[Paper Review] Weak error estimates for trajectories of SPDEs for Spectral Galerkin discretization

Charles-Édouard Bréhier, Martin Hairer+1 more|arXiv (Cornell University)|Feb 12, 2016

Stochastic processes and financial applications16 references6 citations

TL;DR

This paper establishes that for spectral Galerkin discretization of semilinear SPDEs—such as stochastic heat and damped wave equations—the weak error in approximating the full trajectory over [0, T] is twice the strong error rate under two conditions: (1) when the covariance operator commutes with the generator, and (2) in the non-commuting case, via analysis of a commutator term. The key innovation lies in using the Itô map and second-order Taylor expansion to transfer error from stochastic convolution to weak convergence.

ABSTRACT

We consider stochastic semi-linear evolution equations which are driven by additive, spatially correlated, Wiener noise, and in particular consider problems of heat equation (analytic semigroup) and damped-driven wave equations (bounded semigroup) type. We discretize these equations by means of a spectral Galerkin projection, and we study the approximation of the probability distribution of the trajectories: test functions are regular, but depend on the values of the process on the interval $[0,T]$. We introduce a new approach in the context of quantative weak error analysis for discretization of SPDEs. The weak error is formulated using a deterministic function (Itô map) of the stochastic convolution found when the nonlinear term is dropped. The regularity properties of the Itô map are exploited, and in particular second-order Taylor expansions employed, to transfer the error from spectral approximation of the stochastic convolution into the weak error of interest. We prove that the weak rate of convergence is twice the strong rate of convergence in two situations. First, we assume that the covariance operator commutes with the generator of the semigroup: the first order term in the weak error expansion cancels out thanks to an independence property. Second, we remove the commuting assumption, and extend the previous result, thanks to the analysis of a new error term depending on a commutator.

Motivation & Objective

• To establish weak convergence rates for the full trajectory of SPDEs under spectral Galerkin space discretization, rather than just at a fixed time point.
• To resolve the challenge of weak error estimation in the space of continuous trajectories, E = C([0, T], H), which is more complex than pointwise evaluation.
• To extend the well-known 'twice the strong rate' result from finite-dimensional SDEs to infinite-dimensional SPDEs with additive, spatially correlated noise.
• To develop a novel analytical framework based on the Itô map and second-order Taylor expansion to relate weak error to spectral approximation of the stochastic convolution.
• To prove the weak convergence rate remains twice the strong rate even when the covariance operator does not commute with the generator, by analyzing a new commutator-dependent error term.

Proposed method

• Formulate the SPDE in Hilbert space using the stochastic evolution equation framework: dX(t) = AX(t)dt + F(X(t))dt + dW_Q(t).
• Apply spectral Galerkin projection using orthogonal projection P_N onto the first N eigenvectors of the linear operator A, yielding the approximate solution X_N.
• Define the weak error using test functions ϕ in C_b^2(E, R), where E = C([0, T], H) is the space of continuous trajectories.
• Introduce the Itô map, which deterministically maps the stochastic convolution (with F=0) to the full solution, enabling error transfer via Taylor expansion.
• Use second-order Taylor expansion of the Itô map to decompose the weak error into terms involving the spectral approximation error of the stochastic convolution.
• Analyze the commutator [B, P_N] arising in the non-commuting case and derive moment bounds on the resulting error process via factorization and stochastic integration techniques.

Experimental results

Research questions

• RQ1 Can the weak convergence rate for SPDE trajectories under spectral Galerkin discretization be shown to be twice the strong rate, even in the absence of commutativity between the covariance operator and the generator?
• RQ2 How can the weak error in the space of continuous trajectories be analyzed when standard pointwise weak error techniques fail?
• RQ3 What role does the Itô map play in transferring error from the stochastic convolution to the weak error of the full trajectory?
• RQ4 In the non-commuting case, how can the commutator term [B, P_N] be controlled to preserve the weak convergence rate?
• RQ5 What are the precise regularity and integrability conditions on the noise and initial data that allow for this rate doubling?

Key findings

• For SPDEs with additive, spatially correlated noise, the weak error in the space of trajectories C([0, T], H) is bounded by a constant times the sum of three terms: O(λ_N^{-(s_0 - s)/2}), O(λ_N^{-(1 - (s_F + s + ε)/2)}), and O(λ_N^{-(1/2 - s - ε)}), where λ_N is the N-th eigenvalue of the generator.
• When the covariance operator Q commutes with the generator A, the first-order term in the weak error expansion vanishes due to an independence property, leading to a clean rate doubling.
• In the non-commuting case, the rate doubling is preserved by proving that the commutator-induced error term is controlled via a new bound: E∥ρ_N∥^2_{∞,s,T} ≤ C_{ε,s,T} λ_N^{-(1 - s - ε)} for any ε ∈ (0, 1/2 - s).
• The bound on the commutator term is derived using the factorization method and moment estimates on the Gaussian process involving (t - r)^{-ε/2} and the operator [B, P_N].
• The analysis confirms that the weak convergence rate is exactly twice the strong rate in both the commuting and non-commuting settings, under appropriate regularity and decay assumptions on the noise and initial data.
• The result holds for both parabolic (e.g., stochastic heat equation) and hyperbolic (e.g., damped wave equation) SPDEs, provided the initial condition and test functions are sufficiently regular.