[Paper Review] Minimal thinness for subordinate Brownian motion in half space

Panki Kim, Renming Song+1 more|arXiv (Cornell University)|Oct 4, 2010

Stochastic processes and financial applications25 references3 citations

TL;DR

This paper establishes that the same criterion for minimal thinness in the half-space—previously known for Brownian motion—applies universally to a broad class of subordinate Brownian motions, including symmetric $α$-stable processes. Using sharp Green function estimates and the boundary Harnack principle, the authors prove that a set below a Lipschitz graph is minimally thin at the origin if and only if a certain integral condition on the function’s decay holds, independent of the process's stability index.

ABSTRACT

We study minimal thinness in the half-space $H:=\{x=(\wt{x}, x_d):\, \wt{x}\in \R^{d-1}, x_d>0\}$ for a large class of rotationally invariant Lévy processes, including symmetric stable processes and sums of Brownian motion and independent stable processes. We show that the same test for the minimal thinness of a subset of $H$ below the graph of a nonnegative Lipschitz function is valid for all processes in the considered class. In the classical case of Brownian motion this test was proved by Burdzy.

Motivation & Objective

• To extend the classical theory of minimal thinness from Brownian motion to a wide class of rotationally invariant Lévy processes, including symmetric $α$-stable processes.
• To determine whether the same integral test for minimal thinness at the origin in the half-space applies to discontinuous processes.
• To establish a unified criterion for minimal thinness that is independent of the process's characteristic exponent or jump behavior.
• To leverage recent advances in potential theory for subordinate Brownian motions, particularly sharp Green function estimates and the boundary Harnack principle.

Proposed method

• Deriving precise two-sided estimates for the Green function of the half-space for subordinate Brownian motions satisfying a specific scaling condition (H).
• Using the boundary Harnack principle to obtain sharp estimates on the Martin kernel near the boundary and the origin.
• Applying balayage techniques and excessive function constructions to test minimal thinness via comparison with the Martin kernel.
• Establishing a lower bound for the excessive function on the set $A$ below the Lipschitz graph to verify minimal thinness.
• Using integral comparisons involving $|x|^{-d}$ and the Lipschitz function $f$ to relate the divergence of the integral to non-minimal thinness.
• Proving that the same integral condition (1.2) governs minimal thinness across all processes in the class, despite differing path properties.

Experimental results

Research questions

• RQ1 Does the same criterion for minimal thinness at the origin in the half-space apply to subordinate Brownian motions beyond Brownian motion?
• RQ2 Is the integral condition (1.2) sufficient and necessary for minimal thinness of a set below a Lipschitz graph across all processes in the considered class?
• RQ3 Can the boundary Harnack principle and Green function estimates be used to unify minimal thinness criteria for discontinuous Lévy processes?
• RQ4 Why does the stability index $\alpha$ not affect the minimal thinness criterion for $\alpha$-stable processes, contrary to other thinness conditions?

Key findings

• The criterion for minimal thinness of a set $A = \{x = (\widetilde{x}, x_d) \in H : 0 < x_d \leq f(\widetilde{x})\}$ at the origin in the half-space is the same for all subordinate Brownian motions satisfying condition (H), including symmetric $\alpha$-stable processes.
• The integral condition $\int_{\{|ω|<1\}} f(\widetilde{x}) |\widetilde{x}|^{-d} \, d\widetilde{x} < \infty$ is both necessary and sufficient for minimal thinness at 0.
• The result is surprising because the minimal thinness criterion is independent of the process’s stability index $\alpha$, even though other thinness criteria (e.g., for thorns) do depend on $\alpha$.
• The proof relies on sharp Green function estimates and the boundary Harnack principle, which allow uniform control over Martin kernel behavior near the boundary.
• The excessive function $s$ constructed from the Martin kernel and the Lipschitz function $f$ satisfies $\liminf_{x \to 0, x \in A} s(x)/M^H(x,0) > 0$, implying minimal thinness when the integral (1.2) is finite.
• The same argument shows that if the integral (1.2) diverges, then $A$ is not minimally thin, extending the classical Beurling–Dahlberg theorem to this class of processes.