[Paper Review] Stochastic Heat Equation with general noise

Yaozhong Hu, Xiong Wang|arXiv (Cornell University)|Dec 11, 2019

Stochastic processes and financial applications18 references4 citations

TL;DR

This paper establishes the well-posedness of a one-dimensional nonlinear stochastic heat equation driven by rough Gaussian noise with Hurst parameter $ H \in (\frac{1}{4}, \frac{1}{2}) $, removing the restrictive condition $ \sigma(0) = 0 $ previously required in prior work. By introducing a spatially decaying weight $ \lambda(x) = c_H(1 + |x|^2)^{H-1} $, the authors construct a weighted solution space $ \mathcal{Z}_{\lambda,T}^p $ that accommodates non-zero $ \sigma(0) $, and derive exact asymptotic growth rates for the solution and its Hölder coefficients in both time and space as $ t, x \to \infty $.

ABSTRACT

In this paper, we study a nonlinear one spatial dimensional stochastic heat equations driven by Gaussian noise: $\frac{\partial u }{\partial t}=\frac{\partial^2 u }{\partial x^2}+σ(u )\dot{W} $, where $\dot{W} $ is white in time and has the covariance of a fractional Brownian motion with Hurst parameter $H\in(\frac 14,\frac 12)$. We remove a critical and unnatural condition $σ(0)=0$ previously imposed in a recent paper by Hu, Huang, Lê, Nualart and Tindel. The idea is to work on a weighted space $\mathcal{Z}_{λ,T}^p$ for some power decay weight $λ(x)=c_H(1+|x|^2)^{H-1}$. We obtain the weak existence of solution. With additional decay conditions on $σ$ we obtain the existence of strong solution and the pathwise uniqueness of the strong solution. The reason to introduce the weight function is that the solution $u(t,x)$ may explode as $|x| ightarrow \infty$ when the "diffusion coefficient" $σ(u)$ does not satisfy $σ(0)=0$ regardless of the initial condition. This motivates us to study the exact asympotics of the solution $u_{ m add}(t,x)$ as $t$ and $x$ go to infinity when $σ(u)=1$ and when the initial condition $u_0(x)\equiv 0$. In particular, we find the exact growth of $\sup_{|x|\leq L}{|u_{ m add}(t,x)|}$. Furthermore, we find the sharp growth rate for the Hölder coefficients, namely, $\sup_{|x|\leq L} \frac{| u_{ m add}(t,x+h)-u_{ m add}(t,x)|}{|h|^β}$ and $\sup_{|x|\leq L} \frac{| u_{ m add}(t+τ,x)-u_{ m add}(t,x)|}{τ^α}$. These results are interesting and fundamental themselves.

Motivation & Objective

• To remove the technical assumption $ \sigma(0) = 0 $, which was critical in prior work on the stochastic heat equation with rough noise.
• To extend the solution framework to handle general nonlinear diffusion coefficients $ \sigma $, including the affine case $ \sigma(u) \equiv 1 $, by constructing a new solution space.
• To establish pathwise uniqueness and existence of strong solutions in a weighted space $ \mathcal{Z}_{\lambda,T}^p $ with power decay weight $ \lambda(x) = c_H(1 + |x|^2)^{H-1} $.
• To derive exact asymptotic growth rates for the solution $ u_{\text{add}}(t,x) $ and its Hölder coefficients as $ t, x \to \infty $, including suprema over unbounded domains.
• To provide sharp estimates on the growth of $ \sup_{|x|\leq L} |u_{\text{add}}(t,x)| $, and the Hölder norms in space and time, under additive noise.

Proposed method

• Introduce a weighted space $ \mathcal{Z}_{\lambda,T}^p $ with weight $ \lambda(x) = c_H(1 + |x|^2)^{H-1} $ to replace the unweighted space $ \mathcal{Z}_{T}^p $, enabling solutions when $ \sigma(0) \neq 0 $.
• Use weighted heat kernel estimates and majorizing measure techniques to control the irregularity of the rough noise in space and time.
• Decompose the difference of solutions into three terms $ J_1, J_2, J_3 $ based on spatial increments and noise structure, leveraging the self-similarity and covariance structure of the fractional noise.
• Apply Hölder continuity assumptions on $ \sigma $ and its derivative, along with uniform decay conditions, to bound the increments of the solution in space and time.
• Use Gronwall’s inequality on the resulting integral inequalities involving $ I_1(t) $ and $ I_2(t) $, the second moments of solution differences, to prove pathwise uniqueness.
• Establish the existence of a H"older continuous modification of the solution via Theorem 1.5, relying on the moment bounds and decay structure.

Experimental results

Research questions

• RQ1 Can the well-posedness of the nonlinear stochastic heat equation be established without the restrictive assumption $ \sigma(0) = 0 $?
• RQ2 What is the exact asymptotic behavior of the solution $ u_{\text{add}}(t,x) $ as $ t \to \infty $ and $ x \to \infty $?
• RQ3 What are the sharp growth rates for the spatial and temporal Hölder coefficients of the solution over unbounded domains?
• RQ4 How can a weighted solution space $ \mathcal{Z}_{\lambda,T}^p $ be constructed to accommodate non-zero initial and diffusion values when $ \sigma(0) \neq 0 $?
• RQ5 What are the precise asymptotic behaviors of $ \sup_{|x|\leq L} |u_{\text{add}}(t,x)| $, $ \sup_{|x|\leq L} \frac{|u(t,x+h)-u(t,x)|}{|h|^\beta} $, and $ \sup_{|x|\leq L} \frac{|u(t+\tau,x)-u(t,x)|}{\tau^\alpha} $ as $ t, x \to \infty $?

Key findings

• The paper establishes the existence and pathwise uniqueness of a strong solution to the stochastic heat equation with general $ \sigma $, even when $ \sigma(0) \neq 0 $, by introducing a weighted space $ \mathcal{Z}_{\lambda,T}^p $ with $ \lambda(x) = c_H(1 + |x|^2)^{H-1} $.
• For the additive noise case $ \sigma(u) \equiv 1 $, the exact growth rate of $ \sup_{|x|\leq L} |u_{\text{add}}(t,x)| $ as $ t, x \to \infty $ is derived, showing it grows like $ t^{H/2} $ up to logarithmic corrections.
• The sharp growth rate for the spatial Hölder coefficient $ \sup_{|x|\leq L} \frac{|u(t,x+h)-u(t,x)|}{|h|^\beta} $ is found to be $ t^{H/2} $, with $ \beta = H $, indicating the solution is almost surely $ H $-Hölder continuous in space.
• The temporal Hölder coefficient $ \sup_{|x|\leq L} \frac{|u(t+\tau,x)-u(t,x)|}{\tau^\alpha} $ grows like $ t^{H/2} $, with $ \alpha = H/2 $, indicating the solution is $ H/2 $-Hölder continuous in time.
• The solution $ u_{\text{add}}(t,x) $ exhibits exact asymptotic behavior: $ \sup_{|x|\leq L} |u_{\text{add}}(t,x)| \sim t^{H/2} $ as $ t \to \infty $, and the Hölder norms scale accordingly with $ t^{H/2} $.
• The method successfully overcomes the breakdown of standard techniques from [12] by replacing the unweighted space with a weighted one, enabling analysis of non-affine $ \sigma $ and the full class of nonlinearities including $ \sigma(u) \equiv 1 $.