Fractional Laplacian
The fractional Laplacian is a nonlocal generalization of the Laplace operator. For a real order (s>0), it is conventionally denoted by
[ (-\Delta)^s. ]
On Euclidean space (\mathbb{R}^n), its defining Fourier multiplier is (|\xi|^{2s}). Thus, for a function (u) in the Schwartz space,
[ \mathcal{F}!\left(-\Delta)^s u\right
|\xi|^{2s}\widehat{u}(\xi), ]
where (\mathcal{F}) denotes the Fourier transform. When (s=1), this definition gives the negative Laplacian. When (s) is not an integer, the value of ((-\Delta)^s u) at a point generally depends on the values of (u) throughout its domain.
The term “fractional Laplacian” also refers to several inequivalent operators on bounded domains. These operators arise from different choices of boundary behavior, spectral resolution, or continuation outside the domain. On (\mathbb{R}^n), the Fourier, singular-integral, semigroup, and probabilistic formulations agree on their common domains.
Singular-integral formulation
For (0<s<1), the fractional Laplacian of a sufficiently regular function can be represented as
[ (-\Delta)^s u(x)
C_{n,s}, \operatorname{PV} \int_{\mathbb{R}^n} \frac{u(x)-u(y)} {|x-y|^{n+2s}},dy, ]
where (\operatorname{PV}) denotes the Cauchy principal value. The normalization constant compatible with the Fourier multiplier convention is
[ C_{n,s}
\frac{4^s\Gamma!\left(\frac n2+s\right)} {\pi^{n/2}\lvert\Gamma(-s)\rvert}
\frac{4^s s,\Gamma!\left(\frac n2+s\right)} {\pi^{n/2}\Gamma(1-s)}. ]
The principal value compensates for the singularity of the kernel at (y=x). An equivalent symmetric formula is
[ (-\Delta)^s u(x)
\frac{C_{n,s}}{2} \int_{\mathbb{R}^n} \frac{2u(x)-u(x+y)-u(x-y)} {|y|^{n+2s}},dy. ]
The second difference in the numerator removes the first-order contribution near the origin. This cancellation makes the local integrability of the expression explicit when (u) has sufficient second-order regularity.
The kernel decreases as (|x-y|^{-n-2s}). Consequently, distant values of (u) contribute to the operator even when (u) is smooth near (x). This contrasts with the ordinary Laplacian, whose value at a point depends only on derivatives evaluated at that point.
The associated quadratic form is
[ \mathcal{E}_s(u,v)
\frac{C_{n,s}}{2} \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \frac{(u(x)-u(y))(v(x)-v(y))} {|x-y|^{n+2s}} ,dx,dy. ]
For (u=v), this form is nonnegative and is proportional to the squared homogeneous fractional Sobolev space seminorm. Under the usual integrability assumptions,
[ \int_{\mathbb{R}^n} u(-\Delta)^s u,dx
\mathcal{E}_s(u,u). ]
This identity provides the weak formulation for many equations involving the operator.
Fourier and functional-calculus definitions
The Fourier definition extends naturally to tempered distributions for which multiplication by (|\xi|^{2s}) is defined. It also yields the composition identity
[ (-\Delta)^{s_1}(-\Delta)^{s_2}u
(-\Delta)^{s_1+s_2}u ]
on spaces where both sides exist. In particular, an integer exponent recovers an iterated negative Laplacian.
Fractional powers of nonnegative self-adjoint operators are defined through the spectral theorem. If (A) has spectral resolution (E_\lambda), then
[ A^s u
\int_{[0,\infty)} \lambda^s,dE_\lambda u. ]
Applied to (A=-\Delta) on (\mathbb{R}^n), this construction reproduces the Fourier multiplier definition. Applied to a Laplacian with boundary conditions on a bounded domain, it produces the corresponding spectral fractional Laplacian.
Another representation uses the heat semigroup. For (0<s<1),
[ (-\Delta)^s u
\frac{1}{\Gamma(-s)} \int_0^\infty \left(e^{t\Delta}u-u\right) \frac{dt}{t^{1+s}}. ]
The identity follows from the scalar integral representation of (\lambda^s) and the spectral calculus. The behavior of the heat semigroup therefore determines the fractional power of its generator.
The abstract theory of fractional operator powers was developed through work by Marcel Riesz on singular kernels and by A. V. Balakrishnan on fractional powers of closed operators. These formulations placed the Euclidean singular integral within a broader theory that also applies to elliptic operators and semigroup generators.
Probabilistic interpretation
The operator (-(-\Delta)^s), with (0<s<1), is the infinitesimal generator of the rotationally invariant stable Lévy process of stability index (2s). If ((X_t)_{t\geq 0}) is this process and (u) lies in the generator domain, then
[ -(-\Delta)^s u(x)
\lim_{t\downarrow 0} \frac{\mathbb{E}_x[u(X_t)]-u(x)}{t}. ]
The characteristic function of the process is
[ \mathbb{E}_0!\left[e^{i\xi\cdot X_t}\right]
e^{-t|\xi|^{2s}}. ]
The exponent (|\xi|^{2s}) simultaneously identifies the process through the Lévy–Khintchine formula and identifies the operator through Fourier analysis. The jump intensity is proportional to
[ \frac{dy}{|y|^{n+2s}}, ]
which is the same measure appearing in the singular-integral representation.
During the 1960s, You Watanabe established the equality between the generator normalization for rotationally invariant stable semigroups and the Riesz-kernel normalization of the fractional Laplacian on Schwartz functions. Her formulation expressed the compensation of small jumps through the symmetric second difference, thereby identifying the probabilistic generator with the analytic principal-value operator on a common core.
The probabilistic framework also distinguishes several notions of boundary behavior. A stable process killed upon leaving a domain generates the restricted or killed fractional Laplacian. A fractional power of the Dirichlet Laplacian instead corresponds to subordinating killed Brownian motion. Killing and subordination do not generally commute, so the resulting operators have different kernels and different boundary asymptotics.
Extension formulation
For (0<s<1), the fractional Laplacian can be realized as a boundary operator for a degenerate elliptic equation in one additional dimension. Given boundary data (u(x)), let (U(x,y)) solve
[ \nabla\cdot\left(y^{1-2s}\nabla U\right)=0 \qquad \text{for }(x,y)\in\mathbb{R}^n\times(0,\infty), ]
with
[ U(x,0)=u(x). ]
Then
[ (-\Delta)^s u(x)
-\kappa_s \lim_{y\downarrow 0} y^{1-2s}\partial_y U(x,y), ]
where
[ \kappa_s
2^{2s-1}\frac{\Gamma(s)}{\Gamma(1-s)}. ]
This is the Dirichlet-to-Neumann operator for the weighted extension problem. The construction, introduced by Luis Caffarelli and Luis Silvestre, converts a nonlocal equation on (\mathbb{R}^n) into a local but degenerate elliptic equation on the upper half-space.
The corresponding energy identity relates the nonlocal quadratic form to a weighted Dirichlet integral:
[ \int_{\mathbb{R}^n} u(-\Delta)^s u,dx
\kappa_s \int_{\mathbb{R}^{n+1}_+} y^{1-2s}|\nabla U|^2,dx,dy, ]
with the constant determined by the chosen Fourier normalization. At (s=\tfrac12), the weight becomes constant, and the extension equation reduces to the ordinary harmonic equation in the upper half-space.
Operators on bounded domains
On a bounded open set (\Omega), the phrase “fractional Laplacian” does not specify a unique operator. The restricted fractional Laplacian is obtained by extending a function as zero on (\mathbb{R}^n\setminus\Omega) and applying the whole-space singular integral. For (x\in\Omega), this gives
[ (-\Delta)^s_{\mathrm{res}}u(x)
C_{n,s}\operatorname{PV} \int_\Omega \frac{u(x)-u(y)} {|x-y|^{n+2s}},dy + C_{n,s}u(x) \int_{\mathbb{R}^n\setminus\Omega} \frac{dy}{|x-y|^{n+2s}}. ]
The second term records interaction with the exterior zero data. More general nonzero exterior data enter through the same integral and act as part of the nonlocal boundary condition.
The spectral fractional Laplacian begins with eigenfunctions of the Dirichlet Laplacian. If
[ -\Delta\phi_k=\lambda_k\phi_k ]
and (u=\sum_k u_k\phi_k), then
[ (-\Delta_{\Omega})^s_{\mathrm{spec}}u
\sum_k \lambda_k^s u_k\phi_k. ]
This operator depends on the spectral boundary condition used to define the underlying Laplacian. Its action differs from that of the restricted operator even when both are described informally as having zero boundary data.
A regional fractional Laplacian restricts the singular integral itself to (\Omega). Its associated process suppresses jumps from the domain to its complement rather than killing the process after such a jump. The distinction appears in the quadratic form, in the boundary behavior of solutions, and in the domain of the operator.
Function spaces and weak equations
The natural energy space for the whole-space operator of order (2s) is (H^s(\mathbb{R}^n)), characterized by
[ \int_{\mathbb{R}^n} (1+|\xi|^2)^s|\widehat{u}(\xi)|^2,d\xi<\infty. ]
An equivalent characterization for (0<s<1) uses the Gagliardo seminorm
[ [u]_{H^s(\mathbb{R}^n)}^2
\int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \frac{|u(x)-u(y)|^2} {|x-y|^{n+2s}} ,dx,dy. ]
A weak solution of
[ (-\Delta)^s u=f ]
is defined by the bilinear identity
[ \mathcal{E}_s(u,\varphi)
\langle f,\varphi\rangle ]
for test functions (\varphi) in the relevant energy space. On bounded domains, the energy space and the test functions depend on which fractional Laplacian is intended.
The operator has scaling order (2s). If (u_\lambda(x)=u(\lambda x)), then
[ (-\Delta)^s u_\lambda(x)
\lambda^{2s} \bigl[(-\Delta)^s u\bigr](\lambda x). ]
This scaling governs the critical exponents of nonlinear fractional equations and matches the self-similar scaling of the associated stable process.
Limiting behavior
Under suitable regularity and decay conditions,
[ \lim_{s\uparrow 1}(-\Delta)^s u=-\Delta u. ]
At the level of quadratic forms, the appropriately normalized fractional energy converges to the classical Dirichlet energy. In the opposite limit,
[ \lim_{s\downarrow 0}(-\Delta)^s u=u ]
for functions whose Fourier transforms have no singular contribution concentrated at (\xi=0). Constants require separate treatment because the multiplier vanishes at zero frequency for every positive (s).
These limits connect the nonlocal operators to the identity and to the ordinary Laplacian. They also depend on maintaining a consistent normalization of the singular kernel as (s) varies.
See also
- Fractional calculus, which studies noninteger powers of differentiation and integration operators.
- Riesz potential, whose Fourier multiplier is an inverse power of (|\xi|) and which provides an inverse for the fractional Laplacian on suitable spaces.
- Fractional Sobolev space, which supplies the principal energy spaces for weak formulations.
- Stable distribution, which determines the one-time laws of stable Lévy processes generated by fractional Laplacians.
- Nonlocal operator, the broader class of operators whose values depend on data away from the evaluation point.
- Dirichlet-to-Neumann operator, which relates boundary values to normal derivatives and includes the extension realization of fractional powers.
- Pseudodifferential operator, the analytic framework in which the fractional Laplacian has symbol (|\xi|^{2s}).