Recurrent Jump Mechanisms

• Recurrent jump mechanisms are defined as systems where local dynamics and non-local, heavy-tailed jumps interlace to ensure repeated returns to a starting region.
• They are analyzed through Dirichlet forms, overshoot techniques, and Lyapunov criteria, balancing volumetric growth against jump-tail decay.
• Applications span reinforcement models, robotics, and neural computation, demonstrating how controlled non-local moves yield system stability.

A recurrent jump mechanism is any mathematical structure or physical process in which a system evolves through a combination of continuous or local dynamics and non-local, possibly heavy-tailed, jumps, in such a way that the process almost surely returns to its initial neighborhood infinitely often (recurrence). This mode of dynamical evolution is relevant across probability theory, stochastic processes, dynamical systems, reinforcement models, and engineering applications including robotics and neural computation. Recurrence in jump systems is governed by a subtle interplay between local geometry, volumetric growth, kernel tail behavior, and reinforcement or interaction patterns.

1. Symmetric Jump–Diffusions: Dirichlet Form Framework

In the probabilistic analytic formulation, symmetric jump mechanisms are encoded by regular Dirichlet forms of the type

$\E(u,v) = \E^{(c)}(u,v) + \iint_{X\times X \setminus \Delta} (\widetilde u(x) - \widetilde u(y))(\widetilde v(x) - \widetilde v(y)) J(dx,dy),$

where $\E^{(c)}$ is the diffusion (local) component and $J$ is a symmetric Radon measure (jump component). Volume growth $V(x,r)$ and the jump-tail intensity

$$\omega(r) = \sup_{x \in X^{(j)}} \int_{d(x,y) \ge r} (d(x,y) \wedge r)^2\, j(x,dy)$$

quantify the large-scale geometry and energetic cost of long-range jumps. The process is recurrent if for some $x_0 \in X$

$$\liminf_{r \to \infty} \frac{V^{(c)}(x_0, r) + V^{(j)}(x_0, r)\,\omega(r)}{r^2} < \infty,$$

i.e., the combined local and jump mechanism expends only $O(r^2)$ energy to reach distance $r$ from $x_0$ as $\E^{(c)}$0 (Masamune et al., 2012).

Recurrence may persist even on topologically disconnected spaces, provided the jump kernel is sufficiently thick to stochastically “glue” the components together. If the jump tail decays slowly, jumps can efficiently return the process from far away, enforcing global recurrence.

2. Reinforced Jump Processes: Vertex-Reinforced Jump Process (VRJP)

The VRJP represents a canonical class of recurrent jump mechanisms on discrete graphs, where the jump rate from $\E^{(c)}$1 to $\E^{(c)}$2 increases with the local time $\E^{(c)}$3. On $\E^{(c)}$4, the process is recurrent if

$\E^{(c)}$5

for the reinforcement function $\E^{(c)}$6; otherwise, under further monotonicity, the process localizes on exactly three sites. For any initial equal weighting, VRJP on $\E^{(c)}$7 cannot be transient: recurrence or localization is enforced by pathwise arguments and martingale analyses (Collevecchio et al., 2020).

On Galton–Watson trees, a phase transition between recurrence and transience is controlled by the offspring mean $\E^{(c)}$8 and the reinforcement parameter $\E^{(c)}$9 via an explicit martingale law $J$0, with recurrence when $J$1 (Basdevant et al., 2010).

In higher dimensions and in two dimensions, recurrence persists under strong reinforcement, with explicit criteria relating the edge weights $J$2 and exponential or fractional moment bounds for the Green’s function of an associated random Schrödinger operator. The underlying network viewed through the VRJP becomes recurrent (in the electrical sense) when weighted conductances decay sufficiently rapidly (Collevecchio et al., 2018, Sabot, 2019).

3. Recurrence Criteria in Markov Jump Processes and Overshoot Techniques

For general jump Markov processes in continuous time, recurrence at a point is characterized via the sequence of “overshoots”: if the chain of successive up- or down-crossings remains bounded, the process is recurrent; if not, it is transient. For stable-like processes with generator $J$3 where $J$4 for $J$5 and $J$6 for $J$7, the process is recurrent iff $J$8, achieved by explicit overshoot moment calculations and Markov chain embeddings (Böttcher, 2010).

Quasi-stationary distributions for $J$9-recurrent jump processes are characterized by their Q-matrix and associated taboo (exit-avoiding) embedded chains. Existence and explicit construction of QSDs are directly governed by the structure (and finiteness) of the set of killing states and the $V(x,r)$0-recurrence property of the jump semigroup (Du et al., 2024).

4. State-Dependent Jump Processes and Generalizations

A broad class of state-dependent jump mechanisms are governed by SDEs of the form

$V(x,r)$1

with $V(x,r)$2 forming a state-dependent Poisson process of rate $V(x,r)$3. The master equation combines deterministic transport, departure from the current state at rate $V(x,r)$4, and arrival into $V(x,r)$5 via jumps from $V(x,r)$6 at rate $V(x,r)$7. Recurrence or the nature of stationary/steady states relies on the interplay between drift $V(x,r)$8, jump amplitude $V(x,r)$9, and frequency $$\omega(r) = \sup_{x \in X^{(j)}} \int_{d(x,y) \ge r} (d(x,y) \wedge r)^2\, j(x,dy)$$0; analytical solutions are available for specific choices such as exponential mark laws and Stratonovich interpretation (Bartlett et al., 2018).

The diffusive limit ($$\omega(r) = \sup_{x \in X^{(j)}} \int_{d(x,y) \ge r} (d(x,y) \wedge r)^2\, j(x,dy)$$1, $$\omega(r) = \sup_{x \in X^{(j)}} \int_{d(x,y) \ge r} (d(x,y) \wedge r)^2\, j(x,dy)$$2) interpolates recurrent jump processes and continuous diffusions, unifying classical Brownian recurrence and pure jump regimes.

5. Regime-Switching Jump Diffusions and Lyapunov Criteria

For multivariate jump diffusions with regime-switching (i.e., a Markov process combining Brownian motion, state-dependent jumps, and a discrete state Markov chain), recurrence is established by constructing $$\omega(r) = \sup_{x \in X^{(j)}} \int_{d(x,y) \ge r} (d(x,y) \wedge r)^2\, j(x,dy)$$3 Lyapunov functions $$\omega(r) = \sup_{x \in X^{(j)}} \int_{d(x,y) \ge r} (d(x,y) \wedge r)^2\, j(x,dy)$$4 for which the infinitesimal generator $$\omega(r) = \sup_{x \in X^{(j)}} \int_{d(x,y) \ge r} (d(x,y) \wedge r)^2\, j(x,dy)$$5 outside a compact set. The jump mechanism contributes stabilizing or destabilizing effects via compensated Poisson integrals in the generator. Lyapunov techniques extend to positive Harris recurrence and full ergodicity with unique invariant measures (Chen et al., 2018).

Explicit examples demonstrate that jump kernels with suitable heavy-tails can overcome otherwise transient drift/diffusion, restoring recurrence. In one dimension, recurrence is assured if

$$\omega(r) = \sup_{x \in X^{(j)}} \int_{d(x,y) \ge r} (d(x,y) \wedge r)^2\, j(x,dy)$$6

where $$\omega(r) = \sup_{x \in X^{(j)}} \int_{d(x,y) \ge r} (d(x,y) \wedge r)^2\, j(x,dy)$$7 and $$\omega(r) = \sup_{x \in X^{(j)}} \int_{d(x,y) \ge r} (d(x,y) \wedge r)^2\, j(x,dy)$$8 capture drift and jump contributions.

6. Recurrent Jump Mechanisms in Robotics and Artificial Systems

In robotics, recurrent jump mechanisms describe control and planning systems where agents or robots must execute repeated, non-continuous state transitions (e.g., consecutive jumps or hops in locomotion). In quadrupedal locomotion under reduced gravity, a dual-horizon internal model encodes both fast (short-horizon) vertical and slow (long-horizon) horizontal dynamics, outputting a fused representation that enables stable, continuous jumping under long aerial phases (Xu et al., 9 Mar 2026). Recurrent mechanisms here are implemented via multi-scale temporal encoders and phase-adaptive rewards, validated in hardware-in-the-loop platforms with mixed reality (Xu et al., 9 Mar 2026).

For legged robots on stepping stones, a hierarchical architecture combining off-line trajectory optimization with on-line model predictive control enables robust, recurrent jumping across irregular terrain and under model uncertainty. The mechanism is explicitly phase-scheduled, with hybrid resets and re-planning at each landing event (Nguyen et al., 2022).

In neural sequence processing, recurrent jump mechanisms augment LSTM RNNs with agents that learn when to skip or jump over irrelevant subsequences (e.g., via punctuation) to accelerate inference without loss in accuracy. Structural-Jump-LSTM applies this by employing reinforcement learning-based agents for skip and jump, providing up to $$\omega(r) = \sup_{x \in X^{(j)}} \int_{d(x,y) \ge r} (d(x,y) \wedge r)^2\, j(x,dy)$$9 computational savings over vanilla LSTM decoding (Hansen et al., 2019).

7. Synthesis and Structural Features

Recurrent jump mechanisms constitute a structural class of dynamical systems in which non-local moves—implemented via heavy-tailed kernels, reinforcement, regime switching, or agent actions—do not suffice to drive the process to infinity; local revisits are statistically enforced. The analytic core unites Lyapunov–Foster criteria, Green function or fractional-moment bounds (in random-environment representations), and phase-space volume vs. kernel tail comparisons.

Distinguishing features include:

• Energetic dichotomy: Volume growth versus jump-tail decay determines recurrence or transience.
• Heavy-tail stabilization: Jumps decaying slower than polynomial may enforce recurrence even in disconnected or highly inhomogeneous state spaces.
• Reinforcement and localization: Vertex-reinforced or edge-reinforced jump processes can enforce not only recurrence but also strong localization, depending on the reinforcement regime.
• Simulation and control: Recurrent jump control strategies extend to complex engineered systems, both in learned policy architectures and hybrid analytic/planner loops.

Recurrent jump mechanisms thus offer both a unifying probabilistic theory (from Dirichlet forms to random environments) and a diverse set of applications in stochastic processes, robotics, and sequential decision systems (Masamune et al., 2012, Collevecchio et al., 2020, Böttcher, 2010, Xu et al., 9 Mar 2026, Nguyen et al., 2022, Hansen et al., 2019).