Locally Lipschitz Coefficients in Analysis

• Locally Lipschitz coefficients are functions that satisfy Lipschitz conditions on compact sets, ensuring the local existence and uniqueness of solutions in differential equations.
• They enable powerful extension theorems and preserve pointwise Lipschitz constants, which are critical in the analysis of Sobolev spaces on metric spaces.
• Their usage is central in stochastic systems and numerical schemes, where local bounds replace global Lipschitz conditions to manage nonlinearity and approximation accuracy.

A function or map is said to have locally Lipschitz coefficients if it satisfies a Lipschitz condition in a neighborhood of any point, but may not satisfy a uniform Lipschitz bound globally. Locally Lipschitz regularity occurs pervasively in the analysis of differential equations in metric, deterministic, and stochastic settings—including ODEs, SDEs, SPDEs, as well as in the theory of function spaces on metric spaces and interacting particle systems. The property is both a crucial technical tool enabling well-posedness in models where global Lipschitz continuity fails (e.g., under polynomial or super-linear growth) and an essential ingredient in developing approximation, extension, and convergence results for various numerical and analytic schemes.

1. Definitions and General Properties

A function $f : \mathbb{R}^n \to \mathbb{R}^m$ is locally Lipschitz if, for every compact set $K\subset\mathbb{R}^n$, there exists a constant $L_K>0$ such that

$$\|f(x)-f(y)\| \leq L_K \|x-y\| \ \ \forall x,y\in K.$$

This property generalizes to maps $f: C \to Y$ between metric spaces, and to coefficients $b(x), \sigma(x)$ in SDE/SPDE models, expressed as: $$|b(x)-b(y)| + |\sigma(x)-\sigma(y)| \le L_K |x-y|, \quad x, y \in K \quad\text{for each compact }K.$$ Local Lipschitz continuity ensures uniqueness and local existence for ODEs/SDEs/SPDEs up to the (possibly infinite) explosion time, which must be controlled via additional growth or monotonicity (one-sided Lipschitz) conditions to guarantee global solutions (Protter et al., 2017, Tretyakov et al., 2012).

In some contexts, “locally Lipschitz” refers to Lipschitz continuity on arbitrary bounded or compact subsets, with the modulus possibly depending on the chosen subset, and not being uniform over the whole space (Wang et al., 2020).

2. Role in Extensions, Relaxations, and Sobolev Theory

Locally Lipschitz structure is central in the theory of function spaces on metric spaces, and in extension problems:

• Let $(X,d)$ be a metric space, $C \subset X$, and $g: C \to \mathbb{R}$. Define, for $x\in C$,

$$\mathrm{lip}_\mathrm{a}(g,x) := \lim_{r\downarrow 0} \sup_{y_1,y_2 \in C \cap B_r(x)} \frac{|g(y_1) - g(y_2)|}{d(y_1, y_2)}$$

the asymptotic/pointwise Lipschitz constant of $g$. Any extension $f \supset g$ satisfies $$\mathrm{lip}_\mathrm{a}(g,x) \leq \mathrm{lip}_\mathrm{a}(f,x)$$ for $x\in C$.

• Di Marino, Gigli, and Pratelli show that for any locally Lipschitz $g$ defined on $C$, and any $\varepsilon>0$, there is a global extension $f$ to $X$ with global Lipschitz constant $L + \varepsilon$, which preserves the pointwise Lipschitz constant: $$\mathrm{lip}_\mathrm{a}(f,x) = \mathrm{lip}_\mathrm{a}(g,x)$$ for all $x \in C$ (Marino et al., 2020).
• This extension result leads to the invariance (under support changes) of relaxation-based Sobolev spaces (à la Cheeger), showing that Sobolev spaces $\mathrm{Ch}_{p,X}$ defined by asymptotic Lipschitz constants depend only on the support of the measure (Marino et al., 2020).

Locally Lipschitz extension thus underpins the analysis of nonlinear and non-smooth spaces, where global regularity fails, and is essential in the precise definition and manipulation of function spaces—especially in the setting of metric measure spaces.

3. Locally Lipschitz Coefficients in Differential Equations

Stochastic Differential Equations (SDEs)

• Existence and uniqueness of strong solutions for SDEs

$dX_t = b(X_t)dt + \sigma(X_t) dW_t$

can often be established with locally Lipschitz (rather than globally Lipschitz) $b$ and $\sigma$, provided additional monotonicity (dissipativity, “one-sided Lipschitz”) and/or polynomial/superlinear growth conditions are met (Tahmasebi et al., 2013, Anton, 29 May 2024, Tahmasebi, 2013, Zhang et al., 2014, Protter et al., 2017, Tretyakov et al., 2012). Monotonicity conditions typically take the form

$\langle b(x)-b(y), x-y \rangle \le L|x-y|^2$

which, together with locally Lipschitz behavior, precludes explosion and allows the derivation of moment estimates by Gronwall-type arguments.

• Infinite-dimensional or path-dependent generalizations (McKean-Vlasov, delay equations, neutral SDDEs) use analogous local Lipschitz assumptions, controlling dependencies on both the state and the measure/particle law (Erny, 2021, Reis et al., 2023, Tan et al., 2017).
• In models with fractional Brownian motion or combined Brownian/fractional noise, pathwise/semimartingale techniques, truncation, and stopping time arguments are employed to build solutions under local Lipschitz continuity, as in (Wang et al., 2020).

Stochastic Partial Differential Equations (SPDEs)

• For SPDEs driven by space-time white noise, global well-posedness is established for locally Lipschitz drift and diffusion with at most linear growth (uniformly in time), by reducing to globally Lipschitz truncated systems and employing moment/tail estimates and probabilistic cutoff arguments (Guerngar et al., 9 Nov 2025, Foondun et al., 14 Nov 2024). Such results extend to fractional time/space SPDEs, where the behavior of truncated Lipschitz constants (notably for $\sigma$) must be tracked carefully in relation to the regularity properties of the kernel.

4. Analytical and Malliavin Regularity Under Local Lipschitz

Malliavin Calculus and Smoothness of Densities

• For (S)DEs with locally Lipschitz (in space) coefficients, smoothness and nondegeneracy of the law can be recovered by approximation schemes:
  • Construct a sequence of globally Lipschitz approximations—usually via truncation or mollification—with the same initial data (Tahmasebi et al., 2013, Tahmasebi, 2013, Reis et al., 2023).
  • Prove uniform bounds for moments and (higher-order) Malliavin derivatives of the approximating solutions (using Lyapunov functions and/or monotonicity).
  • Pass to the limit, using Banach-Alaoglu or compactness arguments, to obtain the desired properties (e.g., Malliavin differentiability of all orders (Anton, 29 May 2024), smooth density (Tahmasebi et al., 2013, Tahmasebi, 2013)).
• In high-dimensional, interacting, or mean-field settings (McKean–Vlasov SDEs), Malliavin regularity can be “transferred” from the finite-particle system to the limit via propagation of chaos, provided locally Lipschitz and one-sided Lipschitz conditions hold for the drift, and the diffusion is uniformly Lipschitz (Reis et al., 2023).
• In Hörmander-type settings, local Lipschitz together with nondegeneracy conditions (satisfied at a single point, or on a set) suffice for the solution’s law to admit a $C^\infty$-density (Tahmasebi, 2013, Anton, 29 May 2024).

5. Numerical Schemes: Stability and Convergence

Numerical solutions of SDEs/SPDEs demand careful handling of local lipschitzness:

• Euler and Milstein schemes can diverge in the presence of nonglobal Lipschitz coefficients; one must localize or “tame” increments (balanced/tamed schemes) to obtain convergence (Zhang et al., 2014, Tretyakov et al., 2012, Protter et al., 2017).
• Stopping-time/truncation techniques (simulate up to the exit of a compact) allow strong/weak convergence results at the usual rates—provided the underlying continuous equation cannot explode and appropriate moment bounds are maintained (Protter et al., 2017, Tretyakov et al., 2012, Zhang et al., 2014).
• Fundamental mean-square convergence theorems are valid under local Lipschitz conditions with monotonicity and polynomial growth, subject to boundedness of the solution's moments (Tretyakov et al., 2012).

Method/Class	Local Lipschitz Usage	Convergence Principle
Truncated Scheme	Coefficient truncation on compact sets	Converges to original as cutoff increases (Tretyakov et al., 2012)
Balanced/Tamed Euler	Bounded increments via nonlinear functions	Prevents divergence, retains strong order (Zhang et al., 2014)
Implicit θ/Euler-Maruyama	Well-posedness relies on monotonicity/local	Mean-square order $1/2$ (or $1$ if commutativity holds)

6. Applications and Structural Consequences

• Locally Lipschitz coefficients support extension operators and Sobolev space invariance essential in metric measure analysis, particularly over highly singular or fractal spaces (Marino et al., 2020).
• In stochastic models, polynomial or superlinear drift/diffusion (outside globally Lipschitz regime) naturally fit the locally Lipschitz setting—ubiquitous in finance, biology, and physics (e.g., population models, CIR processes, neural models).
• Well-posedness of McKean–Vlasov equations, propagation of chaos, and regularity for mean-field limits are established using truncation, Osgood’s lemma, and monotonicity structures under local Lipschitz conditions (Erny, 2021, Reis et al., 2023).
• Averaging principles in slow-fast systems—whether driven by Brownian, fractional Brownian, or mixed noise—leverage locally Lipschitz assumptions, with global analysis made possible via time discretization, truncation, and stopping-time bounds (Wang et al., 2020, Liu et al., 2018).

7. Open Problems and Extensions

• Sharpness of tail and moment bounds in SPDEs driven by space-time white noise with only local Lipschitz/nonlinear coefficients remains a subject of further scrutiny; growth restrictions on truncated Lipschitz constants (e.g., $L_{N,\sigma}$) have been shown to be essentially optimal (Guerngar et al., 9 Nov 2025).
• Analysis of numerical schemes in the presence of both local Lipschitzness and non-Lipschitz growth conditions continues to motivate new explicit strategies (fully tamed, drift-tamed, balanced, or splitting schemes).
• Generalization to infinite-dimensional, path- or measure-dependent systems (e.g., SPDEs, dissipative PDEs with random coefficients, McKean–Vlasov PDEs) significantly deepens the analytic and probabilistic toolkit required, blending truncation, mollification, Malliavin calculus, and compactness techniques.

In conclusion, locally Lipschitz coefficients play a foundational role in modern analysis of differential and stochastic systems, bridging the gap between the classical theory (based on global regularity) and realistic models where such regularity fails, through the deployment of monotonicity, truncation, localization, and approximation paradigms across deterministic and stochastic frameworks.