[Paper Review] Transition probability estimates for long range random walks

Mathav Murugan, Laurent Salo-Coste|arXiv (Cornell University)|Nov 11, 2014

Stochastic processes and statistical mechanics15 references15 citations

TL;DR

This paper establishes sharp global upper and lower bounds for the $n$-step transition probability density of long-range random walks on uniformly discrete metric measure spaces with volume doubling and regular variation. By extending jump kernel conditions to general regularly varying functions via a metric transformation, it proves heat kernel estimates matching the scaling $n / (V_h(d(x,y)) \phi(d(x,y)))$ and $1/V_h(n^{1/\beta})$, up to multiplicative constants, under a generalized jump kernel condition $JP(\beta)$. The results generalize and unify previous estimates for Lévy-type processes on metric spaces with power-law decay.

ABSTRACT

Let $(M,d,μ)$ be a uniformly discrete metric measure space satisfying space homogeneous volume doubling condition. We consider discrete time Markov chains on $M$ symmetric with respect to $μ$ and whose one-step transition density is comparable to $ (V_h(d(x,y)) ϕ(d(x,y))^{-1}$, where $ϕ$ is a positive continuous regularly varying function with index $β\in (0,2)$ and $V_h$ is the homogeneous volume growth function. Extending several existing work by other authors, we prove global upper and lower bounds for $n$-step transition probability density that are sharp up to constants.

Motivation & Objective

• To derive global upper and lower bounds for the $n$-step transition probability density of long-range random walks on uniformly discrete metric measure spaces.
• To generalize existing heat kernel estimates beyond power-law decay to general regularly varying functions.
• To establish sharp estimates matching the scaling $n / (V_h(d(x,y)) \phi(d(x,y)))$ and $1/V_h(n^{1/\beta})$ up to multiplicative constants.
• To unify and extend prior results on long-range Lévy-type processes in metric measure spaces with volume doubling.

Proposed method

• Use of a uniformly discrete metric measure space $(M,d,\mu)$ satisfying volume doubling and comparability to counting measure.
• Introduction of a generalized jump kernel condition $JP(\beta)$, where $J(x,y) \asymp 1/(V_h(d(x,y)) \phi(d(x,y)))$ for a regularly varying function $\phi$ of index $\beta \in (0,2)$.
• Application of a metric transformation via a concave function $g$ to reduce the generalized case to the standard power-law case.
• Proof of upper and lower heat kernel bounds using the Chapman-Kolmogorov equation and chaining arguments over intermediate points.
• Use of volume doubling and regular variation to control growth of $V_h$ and $V_h^\prime$ under the transformed metric.
• Establishment of sharp bounds via comparison to the standard $n^{1/\beta}$ scaling and $d(x,y)$-dependent decay.

Experimental results

Research questions

• RQ1 Can sharp global upper and lower bounds be established for the $n$-step transition density of long-range random walks on metric measure spaces with volume doubling and general regularly varying jump kernels?
• RQ2 How does the transition density scale with time $n$ and distance $d(x,y)$ under such generalized jump kernels?
• RQ3 What is the precise role of the volume growth function $V_h$ and the regularly varying function $\phi$ in determining the heat kernel behavior?
• RQ4 Can the standard power-law estimates be extended to more general jump kernels using a metric transformation technique?
• RQ5 To what extent do the bounds remain sharp when $\phi$ is not a power function but a general regularly varying function of index $\beta \in (0,2)$?

Key findings

• The $n$-step transition density $h_n(x,y)$ satisfies global upper bounds of the form $h_n(x,y) \leq C \left( \frac{1}{V_h(n^{1/\beta})} \wedge \frac{n}{V_h(d(x,y)) \phi(d(x,y))} \right)$, valid for all $n \in \mathbb{N}^*$ and $x,y \in M$, with $C$ independent of $n,x,y$.
• A matching lower bound holds: $h_n(x,y) \geq c \left( \frac{1}{V_h(n^{1/\beta})} \wedge \frac{n}{V_h(d(x,y)) \phi(d(x,y))} \right)$, proving sharpness up to constants.
• The results extend to jump kernels with $\phi$ a general regularly varying function of index $\beta \in (0,2)$, not just power laws, via a transformation of the metric using a concave function $g$.
• Under the transformed metric $d' = g \circ d$, the jump kernel satisfies the standard $UJP(\delta)$ and $LJP(\delta)$ conditions for some $\delta \in (\beta,2)$, enabling application of known estimates.
• The volume growth under the new metric satisfies $V_h^\prime(r) \asymp V_h(r^{\delta/\beta} l_\#(r^{\delta/\beta}))$, preserving the scaling structure.
• The final bounds are expressed in terms of the original metric and $\phi$, showing that the asymptotic behavior is governed by $V_h(n^{1/\beta} l_\#(n^{1/\beta}))$ and $V_h(d(x,y)) \phi(d(x,y))$.