Unified Lévy Noise Framework

• Unified Lévy noise framework is a rigorous formulation that defines action functionals for SDEs under both finite and infinite jump activity regimes.
• It employs variational and path-integral approaches to derive rate functionals, enabling accurate large deviation analysis and optimal transition path computation.
• The framework supports high-dimensional and infinite-dimensional extensions, addressing technical challenges in non-Gaussian noise for nonequilibrium stochastic dynamics.

A unified framework for Lévy noise addresses the mathematical structure and interpretation of action functionals for stochastic differential equations (SDEs) driven by Lévy processes, including both finite and infinite jump activity regimes. Such action or rate functionals form the foundation of large-deviation theory for jump processes and underpin the analysis of rare events, most-probable paths, and transition phenomena in systems with both continuous (diffusive) and discontinuous (jump) stochastic forcing. Rigorous treatment of these functionals in finite and infinite dimensions, across dynamical, variational, and path-integral perspectives, enables both theoretical development and computational practice in the paper of nonequilibrium stochastic dynamics with Lévy noise.

1. Functional Large Deviations Principles for Lévy-Driven SDEs

Large-deviation theory frames the probability of rare events in path space through action functionals. For SDEs driven by Lévy noise,

$$dX_t = b(X_{t-})\,dt + \sigma(X_{t-})\,dB_t + \eta(X_{t-})\,dL_t,$$

where $L_t$ is a real-valued Lévy process with Lévy triplet, action functionals $I[\phi]$ are constructed as the large-deviation rate functions for scaled path processes. Under the assumption of finite exponential moments for $L$, precise functional large-deviation results can be established in the Skorokhod space $D[0, T]$ equipped with the supremum norm. Specifically, the action functional takes the form

$$I[\phi] = \begin{cases} \int_{0}^{T} L\bigl(\phi(t), \dot\phi(t)\bigr)\,dt, & \phi \in AC[0, T],\, \phi(0) = x_0, \ +\infty, & \text{otherwise}, \end{cases}$$

where $L(x, v)$ is a Lagrangian explicitly given via the Legendre transform of the associated Hamiltonian (Yuan et al., 2019).

2. Lévy Symbol, Hamiltonian, and Variational Structure

The Lévy symbol (generator) encodes the infinitesimal evolution:

$$q(x,\xi) = i\,b(x)\,\xi - \frac{1}{2}\,\sigma^2(x)\,\xi^2 + \int_{y\neq 0} \Big(e^{i\xi\,\eta(x)\,y} - 1 - i\xi\,\eta(x)\,y\,1_{|y|\leq1}\Big)\,\nu(dy),$$

with $b$ drift, $\sigma$ diffusion coefficient, and $\eta$ jump amplitude. The associated Hamiltonian reads

$$H(x, \theta) = q(x, -i\,\theta) = b(x)\theta + \frac{1}{2}\sigma^2(x)\theta^2 + \int_{\mathbb{R} \setminus \{0\}} \Big(e^{\eta(x)\theta y} - 1 - \eta(x)\theta y\,1_{|y|\leq1}\Big)\nu(dy).$$

The Lagrangian is then

$$L(x,v) = \sup_{\theta \in \mathbb{R}}\big\{ \theta v - H(x,\theta) \big\},$$

giving the explicit variational structure for Lévy-driven action principles (Yuan et al., 2019).

3. Probability-Flow Equivalence and Finite vs Infinite Activity

A substantial advance for unified action principles is the probability flow equivalence between jump-diffusions and pure diffusions with modified drift, allowing explicit OM functionals for finite-activity jumps without relying on pathwise Girsanov formulae, which are typically unavailable for processes with jumps. For SDEs with finite jump activity, the OM action for a path $\varphi$ is

$$A[\varphi] = \int_0^T \bigg\{ \frac{1}{2\sigma^2}\big|\dot\varphi(s) - b(\varphi(s)) - \ell_J(\varphi(s))\big|^2 + \nabla\cdot[b(\varphi(s)) + \ell_J(\varphi(s))] + \varepsilon_J(\varphi(s)) \bigg\}\,ds,$$

where $\ell_J(x) = \int_{\mathbb{R}^d} z\,\nu(x, dz)$ is the average jump drift and $\varepsilon_J$ encodes corrections involving the Lévy intensity at the origin (Huang et al., 2 Sep 2024).

For infinite-activity Lévy processes, a continuous-time OM action does not exist due to diverging small-jump intensity. Instead, a time-discretized Onsager–Machlup functional is rigorously constructed for trajectories sampled at partitions $0 = t_0 < t_1 < \ldots < t_n = T$:

$$S^\Delta(x_0, ..., x_n) = \sum_{i=1}^n \Bigg\{ \frac{|x_i - x_{i-1} - b(x_i)\Delta t_i + J_i^{(comp)}|^2}{2\sigma^2\Delta t_i} - \frac{1}{\Delta t_i}\int_{|z|<1} F(x_i, z)\,\nu(x_i, dz)(x_i - x_{i-1}) + \cdots \Bigg\},$$

where $J_i^{(comp)}$ and the nonlocal integral quantify the compensation for infinite small jumps (Huang et al., 2 Sep 2024).

4. High-Dimensional and Infinite-Dimensional Extensions

For vector-valued SDEs,

$dX(t) = f(X(t))\,dt + B\,dW(t) + dL(t)$

with nondegenerate $B$ and Lévy process $L$, the Onsager-Machlup functional, under a (Poincaré) symmetry condition on a modified drift, becomes

$$L(x,v) = \frac{1}{2}|B^{-1}(f(x) - v - \eta)|^2 + \frac{1}{2}\operatorname{Tr}[\nabla_x f(x)],$$

where $\eta = \int_{|z|<1} z\,\nu(dz)$ (Hu et al., 2021). In the context of SPDEs driven by Lévy noise in Hilbert space, the Onsager-Machlup action requires boundedness and symmetry properties of nonlocal drift; the explicit functional includes trace-class corrections due to the infinite-dimensional Itô formula and jump integrals, with the Euler–Lagrange equation taking a Hilbert–space-valued second-order form (Hu et al., 2020).

5. Role of the Action Functional in Rare Events and Variational Principles

The unified Lévy action framework plays a crucial role in giving the exponential scaling of rare event probabilities in small noise regimes:

$$P\{X^\epsilon \approx \phi\} \approx \exp\left(-I[\phi]/\epsilon\right),$$

where minimizers of the functional $I[\phi]$ (subject to boundary conditions) constitute the most-probable or optimal transition paths between prescribed states, generalizing the Freidlin–Wentzell quasipotential to non-Gaussian (jump) settings (Yuan et al., 2019, Hu et al., 2021). The variational characterization enables explicit computation of most-likely paths and yields stochastic Euler–Lagrange equations, which in the jump case feature nontrivial corrections from the Lévy generator.

6. Numerical Schemes and Computational Methods

Numerical computation of minimizers and rare event pathways in Lévy-driven systems requires solution of high-dimensional or path-dependent boundary-value problems for action minimization. Approaches include shooting methods (Newton schemes or NN-based shooting), relaxation and discretization (finite-difference with constraints), and Hamiltonian formulations (solving the corresponding Hamilton’s equations with transversality). These strategies are necessary for practical application of the Lévy action framework to systems of moderate to high dimension, for which analytic minimizers are intractable (Hu et al., 2021).

7. Technical Assumptions and Limitations

Rigorous construction of Lévy action functionals requires several technical conditions:

• Finite exponential moments for the Lévy process ($\mathbb{E}\exp\{\theta|L_1|\} < \infty$) to ensure boundedness and differentiability of the Hamiltonian (Yuan et al., 2019).
• Global Lipschitz and boundedness for SDE coefficients ($b,\sigma,\eta$) with $\sigma$ uniformly elliptic.
• For infinite activity, pathwise action is defined only in discrete time; no continuous-time OM action exists due to divergence of jump intensity at the origin (Huang et al., 2 Sep 2024).

These conditions delimit the class of systems for which the unified OM formalism applies, distinguishing between jump processes of finite and infinite activity and clarifying the transition from continuous-diffusive to strongly non-Gaussian jump behaviour.

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References:

• "Action functionals for stochastic differential equations with Lévy noise" (Yuan et al., 2019)
• "Probability Flow Approach to the Onsager–Machlup Functional for Jump-Diffusion Processes" (Huang et al., 2 Sep 2024)
• "Transition pathways for a class of high dimensional stochastic dynamical systems with Lévy noise" (Hu et al., 2021)
• "Onsager-Machlup action functional for stochastic partial differential equations with Lévy noise" (Hu et al., 2020)