \in (T,\infty)$ and $|x-y| < 4\varepsilon(A)/5$, with a constant $C_{38}$ depending on $d$, $\alpha$, $\beta$, $D$, $A$, $M$, and $T$, and the estimate involves $e^{-\lambda^{b,D_{x}\cup D_{y}}_{1}t}\delta_{D}(x)\delta_{D}(y)$.
• The eigenfunction $\phi$ associated with $\lambda_{1}$ satisfies $c_{2}^{-1}e^{-\lambda_{1}} \leq \phi(x) \leq c_{2}e^{-\lambda_{1}}(1\wedge\delta_{D}(x)^{\alpha/2})$, showing its boundary decay is exactly $\delta_{D}(x)^{\alpha/2}$.
• The Green function of $\mathcal{L}^{b}$ in $D$ admits sharp two-sided estimates that follow directly from the heat kernel bounds, with the same $\delta_{D}(x)^{\alpha/2}\delta_{D}(y)^{\alpha/2}$ behavior.
• The estimates are robust under the condition $b(x,z) \geq -\frac{\mathcal{A}(d,-\alpha)}{\mathcal{A}(d,-\beta)}|z|^{\beta-\alpha}$, ensuring the positivity and conservativeness of the associated Feller process $X^{b}$.