Uniqueness for stochastic differential equations in Hilbert spaces with irregular drift

1. Problem Statement

The paper addresses the strong well-posedness (existence and pathwise uniqueness) of Stochastic Differential Equations (SDEs) in infinite-dimensional separable Hilbert spaces ($H$) with irregular (Hölder continuous) drift.

The specific equation studied is:
$$dX_t = (AX_t + b(X_t))dt + (-A)^{-\gamma/2}dW_t, \quad X_0 = x_0 \in H$$
where:

• $A$ is a self-adjoint, negative definite operator with a purely atomic spectrum (e.g., the Dirichlet Laplacian).
• $W$ is a cylindrical Wiener process.
• $b: H \to H$ is an $\alpha$-Hölder continuous function ($b \in C^\alpha(H, H)$) with $\alpha \in (0, 1]$.
• $\gamma \geq 0$ is a parameter controlling the regularity of the noise (specifically, the stochastic convolution $Z_t$).

The Core Challenge:
In finite dimensions, the "regularization by noise" phenomenon is well-understood: even if the drift $b$ is merely bounded and measurable, additive noise can ensure unique strong solutions. However, in infinite dimensions, the noise is often "rougher" relative to the state space, and previous results required structural assumptions on the drift $b$ (e.g., specific decay rates of its Fourier coefficients) to prove uniqueness. The authors aim to remove these structural constraints, proving uniqueness for general Hölder continuous drifts under optimal conditions on $\alpha$ and $\gamma$.

2. Methodology

The authors develop a novel framework that avoids the classical Zvonkin transformation (which relies on solving infinite-dimensional Kolmogorov equations and imposes strict structural conditions on $b$). Instead, they combine three distinct mathematical tools:

1. Stochastic Sewing in Hilbert Spaces:
   Based on the theory developed by Lê [Lê20, Lê23], this technique allows for the construction of integrals and the analysis of SDEs by controlling the "sewing" of local increments. The authors adapt this to infinite dimensions, overcoming the difficulty that standard Gaussian estimates (like Lipschitz bounds on expectations) fail in infinite dimensions unless perturbations lie in the Cameron-Martin space.

2. Lasry-Lions Approximation:
   To handle the lack of a Gaussian kernel decomposition in infinite dimensions (which is used in finite dimensions to split functions into Lipschitz and bounded parts), the authors employ the Lasry-Lions regularization method [LL86]. This allows them to approximate the Hölder continuous drift $b$ by a sequence of Lipschitz functions, enabling the application of stochastic sewing estimates.

3. Gaussian Analysis and Sobolev Scaling:
   The authors perform a delicate analysis of the stochastic convolution $Z_t$. They measure the regularity of the drift not just in the base Hilbert space $H$, but also in Sobolev scales $H_\rho$ associated with the operator $A$. By "tuning" the regularity parameter $\rho$, they optimize the trade-off between the smoothing effect of the noise and the roughness of the drift.

Key Technical Innovation:
The proof strategy involves a simultaneous contraction in two spaces: $C^\tau([0, 1], H)$ and $C^{\tau-\rho/2}([0, 1], H_\rho)$. This dual-space approach allows the authors to capture the regularization effect of the noise more precisely than previous single-space methods.

3. Key Results

Main Theorem (Theorem 2.4)

The paper establishes that the SDE admits a unique strong solution provided the Hölder exponent $\alpha$ and the noise parameter $\gamma$ satisfy:
$$\alpha > \alpha^*(\gamma)$$
where the critical threshold $\alpha^*(\gamma)$ is defined explicitly as:
$$\alpha^*(\gamma) = \begin{cases} \frac{\gamma}{\gamma + 1} & \text{if } 0 \leq \gamma \leq 1 \\ \frac{1}{2} & \text{if } 1 \leq \gamma \leq 3 \\ \sqrt{\frac{\gamma - 1}{2}} & \text{if } \frac{3}{2} \leq \gamma < 3 \end{cases}$$
Note: The result holds for all $\gamma \in [0, 3)$.

Application to Stochastic Heat Equation (Corollary 2.5)

Applying the main theorem to the stochastic heat equation with Dirichlet boundary conditions in dimension $d$:
$$dX_t = \Delta X_t dt + b(X_t)dt + (-\Delta)^{-\gamma/2}dW_t$$
The authors derive that for dimensions $1 \leq d \leq 7$, there exists a regime of $\alpha$ ensuring a unique strong solution.

• Significance: This significantly extends the range of admissible dimensions compared to previous works (which were often limited to $d=1, 2$ or required $d \leq 3$ with strong drift restrictions).
• Comparison: For $d=3$, the paper requires $\alpha > 1/3$, whereas the recent independent work by Addona et al. [ABOS25] required $\alpha > 2/3$.

4. Comparison with Existing Literature

• Da Prato & Flandoli (2010): Their seminal work established uniqueness but required a structural condition (1.3) on the drift involving the decay of Fourier coefficients. This condition is trivial for $d=1$ but restrictive for $d \geq 2$. The current paper completely removes this structural assumption.
• Recent Works (AB23, AB25, AMP23): These papers relaxed the conditions but still required some decay of the Hölder norms of the drift components for $d \geq 2$. The current work imposes no additional structural assumptions on $b$ beyond Hölder continuity.
• Independent Work [ABOS25]: The authors note that [ABOS25] independently removed the structural assumption but achieved a weaker range for $\alpha$ (e.g., $\alpha > 2/3$ for $d=3$ vs. $\alpha > 1/3$ here) and a more limited range for $\gamma$ (up to 1 vs. up to 3 here).

5. Significance and Future Directions

• Methodological Breakthrough: The paper demonstrates that the Stochastic Sewing framework, combined with Lasry-Lions approximation, is a powerful alternative to the Zvonkin/Kolmogorov equation approach. It avoids the technical bottlenecks of solving infinite-dimensional PDEs.
• Optimality: The derived conditions on $\alpha$ and $\gamma$ are shown to be sharp within the context of the method, covering a broader range of dimensions and drift regularities than any previous result.
• Future Applications:
  • Time-Dependent Drift: The framework naturally extends to time-dependent drifts $b(t, x)$ (e.g., $b \in L^q([0, 1], C^\alpha)$), which is difficult to handle via Kolmogorov equations (requiring a shift from elliptic to parabolic theory).
  • Fractional Noise: The method is adaptable to equations driven by fractional Brownian motion (non-Markovian settings), where Kolmogorov-based methods are generally inapplicable.

Conclusion

This paper represents a substantial advancement in the theory of infinite-dimensional SDEs. By introducing a new analytical toolkit that bypasses the need for structural assumptions on the drift, the authors establish strong well-posedness for a much wider class of stochastic evolution equations, including the stochastic heat equation in dimensions up to 7, with significantly relaxed regularity requirements on the drift.