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Strong error analysis of a stochastic exponential integrator for SPDEs driven by fractional Brownian motion with H<1/2

Antoine Tambue, Aurelien Junior Noupelah, Louis Aime Fono

arXiv:2610.02412v1math.NAcs.NAmath.PR

TL;DR

Strong numerical analysis of semilinear SPDEs in the rough H < 1/2 fractional Brownian regime is delicate because the noise has low temporal regularity. The paper analyzes a finite element stochastic exponential integrator using semigroup, canonical Hilbert-space, and Malliavin-calculus estimates, obtaining an H-dependent strong temporal rate that reaches H + 1/2 under maximal spatial regularity.

  • Problem

    For H < 1/2, negatively correlated fBm increments have low temporal regularity, and estimates used in the smoother regime do not directly apply to strong numerical analysis of semilinear SPDEs.

    Source: §1 · p. 3

  • Method

    The paper combines finite element spatial discretization and a stochastic exponential integrator with analytic semigroup estimates, fBm’s canonical Hilbert-space structure, and Malliavin calculus.

    Source: §1 · p. 4, §2.1 · p. 6, §2.2 · p. 8

  • Results

    The stochastic exponential integrator’s H-dependent strong temporal convergence rate reaches H + 1/2 under maximal spatial regularity.

    Source: §1 · pp. 4–5

  • Takeaways & Limitations

    The analysis establishes strong temporal convergence for the rough fractional regime while allowing a linear operator that need not be self-adjoint.

    Source: §1 · pp. 2, 5

    The stated setting uses H = L2(Λ) on bounded domains Λ ⊂ R^d with d = 1, 2, or 3.

    Source: §1 · p. 2

Abstract

from arXiv · show

Fractional Brownian motion provides a useful framework for modeling random phenomena with anti-persistent temporal correlations that arise in applications such as anomalous diffusion, hydrology, finance, and geophysical processes. In this paper, we study a class of semilinear stochastic partial differential equations driven by additive fractional Brownian motion with Hurst parameter $H\in(0,\frac12)$. We establish the well-posedness and space-time regularity of the mild solution and investigate the strong convergence of a stochastic exponential integrator for the temporal discretization. The analysis allows the linear operator to be non-self-adjoint and combines analytic semigroup estimates with the canonical Hilbert-space structure associated with fractional Brownian motion and Malliavin calculus to handle the low temporal regularity of the noise. We derive an $H$-dependent strong temporal convergence rate, which reaches $H+\frac12$ under maximal spatial regularity. Numerical experiments for a stochastic advection--diffusion--reaction problem with heterogeneous Darcy flow confirm the theoretical temporal convergence rates. We also estimate the mean of the solution with different Hurst parameters $H\in(0,\frac12)$ using the Monte Carlo method.

1. Introduction

The paper analyzes strong temporal approximation of semilinear SPDEs driven by rough fractional Brownian motion, using a stochastic exponential integrator and estimates tailored to the low regularity of the noise. Its convergence rate depends on the Hurst parameter and reaches H + 1/2 under maximal spatial regularity.

1. Introduction

  • The paper presents numerical simulations for a semilinear SPDE, illustrating the theoretical convergence rate of its fully discrete stochastic exponential scheme and estimating the mean with the Monte Carlo method.The setting uses a bounded domain Λ ⊂ Rd, with d = 1, 2, or 3.Source: §1 · pp. 2, 5
  • For H < 1/2, negatively correlated fBm increments have low temporal regularity, and estimates used in the smoother regime do not directly apply.The analysis therefore uses estimates adapted to fBm’s canonical Hilbert space and Malliavin calculus for random integrands.Source: §1 · p. 3
  • The approach combines finite element spatial discretization and a stochastic exponential integrator, allowing analysis with a linear operator that need not be self-adjoint.Analytic semigroup estimates, the canonical Hilbert-space structure of fBm, and Malliavin-calculus arguments address the rough noise.Source: §1 · pp. 2, 4
  • A diagonal–off-diagonal decomposition of fractional covariance terms and summability of the discrete covariance sequence yield a stochastic-convolution quadrature estimate.These auxiliary estimates are tailored to temporal discretization.Source: §1 · p. 4
  • A uniform Malliavin-derivative bound and a cellwise fractional-covariance estimate control random Taylor-expansion integrands as divergence integrals.Together, these estimates control the accumulated nonlinear drift defect without losing temporal order.Source: §1 · pp. 4–5
  • The stochastic exponential integrator has an H-dependent strong temporal convergence rate that reaches H + 1/2 under maximal spatial regularity.The analysis combines finite element spatial discretization with temporal error estimates.Source: §1 · pp. 4–5

2. Functional framework, fractional noise and well-posedness

This section sets the analytic and probabilistic framework for the semilinear SPDE, then establishes existence and uniqueness of its mild solution under stated assumptions.

2.1. Functional analytic setting

  • The operator framework uses a bounded analytic semigroup and fractional domain spaces to describe spatial regularity, without requiring the linear operator to be self-adjoint.The setting includes second-order elliptic operators and semigroup smoothing estimates.Source: §2.1 · p. 6

2.2. Fractional Brownian motion and the canonical Hilbert space

  • The fractional-noise framework uses a canonical Hilbert space whose covariance norm captures fractional increments and supports later cellwise temporal estimates.The paper also represents the Hilbert-space-valued fractional Brownian motion through the covariance operator's eigenpairs.Source: §2.2 · pp. 7–8

2.3. Malliavin derivative and divergence integral

  • Random integrands in the nonlinear Taylor expansion require Malliavin-derivative control as well as the canonical norm to estimate their divergence integrals.This is the mechanism used for random kernels in the accumulated nonlinear drift defect.Source: §2.2 · p. 9, §2.3 · pp. 10–11

2.4. Assumptions and well-posedness

  • The assumptions include measurable initial data, deterministic additive fractional noise, and a twice differentiable nonlinearity with bounded second derivative.The bounded second derivative is used when differentiating random Taylor coefficients with respect to the fractional Gaussian noise.Source: §2.4 · pp. 11–12
  • Under Assumptions 1–3, the SPDE has a unique mild solution, established by a weighted-norm contraction argument.The proof also obtains an a priori bound for the solution.Source: §2.4 · pp. 12, 15

3. Regularity of the mild solution

The section establishes spatial and temporal regularity estimates for the mild solution and states that analogous estimates hold for its semidiscrete counterpart under uniform discrete semigroup estimates.

3. Regularity of the mild solution

  • The section establishes spatial and temporal regularity estimates for the mild solution, which are used in the strong convergence analysis.The result is stated under Assumptions 1–3 and a parameter condition involving β and H.Source: §3 · p. 16
  • Analogous regularity estimates hold for the semidiscrete solution when the discrete semigroup estimates are uniform in the spatial discretization parameter h.Source: §3 · p. 16
  • The temporal regularity proof estimates the deterministic and stochastic convolution terms separately before combining their bounds.Source: §3 · pp. 18–20

4. Space approximation and error estimates

The section introduces a finite-element spatial semidiscretization and establishes a strong mean-square error estimate between its mild solution and the original solution.

4. Space approximation and error estimates

  • The spatial approximation uses continuous piecewise-linear finite elements on a regular triangulation and defines a corresponding semidiscrete evolution problem.Source: §4 · pp. 20–21
  • The spatial error analysis uses the difference between the continuous and discrete semigroups, with an auxiliary estimate whose constant is independent of h and t.Source: §4 · p. 21
  • The section establishes a strong mean-square error estimate between the mild solution and its spatial semidiscrete approximation.The supplied theorem passage does not state the estimate's rate.Source: §4 · p. 21

5. Fully discrete stochastic exponential scheme and strong error estimates

The section introduces a fully discrete stochastic exponential scheme and develops auxiliary estimates for its strong-error analysis. These estimates support the main convergence result for the scheme.

5. Fully discrete stochastic exponential scheme and strong error estimates

  • The fully discrete stochastic exponential approximation is introduced for the semidiscrete problem, with a strong-convergence analysis.The scheme is also represented in an equivalent one-step form.Source: §5 · pp. 24–25

5.1. Auxiliary estimates

  • The auxiliary estimates bound stochastic-convolution quadrature error and the Malliavin derivative, which are used to control stochastic terms in the nonlinear drift defect.The Malliavin-derivative estimate is uniform in the spatial discretization parameter h, the time step, and the time index.Source: §5.1 · pp. 25, 28, Lemma 5 (Cellwise fractional… · p. 32

Lemma 5 (Cellwise fractional covariance estimate). Let H ∈

  • For H < 1/2, the proof invokes the increment characterization of the canonical Hilbert space; on each Ii, the approximation residual satisfies eg(r) = g(r) − gi.Source: Lemma 5 (Cellwise fractional… · p. 31

5.2. Strong convergence of the fully discrete scheme

6. Numerical simulations

The simulations test SETDM1's strong temporal convergence for four Hurst parameters and compare solution means under heterogeneous Darcy flow. The observed convergence orders agree with the theoretical orders, while the mean concentration changes moderately between H = 0.20 and H = 0.40.

Numerical setup

  • The simulations use a stochastic advection–diffusion–reaction problem with heterogeneous Darcy flow and estimate solution means using Monte Carlo.The permeability field is modeled as lognormal with a truncated Karhunen–Loève expansion; 30 modes are retained.Source: §6 · p. 38

Temporal convergence

  • Observed orders 0.72, 0.83, 0.94, and 1.01 for H = 0.20, 0.30, 0.40, and 0.49 agree well with theoretical orders 0.70, 0.80, 0.90, and 0.99, respectively.The errors are measured at T = 1 in the root-mean-square L2(Λ) norm against a reference solution, using 200 shared-path fractional Brownian samples.Source: §6 · p. 39, Figure 2 · p. 40

Solution means

  • The stochastic mean preserves the main transport pathways of the heterogeneous Darcy velocity field, while fractional forcing changes concentration levels along them.The deterministic solution is obtained by suppressing fractional noise while keeping other parameters unchanged.Source: §6 · pp. 38, 40
  • Between H = 0.20 and H = 0.40, the mean concentration shows moderate changes in intensity and spatial distribution.Source: §6 · p. 40
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