Sampling with time-changed Markov processes

Andrea Bertazzi; Giorgos Vasdekis

REVIEW 3 major objections 6 minor 15 references

Sampling with time-changed Markov processes

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Random time-changes turn Markov processes with a law of large numbers into samplers for any reweighted target, with the speed function controlling convergence.

desk verdict Solid unifying framework for time-changed MCMC; the core invariance/LLN theorem is clean, but the stronger ergodicity claims rest on unverified technical assumptions. read the letter →

arxiv 2501.15155 v2 pith:YGETJM3M submitted 2025-01-25 stat.CO math.PR

Andrea Bertazzi , Giorgos Vasdekis This is my paper · ORCID

classification stat.COmath.PR MSC 60J2560J6065C0565C40

keywords time-changedMarkovprocesseschainMonteCarlopiecewisedeterministicZig-ZagprocessgeometricergodicityfunctionalcentrallimittheoremimportancesamplingLangevindiffusion

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that sampling from a target distribution $\mu$ can be reduced to choosing a speed function $s$ for a well-understood Markov process $Y$. The time-changed process $X_t=Y_{r(t)}$, with $r(t)=\int_0^t s(X_u)\,du$, follows the same paths as $Y$ but runs faster where $s$ is large. Theorem 2.3 shows that if $Y$ satisfies a strong law of large numbers for the reweighted distribution $\tilde\mu(dx)\propto s(x)\mu(dx)$, then $X$ has $\mu$ as its unique stationary distribution and satisfies a strong law of large numbers for every $f\in L^1(\mu)$. The authors use this to construct time-changed versions of Langevin diffusions and piecewise deterministic Markov processes, recovering the Speed-Up Zig-Zag process and producing new samplers, and they give conditions under which the time-changed process is geometrically or uniformly ergodic and obeys a functional central limit theorem. A reader should care because it turns a single base process into a tunable sampler for a family of targets, with the speed function as the dial controlling convergence.

What carries the argument

The load-bearing object is the random time-change $r(t)=\int_0^t s(X_u)\,du$ and its inverse relation $Y_t=X_{r^{-1}(t)}$. The paper's key identity is the generator scaling $L=s\tilde L$ on a set $A$ of functions in both generator domains; this is what lets ergodicity and CLT results flow from $Y$ to $X$. The invariance proof itself rests on the change of variables $\frac1T\int_0^T f(X_t)\,dt=\frac{r(T)}T\frac1{r(T)}\int_0^{r(T)} \frac{f(Y_u)}{s(Y_u)}\,du$, where $r(T)/T$ converges almost surely to $\mu(s)$ by the base LLN. For the drift arguments, a Lyapunov function $V$ with $\tilde L V\le -WV+\gamma 1_C$ is turned into $LV\le -sWV+s\gamma 1_C$, so the speed function appears directly in the exponential rate.

What would settle it

Simulate the time-changed Zig-Zag process on the two-component Gaussian mixture of Section 6.1 with speed $s(x)=\mu(x)^{-0.9}$ and measure the long-run fraction of time spent in each component over increasing horizons $T$. Theorem 2.3 predicts the fractions converge to the mixture weights; if the discrepancy does not shrink as $T$ grows, the LLN/invariance claim fails. A sharper test of the ergodicity results: compute $LV$ for the explicit $V$ in equation (22) for a non-quadratic target; the proof of Proposition 4.9 needs $LV=s\tilde L V$ outside a compact set, so any state where this identity fails would refute the claimed geometric ergodicity for that speed.

Watch

Extended reading notes

Core claim

The central discovery is that the time-change is, in essence, a pathwise importance reweighting: speeding up time where $s$ is large is the same as reweighting the base process's path by $1/s$. Because of this, the time-changed process inherits the base process's law of large numbers provided the base process targets the $s$-biased distribution, and its ergodic averages converge to $\mu$. The authors further show that under drift conditions on $Y$ the generator identity $L=s\tilde L$ lets the speed function repair slow convergence: requiring $s(x)\ge\zeta/W(x)$ outside a compact set converts geometric ergodicity, and stronger lower bounds on $s$ give uniform ergodicity even for base processes that only converge polynomially. They also obtain a functional central limit theorem whose asymptotic variance is, up to the constant $\mu(s)$, the base process's variance for the observable $g/s$.

Load-bearing premise

The load-bearing premise is that the drift certificate for the base process—a function $V$ used to prove its ergodicity—lies in the class where the time-changed generator is exactly $s$ times the base generator, and that the time-changed process is aperiodic; the paper assumes both and notes that aperiodicity is only conjectured in general.

Editorial extensions

If this is right

Every continuous-time MCMC algorithm whose base process satisfies an LLN can be re-purposed as a sampler for any absolutely continuous reweighting of its target by choosing a lower-bounded speed function.
Known algorithms such as the Speed-Up Zig-Zag are time-changes of a standard Zig-Zag, so their LLN and ergodicity properties transfer from the base process without ad-hoc proofs.
For a PDMP, the time-changed process is again a PDMP with vector field and rate multiplied by $s$ and unchanged jump kernel; this yields new $\mu$-stationary BPS and RHMC variants.
The FCLT relation $\gamma_g^2=\mu(s)\tilde\gamma_{\tilde g}^2$ makes variance comparisons between samplers a question of choosing $s$, mirroring self-normalised importance sampling.
A jump-process implementation (Algorithm 1) with rate $s$ and a kernel leaving $\tilde\mu$ invariant can be simulated with only exponential clocks and gives asymptotically unbiased estimates.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Going beyond the paper: the self-normalised importance-sampling analogy suggests that the optimal speed for estimating a fixed observable $g$ should be $s(x)\propto|g(x)-\mu(g)|$, an analogue of the optimal IS proposal that the paper mentions but does not prove.
Going beyond the paper: the Section 5 observation that Jacobian determinants act as speed functions points to a systematic source of new speed functions—any diffeomorphism from a compact domain to $\mathbb{R}^d$, such as the stereographic projection, yields a speed function and a corresponding time-changed sampler.
Going beyond the paper: because the speed function can be any positive lower-bounded function, one could tune it adaptively during a run, alternating exploration of modes with sampling of tails; the paper lists this as future work, not a demonstrated method.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The thing you should know: this paper is a genuine unification, not just a repackaging. Theorem 2.3, which says that a time-changed process targets the reweighted measure whenever the base process has an LLN under the tilted measure, is clean and useful. It subsumes the Speed-Up Zig-Zag, time-changed Langevin diffusions, and importance sampling on paths, and it makes the connection to space transformations explicit. The paper is well written, gives a working algorithm for jump processes, and ships code. That part I buy.

The softer spots are where the advertised rate-control results live. Section 4's geometric/uniform ergodicity and FCLT theorems all require V ∈ A (Definition 4.2) and aperiodicity (Assumption 4.3). The paper is honest about these being assumptions rather than theorems: it says "we shall then assume that V ∈ A" and "we conjecture that Assumption 4.3 holds." But the consequence is that Proposition 4.9 and Proposition 4.11, the actual payoff for the Zig-Zag example, inherit those unverified membership conditions. Proposition B.1 only establishes A-membership for bounded functions with bounded s L̃ f, so it does not cover the unbounded exponential Lyapunov functions used in the applications. That's a real gap, though not a fatal one: the invariance and LLN result does not depend on it, and the ergodicity/FCLT conclusions would become theorems if the membership could be checked for these examples.

The numerical sections are illustrative rather than exhaustive. There are no error bars on the trace plots, and the speed exponent a in s(x)=exp(aU(x)) is tuned without a clear adaptive or data-driven rule. Given the theory gap, I'd treat the numerics as proof-of-concept, not as strong empirical evidence.

Overall: this paper deserves a serious referee. It advances the field by giving a common language to algorithms that previously seemed separate, and the central theorem is solid. The referee should push for either verifying V ∈ A and aperiodicity for the Zig-Zag and BPS/RHMC examples, or reformulating the theorems so those conditions become explicit hypotheses whose satisfaction is a concrete open problem. I'd cite it for the framework and the invariance theorem; I'd be more careful about citing the ergodicity claims until the assumptions are sorted out.

Referee Report

3 major / 6 minor

Summary. The paper proposes a general construction for MCMC samplers based on random time changes of a Markov process. Given a base process Y targeting ~μ ∝ s μ with a user-chosen 'speed function' s, the time-changed process X_t = Y_{r(t)} with clock r(t) = ∫_0^t s(X_u) du (Eq. (1)) is shown in Theorem 2.3 to have μ as its unique stationary distribution and to satisfy a strong LLN for every f ∈ L^1(μ), provided Y satisfies an LLN with respect to ~μ. The construction is applied to PDMPs (Zig-Zag, Bouncy Particle Sampler, Randomized HMC), to overdamped and underdamped Langevin diffusions, and to discrete-state jump processes; Proposition 3.4 recovers the speed-up ZZP of Vasdekis and Roberts [2023] and Example 3.8 recovers the time-changed Langevin dynamics of Roberts and Stramer [2002]. Section 4 develops criteria for geometric and uniform ergodicity (Theorems 4.4, 4.5, 4.13) and a functional central limit theorem with an importance-sampling-like variance formula (Theorem 4.6, Eqs. (14)–(15)), all under drift, petite-set, domain-membership (V ∈ A), and aperiodicity conditions. Section 5 relates time changes to space transformations, and Section 6 reports toy simulations for a multimodal Gaussian mixture and a heavy-tailed rare-event problem.

Significance. The construction is conceptually appealing and unifies several known algorithms, connecting time changes with pathwise importance sampling and with Jacobian-based space transformations; the relationship to prior work is drawn carefully, and credit is explicitly given where algorithms are re-discovered. Theorem 2.3 is the clean centerpiece: its assumptions are nearly minimal, and the proof is a short change-of-variables argument that also justifies the reweighted estimators of Section 2.2. The paper is unusually explicit about what is assumed versus proved, flags the V ∈ A condition and the aperiodicity conjecture, and ships reproducible code. If the Section 4 hypotheses are verified for the examples, the rate-control message would be a genuine contribution to continuous-time MCMC; the open questions identified below concern exactly that verification.

major comments (3)

[§4.1 (Definition 4.2), §4.3 (Assumption 4.8), Appendix D.1] The ergodicity statements for the running example hinge on the membership V ∈ A of Definition 4.2, yet this membership is never established; the paper's own Section 4.1 states 'we shall then assume that V ∈ A.' Proposition B.1 only puts bounded functions f with bounded s L~ f into A, whereas the ZZP Lyapunov function in Eq. (22) is unbounded (exponential in α U~ plus a sum of φ terms), and the drift condition (12) forces L~ V to be unbounded as well. The proof of Theorem 4.4 (Appendix D.1) begins with the identity L V = s L~ V, which for the ZZP application is exactly the content of Assumption 4.8 that Proposition 4.9 assumes without verification. Since the advertised rate-control contribution rests on Theorems 4.4–4.6, the paper should either prove V ∈ A under the conditions of Assumption 4.8 or give checkable sufficient conditions, for instance through the PDMP martingale problem, and verify them for the time-changed ZZP.
[§4.2 (Theorem 4.6), §4.3 (Proposition 4.11), Appendix E] The FCLT statement and the variance identity (15) depend on two conditions that Proposition 4.11 does not justify. First, the proof in Appendix E uses 'ĝ ∈ A' to identify the solution of the Poisson equation for the time-changed generator with that of the base generator; for the time-changed ZZP one must check that the solution of g = −L ĝ lies in A, which is not done. Second, the global bound s(x)W(x)V(x) ≥ 1 for all x ∈ E is asserted in Theorem 4.6 but not verified in the ZZP setting with s(x) = exp(βU(x)) and the exponential V of Eq. (22). Until these hypotheses are verified, Proposition 4.11 is conditional on the same unresolved domain-membership issue as Proposition 4.9.
[Assumption 4.3, §3.1.2, Theorems 4.4–4.6] Assumption 4.3 (aperiodicity of the time-changed process) is assumed throughout Section 4, and the paper itself calls it a conjecture in the general case ('We conjecture that Assumption 4.3 holds under aperiodicity of Y and some regularity conditions'), with a proof cited only for the time-changed ZZP. Theorems 4.5 and 4.6 are advertised as applicable to the other PDMP examples of Section 3.1.2 (BPS, RHMC), but no aperiodicity argument is supplied for those processes, so those applications do not yet satisfy the stated hypotheses. The paper should either restrict the general claims, prove Assumption 4.3 under the stated regularity conditions, or mark explicitly which examples satisfy all hypotheses.

minor comments (6)

[§4.1, Eq. (11)] In Eq. (11) the total-variation distance is written with the base process's stationary distribution ~μ; since X targets μ, this should read μ, and the same slip appears in the surrounding text.
[§2.2, Proposition 4.14] There are a few typos: 'streightforward' and 'Howerver' in Section 2.2, and 'FLCT' in Proposition 4.14, which should read 'FCLT'.
[References] Roberts and Stramer [2002a] and [2002b] appear to be the same article listed twice in the bibliography; the citations in the Introduction and in Example 3.8 should point to a single entry.
[§6, Figures 1, 2, 4, 5] In the numerical experiments the speed exponent a is tuned by hand (a = 0.9 in Figure 1 and in Figure 4b, a = 0.3 in Figure 5b), and the illustrative traces are shown without error bars or a sensitivity discussion; a brief explanation of the choice of a and of the stability of the conclusions in a would make the empirical support more convincing, although Figure 5 does quantify MSE over 5000 runs.
[Appendix B.3, Lemma B.3] The notation s_C is used both for the supremum of s over the set D and in the definition of t1 = t0 / s_C, which collides with the lower bound s of Assumption 2.1; the two objects should be given distinct names.
[Abstract] The abstract contains two consecutive sentences announcing numerical simulations ('Throughout the paper we give various visualisations...' and 'Finally, we provide numerical simulations...'); one of the two should be removed to avoid redundancy.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 2.3 is a self-contained change-of-measure argument; the Section 4 conditions (V in A, aperiodicity) are stated assumptions and technical gaps, not circular inputs.

full rationale

The paper's central invariance and LLN result, Theorem 2.3, is not circular: it assumes only that the base process Y satisfies an SLLN with respect to the tilted measure \tilde\mu \propto s\mu, and derives, via the change of variables in Appendix A (Eqs. (19)-(21)), that the time-changed process X has stationary distribution \mu and satisfies an SLLN for every f in L^1(\mu). The target \mu is not used to define the LLN of Y; rather, \tilde\mu is constructed from the user-chosen speed function s and \mu, and the proof is an explicit calculation. The ergodicity and FCLT results in Section 4 are conditional on technical conditions such as V \in A (Definition 4.2) and aperiodicity (Assumption 4.3). These are stated assumptions rather than derived conclusions, and the paper explicitly flags that V \in A is assumed ('Throughout this section, we shall then assume that V \in A') and that aperiodicity is conjectured in general ('We conjecture that Assumption 4.3 holds under aperiodicity of Y'). Such unproved technical hypotheses are correctness risks, not circularity: they are not the same as the conclusions being derived. The citation to Vasdekis and Roberts [2023] for aperiodicity of the time-changed Zig-Zag process is a self-citation, but it invokes an independent published result used to verify one assumption, not a premise that restates the theorem being proved; the same holds for the citation of Bierkens et al. [2019b] for the base process drift and LLN. Numerical tuning of parameters such as a in s(x)=exp(aU(x)) affects only the experiments, not the theoretical statements. No fitted quantity is later renamed as a prediction, and no known algorithm is merely relabelled as a new derivation: the connection to space transformations and importance sampling is explicitly acknowledged. Therefore the derivation chain is self-contained up to its stated assumptions, and no circular step is exhibited.

Assumptions & free parameters 1 free parameters · 7 assumptions · 0 invented entities

The central theorems are conditional on user-chosen speed functions and standard LLN/drift assumptions on the base process. No new physical entities are introduced. The numerical sections add tuning hyperparameters that do not enter the theoretical claims.

free parameters (1)

speed exponent a in s(x)=exp(aU(x)) = a=0.3 chosen by smallest MSE in Sec 6.2; alpha=0.9 in Fig 1
User-chosen tuning parameter for numerical demonstrations; not part of the theoretical claim.

assumptions (7)

domain assumption Assumption 2.1: s continuous, mu(s)<infinity, s bounded below
Ensures time-change is well-defined and non-explosive; used in all main theorems.
domain assumption Assumption 2.2: base process Y satisfies a strong LLN for tilde mu proportional to s mu
The invariance result Theorem 2.3 directly rests on this; verified for ZZP, BPS, Langevin diffusions via cited results.
domain assumption Assumption 4.1: petite-set condition and drift inequality for Y with Lyapunov function V
Used to transfer geometric and uniform ergodicity to X in Theorems 4.4 and 4.5.
ad hoc to paper V in A and bar V in A (Definition 4.2)
The generator of X is s times the generator of Y only on A; proofs assume the Lyapunov function lies in A, with partial justification for bounded functions only.
domain assumption Assumption 4.3: X is aperiodic
Needed for convergence to the stationary distribution; the authors conjecture it follows from mild conditions but do not prove it in general.
domain assumption Assumptions 3.3 and 4.8: regularity and tail-growth conditions on U and s for the Zig-Zag process
Imported from Bierkens et al. 2019 for LLN and drift of the base ZZP.
standard math Monmarche 2016 Eyring-Kramers estimate for the standard ZZP
Used in Proposition F.1 to bound hitting times for the time-changed ZZP; external benchmark result.

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Pith. "Pith review of Sampling with time-changed Markov processes." pith.science (2026). https://pith.science/paper/YGETJM3M

@misc{pith2026250115155,
  author       = {Pith},
  title        = {Pith review of: Sampling with time-changed Markov processes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YGETJM3M}},
  note         = {Machine review of arXiv:2501.15155}
}

read the original abstract

We study time-changed Markov processes to speed up the convergence of Markov chain Monte Carlo (MCMC) algorithms. The time-changed process is defined by adjusting the speed of time of a base process via a user-chosen, state-dependent function. We explore the properties of such transformations and apply this idea to several Markov processes from the MCMC literature, such as Langevin diffusions and piecewise deterministic Markov processes, obtaining novel modifications of classical algorithms and also re-discovering known MCMC algorithms. We prove theoretical properties of the time-changed process under suitable conditions on the base process, focusing on connecting the stationary distributions and qualitative convergence properties such as geometric and uniform ergodicity, as well as a functional central limit theorem. We also provide a comparison with the framework of space transformations, clarifying the similarities between the approaches. Throughout the paper we give various visualisations and numerical simulations on simple tasks to gain intuition on the method and its performance. Finally, we provide numerical simulations to gain intuition on the method and its performance on benchmark problems. Our results indicate a performance improvement in the context of multimodal distributions and rare event simulation.

Figures

Figures reproduced from arXiv: 2501.15155 by the authors.

**Figure 1.** Simulations for a mixture of two, one-dimensional standard normal random variables with means at 10 and −10, and equal weight. The base process is the standard ZZP. The plots on the left show short simulations, highlighting the different dynamics, while the plots on the right show longer runs of the processes. Proposition 3.4 (Time-changed ZZP). Let s : (x, v) 7→ s(x) be a speed function such that s ∈ C1 (R d ). Sup… view at source ↗

**Figure 2.** Simulations for a two-dimensional, isotropic student distribution with mean at the origin and 5 degrees of freedom. The plot on the right shows the logarithm of the velocity in the colourmap. One-dimensional bimodal target. Consider the case where µ is a one-dimensional bimodal density µ(x) ∝ exp(−U(x)), corresponding to a smooth double well potential U with two minima at x0, x2 ∈ R and a local maximum at x1 ∈ R, wh… view at source ↗

**Figure 3.** Comparison between time-change and space transformation, using the standard ZZP as base process and a two-dimensional, standard normal target distribution. The choices of the speed function and diffeomorphism are as in (18) and (17). The colours and the scale on the right of each plot represent the logarithm of the speed of each process, calculated as distance travelled in a small time unit divided by the length of… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Numerical simulations in the context of Section 6.1. The plots in the second row are obtained as follows. First, we simulate a jump process with initial condition at the origin, jump rate s, and jump kernel given by the Metropolis adjusted ZZP [Bertazzi et al., 2023] w…

**Figure 5.** Figure 5: Numerical simulations in the context of Section 6.2. The plots show estimates of the probability of the event {x ∈ R 2 : |x| > 150} obtained simulating 5000 independent runs of the jump process based on the Metropolis-adjusted ZZP for 106 iterations. intuitive to use …

**Figure 6.** Figure 6: Further numerical simulations for a mixture of normal distributions as in Section 6.1. The process in the left plot has time horizon 3 × 104 and initial condition at the origin. The jump process on the right plot is obtained first discretising the continuous path of th…

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Reference graph

Works this paper leans on

15 extracted references · 11 canonical work pages

[1]

Applying the generator of Y to ¯V we find eL ¯V(z) = 1 (1 + V(z))2 ⟨ ˜Φ(z), ∇V(z)⟩ + ˜λ(z) Z ( ¯V(y) − ¯V(z)) eQ(z, dy)

We shall show that ¯V also satisfies a drift condition of the form (12). Applying the generator of Y to ¯V we find eL ¯V(z) = 1 (1 + V(z))2 ⟨ ˜Φ(z), ∇V(z)⟩ + ˜λ(z) Z ( ¯V(y) − ¯V(z)) eQ(z, dy). For the jump part we obtain ˜λ(z) Z ( ¯V(y) − ¯V(z)) eQ(z, dy) = ˜λ(z) Z V(y) − V(z) (1 + V(y))(1 + V(z)) eQ(z, dy) Considering both the case V( y) ≥ V(z) and V( z...

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Using the law of large numbers we get that P Z ∞ 0 1Xt∈A = ∞ = 1, meaning that X is Harris recurrent in the sense of Meyn and Tweedie [1993a], and therefore it possesses a unique invariant measure ν (see e.g. Az´ ema et al. [1969], Meyn and Tweedie [1993a]). From (21) we get that ν = µ and this concludes the proof. Appendix B. General results for time-cha...

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Let τ := inf{t >0 : Xt = x1}, i.e. the time elapsed before the process finally moves from its initial condition ( X0, V0) = ( x0, −1) to x1. In the next proposition we state our result for speed functions of the form sε(x) = s(x)1/ε, where s is a fixed speed function, satisfying the properties discussed in Section 3.1.1. Proposition F.1. Consider a smooth...

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and when s(x) = exp( aU(x)) for a ∈ (0, 1). In the former case we obtain from [Monmarch´ e, 2016, Theorem 1.1] that E(x0,−1)[τ ] ≤ s 8µε U′′(x0) exp U(x1) − U(x0) ε (1 + o (ε)), SAMPLING WITH TIME-CHANGED MARKOV PROCESSES 29 while in the latter case we obtain E[τ ] ≤ s 8µε (1 − a)U′′(x0) exp (1 − a) U(x1) − U(x0) ε (1 + o (ε)). As ε tends to 0, we see tha...

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Consider a bounded set C ⊃ C

establishes that, when γi(x) are lower and upper bounded, any set of the form C = Cx × Cv where Cx is a compact set and Cv ⊂ {±1}d is (t, b(t), ν)-small for the ZZP for any t ≥ t0, where t0 is a large enough time and b(t) is continuous in t. Consider a bounded set C ⊃ C. Inspecting the proof of [Bertazzi and Bierkens, 2022, Lemma 4.1] one can see that Ass...

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Assumption 4.12(3) gives that the process Y satisfies eLV(z) = eQV(z) − V(z) ≤ −(1 − W(z))V(z) + η1 C, that is a drift condition of the form (12)

Finally, we show the drift condition for Y required by Theorem 4.4 holds. Assumption 4.12(3) gives that the process Y satisfies eLV(z) = eQV(z) − V(z) ≤ −(1 − W(z))V(z) + η1 C, that is a drift condition of the form (12). Therefore, we can apply Theorem 4.4 to obtain that X is geometrically ergodic if there exists β >0 such that s(x) ≥ β/1−W(x) for all x /...

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gives that for all f ∈ C∞ c (E), the generator is given by eLf (y) = ⟨˜b(y), ∇f (y)⟩ + 1 2 dX i,j=1 ˜aij(y)∂i∂jf (y), where ˜a(x) = ˜σ(x)˜σ(x)T . Proposition B.1 gives that the time changed process has generator Lf (y) = ⟨s(y)˜b(y), ∇f (y)⟩ + 1 2 dX i,j=1 s(y)˜aij(y)∂i∂jf (y),...

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Applying the generator of Y to ¯V we find eL ¯V(z) = 1 (1 + V(z))2 ⟨ ˜Φ(z), ∇V(z)⟩ + ˜λ(z) Z ( ¯V(y) − ¯V(z)) eQ(z, dy)

We shall show that ¯V also satisfies a drift condition of the form (12). Applying the generator of Y to ¯V we find eL ¯V(z) = 1 (1 + V(z))2 ⟨ ˜Φ(z), ∇V(z)⟩ + ˜λ(z) Z ( ¯V(y) − ¯V(z)) eQ(z, dy). For the jump part we obtain ˜λ(z) Z ( ¯V(y) − ¯V(z)) eQ(z, dy) = ˜λ(z) Z V(y) − V(z) (1 + V(y))(1 + V(z)) eQ(z, dy) Considering both the case V( y) ≥ V(z) and V( z...

work page 1996

[2] [8]

Az´ ema et al

Using the law of large numbers we get that P Z ∞ 0 1Xt∈A = ∞ = 1, meaning that X is Harris recurrent in the sense of Meyn and Tweedie [1993a], and therefore it possesses a unique invariant measure ν (see e.g. Az´ ema et al. [1969], Meyn and Tweedie [1993a]). From (21) we get that ν = µ and this concludes the proof. Appendix B. General results for time-cha...

work page 1969

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Z r(t2) r(t1) 1 s(Yt) 1 Yt∈Adt # ≥ 1 sC∆t Ex

and MCMC algorithms Roberts and Rosenthal [2004]. A set C ⊂ E is ( t0, η, ν)-small for a process Yt if there exist a constant η >0 and a non-trivial probability measure ν on E, i.e. such that ν(E) > 0, such that for all y ∈ C Py(Yt0 ∈ ·) ≥ ην (·). A related, weaker condition is that of petite set : there exists a probability measure α on (0, ∞) such that ...

work page 2004

[4] [12]

the time elapsed before the process finally moves from its initial condition ( X0, V0) = ( x0, −1) to x1

Let τ := inf{t >0 : Xt = x1}, i.e. the time elapsed before the process finally moves from its initial condition ( X0, V0) = ( x0, −1) to x1. In the next proposition we state our result for speed functions of the form sε(x) = s(x)1/ε, where s is a fixed speed function, satisfying the properties discussed in Section 3.1.1. Proposition F.1. Consider a smooth...

work page 2016

[5] [13]

and when s(x) = exp( aU(x)) for a ∈ (0, 1). In the former case we obtain from [Monmarch´ e, 2016, Theorem 1.1] that E(x0,−1)[τ ] ≤ s 8µε U′′(x0) exp U(x1) − U(x0) ε (1 + o (ε)), SAMPLING WITH TIME-CHANGED MARKOV PROCESSES 29 while in the latter case we obtain E[τ ] ≤ s 8µε (1 − a)U′′(x0) exp (1 − a) U(x1) − U(x0) ε (1 + o (ε)). As ε tends to 0, we see tha...

work page 2016

[6] [14]

Consider a bounded set C ⊃ C

establishes that, when γi(x) are lower and upper bounded, any set of the form C = Cx × Cv where Cx is a compact set and Cv ⊂ {±1}d is (t, b(t), ν)-small for the ZZP for any t ≥ t0, where t0 is a large enough time and b(t) is continuous in t. Consider a bounded set C ⊃ C. Inspecting the proof of [Bertazzi and Bierkens, 2022, Lemma 4.1] one can see that Ass...

work page 2022

[7] [15]

Assumption 4.12(3) gives that the process Y satisfies eLV(z) = eQV(z) − V(z) ≤ −(1 − W(z))V(z) + η1 C, that is a drift condition of the form (12)

Finally, we show the drift condition for Y required by Theorem 4.4 holds. Assumption 4.12(3) gives that the process Y satisfies eLV(z) = eQV(z) − V(z) ≤ −(1 − W(z))V(z) + η1 C, that is a drift condition of the form (12). Therefore, we can apply Theorem 4.4 to obtain that X is geometrically ergodic if there exists β >0 such that s(x) ≥ β/1−W(x) for all x /...

work page 2022

[8] [1995]

Geometric ergodicity of the Bouncy Particle Sampler

Alain Durmus, Arnaud Guillin, and Pierre Monmarch´ e. Geometric ergodicity of the Bouncy Particle Sampler. The Annals of Applied Probability , 30(5):2069–2098, 10

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