[Paper Review] On the Cauchy problem for stochastic parabolic equations in Hölder spaces

Kai Du, Jiakun Liu|arXiv (Cornell University)|Nov 9, 2015

Stochastic processes and financial applications15 references3 citations

TL;DR

This paper establishes a sharp $C^{2+eta}$-theory for stochastic parabolic equations in Hölder spaces by proving existence, uniqueness, and optimal regularity of solutions under minimal assumptions on coefficients and data. It introduces a novel framework for stochastic Hölder spaces and proves that solutions inherit the Hölder regularity of the data, resolving an open problem posed by Krylov on the sharpness of the theory.

ABSTRACT

In this paper, we establish a sharp $C^{2+α}$-theory for stochastic partial differential equations of parabolic type in the whole space.

Motivation & Objective

• To resolve an open problem posed by Krylov regarding the sharp $C^{2+eta}$-theory for stochastic parabolic equations.
• To establish a complete and optimal regularity theory in Hölder spaces for the Cauchy problem of second-order SPDEs.
• To define and characterize a suitable class of stochastic Hölder spaces where the solution theory is both necessary and sufficient.
• To prove that the solution space coincides exactly with the data space under minimal regularity assumptions on coefficients and free terms.

Proposed method

• Introduces a notion of quasi-classical solutions in $L_{ ext{loc}}^{ ho}( ext{space-time}; L_{ ext{prob}}^{ ho})$-valued Hölder spaces for SPDEs.
• Defines stochastic Hölder norms $|u|_{m+eta; ho}$ and $|u|_{(m+eta,eta/2); ho}$ using $L^{ ho}( ho o ho)$-norms of spatial and space-time derivatives.
• Imposes uniform parabolicity (1.2) and assumes Hölder continuity of coefficients $a^{ij}, b^i, c, u, u_x, u_x$, and $ u_x$ in space with a uniform bound $K$.
• Applies a mollification technique to approximate solutions and uses the dominated convergence theorem to pass to limits in $L^{ ho}$-norms.
• Employs a priori estimates and compactness arguments to prove existence and uniqueness in the proposed Hölder space framework.
• Uses the Itô formula and stochastic integral representation to derive integral equations satisfied by the solution.

Experimental results

Research questions

• RQ1 Can a sharp $C^{2+eta}$-theory be established for stochastic parabolic equations such that the solution space is exactly the image of the data space under the solution operator?
• RQ2 What are the minimal regularity assumptions on coefficients and data that ensure the solution lies in a stochastic $C^{2+eta}$-space?
• RQ3 How can one define a stochastic Hölder space framework that supports both existence and uniqueness of solutions with optimal regularity?
• RQ4 Does the solution inherit the Hölder regularity of the data $f$ and $g$ in the sense of $L^{ ho}( ho)$-norms?
• RQ5 Can the solution theory be made fully optimal, such that every element in the solution space arises as a solution for some data in the same space?

Key findings

• The paper establishes a sharp $C^{2+eta}$-theory for the Cauchy problem of stochastic parabolic equations in the whole space.
• For $f o C^{eta}_x( ho)$ and $g o C^{1+eta}_x( ho)$, the solution $u$ belongs to $C^{2+eta,eta/2}_x( ho)$, with a norm estimate $|u|_{2+eta,eta/2; ho} o C(|f|_{eta; ho} + |g|_{1+eta; ho})$.
• The solution operator is surjective: every element in $C^{2+eta,eta/2}_x( ho)$ arises as a solution for some $f, g$ in the same data space.
• The coefficients $a^{ij}, b^i, c, u, u_x, u_x$ are assumed to be uniformly $C^{eta}$ in space with a uniform bound $K$, ensuring the sharpness of the theory.
• The mollification method ensures that $u^{ ho} o u$ in $L^{ ho}$-norm and that $D^m u^{ ho} o D^m u$ in $L^{ ho}$-norm, with uniform convergence if $D^m u$ is uniformly strongly continuous.
• The Hölder seminorm of the mollified solution satisfies $[[u^{ ho}]]_{eta; ho} o C(n, ho)[[u]]_{eta; ho}$, and $[[u^{ ho} - u]]_{eta/2; ho} o 0$ as $ ho o 0$.