[Paper Review] Stochastic continuity of random fields governed by a system of stochastic PDEs

Kai Du, Jiakun Liu+1 more|arXiv (Cornell University)|Jun 6, 2017

Stochastic processes and financial applications8 references3 citations

TL;DR

This paper establishes a $C^{2+ heta}$-regularity theory for systems of stochastic PDEs by introducing a modified stochastic parabolicity condition involving $L^p$-norms, ensuring Hölder continuity of solutions in space and time. It proves sharp Schauder-type estimates and solvability results in $L^p$-valued Hölder spaces, extending classical SPDE theory to systems under minimal assumptions on coefficient structures.

ABSTRACT

This paper constructs a solvability theory for a system of stochastic partial differential equations. On account of the Kolmogorov continuity theorem, solutions are looked for in certain Hölder-type classes in which a random field is treated as a space-time function taking values in $L^{p}$-space of random variables. A modified stochastic parabolicity condition involving $p$ is proposed to ensure the finiteness of the associated norm of the solution, which is showed to be sharp by examples. The Schauder-type estimates and the solvability theorem are proved.

Motivation & Objective

• To develop a $C^{2+ heta}$-regularity theory for systems of stochastic PDEs, extending existing $L^p$-theory to systems with non-diagonal diffusion matrices.
• To address the lack of regularity results for systems of SPDEs beyond $p=2$, particularly in Hölder-type spaces.
• To establish sharp conditions under which solutions remain bounded in $L^p$-valued Hölder norms, ensuring stochastic continuity and sample path regularity.
• To generalize the Kolmogorov continuity theorem to random fields taking values in $L^p(\Omega)$, enabling the derivation of pathwise Hölder continuity.

Proposed method

• Introduces a new $L^p$-valued Hölder space $\mathcal{C}_p^\delta$ to treat random fields as space-time functions with values in $L^p(\Omega)$, enabling the analysis of sample path regularity.
• Proposes a modified stochastic parabolicity condition involving $p$ that ensures the finiteness of the solution norm in $\mathcal{C}_p^\delta$, which is shown to be sharp via counterexamples.
• Applies the Kolmogorov continuity theorem in the $L^p(\Omega)$-valued setting to derive stochastic continuity and Hölder regularity of solutions.
• Derives Schauder-type estimates for second-order derivatives of solutions under the new parabolicity condition, extending deterministic methods to the stochastic system context.
• Uses Fourier analysis and SDE techniques to analyze the $L^p$-norm growth of solutions in a model system with space-time white noise, proving blow-up when the parabolicity condition fails.
• Constructs a framework where the choice of auxiliary matrix $\Lambda$ in the parabolicity condition can be optimized for specific coefficient structures, improving the sharpness of the criterion.

Experimental results

Research questions

• RQ1 Can a $C^{2+\theta}$-regularity theory be established for systems of stochastic PDEs under minimal assumptions on the coefficient matrices?
• RQ2 What is the sharp condition on the coefficients that ensures the solution remains bounded in $L^p$-valued Hölder norms, guaranteeing stochastic continuity?
• RQ3 How does the skew-symmetric part of the diffusion matrix affect the $L^p$-norm of the solution, and can this be captured in a generalized parabolicity condition?
• RQ4 Is the proposed stochastic parabolicity condition optimal, and can it be improved by introducing an auxiliary matrix $\Lambda$?
• RQ5 Can the Kolmogorov continuity theorem be adapted to random fields with values in $L^p(\Omega)$ to derive pathwise Hölder continuity of solutions?

Key findings

• The modified stochastic parabolicity condition involving $p$ is shown to be sharp: if it fails, the $L^p$-norm of the solution may blow up in finite time, as demonstrated in Example 6.5 with $p=3$, $\lambda=3$, $\mu=1$.
• For $p \geq 3$, the choice $\Lambda = \frac{\lambda - \mu}{2(p-2)}$ in the parabolicity condition yields the optimal bound, and $\Lambda = 0$ does not always lead to the minimal requirement.
• The solution $u$ and its first and second derivatives belong to the $L^p$-valued Hölder space $\mathcal{C}_p^\delta$ if the data $f$, $g$, and $\partial g$ are in this space, establishing a full Schauder-type estimate.
• When $\delta p > d$, the solution has a version that is almost surely Hölder continuous in space, as guaranteed by the Kolmogorov continuity theorem applied in the $L^p(\Omega)$-valued setting.
• The paper proves a solvability theorem in the $\mathcal{C}_p^\delta$ space, showing existence and uniqueness of solutions under the proposed parabolicity condition.
• In the model system with $A = \text{diag}(1+\lambda^2, 1+\mu^2)$, $B = \begin{bmatrix} 0 & -\mu \\ \lambda & 0 \end{bmatrix}$, the $L^p$-norm of the solution is infinite for $t > 2/\varepsilon$ if $\varepsilon = \lambda^2 + (p-1)\mu^2 - 2 > 0$, confirming the sharpness of the condition.