[Paper Review] Degenerate SDEs in Hilbert Spaces with Rough Drifts

Feng‐Yu Wang, Xicheng Zhang|arXiv (Cornell University)|Jan 17, 2015

Stochastic processes and financial applications5 references3 citations

TL;DR

This paper establishes the existence and uniqueness of mild solutions for degenerate stochastic differential equations (SDEs) in Hilbert spaces with rough drifts, where the drift is Dini continuous in the noisy component and Hölder continuous of order greater than $\frac{2}{3}$ in the non-noisy component. The key contribution is a regularization transform method that overcomes poor gradient estimates in degenerate settings, extending pathwise uniqueness to rough drifts via finite-dimensional approximations and Biharis's inequality.

ABSTRACT

The existence and uniqueness of mild solutions are proved for a class of degenerate stochastic differential equations on Hilbert spaces where the drift is Dini continuous in the component with noise and Hölder continuous of order larger than $\ff 2 3$ in the other component. In the finite-dimensional case the Dini continuity is further weakened. The main results are applied to solve second order stochastic systems driven by space-time white noises.

Motivation & Objective

• To establish existence and uniqueness of mild solutions for degenerate SDEs in Hilbert spaces with irregular drifts.
• To address the challenge of poor gradient estimates in degenerate systems, where $\|\nabla^{(1)}P_t^0\|_{\infty\to\infty} \approx t^{-3/2}$ is not integrable.
• To extend pathwise uniqueness results to drifts with Dini-type continuity in the noisy component and Hölder continuity of order $> \frac{2}{3}$ in the non-noisy component.
• To apply the results to second-order stochastic systems driven by space-time white noise.
• To generalize previous results on non-degenerate SDEs in Hilbert spaces to the degenerate case with rough coefficients.

Proposed method

• Construct a regularization transform to convert the original SDE into one with smoother coefficients, enabling pathwise uniqueness.
• Use finite-dimensional approximations via projections $\pi^{(n)} = (\pi_1^{(n)}, \pi_2^{(n)})$ onto finite-dimensional subspaces $\mathbb{H}_n = \mathbb{H}_1^{(n)} \times \mathbb{H}_2^{(n)}$.
• Apply Biharis's inequality to control the growth of solutions and prove non-explosion under integrability conditions on $\ell$ and $h$.
• Employ the assumption that $-A_2$ is self-adjoint with discrete spectrum satisfying $\sum_{i \geq 1} \frac{1}{\lambda_i^{1-\delta}} < \infty$ for some $\delta \in (0,1)$ to ensure semigroup regularity.
• Use the stopping time $\tau_m = m \land \inf\{t \geq 0 : |(X_t^{(m)}, Y_t^{(m)})| \geq m\}$ to localize the solution and construct a global mild solution via patching.
• Leverage Corollary 4.2 to ensure pathwise uniqueness across approximating solutions, leading to uniqueness of the global solution.

Experimental results

Research questions

• RQ1 Can mild solutions exist and be unique for degenerate SDEs in Hilbert spaces when the drift is only Dini continuous in the noisy component and Hölder continuous of order $> \frac{2}{3}$ in the non-noisy component?
• RQ2 How can pathwise uniqueness be established in degenerate systems where the linearized semigroup has non-integrable gradient estimates ($t^{-3/2}$) in the noisy component?
• RQ3 What conditions on the drift and noise coefficients ensure non-explosion of the mild solution in the degenerate setting?
• RQ4 Can the regularization transform method be adapted to handle rough drifts in infinite-dimensional degenerate SDEs?
• RQ5 To what extent do the results extend to second-order stochastic PDEs driven by space-time white noise?

Key findings

• The mild solution to the degenerate SDE (1.1) exists and is pathwise unique under Dini-type continuity of the drift in the noisy component and Hölder continuity of order $> \frac{2}{3}$ in the non-noisy component.
• Non-explosion of the solution is guaranteed when the drift satisfies $\int_1^\infty \frac{ds}{\ell_t(s)} = \infty$ for increasing functions $\ell, h$, ensuring finite growth via Biharis's inequality.
• The solution is non-explosive almost surely, as $\mathbb{P}(\zeta < \infty) = 0$ is proven by contradiction using the inverse of $\Gamma_T(s) = \int_1^s \frac{dr}{2\ell_T(C + Cr)}$, which remains finite under the given conditions.
• In the finite-dimensional case, pathwise uniqueness holds for drifts satisfying $\sup_{t \in [0,T]} \|b_t\|_{\gamma_\alpha, \phi} < \infty$ with $\alpha \in (\frac{2}{3}, 1)$ and $\phi \in \mathscr{D}_2$, extending results to rougher drifts.
• The method successfully overcomes the $t^{-3/2}$ singularity in gradient estimates by replacing $\|\cdot\|_{\infty \to \infty}$ with alternative norms, enabling the use of regularization techniques.
• The results are applied to second-order stochastic systems driven by space-time white noise, confirming the existence and uniqueness of mild solutions under the stated regularity conditions.