[Paper Review] On the two-dimensional hyperbolic stochastic sine-Gordon equation

Tadahiro Oh, Tristan Robert+2 more|Edinburgh Research Explorer (University of Edinburgh)|Jul 13, 2019

Advanced Mathematical Physics Problems38 references4 citations

TL;DR

This paper establishes local well-posedness of the two-dimensional hyperbolic stochastic sine-Gordon equation with a time-dependent renormalization for any $β^2 > 0$, overcoming the non-polynomial nonlinearity's challenges. Unlike the parabolic case, which requires $\beta^2 < 8\pi$, the authors prove global existence and convergence in probability for the renormalized solution, resolving a key issue in singular SPDEs with trigonometric nonlinearities.

ABSTRACT

We study the two-dimensional stochastic sine-Gordon equation (SSG) in the hyperbolic setting. In particular, by introducing a suitable time-dependent renormalization for the relevant imaginary multiplicative Gaussian chaos, we prove local well-posedness of SSG for any value of a parameter $β^2 &gt; 0$ in the nonlinearity. This exhibits sharp contrast with the parabolic case studied by Hairer and Shen (2016) and Chandra, Hairer, and Shen (2018), where the parameter is restricted to the subcritical range: $0 &lt; β^2 &lt; 8 π$. We also present a triviality result for the unrenormalized SSG.

Motivation & Objective

• To address the lack of well-posedness for the hyperbolic stochastic sine-Gordon equation with general $\beta^2 > 0$ due to the non-polynomial, oscillatory nonlinearity.
• To overcome the degeneracy of $\sin(\beta u)$ in the rough path regime by introducing a time-dependent renormalization of the imaginary Gaussian multiplicative chaos.
• To establish local well-posedness in a suitable function space for the renormalized equation, extending beyond the subcritical range known from parabolic settings.
• To prove triviality of the unrenormalized model, showing that without renormalization, the solution collapses to the linear wave equation.

Proposed method

• Introduce a time-dependent renormalization factor $\gamma_N(t) = \exp\left(-\frac{\beta^2 t}{8\pi} \log N\right)$ to control the growth of the stochastic nonlinearity.
• Use the Duhamel formulation to rewrite the equation and apply Strichartz estimates in the context of the wave equation with rough noise.
• Define the stochastic convolution $\Psi_N$ as the solution to the linearized equation and prove its almost sure convergence in $C([0,T]; W^{-\varepsilon,\infty}(\mathbb{T}^2))$.
• Decompose the solution as $u_N = \Psi_N + v_N$, reducing the problem to proving local well-posedness for $v_N$ in a fixed-point argument on $X^s(T)$.
• Apply Chebyshev’s inequality and uniform bounds in probability to show convergence of $v_N$ to a limit in $X^s(T)$, ensuring the solution exists locally almost surely.
• Use a modified version of the argument with $\gamma_N^{-1}(t)$ to show that the unrenormalized solution converges in probability to the linear wave solution, proving triviality.

Experimental results

Research questions

• RQ1 Can the two-dimensional hyperbolic stochastic sine-Gordon equation be made well-posed for all $\beta^2 > 0$, beyond the subcritical range $\beta^2 < 8\pi$?
• RQ2 How can the non-polynomial, highly oscillatory nonlinearity $\sin(\beta u)$ be renormalized to prevent degeneracy in the presence of space-time white noise?
• RQ3 What is the role of time-dependent renormalization in stabilizing the solution of the stochastic wave equation with trigonometric nonlinearity?
• RQ4 Does the unrenormalized model yield a non-trivial solution, or does it collapse to the linear equation?
• RQ5 Can convergence in probability of the regularized solution to a limit be established under the conditional probability measure $P_T$?

Key findings

• Local well-posedness is established for the renormalized two-dimensional hyperbolic stochastic sine-Gordon equation for any $\beta^2 > 0$, extending beyond the subcritical regime.
• The solution $u_N$ converges in probability to a limit $u$ in $C([0,T]; W^{-\varepsilon,\infty}(\mathbb{T}^2))$ as $N \to \infty$, with convergence holding $P_T$-almost surely.
• The time-dependent renormalization $\gamma_N(t)$ ensures uniform control over the stochastic nonlinearity, enabling the fixed-point argument in $X^s(T)$.
• The unrenormalized model is trivial: $u_N$ converges in probability to the solution of the linear stochastic wave equation, implying no non-trivial dynamics without renormalization.
• The regularity of the main stochastic term depends on both $\beta$ and time $t$, a key difference from polynomial nonlinearities where regularity is time-independent.
• The convergence of $v_N$ to the linear solution is quantified via $\|\gamma_N^{-1}(t)\|_{L^{1/\delta}_T} \lesssim (\log N)^{-1} \to 0$, showing the renormalization is essential for non-triviality.