Piecewise Diffusion Markov Processes

• Piecewise Diffusion Markov Processes (PDifMPs) are stochastic hybrid systems that blend continuous SDE-driven evolution with discrete jump dynamics.
• They are defined by a characteristic triple (ϕ, λ, Q) that replaces deterministic flows with second-order diffusion operators and integrates jump transitions.
• Numerical schemes like the Thinned Euler-Maruyama method enable accurate simulation and parameter inference, addressing challenges in first-passage and ergodicity analyses.

Piecewise Diffusion Markov Processes (PDifMPs) are stochastic hybrid systems in which a continuous component evolves as a diffusion between random jump times, while a discrete component evolves by event-driven jumps; in recent formulations the state is written either as $x_t=(y_t,v_t)$ on $E=E_1\times \mathbf V$ or as $U_t=(X_t,Z_t)$ on $E=D_1\times D_2$. They are routinely described by a characteristic triple $(\phi,\lambda,\mathcal Q)$, consisting of a stochastic flow between jumps, a jump intensity, and a post-jump transition kernel. In this sense, PDifMPs are the diffusion analogue of piecewise-deterministic Markov processes (PDMPs): the deterministic inter-jump flow of a PDMP is replaced by an SDE-driven stochastic flow (Buckwar et al., 2024, Desmettre et al., 14 Nov 2025).

1. Formal structure and state-space decomposition

A standard PDifMP formulation fixes a filtered probability space carrying a Wiener process and a hybrid state space $E=E_1\times \mathbf V$ or $E=D_1\times D_2$, with the continuous component taking values in $E_1\subset \mathbb R^d$ or $D_1\subseteq \mathbb R^{d_1}$, and the jump component taking values in a finite or countable set $\mathbf V$ or $E=E_1\times \mathbf V$0. The discrete component is right-continuous and piecewise constant; its jump times are denoted $E=E_1\times \mathbf V$1 or $E=E_1\times \mathbf V$2 (Buckwar et al., 2024, Desmettre et al., 14 Nov 2025).

Between jumps, the continuous component follows an SDE with coefficients frozen in the current mode. In one formulation, on $E=E_1\times \mathbf V$3 and given $E=E_1\times \mathbf V$4,

$E=E_1\times \mathbf V$5

In another, on $E=E_1\times \mathbf V$6 with $E=E_1\times \mathbf V$7,

$E=E_1\times \mathbf V$8

The stochastic flow is assumed to satisfy a semigroup property, and the usual global Lipschitz and linear-growth assumptions are used to guarantee a unique strong solution on each inter-jump interval (Buckwar et al., 2024, Desmettre et al., 14 Nov 2025).

Jump times are specified through a state-dependent hazard. One common survival function is

$E=E_1\times \mathbf V$9

and the next jump time is obtained from its generalized inverse. The post-jump state is sampled from a kernel $U_t=(X_t,Z_t)$0. In the numerical approximation paper, $U_t=(X_t,Z_t)$1 is restricted so that the continuous state does not jump and only the discrete mode changes. In the later first-passage paper, a generalized PDifMP also allows discontinuities in the continuous component through a jump-size function $U_t=(X_t,Z_t)$2, so that both the mode and the continuous coordinate may change at event times (Buckwar et al., 2024, Desmettre et al., 10 Jul 2025).

2. Local characteristics, generators, and the PDMP comparison

The local characteristic triple $U_t=(X_t,Z_t)$3 plays the same structural role as in Davis-type PDMP theory, but the continuous part is second-order rather than first-order. A representative extended generator, given for the one-dimensional continuous variable in the numerical approximation work, is

$U_t=(X_t,Z_t)$4

This decomposition makes the PDifMP structure explicit: a diffusion operator inside each mode, plus a jump operator coupling the modes (Buckwar et al., 2024).

The same paper develops a compensated-random-measure formulation and a generalized Itô formula. Writing the jump part as a random measure $U_t=(X_t,Z_t)$5 with compensator $U_t=(X_t,Z_t)$6, and $U_t=(X_t,Z_t)$7, it obtains a martingale representation

$U_t=(X_t,Z_t)$8

with $U_t=(X_t,Z_t)$9, where $E=D_1\times D_2$0 is the diffusion operator and $E=D_1\times D_2$1 the jump operator. This is the analytic basis for weak convergence and martingale-problem arguments in the PDifMP setting (Buckwar et al., 2024).

The comparison with PDMPs is direct. In a PDMP, the transport term is $E=D_1\times D_2$2; in a PDifMP, it is replaced by a second-order operator of the form

$E=D_1\times D_2$3

This replacement is not merely cosmetic. It removes explicit deterministic flow maps in general, introduces pathwise noise into the intensity evaluation, and makes boundary and interface questions more delicate. PDMP semigroup theory nevertheless provides a clear template: one PDMP analysis proves Feller property and that $E=D_1\times D_2$4 is a core for the generator under standard assumptions, and explicitly notes that the same semigroup/generator/core machinery would also be central for a piecewise diffusion theory (Holderrieth, 2019).

3. Numerical approximation and path simulation

When explicit inter-jump diffusion flows are unavailable, the main direct numerical framework is the Thinned Euler-Maruyama (TEM) scheme. It combines Poisson thinning for the jump mechanism with Euler-Maruyama for the inter-jump diffusion. A dominating homogeneous Poisson process with intensity $E=D_1\times D_2$5 is used, under the assumption

$E=D_1\times D_2$6

and each proposal time is accepted as a true jump according to the intensity evaluated along the numerically propagated diffusion path (Buckwar et al., 2024).

On a local grid $E=D_1\times D_2$7, the Euler-Maruyama update is

$E=D_1\times D_2$8

with continuous interpolation

$E=D_1\times D_2$9

The resulting approximation is jump-adapted: after each accepted jump, the EM discretization restarts from the new post-jump state on a fresh local mesh (Buckwar et al., 2024).

The convergence analysis has both mean-square and weak components. For bounded observables $(\phi,\lambda,\mathcal Q)$0, the mean-square estimate is

$(\phi,\lambda,\mathcal Q)$1

and under stronger smoothness assumptions the weak expansion is

$(\phi,\lambda,\mathcal Q)$2

The weak proof is generator-based and uses a backward integro-PDE together with generalized Itô formulas for the exact and numerical processes (Buckwar et al., 2024).

A second, model-specific scheme is the Thinned Splitting Method (TSM), introduced for a glioma cell migration model. There the continuous subsystem is decomposed into explicitly solvable subequations and integrated by a Lie-Trotter splitting, while jumps are still handled by thinning. The paper does not prove convergence for TSM, but reports that it appears more stable than TEM at coarse time steps, whereas TEM is the theoretically justified general-purpose method (Buckwar et al., 2024).

A complementary simulation perspective appears in the ABC inference paper. There, inter-jump SDEs are exactly solvable in the benchmark models, so $(\phi,\lambda,\mathcal Q)$3 is simulated exactly on each inter-jump interval, and the jump mechanism is handled either by direct exponential waiting times for constant rates or by thinning for bounded state-dependent rates. This produces explicit path algorithms for PDifMPs with observed $(\phi,\lambda,\mathcal Q)$4-paths and latent jumps (Desmettre et al., 14 Nov 2025).

4. First-passage problems, boundaries, and interface effects

A dedicated first-passage study considers one-dimensional PDifMPs with time-varying thresholds and defines

$(\phi,\lambda,\mathcal Q)$5

On each inter-jump interval, the PDifMP behaves like a diffusion with frozen mode, so the FPT problem can be reduced to an exact diffusion FPT simulation on that interval. The principal technical step is a Lamperti transform that normalizes the diffusion coefficient and yields

$(\phi,\lambda,\mathcal Q)$6

The transformed threshold is $(\phi,\lambda,\mathcal Q)$7, and Girsanov’s theorem is then applied at stopping times to turn the problem into Brownian-motion sampling weighted by explicit exponential functionals (Desmettre et al., 10 Jul 2025).

The paper derives two families of weighting terms. For inter-jump FPT simulation it introduces $(\phi,\lambda,\mathcal Q)$8, $(\phi,\lambda,\mathcal Q)$9, and $E=E_1\times \mathbf V$0, and represents the acceptance weight by Poisson thinning under the graph of $E=E_1\times \mathbf V$1. If crossing does not occur before the next jump, a new conditional problem arises: one must sample the pre-jump diffusion value at the jump time conditional on survival below a time-varying boundary. The paper’s main novelty is a hybrid exact or limit-exact conditional simulation step for this event, based on a conditionally constrained auxiliary Brownian process, a Bessel-bridge reconstruction, and a split Girsanov factorization across $E=E_1\times \mathbf V$2 and $E=E_1\times \mathbf V$3. The final theorem states that, under the stated assumptions and with $E=E_1\times \mathbf V$4, the output of the algorithm converges in law to the distribution of $E=E_1\times \mathbf V$5 as $E=E_1\times \mathbf V$6 (Desmettre et al., 10 Jul 2025).

Boundary questions in PDifMPs also have a broader stationary-distribution aspect. A PDMP Monte Carlo paper on piecewise-smooth densities isolates a boundary-balance law of the form $E=E_1\times \mathbf V$7, where the interface kernel must preserve an incoming/outgoing flux measure, and explicitly argues that the same issue is equally fundamental for piecewise diffusions, sticky/skew/interface diffusions, regime-switching diffusions, and constrained diffusion samplers (Chevallier et al., 2021). This suggests that PDifMP analysis at discontinuity surfaces is not only a matter of specifying local dynamics inside each region; it also requires a correct interface law for the probability current.

5. Inference, ergodicity, and application domains

Likelihood-based inference for PDifMPs is difficult even when inter-jump SDEs are explicitly solvable, because the full likelihood is unclear once jump times are unobserved or only partial information is available. A recent inference framework therefore proposes approximate Bayesian computation (ABC) for PDifMPs, with the observed dataset taken as

$E=E_1\times \mathbf V$8

where $E=E_1\times \mathbf V$9 is the observed time series of the continuous component and $E=D_1\times D_2$0 is the number of jumps on the observation window. In multidimensional examples only the first coordinate of $E=D_1\times D_2$1 is assumed observed. The proposed summary vector is

$E=D_1\times D_2$2

where $E=D_1\times D_2$3 is an estimated invariant density, $E=D_1\times D_2$4 an estimated invariant spectral density, and

$E=D_1\times D_2$5

is the mean quadratic variation. Inference is then performed by SMC-ABC. The framework is designed for PDifMPs that empirically exhibit ergodic behaviour, because in that regime invariant-density and spectral summaries extracted from a single long trajectory are informative (Desmettre et al., 14 Nov 2025).

The same paper studies four representative examples: an OU-PDifMP, a weakly damped stochastic harmonic oscillator PDifMP, a Wiener-process-with-piecewise-drift PDifMP, and a switched stochastic harmonic oscillator PDifMP. It reports that the ABC method reliably recovers model parameters across all examples, including cases with partial observation and cases where parameters enter state-dependent jump-rate functions, while also noting that ergodicity is empirically assessed rather than formally proved (Desmettre et al., 14 Nov 2025).

A direct applied PDifMP model appears in glioma cell migration. There the state is $E=D_1\times D_2$6, with $E=D_1\times D_2$7 the cell position, $E=D_1\times D_2$8 an environmental signal or amount of bound receptors, and $E=D_1\times D_2$9 the cell velocity. Between jumps,

$E_1\subset \mathbb R^d$0

so $E_1\subset \mathbb R^d$1 is constant between jumps and changes only at random turning events. This application shows how PDifMPs encode diffusive migration conditioned on a piecewise constant motility mode, and it serves as the benchmark for comparing TEM and TSM (Buckwar et al., 2024).

6. Adjacent literatures, common misconceptions, and open directions

PDifMPs should not be conflated with the broader PDMP literature. Several recent works are structurally adjacent but not direct PDifMP contributions. One multiscale spatial gene-network paper derives an infinite-dimensional continuous PDMP in which the continuous component becomes a reaction–diffusion PDE and the discrete component remains a jump process; it is best described as a switching PDE or hybrid spatial Markov process, not a piecewise Itô diffusion (Debussche et al., 2020). A survival-extrapolation paper combines a discretised diffusion prior for latent hazards with Piecewise Deterministic Monte Carlo for posterior sampling; it is highly relevant to the interaction of diffusion modeling and PDMP computation, but it does not define a single hybrid stochastic process whose trajectories are piecewise diffusive (Hardcastle et al., 9 May 2025).

The deterministic side of the literature is also informative. Automated simulation of PDMP-based MCMC has focused on adaptive local horizons, grid-based piecewise-constant upper bounds, and thinning for exact event-time simulation when integrated hazards are not analytically tractable (Andral et al., 2024). PDMP generative modeling has developed reverse-time theory in which the reverse characteristics are governed by flux equations and conditional velocity laws rather than diffusion scores (Bertazzi et al., 2024). PDMP approximation theory has reformulated value functionals as fixed points of integral operators and used smoothing plus deterministic cubature to avoid direct integro-differential equations (Kritzer et al., 2017). These are not PDifMP papers, but they clarify the analytic and computational motifs that recur once deterministic inter-jump flow is replaced by diffusion.

A common misconception is therefore that any model mixing diffusions and jumps is already a PDifMP. The direct PDifMP literature is narrower: it requires a diffusion-driven continuous component on inter-jump intervals together with a discrete event mechanism that alters the regime and, in generalized versions, may also create continuous-state discontinuities (Buckwar et al., 2024, Desmettre et al., 10 Jul 2025). The current research frontier remains correspondingly specific. The direct papers leave open rigorous ergodicity theory for representative models, broader inference methods beyond ABC, exact or asymptotically exact treatment of state-dependent first-passage problems in higher dimension, and a more complete boundary/interface theory for stationary laws. The adjacent PDMP interface literature suggests that such a theory will likely be formulated in terms of probability-flux conservation across switching surfaces rather than purely local state densities (Chevallier et al., 2021).