Local-Global Stochastic Sewing Lemmas

• Local-Global Stochastic Sewing Lemmas are probabilistic techniques that reconstruct integrated processes from local increments under both fine-scale control and global decay conditions.
• They enable robust estimation in systems with jumps, singular coefficients, and non-Markovian noise by combining dyadic decompositions with martingale techniques.
• These lemmas support advanced analysis in SDEs, SPDEs, rough paths, and ergodic theory, providing uniform error bounds and stability estimates.

A local-global stochastic sewing lemma is a probabilistic principle for reconstructing additive (integrated) processes from local, possibly irregular, two-parameter increments under a combination of local and global control conditions. Such lemmas extend the classical and stochastic sewing techniques to provide uniform-in-time or global bounds and to handle settings with jumps, singular coefficients, multi-scale behaviour, non-Markovian noise, or multiple parameters. The “local-global” terminology denotes the ability to combine sharp control on small scales (local sewing) with global damping or decay features (e.g., exponential weights), enabling robust estimation across both fine and large scales. These tools have become central in advanced stochastic analysis, particularly for the study of stochastic differential equations (SDEs), stochastic partial differential equations (SPDEs), regularization by noise, rough integration, and ergodic theory under singular and non-Markovian regimes.

1. Algebraic and Probabilistic Setup

Let $(\Omega, \mathcal{F}, (\mathcal{F}_t), \mathbb{P})$ be a filtered probability space. A two-parameter increment $\Xi_{s,t}$ is a family of $\mathcal{F}_t$-measurable Banach space-valued random variables, continuous and null on the diagonal ($\Xi_{s,s}=0$). The fundamental object of interest is the three-point increment

$$\delta \Xi_{s,u,t} = \Xi_{s,t} - \Xi_{s,u} - \Xi_{u,t}$$

and, for mild versions, the mild increment

$$\hat{\delta} \Xi_{r,s,t} = \Xi_{r,t} - \Xi_{s,t} - S_{t-s}\Xi_{r,s}$$

involving a semigroup $S_t$. Assumptions are imposed on the moment growth of $\Xi_{s,t}$ and on conditional expectations of $\delta \Xi_{s,u,t}$, with exponents tailored to the process’ regularity. In the multiparameter case, rectangular increments and associated difference operators generalize the sewing framework to higher-dimensional domains with appropriate filtration structures and compatibility (commuting filtrations).

2. Local, Global, and Local-Global Lemma Statements

Local stochastic sewing lemmas provide convergence of Riemann-type sums for $\Xi_{s,t}$ to a limit as the mesh size tends to zero, under localized moment control on $\delta \Xi_{s,u,t}$. These require conditions such as: for strictly positive $\alpha_{1,i}, \alpha_{2,i}$ with $\alpha_{1,i}+\alpha_{2,i}>1$ and controls $w_{1,i}, w_{2,i}$, for all $0 \leq s \leq u \leq t \leq T$,

$$\|\,\mathbb{E}_s[\delta\Xi_{s,u,t}]\,\|_{L^{qr}} \leq \sum_{i=1}^N w_{1,i}(s,u)^{\alpha_{1,i}} w_{2,i}(u,t)^{\alpha_{2,i}}$$

and similarly for the non-conditional $L^q$-norms with exponents exceeding $1/2$ (Allan et al., 13 Aug 2024). Local sewing applies in regimes with only small-scale control, yielding convergence but not uniform global quantitative estimates.

Global stochastic sewing lemmas (Lê, Beer-Mourrat-Gubinelli, Butkovskiy) integrate over the whole interval and require two-scale bounds: for exponents $\gamma_1 > 1/2$, $\gamma_2 > 1$,

$$\|\delta A_{s,u,t}\|_{L^m} \leq \Gamma_1 |t-s|^{\gamma_1}, \quad \|\mathbb{E}^s[\delta A_{s,u,t}]\|_{L^m} \leq \Gamma_2 |t-s|^{\gamma_2}$$

guaranteeing that for partitions of $[S,T]$, the sum converges to $A_T-A_S$ with explicit bounds (Butkovsky, 14 Oct 2025).

Local-global sewing lemmas interpolate these two, allowing for the combination of localized Hölder gain (on small intervals) with global decay (e.g., exponential weights or uniform-in-time bounds). For example, the local-global lemma in (Mayorcas et al., 25 Nov 2025) assumes for short intervals $|t-s|\leq 1$ and for some $\lambda > 0$:

• $$\|A^T_{s,t}\|_{L^m} \leq e^{-\lambda(T-t)} C_A |t-s|^{1/2+\varepsilon_1}$$ (local gain, global decay),
• analogous bounds on the three-point increment and the martingale term, with remainders of the form $O(2^{-V\theta}/\lambda^\ast) + O(2^{V}C_A)$.

A summary table illustrates the distinctions:

Lemma Type	Main Hypothesis	Main Conclusion
Local	Small-interval bounds on $\delta\Xi$	Existence of limit for Riemann sums
Global	Uniform bounds on all intervals	Process with additive structure and error control
Local-Global	Small-block (local) & large-block (global)/decay	Uniform estimates combining both scales

3. Proof Structure: Dyadic and Martingale Decomposition

All sewing lemmas employ a dyadic decomposition of the interval(s), writing the sewn integral or increment as a telescoping sum over dyadic levels. At each dyadic step $k$, the contribution from $\delta \Xi$ over subintervals of size $2^{-k}(t-s)$ is controlled by the corresponding moment bound. The predictable (conditional mean) and martingale (orthogonal) components are separated using discrete Burkholder–Davis–Gundy (BDG) inequalities. Fine-scale, or “local,” blocks are handled using the high regularity exponents, yielding fast geometric decay, while coarse, or “global,” blocks exploit global damping (e.g., exponential weights or a finite partition) for uniform control.

For shifted sewing lemmas, the innovation is to shift the filtration conditioning back by the interval length. This enables cancellation that controls singular behaviour missed by classical (non-shifted) sewings, especially in non-Gaussian or Lévy-driven systems (Butkovsky, 14 Oct 2025).

In the multiparameter setting, rectangular increments and grid-like partitions furnish analogous telescopic and BDG structures, with one BDG step for each direction and commutation properties from the filtration.

4. Applications in Stochastic Analysis and PDEs

Local-global stochastic sewing lemmas underpin crucial advances in regularization by noise, stability/regularity in singular SPDEs, and stochastic differential systems with non-Markovian or non-semimartingale noise.

• In (Mayorcas et al., 25 Nov 2025), local-global sewing estimates allow tightness and uniform Gaussian moment bounds for solutions to SDEs with fractional Brownian drivers and singular drift, as well as uniform-in-time control on solution-to-noise and Jacobian stability maps, essential for Malliavin calculus, the strong-Feller property, and ergodicity.
• The mild stochastic sewing lemma (Li et al., 2021) extends these techniques to semigroup-driven SPDEs, yielding uniform $L^p$-estimates for mild increments, quantitative regularity enhancements, and a rigorous framework for fractional averaging and SPDEs in random environments.
• For jump processes and rough paths (Allan et al., 13 Aug 2024), local-global sewing builds adapted rough stochastic integrals in the presence of multiple discontinuous controls, producing solutions to rough SDEs with jumps and càdlàg sample paths. It reproduces classical Itô and rough integrals as special cases, with global error estimates and stability to jumps.
• The multiparameter extension (Bechtold et al., 2023) enables sharp regularity of local times for Gaussian sheets, crucial for the analysis of d-parameter SDEs driven by fractional Brownian sheets and for noise-induced regularization effects depending on additive noise directions (“boundary terms”).

5. Distinctions between Local, Global, and Local-Global Frameworks

The central distinction lies in the scale of estimates:

• The local lemma only provides convergence under local moment or conditional expectation bounds on small intervals, without yielding uniform-in-time continuity or error control.
• The global lemma pulls these local properties together, constructing an adapted process with uniform control—essential for establishing robust properties, such as pathwise uniqueness, moment bounds across the whole interval, and global stability, indispensable for stochastic averaging or ergodicity proofs.
• The local-global approach allows patching: by tuning the local scale (block size or dyadic depth) and threshold (e.g., the “V” parameter), one balances local gains (small time Hölder exponents from the regularization by noise or SDE structure) with global decay (e.g., exponential in time, as governed by the semigroup or Ornstein-Uhlenbeck kernel). This hybrid control is instrumental in proving results for processes with both fine- and coarse-scale singularities, or systems where spatial and temporal structures decouple at distinct scales.

6. Extensions and Future Directions

Current research extends the local-global sewing paradigm in several directions:

• Semigroup/Operator-valued sewings: The mild version incorporates analytic semigroups, permitting regularity transfer from time to space via interpolation spaces and semigroup smoothing (Li et al., 2021).
• Non-Gaussian/noise-driven and rough settings: The generalization to stable/Volterra processes and rough stochastic integration with jumps handles cases where classical martingale or semimartingale techniques break down (Allan et al., 13 Aug 2024).
• Multi-parameter and spatial domains: Higher-dimensional sewings reflect the structure of vector-valued/multiparameter fields, such as Gaussian sheets or kinetic equations, addressing boundary regularization effects and strong space-time local non-determinism (Bechtold et al., 2023).
• Patching and maximal inequalities: Combination of patching arguments with shifted sewings allows passage from local into global bounds even when the global sewing fails directly (Butkovsky, 14 Oct 2025).

A plausible implication is that the local-global sewing principle will continue to play a foundational role in the analysis of singular stochastic systems, especially in regimes where classical deterministic or Markovian regularizing mechanisms are absent. The sophisticated dyadic and martingale decomposition framework ensures these lemmas can be adapted to future directions in infinite-dimensional and non-standard stochastic analysis.