Singular Jump Processes: Theory & Applications

• Singular jump processes are stochastic systems characterized by nonlocal, singular jump mechanisms that defy traditional translation-invariant frameworks.
• The analytical framework employs integro-differential operators, fractional Sobolev/Besov estimates, and Dirichlet form theory to address unbounded or anisotropic jump kernels.
• These processes find applications in stochastic filtering, rare event analysis, optimal control, and generalized gradient flows, offering robust tools for complex system modeling.

Singular jump processes are stochastic processes characterized by highly nonlocal, non-smooth or even unbounded transition mechanisms, where the intensity or structure of jumps cannot be recast within classical regular frameworks (e.g., by translation-invariant and integrable kernels). These processes occur across disciplines: in stochastic filtering (where jump-driven nonlocal operators appear), in the analysis of heavy-tailed rare events, in generalized gradient flows, and in optimal control problems where singularities are unavoidable. The core mathematical and probabilistic structure of singular jump processes often involves fractional or nonlocal operators, discontinuous martingale measures, and intricate dependence on space, time, or path-history.

1. Analytical and Probabilistic Frameworks

The rigorous paper of singular jump processes has required extending classical frameworks—both in partial differential equations and stochastic process theory. For nonlocal evolution, the generator is typically an integro-differential operator of the form

$$A^{(\alpha)}u(x) = \int_{\mathbb{R}^d} \bigg[u(x + y) - u(x) - \chi_{\alpha}(y)(\nabla u(x), y)\bigg] m^{(\alpha)}(t,y) \frac{dy}{|y|^{d+\alpha}},$$

where the presence of the singular kernel $|y|^{-d-\alpha}$ (with $0 < \alpha < 2$) induces strong nonlocality and often singularity at $y=0$ (Mikulevicius et al., 2010). The singularity in the jump measure requires careful function space analysis: sharp $L_p$, fractional Sobolev, and Besov space estimates must be established to guarantee existence, uniqueness, and regularity of SPDE solutions driven by Lévy or Poisson-type jump noise (Mikulevicius et al., 2010, Brzeźniak et al., 2010).

In the context of generalized gradient flows on abstract metric spaces, the singularity is encoded in the jump kernel $\kappa(x,dy)$, satisfying only local energy bounds (e.g., $$\sup_{x \in V} \int (1 \wedge d^2(x, y))\,\kappa(x, dy) < \infty$$) and possibly being unbounded or supported in singular sets (Hoeksema et al., 23 Sep 2025). The evolution and continuity equations, and in particular the energy-dissipation balance, must therefore be formulated to accommodate these singularities, requiring nonstandard test function spaces and limiting arguments.

For Markovian jump processes, particularly in mean-field or controlled settings, singular controls often correspond to jump measures that are only weakly (e.g., in WM$_1$ topology) the limit of regular processes, or to jump rates that become unbounded at certain thresholds (Denkert et al., 14 Feb 2024). This is further complicated in pathwise optimal control, where entropy-based running costs and infinite terminal costs naturally induce singular transition rates (Gao et al., 2023).

2. Singular Jump Kernels and Nonlocal Operators

A paradigmatic example is furnished by the class of non-isotropic Markov processes with jump kernels of the form

$$J^{\phi,1}(u,w) = \frac{1}{|u - w|\,\phi(|u - w|)},$$

where $\phi$ is an increasing function satisfying a weak scaling condition between exponents $0 < \underline{\alpha} \leq \overline{\alpha} < 2$ (Kim et al., 2020). After "lifting" to higher dimensions, one obtains anisotropic processes whose jumps occur in only one coordinate at a time, and the operator does not regularize in the classical sense due to the near-diagonal and unbounded nature of the kernel.

The resolution of analytical and probabilistic questions—heat kernel bounds, parabolic Harnack inequalities, regularity of harmonic functions—proceeds via Dirichlet form theory, Nash-type inequalities, and intricate covering and box decomposition arguments. Sharp two-sided heat kernel bounds for such singular processes are established in terms of scaling functions and the geometry induced by the singular kernel (Kim et al., 2020). This framework accommodates strong directional nonlocality and the lack of analyticity seen in isotropic stable processes.

3. Stochastic Calculus, Martingale Problems, and Control

Singular jump processes frequently appear with drift and noise structures that are ill-posed in classical Ito calculus. Modern developments generalize the Langevin equation to jump process settings, define the drift and diffusivity in terms of predictable compensators and the second moment of the jump distribution, and provide sample path bounds, first passage estimates, and response formulas directly analogous to (but distinct from) the diffusion case (Foxall, 2016, Stutzer et al., 6 Aug 2025). For instance,

$$X_t = X_0 + \int_0^t \mu_s(X) ds + M_t(X), \qquad \langle X \rangle_t = \int_0^t \sigma_s^2(X) ds,$$

where $M$ is a martingale and $\sigma^2$ captures jump-induced variance.

In optimal control problems with singular transition rates (e.g., transition path sampling with infinite terminal cost penalties), rigorous analysis uses Gamma convergence to handle the unbounded controls, passing to the limit in closed-form feedback laws involving committor functions that may become arbitrarily small near "forbidden" sets (Gao et al., 2023). The tropism towards Doob-$h$ transforms is generically observed: the optimal controlled jump kernel is

$$L_{AB}(x, dy) = \frac{h_{AB}(y)}{h_{AB}(x)} L(x, dy),$$

where $h_{AB}$ is the committor function, with the singularity at $h_{AB}(x) \to 0$ encoding the nonapproach to undesired states.

4. The Big Jump Principle and Rare Event Structure

A central structural feature for rare-event statistics in heavy-tailed or subexponential jump processes is the dominance of a single large jump in the sum or additive observable—a property codified as the "big jump principle." For IID variables with subexponential tails,

$$\text{Prob}\left(\sum_{i=1}^N x_i > X\right) \simeq N\,\text{Prob}(x > X)$$

for large $X$, i.e., the probability of large fluctuations in the sum coincides with the probability of seeing one exceptionally large jump (Vezzani et al., 2018, Bassanoni et al., 13 Jan 2025). This principle is generalized to correlated processes (e.g., Lévy walks with memory, processes with quenched disorder) by adjusting the effective number of renewal attempts and via infinite density construction. Universal scaling functions for first exit and passage times are governed by the Lévy exponent $\mu$ and are robust under various physical and stochastic settings (Klinger et al., 2022, Bassanoni et al., 13 Jan 2025).

This mechanism directly shapes the analysis of rare event probabilities, anomalous large deviation rate functions, and even the occurrence of dynamical phase transitions in observables for processes such as the Ornstein-Uhlenbeck process or generalized random walks.

5. Generalized Gradient Flows and Reflecting Solutions

In recent variational treatments, singular jump processes are characterized as generalized gradient flows on possibly abstract metric spaces, well beyond settings with translation-invariant kernels. The core is the evolution equation for measures: $$\partial_t \rho + \text{div}\,j = 0, \quad \text{where } j = \partial_2^* (\rho, -D E(\rho)),$$ and $E(\rho)$ is an entropy functional, typically of Boltzmann or Orlicz type (Hoeksema et al., 23 Sep 2025). Singularities in the kernel necessitate the introduction of "reflecting solutions," which enlarge the class of test functions (to a nonlocal Sobolev-or Orlicz-space $\mathcal{X}_2$), enforce a chain rule, and guarantee energy-dissipation equality: $$\int_s^t (R(\rho_r, j_r) + D(\rho_r)) dr + E(\rho_t) = E(\rho_s).$$ Existence, uniqueness, and compactness are secured through regularization and robust approximation via cutoff kernels. The flexibility of the theory supports its extension to configuration spaces and interacting particle systems, capturing both the geometric and variational aspects of singular nonlocal evolution.

6. Applications, Extensions, and Open Directions

Singular jump processes are encountered in nonlinear filtering (where Zakai equations with jump noise are governed by singular nonlocal operators) (Mikulevicius et al., 2010, Leahy et al., 2014), in mean-field games with multi-dimensional singular controls and non-linear jump impacts (where admissibility and reward continuity must be handled via weak topologies and Marcus-type SDEs) (Denkert et al., 14 Feb 2024), and in optimal control and path sampling (where entropy minimization with singular penalties dictates the occurrence of unbounded transition rates) (Gao et al., 2023). Analytical results—such as two-sided heat kernel bounds, parabolic Harnack inequalities, and stability under subordination—extend from the theory of subordinated diffusions and Dirichlet forms to encompass polynomial-type and more general singular kernels (Kim et al., 2020, Liu et al., 2021).

Open mathematical challenges include the handling of optimal control and pathwise observables in high-dimensional or infinite particle systems, the explicit construction of robust reflecting solutions in non-dense subspaces, and the extension of scaling and large deviation results for correlated or memory-laden singular jump processes. The unification of stochastic calculus methods and gradient flow frameworks, now extended to these singular settings, positions the theory for further development in both pure and applied directions.

Table: Key Formulations and Settings for Singular Jump Processes

Analytical Setting	Key Ingredients / Features	Representative Reference(s)
Integro-differential operator	Nonlocal singular kernels, fractional order	(Mikulevicius et al., 2010, Kim et al., 2020)
Gradient flow on metric space	Entropy-dissipation, reflecting solutions	(Hoeksema et al., 23 Sep 2025)
Markov jump control (transition path)	Entropy cost, singular rates, committor	(Gao et al., 2023)
Stochastic calculus for path-observables	Drift, diffusivity, covariance, TURs	(Foxall, 2016, Stutzer et al., 6 Aug 2025)
Rare event analysis	Big jump principle, scaling functions	(Vezzani et al., 2018, Klinger et al., 2022, Bassanoni et al., 13 Jan 2025)
Mean-field game with singular controls	Marcus SDEs, parametrised controls	(Denkert et al., 14 Feb 2024)

The synthesis of analytical, variational, and probabilistic perspectives is fundamental in understanding the deep structure and wide applicability of singular jump processes throughout mathematics, physics, engineering, and computational science.