[Paper Review] Dirichlet Heat Kernel Estimates for Subordinate Brownian Motions with Gaussian Components

Zhen-Qing Chen, Panki Kim+1 more|arXiv (Cornell University)|Mar 26, 2013

Nonlinear Partial Differential Equations22 references4 citations

TL;DR

This paper establishes sharp two-sided estimates for the Dirichlet heat kernel of subordinate Brownian motions with Gaussian components in $C^{1,1}$ open sets in $\mathbb{R}^d$. By leveraging boundary Harnack principles and Lévy system techniques, the authors derive explicit bounds that hold for all $t > 0$ in bounded domains, extending prior results on purely discontinuous processes to hybrid diffusive-jump processes with general subordinators.

ABSTRACT

In this paper, we derive explicit sharp two-sided estimates for the Dirichlet heat kernels, in C^{1,1} open sets D in R^d, of a large class of subordinate Brownian motions with Gaussian components. When D is bounded, our sharp two-sided Dirichlet heat kernel estimates hold for all t>0. Integrating the heat kernel estimates with respect to the time variable t, we obtain sharp two-sided estimates for the Green functions, in bounded C^{1,1} open sets, of such subordinate Brownian motions with Gaussian components.

Motivation & Objective

• To extend sharp two-sided Dirichlet heat kernel estimates to subordinate Brownian motions that include both Gaussian and jump components.
• To establish such estimates in bounded $C^{1,1}$ open sets for all $t > 0$, overcoming challenges from mixed diffusion-jump dynamics near the boundary.
• To derive corresponding sharp Green function estimates by integrating the heat kernel in time.
• To generalize prior results on purely discontinuous processes (e.g., stable processes) to processes with both diffusive and jump behavior.
• To provide a unified framework for heat kernel estimates under general subordinators with complete Bernstein functions and controlled Lévy measures near zero.

Proposed method

• Utilizes a subordinator $S_t$ with positive drift and complete Bernstein function Laplace exponent $\phi(\lambda) = \lambda + \psi(\lambda)$, where $\psi$ is the Laplace exponent of the subordinator.
• Constructs the subordinate Brownian motion $X_t = B_{S_t}$, where $B$ is a standard Brownian motion independent of $S$, leading to an infinitesimal generator $\mathcal{L}^X = \Delta - \psi(-\Delta)$.
• Employs the Lévy system representation to describe the jump dynamics, with Lévy density $J(x) = \int_0^\infty (4\pi t)^{-d/2} e^{-|x|^2/(4t)} \mu(t) dt$, where $\mu$ is the Lévy measure of the subordinator.
• Applies the boundary Harnack principle for $\Delta + \Delta^{\alpha/2}$-type operators in $C^{1,1}$ domains to control boundary decay rates of harmonic functions.
• Derives heat kernel estimates via time-space path integration, splitting the time integral into regimes based on the relative scale of $t$ and $|x-y|^2$.
• Uses scaling and comparison arguments involving the function $u_0 = \delta_D(x)\delta_D(y)/|x-y|^2$ to bound integrals in different dimensions ($d=1,2,d\geq3$) uniformly.

Experimental results

Research questions

• RQ1 What are the sharp two-sided estimates for the Dirichlet heat kernel of a subordinate Brownian motion with Gaussian and jump components in a $C^{1,1}$ domain?
• RQ2 How do the boundary behavior and exit time distributions affect the heat kernel estimates in such hybrid processes?
• RQ3 Can the boundary Harnack principle be extended to non-local operators with mixed Gaussian and jump components?
• RQ4 What is the precise dependence of the heat kernel on the distance to the boundary and the spatial separation $|x-y|$?
• RQ5 How do the estimates for the Green function emerge from integrating the heat kernel over time in bounded domains?

Key findings

• Sharp two-sided Dirichlet heat kernel estimates are established for all $t > 0$ in bounded $C^{1,1}$ open sets for a large class of subordinate Brownian motions with Gaussian components.
• The estimates are explicit and depend on the distance to the boundary $\delta_D(x)$, $\delta_D(y)$, and the spatial distance $|x-y|$, with the form $\asymp \left(1 \wedge \frac{\delta_D(x)}{\sqrt{t}}\right)\left(1 \wedge \frac{\delta_D(y)}{\sqrt{t}}\right) \left(t^{-1} \wedge \frac{t}{|x-y|^4}\right)$ in $d \geq 3$, adjusted for lower dimensions.
• For $d=2$, the heat kernel estimate is bounded by $\asymp \log\left(1 + \frac{\delta_D(x)\delta_D(y)}{|x-y|^2}\right)$, reflecting logarithmic boundary decay.
• For $d=1$, the estimate scales as $\asymp \left(\delta_D(x)\delta_D(y)\right)^{1/2} \wedge \frac{\delta_D(x)\delta_D(y)}{|x-y|}$, indicating a square-root dependence near the boundary.
• Integrating the heat kernel in time yields sharp two-sided estimates for the Green function in bounded $C^{1,1}$ domains, with the same functional form as the heat kernel in the time-integrated regime.
• The results hold under the growth condition $\mu(r) \leq c\mu(2r)$ for $r \in (0,K)$, ensuring regularity of the Lévy measure near zero, which is essential for the boundary Harnack principle to apply.