The fractional Laplacian ∆^{α/2} in a d-dimensional Euclidean space is a non-local integral
operator with the singular kernel of the form |x − y|^{−d−α}, 0 < α < 2. As in the case of
the standard Laplacian ∆, the fundamental solution of the corresponding heat equation is
called the heat kernel. The heat kernel has a probabilistic interpretation: it represents the
transition densities p(t, x, y) from point x to point y in time t for the corresponding strong
Markov process – the isotropic α-stable process.
Unlike the Laplacian, where the heat kernel is given by an explicit formula (the Gaussian
kernel), the closed-form expression for the heat kernel of the fractional Laplacian remains
unknown. Therefore, the best one can hope for are sharp two-sided estimates.
I will begin by introducing the basic properties of the fractional Laplacian, including its
connection to the heat semigroup, the associated Dirichlet form, and the fundamental heat
kernel estimates. A key aspect of these estimates is how they arise naturally from exact
scaling properties.
Next, I will discuss various formulations of the fractional Laplacian on open subsets of
Euclidean space, such as the reflected, restricted, and censored fractional Laplacians. For
each case, I will explain the corresponding Markov process and show the relevant heat kernel
estimates. All these operators share the feature that their singular kernels are either equal
or comparable to |x − y|^{−d−α} —the so-called uniformly elliptic case. The analysis relies on a combination of functional-analytic and probabilistic techniques.
I will conclude with some very recent results for scenarios where the singular kernel may
vanish at the boundary. I will also motivate the study of such degenerate cases. Surprisingly,
the resulting heat kernel estimates exhibit qualitatively different behavior, highlighting new
phenomena in this setting.
