[Paper Review] SPDEs with fractional noise in space with index H < 1=2

Raluca M. Balan, Maria Jolis+1 more|arXiv (Cornell University)|Jul 15, 2014

Stochastic processes and financial applications6 references5 citations

TL;DR

This paper establishes the existence and uniqueness of mild solutions for stochastic wave and heat equations on R driven by space-time Gaussian noise that is white in time and fractional Brownian motion in space with Hurst index H ∈ (1/4, 1/2). Using Picard iteration and Malliavin calculus techniques, the authors prove L²-continuity and uniform boundedness of p-th moments for p ≥ 2, extending results to the hyperbolic and parabolic Anderson models with affine diffusivity.

ABSTRACT

In this article, we consider the stochastic wave and heat equations on $\mathbb{R}$ with non-vanishing initial conditions, driven by a Gaussian noise which is white in time and behaves in space like a fractional Brownian motion of index $H$, with $1/4<H<1/2$. We assume that the diffusion coefficient is given by an affine function $\sigma(x)=ax+b$, and the initial value functions are bounded and H\"older continuous of order $H$. We prove the existence and uniqueness of the mild solution for both equations. We show that the solution is $L^{2}(\Omega)$-continuous and its $p$-th moments are uniformly bounded, for any $p \geq 2$.

Motivation & Objective

• To establish the existence and uniqueness of mild solutions for stochastic wave and heat equations with fractional noise in space with Hurst index H < 1/2.
• To prove that the solution is L²(Ω)-continuous and has uniformly bounded p-th moments for all p ≥ 2.
• To extend the analysis to the Hyperbolic and Parabolic Anderson Models (HAM and PAM) with affine diffusivity σ(x) = ax + b.
• To overcome technical challenges in stochastic integration with long-range dependent noise by using Picard iteration in a carefully constructed function space.
• To provide a rigorous framework for SPDEs driven by spatially homogeneous Gaussian noise with singular covariance structures.

Proposed method

• The authors use the Picard iteration method to construct the solution as a limit of successive approximations in a suitable function space.
• They establish the well-definedness and convergence of the Picard sequence by proving uniform bounds on the p-th moments of the iterates.
• The proof relies on Malliavin calculus techniques and estimates involving the covariance structure of the fractional noise with index H ∈ (1/4, 1/2).
• Key estimates are derived using the spectral measure µ and the Fourier transform of the noise covariance, particularly under the assumption that ∫(1 + |ξ|²)⁻¹µ(dξ) < ∞.
• The authors use a representation of the noise as a martingale measure and apply Itô-type stochastic integration with respect to this measure.
• They verify that the solution satisfies the mild formulation involving the fundamental solution Gt(x) and stochastic convolution with the noise.

Experimental results

Research questions

• RQ1 Does the stochastic wave equation with fractional noise in space (H < 1/2) admit a unique mild solution under affine diffusivity and Hölder-continuous initial data?
• RQ2 Can the p-th moments of the solution be uniformly bounded for all p ≥ 2 in the case of H ∈ (1/4, 1/2)?
• RQ3 How does the solution regularity and moment behavior depend on the Hurst index H < 1/2, particularly when the noise covariance is not locally integrable?
• RQ4 Can the classical Picard iteration method be adapted to SPDEs driven by non-Markovian, long-range dependent Gaussian noise with H < 1/2?
• RQ5 To what extent do the results extend to the Hyperbolic and Parabolic Anderson Models under the same noise and initial conditions?

Key findings

• The stochastic wave and heat equations driven by space-time Gaussian noise with H ∈ (1/4, 1/2) admit a unique mild solution that is L²(Ω)-continuous.
• The p-th moments of the solution are uniformly bounded for all p ≥ 2, with the bound depending only on the initial data and the noise parameters.
• The solution satisfies a key integral inequality involving the spatial increments of the solution and the noise's self-similarity index H.
• The proof establishes that the Picard iteration sequence is well-defined and converges in the L²(Ω) norm, ensuring the existence of a solution.
• The authors show that the solution to the Parabolic Anderson Model (PAM) can be expressed as u + b, where u solves the SHE with σ(x) = x + b and zero initial data.
• The method avoids the use of multiple stochastic integrals and instead relies on Picard iteration, making it applicable even when martingale techniques fail due to long-range dependence.