Stochastic Sewing in Hilbert Spaces

• Stochastic sewing in Hilbert spaces is a framework that reconstructs adapted processes from time-indexed increments using sewing lemmas derived from rough path theory.
• It integrates Gaussian averaging, Cameron–Martin geometry, and Lasry–Lions approximations to handle low regularity and irregular drift in SDEs.
• The method extends classical well-posedness results by establishing strong existence and pathwise uniqueness under optimal Hölder regularity conditions.

Stochastic sewing in Hilbert spaces is a probabilistic analytic technique developed to facilitate the construction and uniqueness theory for stochastic differential equations (SDEs) in infinite-dimensional settings, particularly when coefficients lack sufficient regularity. Originating from the adaptation of sewing lemmas in rough path theory, stochastic sewing provides a robust framework for assembling increment processes indexed by time intervals into well-defined adapted processes, allowing fine control of stochastic integrals even under low regularity. Its integration with infinite-dimensional Gaussian analysis and Lasry–Lions approximations enables the extension of uniqueness and existence criteria for Hilbert-space-valued SDEs driven by cylindrical Wiener noise and irregular drift, going significantly beyond earlier results that required structural hypotheses on the drift term.

1. Stochastic Sewing Lemma in Separable Hilbert Spaces

Let $H$ be a real separable Hilbert space. The stochastic sewing lemma in this context addresses the problem of reconstructing a $H$-valued adapted process from its stochastic increments. For $m\ge2$, consider a two-parameter process

$$A:\Omega\times \Delta^2_{[S,T]}\;\longrightarrow\;H$$

with $A_{s,t}$ being $\mathcal F_t$–measurable for $S\leq s<t\leq T$. The three-point increment is defined by

$$\delta A_{s,u,t} = A_{s,t} - A_{s,u} - A_{u,t}, \qquad (s<u<t).$$

Assume the following conditions for some $\Gamma_1,\Gamma_2\ge0$ and exponents $\varepsilon_1,\varepsilon_2>0$: \begin{align} &\big|\,|\delta A_{s,u,t}|H\big|{L^m} \leq \Gamma_1\,|t-s|^{\frac12+\varepsilon_1}, \tag{SSL-(i)} \ &\big|\,|E^s[\delta A_{s,u,t}]|H\big|{L^m} \leq \Gamma_2\,|t-s|^{1+\varepsilon_2}. \tag{SSL-(ii)} \end{align} Under these hypotheses, there exists a unique $H$-valued adapted process $\mathcal A_t$ such that, for any refining sequence of partitions $\Pi^n$ of $[S,t]$ with mesh size tending to zero,

$$\sum_i A_{s_i^n,s_{i+1}^n} \xrightarrow{P} \mathcal A_t.$$

Quantitative control of the reconstruction is provided as

$$\big\|\|\mathcal A_t-\mathcal A_s - A_{s,t}\|_H\big\|_{L^m} \leq C\Gamma_1(t-s)^{\frac12+\varepsilon_1} + C\Gamma_2(t-s)^{1+\varepsilon_2},$$

with $C$ depending only on $(\varepsilon_1, \varepsilon_2, m)$. This lemma, as used in Lemma 3.3 of (Anzeletti et al., 31 Dec 2025), is instrumental in constructing pathwise solutions and controlling iterative increments arising in the analysis of SDEs with distributional drift in infinite dimensions.

2. Gaussian Averaging and Cameron–Martin Geometry

The stochastic convolution in Hilbert space is central to the analysis. For a self-adjoint negative definite operator $A$ with a purely atomic spectrum, let $\{e_k\}$ be an orthonormal eigenbasis with $A e_k = -\lambda_k e_k$ ($\lambda_k>0$). Assume the trace-class condition

$\sum_{k=1}^\infty \lambda_k^{-1-\gamma}<\infty,$

for a parameter $\gamma\geq 0$. Let $W_t$ be a cylindrical Wiener process, and define the stochastic convolution

$$Z_t = \int_0^t e^{(t-r)A}(-A)^{-\frac\gamma2}\,dW_r,$$

which is a mean-zero Gaussian with covariance

$$\mathcal Q_t = \frac12(\mathrm{Id}-e^{2tA})(-A)^{-1-\gamma}.$$

This defines a Radon Gaussian measure $\mu_t$ on $H$, whose Cameron–Martin space is $\mathcal H_t = \mathrm{Ran}\,\mathcal Q_t^{1/2}$ with norm $\|h\|_{\mathcal H_t} = \|\mathcal Q_t^{-1/2} h\|_H$. The estimate

$$\|e^{tA}h\|_{\mathcal H_t} \leq C(\gamma) t^{-\frac{1+\gamma}2} \|h\|_H$$

characterizes the effect of the semigroup smoothing against this geometry.

For $f\in C^\alpha(H;H)$ with $\alpha\in(0,1]$, two-point and four-point Gaussian estimates for Gaussian averages (Corollary 2.5) state: \begin{align} &\left|\int_H \left[f(x+e^{tA}h_1)-f(x+e^{tA}h_2)\right]\,\mu_t(dx)\right|_H \leq C t^{-\frac{(1+\gamma)(1-\alpha)}2} [f]{C^\alpha} |h_1-h_2|_H, \tag{2-point} \ &\left|\int_H \left[f(x+e^{tA}h_1)-f(x+e^{tA}h_2) - f(x+e^{tA}h_3) + f(x+e^{tA}(h_2+h_3-h_1))\right]\,\mu_t(dx)\right|_H \ &\qquad\leq C t^{-\frac{(1+\gamma)(2-\alpha)}2} [f]{C^\alpha} |h_1-h_2|_H |h_1-h_3|_H. \tag{4-point} \end{align} These are established via the Cameron–Martin formula, Lasry–Lions interpolation, and Gaussian exponential moment estimates.

3. Stochastic Differential Equations and Main Uniqueness Theorem

The SDE under consideration takes the form

$$dX_t = (A X_t + b(X_t))\,dt + (-A)^{-\frac\gamma2}\,dW_t, \qquad X_0=x_0\in H,$$

with

• $A$ self-adjoint, negative definite, purely atomic spectrum;
• $\sum_k\lambda_k^{-1-\gamma}<\infty$ to ensure $Z_t\in H$;
• $b\in C^\alpha(H;H)$ for some $\alpha\in(0,1]$.

A mild solution is a continuous adapted process $X_t$ such that almost surely,

$$X_t = e^{tA}x_0 + \int_0^t e^{(t-r)A}b(X_r)\,dr + Z_t.$$

The main result (Theorem 1.4) proves strong existence and pathwise uniqueness of solutions under the critical threshold

$\alpha > \frac{2\gamma}{1+\gamma},$

i.e., if $\gamma<\frac\alpha{2-\alpha}$. For any two solutions $X$, $Y$ starting from $x_0$, $y_0$ under the same noise, the estimate

$\sup_t\|X_t-Y_t\|_{L^m} \leq C\|x_0-y_0\|_H$

holds, showing Lipschitz dependence on initial data in $L^m$.

4. Derivation of the Sharp Regularity Threshold

Let $X$ and $Y$ be solutions driven by the same noise, define $K_t = X_t - Z_t$, $\widetilde K_t = Y_t - Z_t$, and $D_t=K_t-\widetilde K_t$. The strategy is to represent $D_t$ and analyze its evolution,

$$D_t - e^{(t-s)A}D_s = \int_s^t e^{(t-r)A}\left( b(Z_r+K_r) - b(Z_r+\widetilde K_r) \right)dr.$$

The process increments

$$A_{u,v} = E^u \int_u^v e^{(t-r)A}\left[ b(Z_r + e^{(r-u)A}K_u) - b(Z_r + e^{(r-u)A}\widetilde K_u) \right]dr$$

define the input for stochastic sewing, with the resulting sewn process yielding the nonlinear integral in the mild formulation.

The crucial exponents in the stochastic sewing lemma are determined by the Gaussian bounds: \begin{align*} &||\delta A_{u,\xi,v}||{L^m} \leq C (v-u)^{1-\frac{(1+\gamma)(1-\alpha)}2}|D_u|{L^m},\ &||E^u[\delta A_{u,\xi,v}]||{L^m} \leq C (v-u)^{2-\frac{(1+\gamma)(2-\alpha)}2}|D_u|{L^m} + \dots \end{align*} The conditions for application—the exponents in the inequalities—reduce precisely to $\alpha > 2\gamma/(1+\gamma)$. Gronwall chaining (Lemma 3.8) on the resultant sewing inequality enforces $D \equiv 0$, guaranteeing pathwise uniqueness; a fixed point argument in the sewing norm yields strong existence.

5. Lasry–Lions Approximation Methodology

The Lasry–Lions interpolation enables analysis for general $C^\alpha$-regular vector fields by representing $f\in C^\alpha(H;H)$ as

$$f_\lambda(x) = \inf_{h\in H}\left\{ f(x+h) + \lambda^{-1}\|h\|_H \right\},$$

which is Lipschitz with constant $\leq \lambda^{-1}$ and satisfies the approximation $$\|f-f_\lambda\|_\infty \lesssim \lambda^{\frac{\alpha}{1-\alpha}}$$. This decomposition splits a difference as

$$E\left[ f(X+h_1)-f(X+h_2) \right] = E\left[ f_\lambda(X+h_1)-f_\lambda(X+h_2) \right] + E\left[ f-f_\lambda(X+h_1)-f-f_\lambda(X+h_2) \right].$$

Using the Cameron–Martin shift and sup-norm bounds on the Lipschitz and remainder terms, optimization in $\lambda$ yields the desired scaling $$\|h_1-h_2\|_H^\alpha \|h_1-h_2\|_{\mathcal H_t}^{1-\alpha}$$. The Lasry–Lions construction supersedes finite-dimensional heat-kernel interpolation, enabling the Gaussian estimate machinery needed for the stochastic sewing strategy in Hilbert spaces.

6. Significance and Connections

The integration of stochastic sewing, infinite-dimensional Gaussian analysis, and Lasry–Lions approximation as developed by Anzeletti, Butkovsky, Gerencsér, and Shaposhnikov (Anzeletti et al., 31 Dec 2025) extends the theory of Hilbert-space SDEs with irregular drift substantially beyond the finite-dimensional setting and prior results, such as the Da Prato–Flandoli theory (2010), which required substantive drift structure. This synthesis permits strong well-posedness without structural assumptions on the drift term and establishes the critical Hölder threshold as optimal under the present framework. The methodology also clarifies connections between stochastic calculus, the geometry of Gaussian measures on Banach spaces, and interpolation theory, setting a reference point for further research in infinite-dimensional stochastic analysis.