Stochastic Curve Shortening Flow

• Stochastic curve shortening flow is a geometric evolution equation that combines curvature-driven motion with noise whose amplitude scales with the curve's length, capturing realistic fluctuations in interface dynamics.
• The flow is reformulated as a quasilinear SPDE and stochastic Stefan problem, allowing rigorous analysis using maximal L^p-regularity methods in Sobolev–Besov spaces.
• This model reveals novel behaviors including finite-time blow-up due to either curvature divergence or extreme length changes, highlighting its impact on understanding stochastic interface evolution.

Stochastic curve shortening flow (SCSF) is a geometric evolution equation for planar curves in which the normal velocity is driven by both mean curvature and a randomly fluctuating perturbation whose amplitude explicitly depends on the current geometric scale, specifically the total curve length. This stochastic partial differential equation (SPDE) arises in physical and mathematical models where interfaces are subject to both dissipative curvature-driven motion and random forces whose intensity reflects the instantaneous size of the evolving curve. SCSF generalizes classical deterministic curve shortening by incorporating “scale-dependent noise,” yielding novel regularity, well-posedness, and blow-up phenomena that require advanced quasilinear stochastic analysis for rigorous treatment (Yan, 26 Nov 2025).

1. Formulation of Stochastic Curve Shortening Flow with Scale-Dependent Noise

The classical (deterministic) curve shortening flow evolves a smooth embedded curve $\gamma: S^1 \times [0,T) \to \mathbb R^2$ according to

$\partial_t \gamma(x,t) = -k(x,t)\,n(x,t)$

where $k(x,t)$ is the scalar curvature and $n(x,t)$ the inward unit normal. To model physically realistic fluctuations in interface evolution, SCSF augments the normal velocity with a Brownian perturbation proportional to the curve’s instantaneous length $L(t)$. The evolution of the curve then becomes, in Stratonovich form,

$$d\gamma_t = -\left[k\,dt + \sigma L(t) \circ dW_t \right] n$$

where $\sigma>0$ is the noise intensity and $W_t$ denotes a standard Brownian motion. The key scale-dependence is encoded by the $L(t)$ factor: as the interface length increases, so does the stochastic forcing amplitude, reflecting the accumulation of microscopic random effects over a larger domain (Yan, 26 Nov 2025).

2. Quasilinear SPDE Reformulation and Stefan Problem Equivalence

Direct analysis of the geometric SCSF is analytically intractable due to the nonlinear, nonlocal interaction between the geometry and the noise. To render the problem accessible to rigorous SPDE methods, the SCSF is equivalently formulated as a one-phase stochastic Stefan problem involving the pair $(k(s,t), L(t))$, where $k$ is parameterized by the arclength $s \in [0, L(t)]$. The coupled $(k,L)$ system is

$$\begin{aligned} dk(s, t) & = [\partial_{ss}k + k^3]\,dt + \sigma k^2 L(t) \circ dW_t,\quad 0<s<L(t) \ dL(t) & = -\int_0^{L(t)} k^2\,ds\,dt - \sigma L(t)\int_0^{L(t)}k\,ds \circ dW_t \end{aligned}$$

For a simple closed curve, $\int_0^{L} k\,ds = 2\pi$, which simplifies the $L$ equation. This system is recast in fixed domain variables $r \in \mathbb T = \mathbb R/\mathbb Z$, $f(r,t) = k(rL(t), t)$, giving a quasilinear SPDE on a product space: $$\begin{aligned} df &= [ (2\sigma^2\pi^2 r^2 + 1/L^2) f - 4\sigma^2\pi r L f^2 + 2\sigma^2\pi^2 r f \ &\ \ \ - r f \int_\mathbb T f^2 dr + f^3 + \sigma^2 f^3 L^2 - \sigma^2\pi f^2 L ]\,dt \ &\qquad + \sigma (f^2 L - 2\pi r f) dW_t \ dL & = L [2\sigma^2\pi^2 - \int_\mathbb T f^2 dr ] dt - 2\sigma\pi L dW_t \end{aligned}$$ (Yan, 26 Nov 2025). This formulation resolves the degeneracy in the geometric description by recasting the free-boundary evolution in an analytic form suitable for stochastic maximal $L^p$-regularity methods.

3. Local Well-Posedness and Regularity Theory

The primary mathematical challenge is to establish maximal local strong solutions in Sobolev–Besov–UMD Banach space settings for a system of coupled, nonlinear, multiplicative-noise-driven SPDEs. The Agresti–Veraar theory for quasilinear stochastic evolution equations is invoked: for state spaces $X = L^q(\mathbb T) \times \mathbb R$, $X_1 = W^{2,q}(\mathbb T) \times \mathbb R$, and interpolation spaces $X_p = B^{2-2/p}_{q,p}(\mathbb T) \times \mathbb R$, one obtains existence and uniqueness of local (in time) solutions up to a maximal stopping time $\tau$,

$$U = (f, L) \in L^p(\Omega; H^{\theta,p}([0, \tau_n]; H^{2(1-\theta), q}(\mathbb T)\times\mathbb R)) \cap L^p(\Omega; C([0, \tau_n]; B^{2-2/p}_{q,p}(\mathbb T)\times\mathbb R))$$

with blow-up (maximality) characterized by either divergence of curvature, collapse ($L(t) \to 0$), or unphysical expansion ($L(t) \to \infty$) (Yan, 26 Nov 2025). This establishes local strong well-posedness for SCSF with scale-dependent noise under physically meaningful initial data, resolving the analytical control of the scale-multiplicative stochastic perturbation.

4. Geometric and Analytical Structure of Scale-Dependent Forcing

The use of “scale-dependent” noise—noise intensity depending on the evolving length $L(t)$—is physically motivated by the cumulative effect of independent random fluctuations acting over a spatially extended interface. In the SCSF, every point of the evolving curve experiences a stochastic normal displacement of form $\sigma L(t)\,dW_t$, which is spatially homogeneous but temporally modulated by $L(t)$. This contrasts with classical additive (scale-independent) noise in standard SPDEs. Notably, the length process $L(t)$ evolves as a geometric Brownian motion modulated by the total squared curvature: $$L(t) = L_0 \exp\left[ -\int_0^t \int_\mathbb T f^2(r,s) dr ds - 2\sigma\pi W_t \right]$$ (Yan, 26 Nov 2025). This closed-form reveals explicit stochastic exponential damping and amplification of the interface length, a distinctive feature of scale-dependent SCSF.

5. Connections to Physical Models and Interface Stochasticity

Scale-dependent noise is a recurring principle in physical models where the interface “collects” random kicks per unit length or area—e.g., molecular dynamics of fluctuating interfaces, stochastic dynamics of front propagation, or crystal facet evolution. In such models, the total stochastic forcing typically scales with the system’s current geometric measure (e.g., length for curves, area for surfaces). The SCSF serves as a rigorous geometric-probabilistic framework for these phenomena, extending classical curvature-driven models to regimes where stochastic effects are not negligible and are fundamentally correlated with geometric scale (Yan, 26 Nov 2025).

6. Mathematical Implications and Future Directions

The SCSF with scale-dependent noise exhibits several features absent in deterministic or scale-independent stochastic geometric flows:

• The scaling of the noise with length profoundly alters both the regularity and the possible singularity formation. Blow-up in finite time can stem not only from classical curvature singularities but also from stochastic fluctuations driving $L(t) \to 0$ or $L(t) \to \infty$.
• The system’s reformulation as a stochastic Stefan problem suggests possible generalizations to moving boundary problems and multidimensional mean curvature flows with scale-multiplied noise.
• Open problems include long-time behavior, statistical steady states, behavior under multiplicative and spatially inhomogeneous noise, and continuum limits for interfaces in high-noise regimes.
• The quasilinear SPDE analysis techniques developed in this context may extend to stochastic geometric flows for higher-dimensional hypersurfaces or for anisotropic mean curvature models.

7. Summary Table: Core Components of SCSF with Scale-Dependent Noise

Component	Mathematical Formulation	Key Reference
Geometric evolution	$d\gamma_t = -[ k\,dt + \sigma L(t) \circ dW_t ] n$	(Yan, 26 Nov 2025)
Curvature/length SPDE	$(k,L)$ system with multiplicative $L(t)$ in noise and instantaneous mean curvature dissipation	(Yan, 26 Nov 2025)
Quasilinear reformulation	SPDE for $(f,L)$ in $B^{2-2/p}_{q,p}(\mathbb T)\times\mathbb R$ with Stratonovich-to-Itô correction	(Yan, 26 Nov 2025)
Well-posedness framework	Agresti–Veraar stochastic maximal $L^p$-regularity for quasilinear SPDEs in UMD Banach spaces	(Yan, 26 Nov 2025)
Blow-up criterion	Finite time singularity if $L(t)\to 0$, $L(t)\to\infty$, or curvature norm diverges	(Yan, 26 Nov 2025)
Explicit law for $L(t)$	$$L(t) = L_0 \exp ( -\int_0^t \int_\mathbb T f^2 dr ds - 2\sigma\pi W_t )$$	(Yan, 26 Nov 2025)

This framework rigorously defines and analyzes the stochastic curve shortening flow under physically realistic, scale-modulated stochastic forcing, and provides a mathematically precise link between geometric interface evolution and scale-dependent probabilistic effects.