Discontinuous Symmetric Markov Processes

• Discontinuous symmetric Markov processes are stochastic models with jumps that exhibit time-reversal invariance and are constructed via regular symmetric Dirichlet forms.
• They employ advanced techniques such as nonlocal generators, martingale decompositions, and heat kernel estimates to analyze discontinuities and boundary behaviors.
• Applications include solving nonlocal PDEs, modeling anomalous diffusion in physics, and designing financial models with jump-dominated dynamics.

Discontinuous symmetric Markov processes form a broad class of stochastic models whose sample paths possess jumps and are invariant under time reversal in law. They are constructed via regular symmetric Dirichlet forms, often pure-jump, with a symmetric jump kernel determining the discontinuity structure. Such processes include symmetric stable Lévy processes, their generalizations, processes with nonlocal boundary conditions, quasidiffusions with discontinuous scale, and Markov processes with darning operations that collapse portions of the state space. The rigorous analysis employs nonlocal generators, martingale techniques, stochastic calculus tailored to nonlocal forms, heat kernel estimates, and convergence methods.

1. Symmetric α-Stable and Stable-Like Markov Processes

Symmetric α-stable processes with $α∈(0,2)$ and $d≥1$ are paradigmatic examples, defined via their characteristic exponent

$$\mathbb{E}[e^{iξ·Z_t}] = \exp[-tψ(ξ)], \quad ψ(ξ) = \int_{\mathbb{R}^d} (1-e^{iξ·y} + i 1_{|y|\leq 1} ξ·y) ν(dy)$$

with Lévy measure $ν(dy)=c_{d,α}|y|^{-d-α}dy$ and a non-degenerate spectral measure on $S^{d-1}$—a condition ensuring rich multidimensional jump geometry [(Zhang, 2010)]. These processes possess the generator

$$Lf(x) = \int_{\mathbb{R}^d} [f(x+y)-f(x)-1_{|y|\leq 1} y·∇f(x)] ν(dy)$$

and admit Lévy–Itô decomposition into compensated small jumps and Poissonian large jumps.

Stable-like and relativistic variants, defined through jump kernels

$J(x,y)=\kappa(x,y)\psi(|x-y|)^{-1}|x-y|^{-d-α}$

with spatially varying $\kappa(x,y)$ and tempering $\psi(\cdot)$, generalize the pure stable case, exhibiting power-law or tempered tails and robustness to boundary regularity [(Kim et al., 2014, Kim, 2015)]. Pure-jump Hunt processes associated to these Dirichlet forms are symmetric in law and encode discontinuities in both transition densities and Green functions. Boundary irregularities and domain geometry (e.g. $C^{1,\eta}$ regularity, Lipschitz, Dini conditions) play a critical role in the asymptotics of the killed process and in local singularities of the heat kernel.

2. Stochastic Differential Equations with Discontinuous Drift

Discontinuous symmetric Markov processes emerge as solutions of SDEs driven by α-stable symmetric Lévy processes with time-dependent Sobolev drifts $(b)$ admitting genuine jump-discontinuities. For $1<α<2$ and $b\in L^q_tW^{β,p}_x$ with $2/p+α/q<1$ and $β$ possibly below continuity threshold, pathwise uniqueness holds up to explosion times [(Zhang, 2010)]:

$dX_t = b(t, X_{t-})dt + dL_t$

A key technical innovation is the extension of Krylov-type $L^q_tL^p_x$ estimates to purely discontinuous semimartingales, enabling control of occupation times even when $b$ is discontinuous. Zvonkin's transformation is generalized via backward PIDEs to smooth irregular drift into an effective, Lipschitz drift in transformed coordinates, yielding unique strong solutions. Localization and stopping times are essential for controlling explosion due to merely local Sobolev regularity.

These sharp results expand the classical theory from the Brownian (continuous path) case to the jump-dominated regime, underpinning well-posedness of the martingale problem and the corresponding nonlocal Fokker–Planck equations.

3. Dirichlet Forms, Darning, and Quasidiffusions

Symmetric Markov processes with jumps can be manipulated via Dirichlet form theory, including darning (collapsing subsets into single points) for both continuous and discontinuous processes [(Chen et al., 2017)]. The construction utilizes the restriction of the original Dirichlet form $(\mathcal{E},\mathcal{F})$ to functions constant on the sets to be collapsed, preserving symmetry and jump structure. The generator and semigroups of the darned process inherit the jump and killing measures on the new, reduced state space. In the approximation regime, processes with additional fast (large-intensity) jumps in the collapsing sets converge to the darning process via Mosco convergence.

More generally, Markov processes associated to one-dimensional Dirichlet forms with discontinuous scale functions lead to quasidiffusions [(Li, 2023)]. The process may not possess the strong Markov property in its naive form, but regularization—via Fukushima's canonical representation or Ray–Knight compactification—yields a Hunt process that is symmetric and skip-free, homeomorphic to a time-changed Brownian motion.

4. Heat Kernel and Green Function Analysis

Discontinuous symmetric Markov processes admit sharp two-sided heat kernel estimates, crucial for analyzing their fine properties. In $C^{1,\eta}$ open domains, for a broad class of stable-like jump kernels,

$p_D(t,x,y) \asymp \big(1 \wedge \tfrac{\delta_D(x)}{t^{1/\alpha}}\big)^{\alpha/2} \big(1 \wedge \tfrac{\delta_D(y)}{t^{1/\alpha}}\big)^{\alpha/2} \left(t^{-d/\alpha} \wedge \tfrac{t}{|x-y|^{d+\alpha}}\right) \quad\text{(for small $t$)}$

with modifications for relativistic, finite-range, or tempered-stable regimes [(Kim et al., 2014, Kim, 2015)]. In bounded smooth domains with boundary-degenerating or -blowing-up jump kernels, estimates incorporate boundary decay/explosion factors, and in killed processes, exponential factors tied to Dirichlet principal eigenvalues [(Cho et al., 15 Dec 2025, Cho et al., 31 Dec 2025)]. Off-diagonal and long-time behavior are influenced by domain geometry, jump intensity regularity, and boundary singularity exponents.

Green function estimates connect with potential theory, capacity, and the hitting probabilities for small sets, and are sensitive to both local and nonlocal features. These analyses are essential for verifying transience, recurrence, boundary Harnack principles, and ergodicity.

5. Stochastic Calculus and Martingale Decomposition

Stochastic calculus for discontinuous symmetric Markov processes leverages Dirichlet forms and martingale additive functionals (MAFs). The Fukushima decomposition splits functionals into a local martingale part and a zero-energy continuous additive functional, inherently extending to jump processes. The extended Nakao operator yields a divergence-like functional, with well-defined stochastic integrals for functions in the appropriate local domain [(Kuwae, 2010)].

The generalized Itô formula includes both continuous and jump contributions, applicable pathwise for quasi-every starting point. This framework supports advanced probabilistic analysis for stable Lévy processes, mixed jump-diffusions, and symmetric Markov models in statistical physics, nonlocal PDEs, or fractal state spaces.

6. Path Properties, Rate Functions, and Zero-One Laws

Advanced heat kernel analysis underpins integral tests for sample-path range and quantile growth rates of symmetric Markov processes. Upper and lower rate functions, defined probabilistically via almost sure confinement or escape rates, admit sharp zero-one laws governed by heat kernel and volume growth asymptotics [(Shiozawa et al., 2015)]. Subcritical/transient and critical/recurrent regimes are distinguished via scaling exponents and log-iterated integrals, generalizing classical Kolmogorov, Spitzer, and Dvoretzky–Erdős laws.

This theory applies to stable-like pure jump processes on Ahlfors regular spaces, Gaussian-type diffusions, and mixed local-nonlocal processes, offering a unified methodology for pathwise probabilistic behavior beyond the continuous-path setting.

7. Applications and Generalizations

Discontinuous symmetric Markov processes arise in numerous domains:

• Nonlocal PDEs with Dirichlet or Neumann conditions: e.g., boundary singularity-sensitive or degenerate Markov jump processes [(Cho et al., 31 Dec 2025, Cho et al., 15 Dec 2025)].
• Finance: models such as variance-gamma, tempered-stable jumps.
• Physics: Lévy flights, anomalous diffusion, gap diffusions in disordered or fractal media.
• Geometry: planar BM with flag poles, mixed local-jump models on manifolds.
• Potential theory: spectral theory, eigenvalue estimates, and long-time asymptotics.

The theory supports extensions to variable order jump intensities, mixed local-nonlocal generators, non-symmetric processes, and highly irregular spaces, provided sufficient control over the Dirichlet form and corresponding kernel properties. Advanced functional analytic and probabilistic tools (Mosco convergence, Nash inequalities, regular representations) facilitate rigorous construction and analysis across these generalizations.