Censored Fractional Laplacian

• The censored fractional Laplacian is a nonlocal operator defined by a principal value integral over a bounded domain, suppressing jumps that would leave the domain.
• It generates a censored symmetric α-stable Lévy process with distinct probabilistic behaviors, exhibiting finite lifetime for α > 1/2 and conservative dynamics for α ≤ 1/2.
• Spectral analysis reveals discrete eigenvalues with Weyl-type asymptotics and requires nonlocal Neumann-type boundary conditions for α in [1, 2), which is critical for nonlocal PDE applications.

The censored fractional Laplacian, also known as the regional fractional Laplacian, is a nonlocal differential operator that serves as the infinitesimal generator of the so-called censored or regional symmetric $\alpha$-stable Lévy process on a bounded domain. Distinguished from the more familiar killed (Dirichlet) and reflected versions of the fractional Laplacian, the censored operator modifies the jump process by suppressing all jumps that would leave the domain, thereby generating a nonlocal process that remains inside the domain without absorption or killing at the boundary. This operator plays a fundamental role in nonlocal potential theory, stochastic processes, and the analysis of elliptic and parabolic equations with nonlocal operators in bounded domains (Chen et al., 2020, Garbaczewski et al., 2018).

1. Analytical Definition

Let $D\subset\mathbb{R}^n$ be a bounded, sufficiently regular open set, and $0<\alpha<2$. The censored (regional) fractional Laplacian acting on a suitable function $f$ is defined by

$$(-\Delta)^{\alpha/2}_{D,\mathrm{cens}}\,f(x) = C_{n,\alpha}\,\mathrm{P.V.}\int_{D} \frac{f(x) - f(y)}{|x-y|^{n+\alpha}}\,dy, \quad x \in D,$$

where $C_{n,\alpha}$ is the normalization constant,

$$C_{n,\alpha} = \frac{\alpha\,2^{\alpha-1}\,\Gamma\left(\frac{n+\alpha}{2}\right)}{\pi^{n/2}\,\Gamma\left(1-\frac{\alpha}{2}\right)} > 0,$$

and $\mathrm{P.V.}$ denotes the Cauchy principal value at $y = x$. This operator, unlike the Dirichlet-restricted (killed) version, only integrates over $y$ within $D$; would-be jumps outside $D$ are suppressed (not killed). In one spatial dimension, for $D = (-1,1)$, the formula takes the form

$$(-\Delta)^{\alpha/2}_{D,\mathrm{cens}}\psi(x) = -A_\alpha \int_{-1}^1 \frac{\psi(u) - \psi(x)}{|u-x|^{1+\alpha}}\,du,$$

where $$A_\alpha = \frac{\Gamma(1+\alpha)}{\pi} \sin \frac{\pi\alpha}{2}$$ (Garbaczewski et al., 2018).

For $0<\alpha<1$, the operation is well-defined up to and on $\partial D$ without further boundary conditions. For $1\leq\alpha<2$, a nonlocal Neumann-type boundary condition, interpreted as vanishing of a suitably defined inward normal derivative on $\partial D$, is both necessary and sufficient for finiteness at the boundary.

2. Probabilistic Interpretation

The censored fractional Laplacian corresponds to the infinitesimal generator of the censored $\alpha$-stable process—also called the censored symmetric $2\alpha$-stable Lévy process—confined to $D$. The process is constructed from the free symmetric $\alpha$-stable process by canceling any jumps that would exit $D$. This implementation, originally attributed to Bogdan, Burdzy, and Chen, leads to a strong Markov process with the following properties:

• For $\alpha \in (1/2,1)$, the process has finite lifetime and, with probability 1, will reach the boundary $\partial D$ in finite time.
• For $\alpha \in (0,1/2]$, the process is conservative (infinite lifetime) and almost surely will not approach $\partial D$ (Chen et al., 2020).
• The associated Dirichlet form on $L^2(D)$ is

$$\mathcal{E}(u,v) = C_{n,\alpha}\,\iint_{D \times D} \frac{(u(x) - u(y))(v(x) - v(y))}{|x-y|^{n+\alpha}}\,dx\,dy,$$

with no killing term. This structure underpins the analytic and spectral properties of the corresponding operator (Chen et al., 2020, Garbaczewski et al., 2018).

3. Spectral Theory

The spectral problem for the censored fractional Laplacian is formulated as

$$(-\Delta)^{\alpha/2}_{D,\mathrm{cens}}\varphi_k(x) = \lambda_k\,\varphi_k(x),\quad x\in D,$$

where $\{\varphi_k\}$ form a complete orthonormal set in $L^2(D)$ and $$0=\lambda_0<\lambda_1\leq\lambda_2\leq\cdots\to\infty$$. For $0<\alpha<1$, the spectrum is discrete without boundary conditions; for $1 \leq \alpha < 2$, the nonlocal Neumann boundary condition is imposed.

Weyl-type asymptotics govern the distribution of the eigenvalues: $\lambda_k \sim C\,k^{2\alpha/n},$ where the constant $C$ depends on both $D$ and $\alpha$. The zero mode represents the constant function, signifying perpetual occupancy of $D$ by the reflecting (conservative) process in the Neumann case (Garbaczewski et al., 2018).

Intrinsic ultracontractivity (IU) holds for the associated semigroups. This ensures that the long-time behavior of the transition kernel approaches a Gaussian-like spatial distribution, analogous to the IU property for the killed process (Garbaczewski et al., 2018).

4. Comparison with Other Fractional Laplacians

The censored fractional Laplacian is one among several non-equivalent boundary-respecting nonlocal Laplacians defined on bounded domains. The principal alternatives are:

Operator (Editor's term)	Definition domain	Boundary interaction
Killed (Restricted, Dirichlet)	$\int_{\mathbb{R}^{n}}$	Process killed on exit; $f\equiv0$ on $\mathbb{R}^{n}\setminus D$
Censored (Regional)	$\int_{D}$	Jumps outside $D$ suppressed; process is not killed
Reflected (Regional on $\overline{D}$)	$\int_{\overline{D}}$	All jumps stay in $\overline{D}$; suitable Neumann data
Spectral (Eigenfunction)	Series expansion	Dirichlet basis; $f$ decomposed using Laplacian eigenbasis
Taboo (Conditioned)	Doob $h$-transform	Conditioned never to reach boundary; specific invariant measure

The censored operator differs critically from the killed Laplacian, which incorporates a killing term and requires $f=0$ off $D$, creating strictly positive spectra. The taboo Laplacian (via Doob’s $h$-transform with ground-state $\phi_1$) produces a Markov generator generically different from the regional one, with invariant density $\rho = \phi_1^2$ (Garbaczewski et al., 2018, Chen et al., 2020).

5. Boundary Behavior, Function Spaces, and Operator Domains

The censored fractional Laplacian's behavior near the domain boundary is distinct:

• For the censored process with $\alpha > 1/2$, hitting the boundary occurs almost surely in finite time (form not conservative).
• For $\alpha \leq 1/2$, the process never attains the boundary; the form is conservative (Chen et al., 2020).
• Vanishing of the nonlocal normal derivative on $\partial D$ is required for $1\leq\alpha<2$ (Garbaczewski et al., 2018).

Function spaces used include the energy space $H^{\alpha/2}(D)$, defined as the closure of $C_c^\infty(D)$ under the norm

$$\|u\|_{H^{\alpha/2}(D)}^2 = \|u\|_{L^2(D)}^2 + \mathcal{E}(u,u).$$

The operator domain is typically those $u\in H^{\alpha/2}(D)$ with $(-\Delta)^{\alpha/2}_{D,\mathrm{cens}}u\in L^2(D)$, or smooth ($C^{2\alpha+\varepsilon}$) functions for pointwise definitions (Chen et al., 2020).

If $u$ vanishes outside $D$, there is a precise relation: $$(-\Delta)^\alpha u(x) = (-\Delta)^\alpha_{D, \mathrm{cens}} u(x) + k_\alpha(x)\,u(x),\quad k_\alpha(x) = C_{n,\alpha}\int_{\mathbb{R}^{n}\setminus D} |x-y|^{-n-\alpha} dy.$$ Near the boundary, $k_\alpha(x)\sim d_\alpha\,p(x)^{-\alpha}$ with $p(x)=\mathrm{dist}(x,\partial D)$ (Chen et al., 2020).

6. Qualitative Properties and Applications

Key qualitative phenomena stem from the probabilistic nature of the censored process:

• For $\alpha \leq 1/2$, no continuous superharmonic function with boundary blow-up exists; explicitly, there is no nontrivial $w$ solving

$$(-\Delta)^\alpha_{D,\mathrm{cens}} w \geq 0 \text{ in } D, \quad \lim_{x\to \partial D} w(x) = +\infty,$$

in striking contrast to the killed Laplacian, where boundary blow-up is generic (Chen et al., 2020).

• For the Poisson problem $(-\Delta)^\alpha_{D,\mathrm{cens}} u=1$ in $D$ (no boundary data), there are no viscosity solutions bounded above or below when $\alpha\in(0,1/2]$.
• For the Lane–Emden equation $(-\Delta)^\alpha_{D,\mathrm{cens}} u=u^p$ in $D$ with zero boundary data, there are no nonnegative nontrivial solutions for any $p>0$ if $\alpha\in(0,1/2]$. This is a direct reflection of the process’ inability to interact with the boundary (Chen et al., 2020).

Applications discussed include nonlocal elliptic and semilinear problems, spectral problems with nonlocal Neumann conditions, and physical models such as disordered semiconducting heterojunctions under Neumann-type constraints (Garbaczewski et al., 2018).

7. Summary and Research Directions

The censored fractional Laplacian, defining a nonlocal process that remains in bounded domains by suppressing exiting jumps, manifests a rich array of analytic, probabilistic, and spectral phenomena not shared by other fractional Laplacians. Its unique boundary behavior underpins Liouville-type nonexistence theorems and nontrivial differences in spectral properties, with consequences for the analysis of bounded-domain nonlocal PDEs and boundary-value problems (Garbaczewski et al., 2018, Chen et al., 2020). Ongoing research includes the refinement of boundary conditions, spectral characterization in more general domains, and applications to physical stochastic processes where boundary interactions are non-absorptive.