[Paper Review] On the uniqueness class, stochastic completeness and volume growth for graphs

Xueping Huang, Matthias Keller+1 more|arXiv (Cornell University)|Dec 13, 2018

Geometric Analysis and Curvature Flows12 references4 citations

TL;DR

This paper establishes an optimal volume growth condition for stochastic completeness of graphs by proving a Grigor'yan-type uniqueness class criterion for the heat equation on a new class of graphs called globally local (GL) graphs. The authors show that stochastic completeness holds under an intrinsic metric with finite distance balls, and the uniqueness class result is sharp—failure of the GL condition by even a logarithmic factor allows non-trivial solutions, demonstrating optimality.

ABSTRACT

In this note we prove an optimal volume growth condition for stochastic completeness of graphs under very mild assumptions. This is realized by proving a uniqueness class criterion for the heat equation which is an analogue to a corresponding result of Grigor'yan on manifolds. This uniqueness class criterion is shown to hold for graphs that we call globally local, i.e., graphs where we control the jump size far outside. The transfer from general graphs to globally local graphs is then carried out via so called refinements.

Motivation & Objective

• To establish a sharp uniqueness class criterion for the heat equation on graphs, analogous to Grigor'yan’s result on manifolds.
• To identify a natural class of graphs—globally local (GL) graphs—where Grigor'yan’s inequality holds despite non-locality.
• To show that stochastic completeness is guaranteed under an intrinsic metric with finite distance balls, without requiring local finiteness or bounded jump sizes.
• To demonstrate the sharpness of the globally local condition by constructing a counterexample where the condition fails by a logarithmic factor, yielding non-trivial solutions.
• To unify and extend prior results on stochastic completeness by removing restrictive assumptions such as uniform lower bounds on measure or finite jump size.

Proposed method

• Introduce the class of globally local (GL) graphs, where jump sizes decay with distance from a reference point under an intrinsic metric.
• Prove Grigor'yan’s inequality for GL graphs using a modified version of Grigor'yan’s original proof strategy, adapted to non-locality.
• Use graph refinements—inserting vertices on edges—to transform arbitrary graphs into GL graphs while preserving stochastic completeness.
• Establish a stability result for stochastic completeness under refinement, relying on the discreteness of the graph space.
• Construct a non-trivial solution to the heat equation on a non-GL graph that satisfies the growth bound of the uniqueness class theorem but violates the GL condition by a logarithmic factor.
• Use intrinsic metrics adapted to the Dirichlet form, particularly the intrinsic pseudo-metric, to define distance balls and volume growth.

Experimental results

Research questions

• RQ1 Can Grigor'yan’s uniqueness class criterion for the heat equation be extended to non-local graphs, and if so, under what conditions?
• RQ2 What structural condition on graphs ensures that the uniqueness class result holds, despite the failure of such results in general graphs?
• RQ3 Is the globally local condition necessary and optimal for the uniqueness class criterion, and how does it compare to volume growth in combinatorial distance?
• RQ4 Can stochastic completeness be guaranteed under minimal assumptions, such as finite distance balls under an intrinsic metric, without requiring local finiteness or bounded jump sizes?
• RQ5 What happens when the globally local condition is weakened by a logarithmic factor—does this allow non-trivial solutions to the heat equation under the growth bound?

Key findings

• The uniqueness class criterion for the heat equation holds on globally local graphs, extending Grigor'yan’s result from manifolds to graphs.
• Stochastic completeness of a graph is guaranteed if there exists an intrinsic pseudo-metric such that all distance balls are finite.
• The globally local condition is sharp: if it fails by a logarithmic factor, then there exist non-trivial solutions to the heat equation satisfying the growth bound of the uniqueness class theorem.
• The counterexample constructed on the integer line with $ b(n-1,n) = n $ for $ n \geq 1 $ shows that the condition cannot be weakened further.
• The growth bound $ \int_0^T \sum_{x \in B_r(0)} |u_t(x)|^2 m(x) dt \leq C e^{c r^2} $ ensures triviality of solutions only if the graph is globally local.
• The proof technique, based on graph refinements and stability of stochastic completeness, allows extending results to general graphs by reducing them to the GL case.