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Domain preserving splitting schemes for a class of SPDEs driven by a standard Brownian motion

Charles-Edouard Bréhier, David Cohen, Gijs Custers

arXiv:2609.29950v1math.NAcs.NAmath.PR

TL;DR

For SPDEs with solutions constrained to [-1,1], the paper proposes a domain-preserving Lie–Trotter splitting scheme. It proves preservation for every time-step size and mean-square convergence at rate 1/2-.

  • Problem

    Standard time integrators may leave the bounded interval that the SPDE solution occupies, motivating numerical schemes that preserve the interval almost surely.

    Source: §1 · p. 1

  • Method

    The paper constructs a domain-preserving Lie–Trotter splitting scheme for the SPDE and proves its properties using an auxiliary equation with extended coefficients.

    Source: §1 · pp. 1–2

  • Results

    The DPLT scheme preserves [-1,1] almost surely for every time-step size and converges in mean square with rate 1/2-.

    Source: §1 · p. 2, §3.2 · p. 9

  • Takeaways & Limitations

    The scheme provides a strongly convergent time integrator that keeps its numerical solution in the prescribed interval almost surely.

    Source: §1 · p. 2

    The stated setting assumes a deterministic initial value; random F0-measurable initial conditions require minor modifications to the setting and proofs.

    Source: §2.1 · p. 3

Abstract

from arXiv · show

We consider a class of SPDEs driven by a standard real-valued Brownian motion, with drift and diffusion coefficients such that there exists a unique mild solution taking values in the interval $[-1,1]$ almost surely. To preserve this qualitative property of the exact solution, we propose a domain preserving Lie--Trotter splitting scheme: for any choice of the time-step size, the numerical solution takes values in the interval $[-1,1]$ almost surely. Furthermore, we prove mean-square convergence with rate $1/2-$ for the domain preserving Lie--Trotter scheme. These theoretical results are illustrated with numerical experiments.

1. Introduction

The paper develops a domain-preserving Lie–Trotter scheme for SPDEs whose solutions stay in [-1,1], addressing the risk that standard integrators leave that interval. It proves almost-sure domain preservation for every time-step size and mean-square convergence at rate 1/2-.

1. Introduction

  • Keeping approximations in the prescribed interval matters because standard semi-implicit Euler–Maruyama and stochastic exponential Euler schemes may leave it.The motivating examples include stochastic Nagumo and Fisher–KPP equations with bounded solution ranges.Source: §1 · p. 1
  • The Lie–Trotter scheme applies a domain-preserving numerical method to the position-parameterized stochastic subsystem and solves the linear heat equation exactly.The scheme is given in equation (22).Source: §1 · p. 2
  • The DPLT scheme preserves [-1,1] almost surely for every time-step size and converges in mean square with rate 1/2-.The results are established for the scheme using the original coefficients.Source: §1 · p. 2
  • The proof extends the drift and diffusion beyond [-1,1], establishes convergence for the modified SPDE, and uses domain preservation to recover the original scheme and solution.The original coefficients are defined only on [-1,1].Source: §1 · p. 2
  • The paper establishes strong convergence for a domain-preserving scheme in the stated general SPDE class and also studies Stratonovich noise, multidimensional noise, and systems of SPDEs.For multidimensional noise and systems, the analysis is omitted for notational simplicity and the contexts are illustrated numerically.Source: §1 · pp. 2–3

2. Setting

The setting is an Itô semilinear SPDE on a cube with Dirichlet boundary conditions and interval-valued data and solutions. The section establishes heat-kernel and mild-solution foundations, including well-posedness through an auxiliary equation with extended coefficients.

2. Setting

  • The problem is posed over a finite time horizon on D=(0,1)^d, with homogeneous Dirichlet boundary conditions and a standard real-valued Brownian driver.The initial value is assumed deterministic, continuous, boundary-zero, and valued in [-1,1].Source: §2.1 · p. 3

2.1. Introducing the SPDE.

  • The coefficient assumptions make the drift and diffusion vanish at ±1 and yield the factorization b=aσ and g=fσ, where σ(u)=(u-1)(u+1).The coefficients also satisfy time regularity and spatial-value smoothness conditions.Source: §2.1 · pp. 3–4

2.2. Heat kernel.

  • The Dirichlet heat kernel is nonnegative, has spatial integral at most one, satisfies the semigroup property, and obeys a stated time-regularity bound.These properties are used in the mild-solution and numerical analyses.Source: §2.2 · p. 4

2.3. Mild solutions and auxiliary SPDE.

  • Under the assumptions, the original SPDE has a unique mild solution, whose existence requires proving preservation of [-1,1] because its coefficients are defined only there.The mild formulation is given through the heat kernel and stochastic and deterministic integral terms.Source: §2.3 · p. 5
  • The proof extends the coefficients to globally Lipschitz functions on R that agree with the originals on [-1,1], then uses the modified equation and domain preservation to obtain the original mild solution.The extended equation has a unique mild solution, and the paper shows that it remains in the interval.Source: §2.3 · pp. 5, 7

3. Domain preserving Lie–Trotter splitting scheme

This section presents the domain-preserving Lie–Trotter scheme, which combines stochastic integration with heat evolution to keep approximations in [-1,1] for any time-step size, and its mean-square convergence to solutions of the modified and original SPDEs. Numerical experiments in one and two spatial dimensions assess its convergence order and domain preservation.

3.1. A class of explicit domain preserving schemes.

  • The DPLT scheme combines an explicit domain-preserving integrator for an auxiliary stochastic subsystem with exact homogeneous heat evolution in a Lie–Trotter step.The stochastic subsystem is advanced first, followed by the heat equation over the same time interval.Source: §3.1 · pp. 7–8

3.2. Main results on the DPLT splitting scheme.

  • For every time-step size, the DPLT scheme (19) takes values in [-1,1] almost surely at every grid point and spatial location.The proposition also states that the numerical solution is almost surely continuous in space.Source: §3.2 · p. 9
  • The original-level scheme (22), using the original coefficients, also preserves [-1,1] almost surely and converges in mean square at order 1/2− to the original SPDE solution.The original-level formulation can replace the extended coefficients because the scheme’s approximations remain in the domain.Source: §3.2 · pp. 9–10
  • Under Assumptions 1–3, the DPLT scheme (19) converges in mean square at order 1/2− to the solution of the modified SPDE.The theorem’s estimate holds for all time-step sizes τ = T/M and every ε in (0, 1/2).Source: §3.2 · p. 10

3.3. Numerical experiments.

  • The numerical experiments are described as illustrating DPLT’s qualitative behavior and mean-square convergence, with comparisons against classical SPDE schemes.The listed comparators include Euler–Maruyama under a CFL condition and semi-implicit Euler–Maruyama.Source: §3.3 · pp. 10–11

3.3. Numerical experiments

  • All simulated DPLT realizations remain in [-1,1], while some realizations from classical time integrators exit the domain.Table 1 compares DPLT, Euler–Maruyama, semi-implicit Euler–Maruyama, and stochastic exponential Euler over 100 simulated paths.Source: §3.3 · p. 11, Table 1 · p. 11
  • For the convergence experiments, Euler–Maruyama is omitted, and the sEM and SEXP coefficients are extended to ensure mean-square convergence.The extensions ensure global Lipschitz continuity; the experiments consider a = 0, f = 2e^u and a = u, f = 5u.Source: §3.3 · pp. 11–12
  • The observed maximum mean-square error convergence order is 1/2 in the one-dimensional experiment, confirming Theorem 11.The experiment used h = 2^-8, time steps τ = 2^-4 to 2^-15, and 200 samples; Figure 1 plots results against a DPLT reference with τ_ref = 2^-16.Source: §3.3 · p. 12, Figure 1 · p. 12
  • The observed maximum mean-square error convergence order is 1/2 in the two-dimensional experiment, as stated in Theorem 11.The experiment used h = 2^-6 and 200 samples, with time steps τ = 2^-4 to 2^-15 and a DPLT reference with τ_ref = 2^-16.Source: §3.3 · p. 12
  • The maximum mean-square errors are reported to be independent of space-mesh size in both one and two spatial dimensions.The one-dimensional experiment used h = 2^-4, 2^-6, 2^-8, and 2^-10; the two-dimensional experiment used h = 2^-4, 2^-6, and 2^-8.Source: §3.3 · p. 12

4. Proof of the main results

Section 4 proves the main results on existence and uniqueness for the SPDE and convergence of the DPLT scheme. The arguments establish regularity and boundedness properties of auxiliary processes, derive strong mean-square error bounds, and connect the constrained mild solution and auxiliary-process estimates to the SPDE and numerical grid values.

4.1. Auxiliary processes. `

  • Proposition 14 establishes temporal regularity properties for the auxiliary process rZ, including estimates on its temporal increments.The proposition states bounds for all moment orders p, with constants depending on T.Source: §4.1 · pp. 14, 18
  • The continuous-time auxiliary process uDP coincides with the DPLT scheme at grid points and has a within-step representation using another auxiliary process.Proposition 13 gives both the grid-point identity and the representation for arbitrary times.Source: §4.1 · p. 15

4.1. Auxiliary processes

  • For the auxiliary process rZ, the proof establishes temporal regularity, moment bounds, and upper bounds (39).The temporal regularity follows from bounded coefficients and Gaussian moment bounds; the supplied passages do not preserve the displayed rates.Source: §4.1 · pp. 18–19
  • The auxiliary process uDP also satisfies a mean-square temporal regularity result.The stated bound applies within time steps for m = 1, …, M − 1; the displayed exponent and rate are not legible in the supplied passages.Source: §4.1 · pp. 19–20
  • uDP is almost surely continuous on [0,T] × D and takes values in [-1,1] almost surely.The domain-preservation proof treats both time-grid points and times between grid points.Source: §4.1 · pp. 19, 21

4.1. Auxiliary processes. `

  • The cited estimates bound the first indexed sum by E2,1 + E3,1 + E4,1 ≤ C(T)|t−tm| and the second indexed sum by E2,2 + E3,2 + E4,2 ≤ C(T)|t−tm|^(1−2ε).Source: §4.1 · pp. 24–25

4.2. Proof of Theorem 11. `

  • Theorem 11’s mean-square error proof decomposes u*−uDP into e1 and e2, then splits e1 into five terms; estimating e1,3 requires addressing when ℓ(s)≥τ.The passage states that the proof establishes the mean-square estimates (25), but does not give their values here.Source: §4.2 · pp. 25, 26, 28
  • For e1,3, the proof handles the ℓ(s)≥τ restriction by cases, using uDP∈[−1,1] and, when t≥τ, temporal regularity with ℓ(s)≥s−τ.The auxiliary process takes values in [−1,1] almost surely.Source: §4.2 · pp. 28–29
  • Temporal regularity estimates and Grönwall yield the auxiliary-process error bound; equality with the exact solution at grid points then gives Theorem 11’s strong error estimates.Source: §4.2 · pp. 29, 31

4.3. Proof of Proposition 6.

  • Almost surely, the mild solution remains in [-1,1] for all times and spatial points and is the unique mild solution to the original SPDE.The proof uses almost-sure continuity to strengthen pointwise domain preservation to a simultaneous statement for all times and spatial points.Source: §4.3 · p. 32

4.4. Proof of Corollary 12.

  • The DPLT scheme’s grid values coincide with the auxiliary and exact solutions, yielding strong error bounds of order τ^(1/2-ε) for every ε∈(0,1/2).The bound is stated with a constant Cε(T).Source: §4.4 · p. 32

5. Generalisations

This section sketches and numerically illustrates extensions of the domain-preserving scheme to Stratonovich noise, multidimensional Itô noise, and systems, noting that detailed analysis is omitted and the extensions can be combined. In the systems experiment, DPLT converges at the predicted rate of 1/2 for both independent and equal Brownian motions, with comparisons supporting its consistency with the SPDE system.

5.1. Generalisation to Stratonovich SPDE.

  • For Stratonovich SPDEs, applying DPLT to the equivalent Itô formulation preserves [-1,1] for every time-step size and has mean-square convergence rate 1/2-.The result assumes the stated conditions on the initial value, coefficients, and integrator, including supplementary conditions ensuring the equivalent Itô formulation is covered.Source: §5.1 · pp. 33–34

5.2. Generalisation to multidimensional noise.

  • For Itô SPDEs driven by K-dimensional noise, the adapted DPLT scheme preserves [-1,1] for any time-step size and generalizes the mean-square convergence rate 1/2-.The diffusion coefficients are factored as g_k = f_kσ; a unique mild solution also remains in [-1,1] almost surely under the stated assumptions.Source: §5.2 · pp. 35–36

5.3. Generalisation to systems.

  • For two-component SPDE systems, the adapted DPLT scheme preserves [-1,1]^2 for any time-step size and generalizes the mean-square convergence rate 1/2-.The systems may use equal or independent Brownian motions; the paper says extension to arbitrary dimension is straightforward.Source: §5.3 · pp. 37–38
  • For independent and equal Brownian motions, Figure 7 shows DPLT convergence at rate 1/2, matching theory; comparisons with sEM and SEXP support consistency with the SPDE system.The experiment uses identical sine initial conditions, a finite-difference mesh h=2^-8, and 200 samples.Source: §5.3 · p. 39
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