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Weak approximation of stochastic partial differential equations: the non linear case
Arnaud Debussche
TL;DR
The paper studies Euler approximation of stochastic partial differential equations, focusing on weak error in the solution’s fixed-time law. It uses Kolmogorov equations and Malliavin integration by parts to handle irregular error terms, and proves weak order twice strong order under a technical condition on σ.
Problem
Method
The proof decomposes weak error over time steps using Kolmogorov equations and applies Malliavin calculus with integration by parts to remove an irregular stochastic integral.
Source: §1 · p. 3, §3 · p. 10Results
The authors prove that the Euler scheme’s weak order is twice its strong order, subject to a technical condition on σ.
Source: §1 · p. 3Takeaways & Limitations
The method is applied to a nonlinear stochastic heat equation driven by space-time white noise.
Source: §1 · p. 3Assumption (2.4) is restrictive, though it is void for additive noise or noise of the form BX dW.
Source: §2 · p. 7
Abstract
from arXiv · showhide
We study the error of the Euler scheme applied to a stochastic partial differential equation. We prove that as it is often the case, the weak order of convergence is twice the strong order. A key ingredient in our proof is Malliavin calculus which enables us to get rid of the irregular terms of the error. We apply our method to the case a semilinear stochastic heat equation driven by a space-time white noise.
1. Introduction
The introduction distinguishes pathwise strong error from weak error in the solution’s fixed-time law, noting that weak approximation of SPDEs has received little study. It presents a Malliavin-calculus approach for the Euler scheme and applies it to a nonlinear stochastic heat equation.
1. Introduction
- Weak error concerns approximation of the solution’s law at a fixed time, and few studies had addressed weak approximation of SPDEs.Strong error instead measures pathwise approximation.Source: §1 · p. 1
- The application is a nonlinear heat equation on a bounded interval with Dirichlet boundary conditions and space-time white noise.The nonlinear functions f and σ are assumed smooth and Lipschitz.Source: §1 · pp. 1–2
- Extending the classical SDE proof is difficult because the SPDE Kolmogorov equation has infinitely many variables and unbounded operators.Earlier SPDE approaches used restricted test functions or transformations unavailable for the heat equation considered here.Source: §1 · p. 2
- The method decomposes weak error using the Kolmogorov equation at each time step, then uses Malliavin calculus and integration by parts to remove an irregular stochastic integral.The authors prove that weak order is twice strong order, subject to a technical condition on σ.Source: §1 · pp. 2–3
2. Preliminaries and main result
The section sets up an implicit Euler approximation for an abstract stochastic partial differential equation and targets the error in smooth-test-function expectations. It states a weak-error estimate under a restrictive condition on the noise coefficient.
Setting and assumptions
- The equation is posed in a Hilbert space with a negative self-adjoint operator A having compact inverse, and the nonlinear function f is C3 with bounded derivatives up to order 3.Source: §2 · pp. 3–4
- The noise coefficient σ is C3 with bounded derivatives and satisfies growth and Lipschitz conditions; the stronger condition (2.4) substantially restricts its form.Source: §2 · p. 4
Objective
- The estimate targets the difference between expected values of a smooth test function at the exact solution and the Euler approximation at terminal time.Source: §2 · p. 6
Main result
- The main result gives a weak-error estimate for the Euler scheme under the stated conditions, including a restrictive assumption on the noise coefficient.The supplied theorem passage does not preserve the estimate’s formula or rate.Source: §2 · pp. 4, 7
Scope of the result
- Condition (2.4) is void for additive noise or noise of the form BX dW; otherwise, the noise must be a perturbation of such noise.The authors identify this condition as crucial to their proof.Source: §2 · p. 7
- The authors describe the result as optimal apart from the restrictive point and say nondegeneracy assumptions could permit weaker test-function smoothness in future work.Source: §2 · p. 7
3. Proof of the main result
The proof controls the weak error by decomposing it into terms and estimating them with stochastic-calculus tools. Galerkin approximations make the infinite-dimensional argument rigorous by allowing estimates with constants independent of the approximation dimension.
Proof strategy
- Galerkin approximations reduce the proof to finite-dimensional equations, with estimates uniform in the dimension before taking the limit.The constant does not depend on m, so the estimate passes to the infinite-dimensional equation as m→∞.Source: §3 · pp. 8, 24
Steps 1–2
- A continuous interpolation of the discrete Euler solution and Kolmogorov-equation and Itô-formula arguments organize the weak-error decomposition.The error is split into contributions denoted a_k, b_k, and c_k.Source: §3 · pp. 8–9
Step 4: Estimate of a_k
- Malliavin calculus handles stochastic-integral terms whose irregularity prevents directly compensating for the unbounded operator A^2.The error contribution a_k is split into parts; its stochastic-integral part is estimated using Malliavin calculus.Source: §3 · pp. 10–12
Step 5: Estimate of b_k
- For the nonlinear error term b_k, Itô’s formula generates additional terms, some of which also require Malliavin integration by parts.The absence of the unbounded operator A does not remove the difficulty introduced by the nonlinear term.Source: §3 · p. 14
Step 7: Conclusion
- The proof gathers estimates for the decomposed error terms and concludes the main error estimate.The conclusion combines the preceding estimates in (3.7).Source: §3 · pp. 13, 24
4. Auxiliary Lemmas
The auxiliary lemmas supply estimates used in the main proof, including regularity bounds and bounds for derivatives of the discrete solution. Their proofs use Galerkin approximations and pass to the limit to justify the results for the infinite-dimensional equation.
Proof approach
- The technical lemmas are proved on Galerkin approximations, with the limit m→∞ taken to obtain results for the infinite-dimensional equation.The section notes that infinite-dimensional estimates can be difficult to justify directly.Source: §4 · p. 24
Lemma 4.3
- Lemma 4.3 states an estimate for the Malliavin derivative of X_k for β in [0, 1/4), derived using the chain rule and discrete Gronwall.The proof begins from a chain-rule formula for the Malliavin derivative of X_k.Source: §4 · pp. 24–25
Derivative estimates
- For the first derivative, the cited passage gives |Du(t, x) · h| ≤ c∥ϕ∥1t−β|h|−β and says taking the supremum over h yields the result.Source: §4 · p. 26