[Paper Review] Boundary behavior and interior Hölder regularity of solution to nonlinear stochastic partial differential equations driven by space-time white noise

Beom-Seok Han, Kyeong-Hun Kim|arXiv (Cornell University)|May 28, 2019

Stochastic processes and financial applications13 references4 citations

TL;DR

This paper establishes existence and uniqueness of solutions in weighted Sobolev spaces for a nonlinear stochastic PDE with space-time white noise on a bounded interval, proving sharp interior Hölder regularity and precise boundary behavior. The key result shows that the solution satisfies $\rho^{-1/2-\kappa}u \in C^{\frac{1}{4}-\frac{\kappa}{2}-\varepsilon,\frac{1}{2}-\kappa-\varepsilon}_{t,x}([0,T]\times(0,1))$ for $\kappa \in (\lambda, 1/2)$, with optimal Hölder exponents depending on the nonlinearity parameter $\lambda$. The analysis accounts for general random coefficients and boundary decay via weighted norms and Hardy-type inequalities.

ABSTRACT

We present uniqueness and existence in weighted Sobolev spaces of the equation $$ u_t=(au_{xx}+bu_x+cu)+ ξ|u|^{1+λ} {\dot{B}}, \quad\,\, t>0, \, x\in (0,1) $$ with initial data $u(0,\cdot)=u_0$ and zero boundary data. Here $λ\in [0,1/2)$, $\dot{B}$ is a space-time white noise, and the coefficients $a,b,c$ and the function $ξ$ depend on $(ω,t,x)$ and the initial data $u_0$ depends on $(ω,x)$. More importantly, we obtain various interior Hölder regularities and boundary behaviors of the solution. For instance, if the initial data is in appropriate $L_p$ spaces, then for any small $\varepsilon>0$ and $T

Motivation & Objective

• To establish a unique solvability theory in weighted Sobolev spaces for a class of nonlinear SPDEs with space-time white noise on a bounded domain.
• To characterize the interior Hölder regularity of the solution in both time and space variables, accounting for the influence of the nonlinearity parameter $\lambda \in [0, 1/2)$.
• To analyze the boundary behavior of the solution, particularly the decay rate near the boundary, using distance-weighted norms.
• To extend existing results on SPDEs with additive noise to the case of multiplicative noise with random, time- and space-dependent coefficients.
• To provide a general framework for SPDEs with non-trivial boundary effects, filling a gap in the literature for domains with non-empty boundaries.

Proposed method

• The authors use weighted Sobolev spaces $\mathfrak{H}^{1/2-\kappa}_{p,\theta,\text{loc}}(I,\infty)$ to handle the solution's regularity and boundary decay, where $\theta$ controls the weight near the boundary.
• They apply a localization procedure via stopping times $\tau_m^R$ to control the solution's growth and ensure pathwise uniqueness in the localized domain.
• A key step involves constructing a sequence of approximating solutions $u_m$ using truncated nonlinearities $|u \wedge m|^{1+\lambda}$ to ensure boundedness and apply a priori estimates.
• The proof relies on a stochastic maximal regularity estimate and a maximum principle argument to preserve non-negativity and ensure pathwise uniqueness of solutions.
• Hardy’s inequality is used to justify the integrability of initial data in weighted $L_p$ spaces, enabling the derivation of Hölder continuity in the weighted norm.
• The convergence of stopping times $\tau_m^m \to \infty$ a.s. ensures the global existence of the solution in the local weighted Sobolev space.

Experimental results

Research questions

• RQ1 What is the optimal Hölder regularity in time and space for the solution of a nonlinear SPDE with space-time white noise and multiplicative noise of order $1+\lambda$?
• RQ2 How does the solution behave near the boundary of a bounded domain, and what is the precise decay rate in terms of the distance to the boundary?
• RQ3 Can existence and uniqueness be established in weighted Sobolev spaces for SPDEs with random, time- and space-dependent coefficients and multiplicative noise?
• RQ4 What is the sharp dependence of the Hölder exponents on the nonlinearity parameter $\lambda$?
• RQ5 How do weighted norms and Hardy-type inequalities contribute to the regularity and integrability of the solution and its initial data?

Key findings

• The solution satisfies $\rho^{-1/2-\kappa}u \in C^{\frac{1}{4}-\frac{\kappa}{2}-\varepsilon,\frac{1}{2}-\kappa-\varepsilon}_{t,x}([0,T]\times(0,1))$ almost surely for any $\kappa \in (\lambda, 1/2)$ and small $\varepsilon > 0$, giving the maximal Hölder exponents $\frac{1}{4} - \frac{\lambda}{2} - \varepsilon$ in time and $\frac{1}{2} - \lambda - \varepsilon$ in space.
• For any $\varepsilon > 0$, the solution satisfies $\sup_{t \leq T} |u(t,x)| \leq N(\omega) \rho^{1-\varepsilon}(x)$ almost surely, indicating strong decay near the boundary.
• If the initial data $u_0$ belongs to $L_p(\Omega; \mathring{W}^1_p(I))$ for $p > 1$, then the condition $\rho^{\frac{1+\theta}{p}-1}u_0, \rho^{\frac{1+\theta}{p}}D_x u_0 \in L_p(\Omega \times I)$ holds for any $\theta > 0$, enabling the use of Hardy’s inequality to control the weighted norms.
• The solution is globally defined in time, as $\tau_m^m \to \infty$ almost surely, ensuring the existence of a unique solution in $\mathfrak{H}^{1/2-\kappa}_{p,\theta,\text{loc}}(I,\infty)$.
• The method of truncation and stopping times ensures pathwise uniqueness and convergence, allowing the construction of a global solution via approximation.
• The result extends to general coefficients $a,b,c,\xi$ depending on $(\omega,t,x)$, making the framework applicable to a broad class of nonlinear SPDEs with boundary effects.