[Paper Review] Moment bounds in spde's with application to the stochastic wave equation

Le Chen, Robert C. Dalang|arXiv (Cornell University)|Jan 25, 2014

Stochastic processes and financial applications33 references3 citations

TL;DR

This paper establishes a general framework for proving existence, uniqueness, and moment bounds for solutions to stochastic partial differential equations (SPDEs), with a focus on the stochastic wave equation and hyperbolic Anderson model. By linking moment estimates to properties of the associated deterministic PDE's fundamental solution, the authors derive explicit formulas for second moments, prove weak intermittency, and obtain sharp Hölder continuity exponents based on initial data integrability.

ABSTRACT

We exhibit a class of properties of an spde that guarantees existence, uniqueness and bounds on moments of the solution. These moment bounds are expressed in terms of quantities related to the associated deterministic homogeneous p.d.e. With these, we can, for instance, obtain solutions to the stochastic heat equation on the real line for initial data that falls in a certain class of Schwartz distributions, but our main focus is the stochastic wave equation on the real line with irregular initial data. We give bounds on higher moments, and for the hyperbolic Anderson model, explicit formulas for second moments. We establish weak intermittency and obtain sharp bounds on exponential growth indices for certain classes of initial conditions with unbounded support. Finally, we relate Hölder-continuity properties of the stochastic integral part of the solution to the stochastic wave equation to integrability properties of the initial data, obtaining the optimal Hölder exponent.

Motivation & Objective

• To develop a unified framework for proving existence, uniqueness, and moment estimates for a broad class of SPDEs, including the stochastic wave and heat equations.
• To extend prior results on the parabolic Anderson model to irregular initial data, including Schwartz distributions and signed measures.
• To characterize the Hölder-continuity of the stochastic integral part of the solution in terms of integrability properties of the initial data.
• To establish sharp bounds on exponential growth indices and moment Lyapunov exponents for the hyperbolic Anderson model.
• To provide explicit formulas for second moments and higher-order moment estimates in the stochastic wave equation with irregular initial conditions.

Proposed method

• The authors formulate the SPDE as a stochastic integral equation involving the fundamental solution $ G(t,x) $ and an initial condition $ J_0(t,x) $, using Walsh's stochastic integration theory.
• They introduce a set of general assumptions on $ G $ and $ J_0 $, including $ L^2 $-continuity, tail control, and integrability, to ensure solution existence and moment bounds.
• The core technique involves analyzing $ n $-fold convolutions of $ G^2 $, leading to a function $ \mathcal{K} $ that controls the growth of moments via its $ L^1 $-norm and convolution structure.
• The authors derive explicit expressions for second moments using the function $ \mathcal{K} $, particularly in the case of the hyperbolic Anderson model with $ \theta \equiv 1 $.
• They apply asymptotic analysis and integral identities (e.g., from Bessel functions and hyperbolic trigonometric integrals) to bound $ \|I(t,x) - I(t',x')\|_p^2 $, leading to Hölder continuity estimates.
• The framework connects the Hölder exponent of the solution to the local integrability of the initial data, with optimal exponents derived via comparison to $ G $'s decay and regularity.

Experimental results

Research questions

• RQ1 Under what conditions on the fundamental solution $ G $ and initial condition $ J_0 $ does the SPDE admit a unique solution with finite moments?
• RQ2 What are the sharp bounds on the exponential growth indices $ \underline{\lambda}(p) $ and $ \overline{\lambda}(p) $ for the stochastic wave equation with unbounded initial data?
• RQ3 How do the moment Lyapounov exponents $ \overline{m}_p $ and $ \underline{m}_p $ behave for the hyperbolic Anderson model, and what conditions imply weak intermittency?
• RQ4 What is the optimal Hölder exponent for the stochastic integral part of the solution to the stochastic wave equation, and how does it relate to the $ L^p $-integrability of the initial data?
• RQ5 Can explicit formulas for second moments be derived for the hyperbolic Anderson model, and how do they depend on the initial condition and noise intensity?

Key findings

• For the hyperbolic Anderson model, the second moment is explicitly given by $ \mathbb{E}[|u(t,x)|^2] = \left(J_0^2 \star G_\kappa^2\right)(t,x) + \lambda^2 \int_0^t \int_{\mathbb{R}} G_\kappa(t-s,x-y)^2 \, dy \, ds $, with $ G_\kappa $ the wave kernel.
• The solution to the stochastic wave equation exists and is unique for initial data in a class of Schwartz distributions, including $ \delta_0 $ and $ \delta_0' $, under the proposed assumptions on $ G $ and $ J_0 $.
• Weak intermittency is established for the hyperbolic Anderson model, with $ \overline{m}_2 > 0 $ and $ \overline{m}_p < \infty $ for all $ p \geq 2 $, under suitable initial data conditions.
• The optimal Hölder exponent for the stochastic integral part of the solution is $ \frac{1-2a}{2} $, where $ a \in (0, \frac{1}{2}) $, and this exponent is directly linked to the $ L^p $-integrability of the initial data.
• Sharp bounds on the exponential growth indices are obtained: $ \underline{\lambda}(p) = \frac{1}{\kappa} \left( \frac{1}{2} \right)^{1-2a} $ and $ \overline{\lambda}(p) = \frac{1}{\kappa} \left( \frac{1}{2} \right)^{1-2a} $, with $ \kappa $ the wave speed and $ a $ related to the decay of $ G $.
• For time increments, the $ L^p $-norm of the increment satisfies $ \|I(t,x) - I(t',x)\|_p^2 \geq \frac{\lambda^2 |x|}{16\kappa^{2a}(1-2a)} h^{1-2a} $, with $ h = t' - |x|/\kappa $, proving the optimal Hölder exponent $ \frac{1-2a}{2} $.