Mosco-Consistent Perturbations Analysis

• Mosco-consistent perturbations are defined as families of changes ensuring the Mosco convergence of forms or functionals, thereby stabilizing solutions in variational and probabilistic frameworks.
• They guarantee strong L2-convergence of semigroups and resolvents while highlighting limitations in preserving global path properties of the underlying processes.
• This framework finds applications in PDEs, stochastic variational inequalities, and machine learning, with robust techniques ensuring reliable approximations under perturbations.

Mosco-consistent perturbations are families of changes to the coefficients, constraint sets, domains, or measures in variational and probabilistic frameworks that ensure Mosco convergence of the associated forms or functionals. This property underpins robust stability of solutions and semigroups in the presence of perturbations, without guaranteeing conservation of global path properties of the associated Markov processes. The concept is central to modern analysis of PDEs, stochastic equations, variational inequalities, and machine learning, providing both necessary and sufficient conditions for the stable solvability and convergence of a broad range of problems under perturbation.

1. Fundamental Definition and Abstract Framework

Mosco convergence refers to a specific mode of convergence for closed quadratic forms (or more generally, convex functionals, or closed subspaces) on a Hilbert or Banach space. A sequence $(\mathcal{E}_n, \mathcal{F}_n)$ converges in the sense of Mosco to $(\mathcal{E}, \mathcal{F})$ if it satisfies two canonical conditions:

• (M1) Lower bound under weak convergence: For every $u \in H$ and every sequence $u_n \rightharpoonup u$ in $H$,

$$\liminf_{n\to\infty} \mathcal{E}_n(u_n, u_n) \geq \mathcal{E}(u, u).$$

• (M2) Existence of recovery sequences: For every $u \in H$ there exists $u_n \to u$ strongly in $H$ such that

$$\limsup_{n\to\infty} \mathcal{E}_n(u_n, u_n) \leq \mathcal{E}(u, u).$$

When the perturbations in the underlying coefficients, constraint sets, domains, or measures guarantee both (M1) and (M2), they are termed Mosco-consistent perturbations (Suzuki et al., 2014). Mosco consistency is characterized as a minimal condition, ensuring strong $L^2$-convergence of semigroups and resolvents, as well as continuous dependence of minimizers and solutions on the perturbation parameter.

2. Sufficient and Necessary Conditions across Models

The Mosco framework admits precise criteria for convergence in diverse settings:

• Diffusions: If $A_n(x)$ are symmetric diffusion matrices satisfying local uniform ellipticity, and $A_n \to A$ locally in $L^1(\mathbb{R}^d)$, then the forms $$\mathcal{E}_n(u,u) = \int A_n(x) |\nabla u(x)|^2 dx$$ Mosco-converge to the limiting form (Suzuki et al., 2014).
• Lévy processes: Taking characteristic exponents $\varphi_n \to \varphi$ locally in $L^1$, the corresponding forms in Fourier space satisfy Mosco convergence (Suzuki et al., 2014).
• Jump processes: Under local $L^1$ convergence and majorization of jump kernels $J_n(x,y) \leq J(x,y)$, the associated forms are Mosco-consistent (Suzuki et al., 2014).
• Variational inequalities with convex constraint sets: Mosco-convergence of sets $C_n\to C$ requires any weak limit from $C_n$ to be in $C$ and strong approximability of $C$ by $C_n$, which ensures the strong convergence of solutions to variational inequalities posed on such sets (Boccardo et al., 9 May 2025, Ngiamsunthorn, 2011).
• Sobolev and $H(\text{curl})$ spaces under domain perturbation: Lipschitz, FR-class domain boundaries and appropriate Hausdorff convergence of domains ensure Mosco-consistency for Sobolev spaces and valid Sobolev inequalities (Fornoni et al., 2022, Liu et al., 2016).

In empirical risk minimization (ERM), Mosco-consistency for convex, l.s.c. risk functionals $L_{D_n}$ is both necessary and sufficient for intrinsic well-posedness of the minimizer map, as characterized by Painlevé-Kuratowski upper semicontinuity (Bounja et al., 25 Jan 2026).

3. Recipes and Techniques for Constructing Mosco-Consistent Perturbations

A generic process for building Mosco-consistent perturbations involves:

1. Base form selection: Choose a quadratic form or functional $\mathcal{E}$ with coefficient(s) $c(x)$.
2. Perturbation design: Modify $c$ to $c_n$, ensuring local $L^1$ convergence ($c_n \to c$ in $L^1_{\mathrm{loc}}$) and maintenance of minimal ellipticity or majorization—e.g., uniform nondegeneracy for diffusions or upper domination for jumps.
3. Verification of Mosco conditions: For each $u \in H$, construct a recovery sequence; for every weak limit from the perturbed forms, ensure the lower bound holds.

Examples of this approach span from thin domains (using connecting operators and weighted spaces) and dynamic boundary layers to discrete-to-continuum graph approximations, always checking reconstruction and lower-bound conditions (Giga et al., 20 Nov 2025). Finite-element tent functions and Muckenhoupt-type bounds support analogous procedures for forms on variable measures, including non-convex interaction potentials (Grothaus et al., 2021).

4. Instability of Global Path Properties

While Mosco convergence provides strong $L^2$-convergence of semigroups, resolvents, and finite-dimensional laws for Markov processes, critical global path properties such as recurrence/transience, conservativeness/explosion, and even the nature (jump vs. diffusion) of the limiting process need not be preserved. Explicit counterexamples illustrate:

• Diffusions can switch between explosion and conservativeness in the limiting process, despite Mosco convergence (Suzuki et al., 2014).
• Lévy-type processes can flip between recurrence and transience as exponents or jump intensities change (Suzuki et al., 2014).
• Similarly, jump process families may undergo analogous transitions.

This suggests that to guarantee preservation of such properties, nonlocal, stronger controls are needed (spectral gaps, Hardy-type conditions, or uniform heat kernel bounds), far beyond the scope of local $L^1$ convergence or Mosco-consistency (Suzuki et al., 2014).

5. Applications in PDEs, Stochastic and Machine Learning Models

Mosco-consistent perturbations play a crucial role in:

• Parabolic and elliptic PDEs: Stability of solutions under domain perturbations follows directly from Mosco-consistency in the underlying function spaces (Ngiamsunthorn, 2011, Fornoni et al., 2022). Both the construction and semicontinuity of invariant manifolds near equilibria are guaranteed under Mosco domain convergence (Ngiamsunthorn, 2011).
• SPDEs and SVI: Random Mosco convergence characterizes stability in stochastic variational inequalities, allowing robust approximation by Trotter-type (p-Laplace), homogenization, and nonlocal-to-local models (Gess et al., 2015). The limiting behavior is governed by Mosco-consistency, regardless of the structural complexity of the drift potentials.
• Nonlocal-to-local transitions: Families of nonlocal quadratic forms (fractional, jump-type) can be shown to converge to gradient-type local forms, facilitating diffusion approximations of jump processes and rigorous BBM-type limits (Gounoue et al., 2019, Yang, 2019).
• Machine learning and ERM: For general convex empirical risk minimization, Mosco-consistent perturbations provide the minimal stability regime, ruling out pathological behaviors such as spurious minimizers or blow-up phenomena (Bounja et al., 25 Jan 2026).

6. Practical Criteria and Limitations

Typical indicators for Mosco-consistency include:

• Local $L^1$ convergence for coefficients, exponents, and jump densities (forms).
• Weak convergence of measures with residuum and perturbation bounds (finite-element approximate schemes).
• Pointwise recovery and weak-lim inf criteria for convex constraint sets or domain subspaces (Banach or Hilbert spaces).
• Validity of abstract conditions on connecting operators, domain regularity, and functional energetic bounds.

Despite its broad applicability, it must be noted that Mosco-consistency does not imply conservation of deeper probabilistic or qualitative spectral features. Counterexamples elucidate the failure of persistence for global features under Mosco-consistent perturbations (Suzuki et al., 2014).

7. Summary Table: Mosco-Consistent Perturbation Regimes

Model Context	Sufficient Condition	Limitation
Elliptic Diffusions	$A_n \to A$ locally in $L^1$, ellipticity	No guarantee for explosion
Lévy/Jump Processes	$\varphi_n \to \varphi$/$J_n \to J$ in $L^1$	Recurrence/transience may change
Convex Constraint Sets	Weak-lim inf/SAA for sets	No structural path guarantee
Domain Variations	Hausdorff convergence, FR-class regularity	Might lose topological properties
ERM/Stochastic Optimization	Mosco for loss functionals, bounded minimizers	Solution set can change size

Mosco-consistent perturbations constitute the minimal, robust approach for the stability and convergence of solutions in nonlinear analysis, stochastic equations, inverse problems, and optimization under highly nontrivial perturbations. Their limitations arise only when strictly global, qualitative behaviors of the underlying processes are of interest, where additional structural controls must be imposed (Suzuki et al., 2014, Bounja et al., 25 Jan 2026, Fornoni et al., 2022).