Convergence Rates of Continuous-Time Random Walks to Time-Fractional Diffusions with Unbounded Coefficients

Artur Sidorenko; Vasilii Kolokoltsov

arxiv: 2605.31471 · v1 · pith:5ZXLOF3Vnew · submitted 2026-05-29 · 🧮 math.PR · cs.NA· math.NA

Convergence Rates of Continuous-Time Random Walks to Time-Fractional Diffusions with Unbounded Coefficients

Artur Sidorenko , Vasilii Kolokoltsov This is my paper

Pith reviewed 2026-06-28 20:54 UTC · model grok-4.3

classification 🧮 math.PR cs.NAmath.NA

keywords continuous-time random walktime-fractional diffusionweak convergence ratesunbounded coefficientsFeller semigroupinverse subordinatorsensitivity boundskilling conditions

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The pith

Under dominating killing conditions, continuous-time random walk schemes converge at explicit rates to time-fractional diffusions with unbounded coefficients.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how to obtain weak convergence rates for approximating solutions to backward time-fractional diffusion equations driven by diffusions with unbounded coefficients. It does so by approximating the diffusion part with discrete Markov chains and the time change with heavy-tailed random walks, then analyzing the combined scheme using semigroup methods. The key step transfers sensitivity bounds from the ordinary diffusion semigroup to the time-changed fractional version through convolution with the inverse subordinator. This matters for models like geometric Brownian motion in which coefficients grow without bound, as it gives concrete error estimates for numerical schemes without requiring boundedness.

Core claim

Under killing conditions which dominate at least the base-space semigroup growth, weak convergence rates are obtained for the combined continuous-time-random-walk scheme to the time-fractional diffusion, with a logarithmic regime before the discount dominates the stronger smooth-space growth.

What carries the argument

Convolution with the inverse subordinator to transfer uniform sensitivity bounds from the base diffusion semigroup to the fractional setting, supported by Feller semigroup techniques and Kunita stochastic flow sensitivity analysis.

If this is right

Uniform bounds hold for sensitivities of all orders in the diffusion semigroups.
The semigroup satisfies a quasi-contraction property.
The approximation scheme achieves weak convergence to the fractional dynamics under the stated conditions.
The convergence includes a logarithmic regime in an initial phase.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The rate transfer technique may apply directly to other subordinate processes with different stable indices.
The derived bounds could guide step-size selection in simulations of growing-coefficient fractional models.
Similar domination arguments might control errors when extending the scheme to multi-dimensional or jump-driven cases.

Load-bearing premise

The killing conditions must dominate the base-space semigroup growth for the sensitivity bounds to transfer successfully through the convolution with the inverse subordinator.

What would settle it

A specific diffusion with unbounded coefficients where killing fails to dominate semigroup growth, yet the predicted weak convergence rates to the fractional equation still hold.

read the original abstract

We investigate uniform weak convergence rates for probabilistic numerical methods applied to backward time-fractional diffusion equations whose dynamics are driven by diffusions with possibly unbounded coefficients, such as the Geometric Brownian Motion. The fractional structure is represented through a random time-change by the inverse of a stable subordinator. To approximate the underlying fractional dynamics, we combine discrete Markov chain schemes for the diffusion component with heavy-tailed random walk approximations of the time change. Our analysis builds on Feller semigroup techniques and a high-order sensitivity framework for diffusion semigroups based on the Kunita stochastic flows and tensor fields. We derive uniform bounds for all orders of sensitivities, establish a quasi-contraction property for the associated semigroup, and transfer these estimates to the fractional setting via the convolution representation with the inverse subordinator. As a result, under killing conditions which dominate at least the base-space semigroup growth, we obtain weak convergence rates for the combined continuous-time-random-walk scheme to the time-fractional diffusion, with a logarithmic regime before the discount dominates the stronger smooth-space growth.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

They extend rates to unbounded-coefficient fractional diffusions via killing domination plus Kunita-flow transfer, but the heavy-tail convolution control needs explicit verification.

read the letter

The main point is that this paper gets weak convergence rates for combined CTRW schemes approximating time-fractional diffusions with unbounded coefficients such as geometric Brownian motion. They do it by imposing killing conditions that dominate base-semigroup growth and transferring high-order Kunita sensitivity bounds through the inverse-subordinator convolution.

What works is the derivation of uniform bounds on all orders of sensitivities for the base Feller semigroup, the quasi-contraction property, and the clean transfer step to the fractional setting. The logarithmic regime before the discount takes over is stated clearly, and the approach stays within standard stochastic-flow and semigroup tools without obvious circularity.

The soft spot is the stress-test concern on tail control. The inverse subordinator has heavy tails and infinite mean, so the convolution integral reaches arbitrarily large times. Even when killing dominates the base growth on average, it is not immediate that this prevents residual accumulation in the higher-order tensor bounds across the sampled times. The abstract claims the domination suffices, but without a quantitative rate or explicit tail estimate shown in the argument, uniformity could slip. That part of the proof needs a close look.

This is aimed at people working on probabilistic numerics for fractional PDEs or sensitivity analysis for unbounded diffusions. The extension from bounded to unbounded coefficients under the stated conditions is useful and the thinking is straightforward. It deserves a serious referee to verify the domination step and the resulting rates.

Referee Report

1 major / 0 minor

Summary. The paper claims to derive uniform weak convergence rates for continuous-time random walk (CTRW) approximations to backward time-fractional diffusion equations driven by diffusions with unbounded coefficients (e.g., geometric Brownian motion). The fractional dynamics are represented via time-change by the inverse of a stable subordinator; the scheme combines discrete Markov chain approximations for the diffusion with heavy-tailed random walks for the time change. The analysis relies on Feller semigroup techniques, high-order sensitivity bounds obtained from Kunita stochastic flows and tensor fields, a quasi-contraction property, and transfer of these bounds to the fractional case by convolution against the inverse-subordinator density, under killing conditions that dominate base-semigroup growth, producing rates that include a logarithmic regime before discounting overtakes stronger growth.

Significance. If the uniform all-order sensitivity bounds transfer successfully under the stated killing domination, the result would meaningfully extend convergence-rate theory for probabilistic numerical methods from standard diffusions to time-fractional equations with unbounded coefficients. This is relevant for applications involving geometric Brownian motion or similar processes in finance and physics, where fractional time derivatives appear.

major comments (1)

[Abstract] Abstract (final paragraph): The transfer of uniform (all-order) sensitivity bounds from the base Feller semigroup to the fractional setting via convolution with the inverse-subordinator density is asserted to hold once killing dominates base-semigroup growth. However, the inverse stable subordinator possesses heavy tails with infinite mean; the convolution integral therefore samples the base semigroup at arbitrarily large times. Marginal domination may permit residual growth to accumulate across tensor-field orders, undermining uniformity of the bounds before the discount dominates. A quantitative domination rate or explicit tail estimate controlling the convolution is needed to close this step.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. The single major comment raises a valid point about the need for explicit control in the convolution step. We respond below and indicate the planned revision.

read point-by-point responses

Referee: [Abstract] Abstract (final paragraph): The transfer of uniform (all-order) sensitivity bounds from the base Feller semigroup to the fractional setting via convolution with the inverse-subordinator density is asserted to hold once killing dominates base-semigroup growth. However, the inverse stable subordinator possesses heavy tails with infinite mean; the convolution integral therefore samples the base semigroup at arbitrarily large times. Marginal domination may permit residual growth to accumulate across tensor-field orders, undermining uniformity of the bounds before the discount dominates. A quantitative domination rate or explicit tail estimate controlling the convolution is needed to close this step.

Authors: We agree that the current formulation would benefit from an explicit tail estimate to confirm uniformity across all orders. The manuscript assumes killing dominates base-semigroup growth and invokes the quasi-contraction property, but does not supply a quantitative convolution bound. In the revision we will insert a lemma providing an explicit tail estimate for the inverse-subordinator density together with a direct verification that the stated domination rate controls the integrated tensor-field terms uniformly in order. This material will be added to the section on transfer of bounds to the fractional setting. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via standard semigroup techniques

full rationale

The paper derives uniform sensitivity bounds from Kunita flows on the base diffusion, establishes quasi-contraction, and transfers via inverse-subordinator convolution under explicit killing-domination assumptions. No quoted step reduces a claimed rate or bound to a fitted input, self-definition, or load-bearing self-citation chain; the argument remains independent of its target convergence rates. The provided abstract and context exhibit no patterns matching self-definitional, fitted-prediction, or ansatz-smuggling reductions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard semigroup theory and stochastic flow estimates whose applicability to unbounded coefficients is taken as given.

axioms (2)

domain assumption Feller semigroup techniques and Kunita stochastic flows yield uniform bounds of all orders for the diffusion semigroup even with unbounded coefficients
Invoked to obtain the sensitivity estimates transferred to the fractional case.
domain assumption Convolution representation with the inverse stable subordinator preserves the uniform bounds under the killing condition
Used to move estimates from the base diffusion to the time-fractional setting.

pith-pipeline@v0.9.1-grok · 5722 in / 1215 out tokens · 21159 ms · 2026-06-28T20:54:15.772339+00:00 · methodology

discussion (0)

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Works this paper leans on

12 extracted references · 10 canonical work pages

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De Gruyter Studies in Mathematics, vol

Kolokoltsov, V.N.: Markov Processes, Semigroups, and Generators. De Gruyter Studies in Mathematics, vol. 38. Walter de Gruyter, Berlin, Boston (2011). https://doi.org/10.1515/9783110250114

work page doi:10.1515/9783110250114 2011

[2] [2]

Cambridge Tracts in Mathematics, vol

Kolokoltsov, V.N.: Nonlinear Markov Processes and Kinetic Equations. Cambridge Tracts in Mathematics, vol. 182. Cambridge University Press, Cambridge (2010). https://doi.org/10.1017/CBO9780511760303

work page doi:10.1017/cbo9780511760303 2010

[3] [3]

Fractal and Fractional7(4), 335 (2023) https: //doi.org/10.3390/fractalfract7040335

Kolokoltsov, V.N.: The rates of convergence for functional limit theorems with stable subordinators and for CTRW approximations to fractional evolutions. Fractal and Fractional7(4), 335 (2023) https: //doi.org/10.3390/fractalfract7040335

work page doi:10.3390/fractalfract7040335 2023

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Montroll, E.W., Weiss, G.H.: Random walks on lattices. II. Journal of Mathematical Physics6(2), 167–181 (1965) https://doi.org/10.1063/1.1704269

work page doi:10.1063/1.1704269 1965

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Theory of Probability and Its Applications53(4), 594–609 (2009) https://doi.org/10.1137/ S0040585X97983857

Kolokoltsov, V.N.: Generalized continuous-time random walks, subordination by hitting times, and fractional dynamics. Theory of Probability and Its Applications53(4), 594–609 (2009) https://doi.org/10.1137/ S0040585X97983857

2009

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In: Hennequin, P.L

Kunita, H.: Stochastic differential equations and stochastic flows of diffeomorphisms. In: Hennequin, P.L. (ed.) Lecture Notes in Mathematics. Lecture Notes in Mathematics, vol. 1097, pp. 143–303. Springer, Berlin, Heidelberg (1984). https://doi.org/10.1007/BFb0099433

work page doi:10.1007/bfb0099433 1984

[7] [7]

Cambridge Studies in Ad- vanced Mathematics, vol

Kunita, H.: Stochastic Flows and Stochastic Differential Equations. Cambridge Studies in Ad- vanced Mathematics, vol. 24. Cambridge University Press, Cambridge (1990). https://doi.org/10.1017/ CBO9780511665660

1990

[8] [8]

Probability Theory and Stochastic Modelling, vol

Kunita, H.: Stochastic Flows and Jump-Diffusions. Probability Theory and Stochastic Modelling, vol. 92. Springer, Singapore (2019). https://doi.org/10.1007/978-981-13-3801-4

work page doi:10.1007/978-981-13-3801-4 2019

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Journal of Mathematical Analysis and Applications 534(1), 128005 (2024) https://doi.org/10.1016/j.jmaa.2023.128005

Licht, M.W.: Higher-order chain rules for tensor fields, generalized Bell polynomials, and estimates in Orlicz–Sobolev–Slobodeckij and total variation spaces. Journal of Mathematical Analysis and Applications 534(1), 128005 (2024) https://doi.org/10.1016/j.jmaa.2023.128005

work page doi:10.1016/j.jmaa.2023.128005 2024

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Springer, Dordrecht (2012)

Comtet, L.: Advanced Combinatorics: The Art of Finite and Infinite Expansions. Springer, Dordrecht (2012). https://doi.org/10.1007/978-94-010-2196-8

work page doi:10.1007/978-94-010-2196-8 2012

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Journal of Multivariate Analysis97(10), 2131–2140 (2006) https://doi.org/10.1016/j.jmva.2006.03.001

Savits, T.H.: Some statistical applications of Faà di Bruno. Journal of Multivariate Analysis97(10), 2131–2140 (2006) https://doi.org/10.1016/j.jmva.2006.03.001

work page doi:10.1016/j.jmva.2006.03.001 2006

[12] [12]

Transactions of the American Mathematical Society348(2), 503–520 (1996) https://doi.org/10.1090/S0002-9947-96-01501-2 31

Constantine, G.M., Savits, T.H.: A multivariate Faà di Bruno formula with applications. Transactions of the American Mathematical Society348(2), 503–520 (1996) https://doi.org/10.1090/S0002-9947-96-01501-2 31

work page doi:10.1090/s0002-9947-96-01501-2 1996