[Paper Review] 3 Isotropic Markov semigroups on ultra-metric spaces∗

Alexander Bendikov, Alexander Grigor'yan+3 more|arXiv (Cornell University)|Aug 18, 2016

advanced mathematical theories71 references66 citations

TL;DR

This paper constructs isotropic Markov semigroups on ultra-metric spaces with compact balls using a reference measure and distance distribution function, establishing sharp bounds on transition densities and Green functions, and proving the generator has pure point spectrum. A key result is the duality between random walks on p-adic trees and isotropic jump processes on their boundaries, showing that the Taibleson and Vladimirov Laplacians arise as generators of such processes, with explicit time-changed relations to fractional derivatives on Qp.

ABSTRACT

Let (X,d) be a locally compact separable ultra-metric space. Given a reference measure \mu\ on X and a step length distribution on the non-negative reals, we construct a symmetric Markov semigroup P^t acting in L^2(X,\mu). We study the corresponding Markov process. We obtain upper and lower bounds of its transition density and its Green function, give a transience criterion, estimate its moments and describe the Markov generator and its spectrum, which is pure point. In the particular case when X is the field of p-adic numbers, our construction recovers fractional derivative and the Taibleson Laplacian (spectral multiplier), and we can also apply our theory to the study of the Vladimirov Laplacian which is closely related to the concept of p-adic Quantum Mechanics. Even in this well established setting, several of our results are new. We also elaborate the relation between our processes and Kigami's jump processes on the boundary of a tree which are induced by a random walk. In conclusion, we provide examples illustrating the interplay between the fractional derivatives and random walks.

Motivation & Objective

• To construct symmetric Markov semigroups on separable ultra-metric spaces with compact balls using a reference measure and distance distribution function.
• To derive sharp upper and lower bounds for the transition density and Green function of the associated Markov process.
• To characterize the spectrum of the Markov generator as pure point and relate it to the distribution of hitting times and Green functions.
• To establish a duality between nearest-neighbor random walks on rooted trees and isotropic jump processes on their boundaries, particularly in the p-adic setting.
• To recover and extend known results on the Taibleson and Vladimirov Laplacians in the context of p-adic analysis and fractional derivatives.

Proposed method

• Define a symmetric Markov semigroup {P_t} on L²(X, µ) via a distance distribution function σ and reference measure µ on an ultra-metric space (X, d) with compact balls.
• Use subordination techniques to relate the semigroup to a Lévy process and derive spectral distribution functions.
• Apply the theory of Dirichlet forms and jump kernels to characterize the generator L and its domain.
• Establish the pure point spectrum of L by analyzing the range of the function v ↦ 1/G(v, o), where G is the Green function.
• Use tree-based random walks with transition probabilities depending on p-adic norms to model the boundary process.
• Derive explicit formulas for hitting probabilities F(v, o), Green functions G(v, o), and the Naïm kernel Θ_o(x, y) via recursive relations on tree levels.

Experimental results

Research questions

• RQ1 How can symmetric Markov semigroups be systematically constructed on ultra-metric spaces with compact balls using a reference measure and distance distribution?
• RQ2 What are the sharp upper and lower bounds for the transition density and Green function of such processes?
• RQ3 How is the spectrum of the Markov generator L related to the geometry of the underlying ultra-metric space and the hitting times of the process?
• RQ4 What is the precise relationship between nearest-neighbor random walks on p-adic trees and isotropic jump processes on their boundaries?
• RQ5 How do the Taibleson and Vladimirov Laplacians on Qp emerge as generators of such processes, and how are they related to fractional derivatives?

Key findings

• The Markov generator L has a pure point spectrum, with eigenvalues given by the range of v ↦ 1/G(v, o) together with 0.
• For the random walk on the p-adic tree T_p with parameter c = (1 + p^{-α})^{-1}, the hitting probability F(v, o) = p^{-α|v|} and the Green function G(v, o) = p^{-α|v|}/(1 - p^{-α}).
• The boundary process induced by the random walk is an isotropic jump process on Z_p with jump kernel proportional to the kernel of the p-adic fractional derivative D_α.
• The boundary process and the D_α-driven process are related by a linear time change: X_t / C = X_α_t with C = p^{-α}(1 - p^{-α}).
• The jump kernel J_α of D_α and the kernel Θ_o of the boundary generator L satisfy J_α(x, y) = (p^α / (1 - p^{-α})) Θ_o(x, y).
• For the non-compact case with reference end ̟, the intrinsic metric and jump kernel of the boundary process are d_φ(x, y) = ((1 - p^{-α}) / (1 + p^{-α})) ||x - y||_p^α and J(x, y) = ((1 - p^{-α})^2 / ((1 + p^{-α})(1 - p^{-α-1}))) ||x - y||_p^{-(α+1)}, respectively, with J(x, y) = ((1 - p^{-α}) / (p^α + 1)) J_α(x, y).