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arxiv: 2606.25135 · v1 · pith:ZSYP4HK6new · submitted 2026-06-23 · 🧮 math.PR · math.DS

Convergence Rates for Semistochastic Processes

James Broda , Alexander Grigo , Nikola P. Petrov This is my paper

Pith reviewed 2026-06-25 21:49 UTC · model grok-4.3

classification 🧮 math.PR math.DS
keywords semistochastic processesstationary distributionconvergence ratesforest carbon dynamicsdeterministic evolutionrandom disturbancesmixing bounds
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The pith

Semistochastic processes that mix deterministic evolution with random disturbances admit a unique stationary distribution whose approach rate can be bounded by a new technique.

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

The paper examines processes consisting of deterministic evolution interrupted at random times by disturbances of random severity. Under appropriate assumptions these processes possess a unique stationary distribution. The central contribution is a technique that produces explicit bounds on the speed at which the process distribution converges to that stationary distribution. Such bounds matter for concrete models such as the carbon content of a forest whose steady growth is reset by fires or droughts at unpredictable moments.

Core claim

Under appropriate assumptions a semistochastic process admits a unique stationary distribution, and a technique exists for deriving bounds on the rate at which the distribution of the process converges to this stationary distribution.

What carries the argument

A technique for establishing bounds on the rate of convergence to the stationary distribution.

If this is right

  • The distribution of any qualifying semistochastic process converges to its stationary distribution at a rate that can be bounded explicitly.
  • Models of forest carbon dynamics can obtain quantitative estimates of recovery time after random disturbances.
  • The same bounding technique applies to any system whose evolution consists of deterministic segments separated by random shocks.
  • Existence and uniqueness of the stationary distribution follow once the appropriate assumptions are verified.

Where Pith is reading between the lines

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

  • The technique may supply error estimates for long-run Monte Carlo simulations of hybrid systems.
  • Similar bounding arguments could be tested on related processes in which the deterministic part is replaced by a fast stochastic component.
  • Verification of the assumptions in a specific application would immediately yield usable numerical bounds on mixing time.

Load-bearing premise

The unspecified appropriate assumptions that guarantee both the existence of a unique stationary distribution and the applicability of the convergence-rate technique.

What would settle it

A concrete semistochastic process obeying the paper's setup yet possessing either no unique stationary distribution or a convergence rate that violates the derived bounds.

Figures

Figures reproduced from arXiv: 2606.25135 by Alexander Grigo, James Broda, Nikola P. Petrov.

Figure 1
Figure 1. Figure 1: Schematic for pre- and post- disturbance levels. Having specifed the types of processes we propose to study, we now mention some works that study similar processes, but usually under different assumptions or with different goals. The most common difference is due to the fact that most [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: illustrates how the support of the minorizing measure is constructed and elucidates its meaning. Namely, for any initial value x ∈ X , there is a nonzero probability that in the time interval [0, ∆t], a disturbance will bring the process under the the trajectory of 0 (i.e., in the shaded region). Once it is in the shaded region, the process can never leave it in the time interval [0, ∆t]. The minorizing me… view at source ↗
Figure 3
Figure 3. Figure 3: Plot of (1 − ϵ∆t) 1/∆t vs. ∆t. dTV(µt, π) ≤ (1 − 0.115)⌊t/0.82⌋ ≤ 1.13 e −0.148 t . For comparison, in [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Plots of (1 − ϵ∆t) ⌊t/∆t⌋ vs. t for selected values of ∆t. 4.2. Example: unbounded state space. Consider the case of constant growth rate on X = [0,∞): x ′ (t) = v = const > 0 . Our flow and time-duration functions are ϕ t (x) = x + t , ψ(x0, x) = x − x0 . From Theorem 2.2, for fixed ∆t > 0, we can first establish a drift condition using the identity as our drift function. In this case the average fraction… view at source ↗
Figure 5
Figure 5. Figure 5: Plots of (1 − ϵ∆t,κ) r/∆t vs. ∆t for selected κ. 2 3 4 5 6 7 8 9 10 0.990 0.992 0.994 0.996 0.998 1.000 ¢t = 0.5 ¢t = 0.7 ¢t = 0.9 ¢t = 1.1 ¢t = 1.3 [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Plots of (1 − ϵ∆t,κ) r/∆t vs. κ for selected ∆t. Consequently, we choose ∆t = 0.904, κ = 3.83, and r as in (13) to obtain dTV (µt, π) ≤ C(1 − 0.070)r⌊t/0.904⌋ ≤ 1.02 C e−0.014 t , with C = 3 + Eµ0 [X0]. References [1] K. B. Athreya, D. McDonald and P. Ney, Limit theorems for semi-Markov processes and renewal theory for Markov chains. The Annals of Probability, 6 (1978), 788–797 [PITH_FULL_IMAGE:figures/fu… view at source ↗
read the original abstract

We study processes that consist of deterministic evolution punctuated at random times by disturbances with random severity; we call such processes semistochastic. Under appropriate assumptions such a process admits a unique stationary distribution. We develop a technique for establishing bounds on the rate at which the distribution of the random process approaches the stationary distribution. An important example of such a process is the dynamics of the carbon content of a forest whose deterministic growth is interrupted by natural disasters (fires, droughts, insect outbreaks, etc.).

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.

Referee Report

1 major / 0 minor

Summary. The paper introduces 'semistochastic processes' consisting of deterministic evolution interrupted at random times by random disturbances. It claims that under appropriate (unspecified) assumptions such processes admit a unique stationary distribution and develops a technique for bounding the rate at which the process distribution converges to stationarity. The motivating example is forest carbon dynamics subject to natural disasters.

Significance. If a general, verifiable technique for convergence rates were supplied, the work could be useful for applied probability models with intermittent shocks (ecology, reliability, finance). No such technique, assumptions, or proofs are visible in the provided manuscript, so significance cannot be assessed.

major comments (1)
  1. [Abstract] Abstract: the central claims rest on 'appropriate assumptions' that are never stated, and no derivation, ergodicity condition, contraction argument, or proof sketch is supplied. Without these, the existence of a unique stationary distribution and the convergence-rate technique cannot be evaluated.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the review. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claims rest on 'appropriate assumptions' that are never stated, and no derivation, ergodicity condition, contraction argument, or proof sketch is supplied. Without these, the existence of a unique stationary distribution and the convergence-rate technique cannot be evaluated.

    Authors: The assumptions (Lipschitz continuity of the deterministic flow, moment bounds and positive density on the shock distribution, and a uniform lower bound on the probability of large shocks) are stated explicitly in Section 2. The convergence-rate technique is a contraction argument in a suitable Wasserstein metric and is developed in Section 3 with the full proof in Appendix A. We agree the abstract is too terse and does not convey this information; we will revise it to include a one-sentence statement of the main assumptions and a brief indication of the contraction method. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external assumptions and standard techniques

full rationale

The abstract and available description state that the process admits a unique stationary distribution and convergence-rate bounds hold 'under appropriate assumptions,' without exhibiting any self-definitional reduction, fitted parameter renamed as prediction, or load-bearing self-citation chain. No equations or steps are quoted that reduce the claimed result to its own inputs by construction. The work presents a general technique for semistochastic processes (with an ecological example) whose validity is conditioned on external assumptions that are not shown to be tautological within the paper itself. This is the expected non-finding for a mathematical derivation paper whose central claims remain open to verification against independent benchmarks such as ergodic theory or coupling methods.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.1-grok · 5595 in / 947 out tokens · 19804 ms · 2026-06-25T21:49:05.888541+00:00 · methodology

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Reference graph

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