Non-Smooth Random Curves: Theory & Applications

• Non-smooth random curves are irregular geometric objects with discontinuities and singularities, central to algebraic geometry, stochastic analysis, and optimization.
• Weak frame theory and polygonal approximations extend classical invariants, enabling measure-valued characterizations of curvature and torsion in low-regularity settings.
• Analytic and probabilistic methods applied to non-smooth curves underpin advances in spectral analysis and randomized optimization algorithms with optimal convergence guarantees.

Non-smooth random curves are geometric objects lacking classical smoothness—meaning their tangent vector may not be well-defined everywhere or may exhibit discontinuities and singularities—yet they arise naturally in diverse settings such as algebraic geometry (singular fibers in positive characteristic), stochastic analysis (Brownian trajectories), optimization (irregular descent paths), and spectral theory (interfaces with low regularity). Their paper encompasses probabilistic, analytic, algebraic, and geometric techniques for quantifying, classifying, and manipulating the resulting structures.

1. Algebraic and Geometric Foundations

Non-smooth curves frequently appear as singular fibers in algebraic geometry, especially over fields of positive characteristic. In (Cañate et al., 2014), non-smooth projective curves of arithmetic genus two in characteristic two serve as fibers in proper separable morphisms of smooth varieties, with singularities such as cusps or tacnodes. The minimal equation for such fibers,

$y^2 + a(x)y + b(x) = 0$

with $a(x), b(x)$ of prescribed degrees, encapsulates both smooth and non-smooth fiber types depending on separability and characteristic.

Failures of classical theorems (e.g., Bertini–Sard) in this context manifest as "moving singularities," quantifiable by discrete invariants and genus drop formulas; for instance, Rosenlicht's

$$g - \bar{g} = \sum \dim(\tilde{\mathcal{O}_p}/\mathcal{O}_p)$$

tracks singular primes across a fibration. Classification is achieved via birational equivalence to universal models, reducing families of non-smooth curves to base changes from a fixed fibration.

In characteristic three, (Borelli et al., 29 Jan 2025) applies a "descent approach," extending Katz's theory of connections with zero $p$-curvature, to classify non-smooth regular curves. Singularities are identified by local invariants—the differential degree and associated semigroups—enabling precise arithmetic measurement and concrete classification in moduli spaces, particularly for genus 3 non-hyperelliptic quartics.

2. Differential Geometry and Weak Frame Theory

Non-smooth curves challenge classical Frenet–Serret formalism. Recent work reformulates frame theory in the weak sense, characterizing curves with finite total curvature and total absolute torsion (Mucci et al., 2019, Bevilacqua et al., 2023, Mucci et al., 14 Jun 2025).

Polygonal approximation techniques extend differential geometric invariants to non-smooth curves through limiting processes. The "weak Frenet frame" in projective space (RP²) comprises weak tangent, normal, and binormal indicatrices obtained via piecewise linear constructions and density arguments. The total length of binormal (resp. normal) indicatrix equals the total absolute torsion (resp. curvature plus torsion), providing robust invariants even when the classical derivatives fail.

For curves in Sobolev spaces ($W^{2,2}$), the relatively parallel adapted frame (RPAF) (Bevilacqua et al., 2023) replaces pointwise classical invariants with distributional measures. Curvature is $\kappa = \sqrt{u_1^2+u_2^2}$, with $u_1, u_2$ flexural densities, while torsion is realized as the (distributional) derivative of the associated phase; ambiguity up to a constant rotation in the normal plane must be accounted for. This framework supports variational problems for elastic rods and provides tools for analysis in low-regularity/random curve ensembles.

Measure-valued generalizations (Mucci et al., 14 Jun 2025) recast the Frenet–Serret system for non-smooth curves, prescribing curvature and torsion as Borel measures (distributional derivatives of BV functions), so the fundamental existence and uniqueness theorem, rectifiability, and total curvature/torsion carry over from smooth to non-smooth settings.

3. Stochastic and Probabilistic Analysis

Brownian trajectories and more general Gaussian processes are canonical examples of non-smooth random curves (Wang et al., 22 Aug 2025). Almost surely, Brownian paths are nowhere differentiable, yet admissible geometric descriptors exist:

• Visitation Measures: Quantify time spent by the curve in subsets of space, e.g.,

$$\mu_t(\Delta) = \int_0^1 1_\Delta(x(\xi(s), t)) ds,$$

where $x(\cdot, t)$ is the evolved curve.

• Self-intersection Local Time (SILT): Measures density of self-intersections,

$$L = \int_0^1\int_0^1 \delta_0(W(t_2) - W(t_1))\, dt_1 dt_2,$$

with conditional expectations computed on polygonal approximations:

$$\mathbf{E}[L |\,\text{fixed points}] \sim \frac{(s^*)^2}{\pi u v^*} \exp\left(-\frac{\sqrt{2}uv^*}{s^*}\right), \quad v^* \to \infty.$$

• Polygonal Approximations: The probability that a polygonal line $\Gamma(\vec{u}_1, ..., \vec{u}_n)$ is $\varepsilon$-inscribed in a Brownian trajectory decays polynomially in $\varepsilon$ and the inter-point distances.

Interacting stochastic flows induce intermittency: evolved visitation density concentrates in rare "hot-spots," modeled via stochastic differential equations with interaction terms and characterized by limiting behavior of occupation measures.

4. Optimization on Non-smooth Random Curves

Randomized algorithms for non-smooth, non-convex optimization are critical for training high-dimensional models, where the objective function's minimization trajectory is itself a non-smooth random curve (Zhang et al., 16 May 2024, Xia et al., 19 Aug 2025). The standard stochastic gradient descent with momentum (SGDM) fails theoretically outside convex/smooth loss functions.

Recent advances show that scaling the update step by an exponential random variable resolves this barrier, yielding unbiased estimates of progress and optimal convergence rates,

$\mathcal{O}(c^{1/2} \epsilon^{-7/2}),$

even for Lipschitz and non-smooth landscapes. This connects online convex optimization to non-convex cases via careful regret conversion and probabilistic averaging.

Randomized smoothing techniques generalize gradient-based optimization for functions not globally Lipschitz, introducing an $(\alpha, \beta)$ subgradient growth condition encoding local, rather than global, regularity (Xia et al., 19 Aug 2025). The smoothed function

$$f_\delta(x) = \mathbb{E}_{w \sim U(\mathbb{S}^{d-1})}[f(x + \delta w)]$$

is continuously differentiable, admits generalized smoothness,

$$\|\nabla f_\delta(x) - \nabla f_\delta(y)\| \leq \frac{c\sqrt{d}}{2\delta} [2\alpha(x) + \beta(x, \delta) + \beta(x, \|x-y\|+\delta)] \|x-y\|,$$

and supports convergence to $(\delta, \epsilon)$-Goldstein stationary points with sample complexity $$\tilde{\mathcal{O}}(d^{3/2}\delta^{-1}\epsilon^{-3})$$ in optimized variants.

5. Spectral and Analytical Theory for Non-smooth Interfaces

In quantum mechanics and PDE theory, random or non-smooth curves may arise as interaction supports (interfaces/boundaries). Schrödinger operators with oblique transmission conditions defined on Lipschitz curves (Benhellal et al., 19 Aug 2024) extend known results for smooth cases.

Operators of the form

$H_a(f_+, f_-) = (-\Delta f_+, -\Delta f_-)$

with boundary conditions coupling the traces and derivatives across $E$, the interface, remain self-adjoint, and the resolvent formula persists. However, the lower regularity modifies eigenvalue asymptotics:

$\lambda_n(H_a) = -\frac{B}{2}a^2 + o(a^2) \quad \text{(corners, $B \in (1,4)$)},$

compared to smooth case $\lambda_n(H_a) = -(a^2/4) + o(a^2)$. This sensitivity to geometric roughness impacts localization and spectral properties in physical models.

6. Classification, Deformation, and Singularities

Systematic classification of non-smooth random curves arises in the context of deformation theory and moduli spaces. Some singularities are non-smoothable: i.e., they cannot be deformed to smooth curves (Stevens, 1 Apr 2025). Two main detection methods are used:

• Dimension Count: If a family of singularities has dimension exceeding the smoothing (Deligne) number,

$e = p + t - 1,$

(with $p$ Milnor number, $t$ Cohen–Macaulay type), it cannot lie in the closure of smooth curves.

• Dedekind Different Semicontinuity Criterion (Buchweitz): If an invariant $d_k(C) > 2k\delta$, where $\delta$ is the singularity's delta-invariant, the singularity is not smoothable.

Examples include monomial curves, cones over self-associated point sets, and explicit Gorenstein singularities. Computational algebra systems (e.g., Singular, Macaulay2) enable concrete calculations of $T^1$, $T^2$, deformation spaces, invariants, and verification of smoothability conditions.

7. Broader Implications and Applications

Non-smooth random curves form a unifying concept across algebraic geometry, stochastic processes, optimization theory, and mathematical physics. Their analysis leverages tools from measure theory, numerically effective invariants, probabilistic methods, and advanced computational algebra. Applications span classification of algebraic varieties in positive characteristic, analysis of polymer chains and stochastic flows, development of robust algorithms for AI and machine learning, modeling of quantum systems with irregular interfaces, and the paper of deformation spaces and moduli of singularities.

These advances illustrate that the lack of smoothness does not preclude rigorous geometric, analytic, or algebraic paper; instead, it motivates new invariants, probabilistic characterizations, and measure-theoretic frameworks that reveal deep insights into the structure and evolution of random curves across mathematics and applied sciences.