Integro-Differential Operators with Unbounded Coefficients

• The topic is defined as an operator combining differential and integral terms with unbounded coefficients that create unique analytical challenges.
• It applies to quantum theory, stochastic control, and finance, employing techniques like Carleman estimates, spectral schemes, and deep BSDEs.
• Numerical and analytical methods, including operator exponentials and variational approaches, ensure well-posedness and effective control in complex models.

An integro-differential operator with unbounded coefficients is a mathematical operator combining both differential and integral terms, where the coefficients involved may be unbounded (i.e., not uniformly bounded on the underlying domain). Such operators arise naturally in diverse fields—including quantum measurement theory, stochastic control, kinetic equations, nonlocal models, and mathematical finance—where modeling of memory, jumps, or spatially heterogeneous (possibly singular) dynamics is required. The unboundedness of coefficients introduces profound analytic and computational challenges, fundamentally affecting the formulation of well-posedness, regularity, numerical approximation, and control.

1. Definitions and Canonical Structures

An integro-differential operator $\mathcal{L}$ typically acts on a function $u$ as

$$\mathcal{L}u(x) = \sum_{i,j} a_{ij}(x) \frac{\partial^2 u}{\partial x_i \partial x_j} + \sum_{j} b_j(x) \frac{\partial u}{\partial x_j} + c(x)u(x) + \int_{\mathbb{R}^d} K(x, y)\left[u(y) - u(x)\right]\,\nu(dy)$$

where coefficients $a_{ij}$, $b_j$, $c$, or the kernel $K$ (or the Lévy measure $\nu$) can be unbounded.

Examples of operators in key settings:

• Elliptic/Parabolic Partial Integro-Differential Operators: Nonlocal terms model jumps or memory (e.g., Lévy processes, fractional Laplacians).
• Quantum Stochastic Differential Operators: Unboundedness enters through creation/annihilation or coupling operators (e.g., Hudson-Parthasarathy QSDEs) (Santis et al., 2010).
• Stochastic Control and HJB Equations: Unbounded coefficients reflect the high activity of jumps or growth in controlled SDEs (Moreno-Franco, 2016, Jakobsen et al., 12 Jul 2024).
• Fractional p-Laplacians with Measurable Kernels: Kernel $K(x, y)$ may be merely measurable or have power-law singularity, yielding unboundedness (Korvenpaa et al., 2016).

The unboundedness can occur in differential coefficients (e.g., $a_{ij}$ or $b_j$ growing as $|x|\to\infty$), in nonlocal kernels with singularities, or through unbounded operator-valued coefficients (e.g., in Banach or Hilbert space settings relevant for quantum/probabilistic or PDE control formulations).

2. Analytical Frameworks: Existence and Uniqueness

The unboundedness of coefficients invalidates straightforward use of classical theory meant for bounded settings (semigroup generation, monotone/variational methods, or fixed-point theory). Several analytical strategies have been developed:

• Hudson-Parthasarathy Framework (Quantum Control, Continuous Measurement):
  • Choosing dense, invariant cores (domains of essential self-adjointness).
  • Ensuring closability and dissipativity for operator coefficients $K, R_i, N_j$.
  • Employing a positive self-adjoint operator $C$ so that $F_{ij}(1+\varepsilon C)^{-1/2}$ is bounded (regularizing the unbounded coefficients) (Santis et al., 2010).
• Degenerate Parabolic/Elliptic Problems:

Carleman estimates with weights tailored to unbounded or degenerate coefficients establish uniqueness and continuous dependence on data (Lorenzi et al., 2012). Weighted Hölder/Sobolev spaces are used to treat degenerate diffusions or drift-dominated problems, often employing Lyapunov function methods to control explosion at infinity (Feehan et al., 2012, Angiuli et al., 2018).

• Fractional Nonlocal and Nonlinear Operators:
  • Monotonicity and coercivity of the operator, even with unbounded measurable coefficients.
  • Regularity and boundary behavior are established through nonlocal Caccioppoli inequalities and tail estimates.
• ABP Maximum Principle and Comparison Techniques:

Maximum and comparison principles are extended for fully nonlinear integro-differential operators, even when the inhomogeneous term is unbounded (relaxing previous dependence on $L^\infty$ norms to $L^n$ norms) (Kitano, 2022), and for nonlinear equations with nonlocal terms on unbounded domains, provided suitable monotonicity and regularity conditions for coefficients and kernels (Ladas et al., 2020).

• Variational and Integrated Semigroup Approaches:

For non-densely defined/unbounded operators in Banach spaces (e.g., in inclusion problems or with nonlocal initial/boundary conditions), integrated semigroup theory is used to reconstruct well-posedness, leveraging measures of noncompactness, weak sequential continuity, and fixed-point theorems (Pietkun, 2018).

3. Computational Methods and Numerical Analysis

Solving equations involving integro-differential operators with unbounded coefficients poses particular challenges:

Spectral and Collocation Schemes:

• Exponentially convergent schemes based on Duhamel–type integral formulations allow discretization in time for variable domain/unbounded coefficient problems. Chebyshev–Gauss–Lobatto nodes are used for spectral convergence, and unbounded coefficients are handled by transforming variable domains into fixed ones, followed by operator exponential approximations (Bohonova et al., 2010).
• Operator exponential terms are computed via Dunford–Cauchy integrals along hyperbolas, enveloping the (possibly unbounded) spectrum of the operator, with Sinc and Gauss quadratures ensuring exponential accuracy (Vasylyk, 2013).

Finite Difference Methods:

• Standard truncation or splitting approaches (treating the Lévy operator as second-order inside a small ball and lower order outside) suffer from blowing-up error constants as the truncation radius shrinks. Newer schemes treat the entire integro-differential operator as genuinely second-order, thereby obtaining error estimates independent of artificial truncation parameters, and requiring no additional regularity on the Lévy measure (Dareiotis, 2016).

Probabilistic and Machine Learning Approaches:

• Deep BSDE schemes encode the solution to nonlocal, possibly unbounded-coefficient integro-PDEs through FBSDEs. Infinite activity (unboundedness from jumps) is handled by replacing small jumps with a Gaussian correction, simulating large jumps explicitly, and employing deep neural network regression for the backward (conditional expectation) component. The schemes provide convergence even in the infinite activity (unbounded) case and are scalable to high dimension, avoiding the curse of dimensionality (Jakobsen et al., 12 Jul 2024).
• Neural operator regression employs neural nets parameterizing both frequency- and field-dependent components in Fourier space, enabling flexible discovery of (possibly unbounded) operator coefficients and scaling from observed flow data. This approach is effective for systems such as the fractional heat equation or Kuramoto–Sivashinsky, even when the underlying coefficient grows without bound (Patel et al., 2018).

4. Inverse and Control Problems

For operators with unbounded coefficients, the inverse recovery and control theory require advanced tools:

• Nodal and Spectral Data Inverse Theory:

Asymptotic expansions for the nodal points of Dirac-type operators with Volterra nonlocal perturbations enable full recovery of both the local (potential) and nonlocal (integral kernel) coefficients. This remains feasible even when the integral kernel may induce unboundedness, provided the necessary nodal sequences are available (Keskin et al., 2016, Keskin et al., 2017).

• Riccati Equations and Control for PDEs with Unbounded Operators:

The Riccati equation arising in boundary control problems for coupled PDEs with unbounded operators (e.g., boundary input for hyperbolic-parabolic systems) exhibits uniqueness and properness of feedback laws under singular estimates on operator semigroups and the control/observation maps (Acquistapace et al., 2020). Exponential stabilization of parabolic integro-differential equations (e.g., viscoelastic fluids) with unbounded observation operators is achieved through finite-dimensional feedback control constructed via the solution of an algebraic Riccati equation; energy-based arguments are critical for control design (Dharmatti et al., 2017).

5. Special Applications and Extensions

• Quantum Stochastic Differential Equations and Continuous Quantum Measurement:

Full master equations derived from quantum stochastic calculus with unbounded operator coefficients (creation/annihilation) yield integro-differential evolution for reduced characteristics (e.g., for measured quantum observables). Technical conditions guarantee unitary solution cocycles and Markov evolution even in the physical presence of unbounded system–field couplings (Santis et al., 2010).

• Obstacle and Free-Boundary Problems for Nonlocal Nonlinear Operators:

For fractional $p$-Laplacian–type operators with unbounded (measurable) kernels, variational methods demonstrate that solution regularity (boundedness, Hölder, continuity) can be inherited from obstacle functions even when the kernel introduces singularities (Korvenpaa et al., 2016).

• Nonlinear Dirichlet Problems and Orthogonal Projection:

The method of orthogonal projection in the space associated to symmetric Dirichlet forms, robust to unbounded coefficients and measure data, yields existence (and precise interpretation) for Dirichlet problems with irregular boundary data, via decomposition into interior and harmonic components (Klimsiak et al., 2023).

• Comparison Principles and Maximum Norms on Unbounded Domains:

For fully nonlinear integro-differential problems and equations posed on unbounded domains, maximum and comparison principles are extended—relaxing regularity and integrability conditions required of the coefficients—by careful weighting techniques and control on the tails of the measure or kernel (Ladas et al., 2020, Kitano, 2022).

6. Open Problems and Future Directions

The rigorous treatment and simulation of integro-differential operators with unbounded coefficients remains a challenging and highly active research area. Directions of particular importance include:

• Enhanced regularity theory and sharp maximum principles for nonlocal, nonlinear equations in multidimensional and non-Euclidean settings.
• High-dimensional numerical methods (esp. for unbounded jump activity)—with ongoing development in machine learning, stochastic approximation, and spectral techniques.
• Inverse spectral and control identification in systems with unbounded nonlocalities, especially in applications to quantum systems and statistical physics.

Summary Table: Analytical and Computational Approaches

Approach	Core Idea	Setting/Assumptions
Dissipative / Core Techniques (Santis et al., 2010)	Dense domains, dissipativity, boundedness under regularization	QSDEs, unbounded operator coefficients
Generalized Duhamel/Operator Exponential (1003.25371304.1271)	Integral transforms, spectral/quadrature methods	Banach space, time-dependent/unbounded domain
Nonlocal Variational & Monotone Operator (Korvenpaa et al., 2016)	Nonlocal Caccioppoli, coercivity via tail terms	Fractional p-Laplacian, measurable kernels
Deep BSDE / Neural Regression (Jakobsen et al., 12 Jul 2024Patel et al., 2018)	FBSDE stochastic simulation, small-jump–to–Gaussian approximation, neural nets for regression	Nonlinear (integro-)PDE, high dimension, unbounded nonlocal operators
Carleman/Comparison Principle Techniques (1210.20652011.15058Kitano, 2022)	Weighted identities, minimum principles, sharp ABP estimates	Unbounded spatial domains, unbounded coefficients

The paper and application of integro-differential operators with unbounded coefficients unites advanced operator theory, functional analysis, stochastic theory, harmonic analysis, and computational methods, and underpins state-of-the-art modeling in physics, finance, control theory, and machine learning for PDEs.