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Milstein-type Schemes for Hyperbolic SPDEs

Felix Kastner, Katharina Klioba

arXiv:2512.19647v5math.NAcs.NAmath.APmath.FAmath.PR

TL;DR

Hyperbolic SPDEs lack the semigroup regularization used in parabolic convergence analyses, leaving higher-order Milstein convergence an open problem. This paper analyzes rational and exponential Milstein schemes and proves optimal pathwise uniform rates under sufficient regularity. Rational schemes have a logarithmic correction, whereas exponential schemes attain the rate without it.

  • Problem

    For hyperbolic SPDEs with contractive semigroups, higher-order Milstein convergence had remained an open research direction beyond the available analyses of other schemes.

    Source: The Milstein scheme for SPDEs · p. 2

  • Method

    The paper analyzes Milstein schemes, including rational semigroup approximations, in an abstract framework for hyperbolic SPDEs.

    Source: §1 · p. 1, Main result · p. 4

  • Results

    With sufficient regularity, the optimal rate is 1; rational Milstein schemes have a logarithmic correction, while exponential Milstein schemes do not.

    Source: Main result · pp. 3–4

  • Takeaways & Limitations

    The work gives a rigorous error analysis for hyperbolic Milstein schemes and establishes pathwise uniform convergence rates for rational and exponential variants.

    Source: Main result · p. 4

    Piecewise constant extensions of the rational scheme, and piecewise affine extensions in the scalar Brownian example, attain only the lower rate h^(1/2)√log(T/h).

    Source: §5.2 · p. 30

Abstract

from arXiv · show

This article studies the temporal approximation of hyperbolic semilinear stochastic evolution equations with multiplicative Gaussian noise by Milstein-type schemes. We take the term hyperbolic to mean that the leading operator generates a contractive, not necessarily analytic $C_0$-semigroup. Optimal convergence rates are derived for the pathwise uniform strong error \[ E_h^\infty := \Big(\mathbb{E}\Big[\max_{1\le j \le M}\|U_{t_j}-u_j\|_X^p\Big]\Big)^{1/p} \] on a Hilbert space $X$ for $p\in [2,\infty)$. Here, $U$ is the mild solution and $u_j$ its Milstein approximation at time $t_j=jh$ with step size $h>0$ and final time $T=Mh>0$. For sufficiently regular nonlinearity and noise, we establish strong convergence of order one, with the error satisfying $E_h^\infty\lesssim h\sqrt{\log(T/h)}$ for rational Milstein schemes and $E_h^\infty \lesssim h$ for exponential Milstein schemes. This extends previous results from parabolic to hyperbolic SPDEs and from exponential to rational Milstein schemes. Moreover, root-mean-square error estimates are strengthened to pathwise uniform estimates. Numerical experiments validate the convergence rates for the stochastic Schr\"odinger equation. Further applications to Maxwell's and transport equations are included.

1. Introduction

The paper develops Milstein time discretizations for hyperbolic stochastic evolution equations and establishes optimal pathwise uniform convergence rates under regularity assumptions. It extends the analysis to rational schemes and explains how the proof handles nonlinearities and uniform-in-time errors.

1. Introduction

  • The introduction frames the problem for contractive, not necessarily analytic semigroups and identifies regularity conditions for attaining order one.Earlier results for hyperbolic equations include exponential Euler and rational schemes, while higher rates had remained an open direction for Milstein schemes.Source: §1 · p. 1, The Milstein scheme for SPDEs · p. 2

The Milstein scheme for SPDEs.

  • Milstein schemes target rates above 1/2 by incorporating higher-order stochastic information, extending earlier work on hyperbolic equations beyond exponential Euler and rational schemes.The scheme uses an iterated stochastic integral; the exponential scheme takes the semigroup itself as its time-step approximation.Source: §1 · p. 1, The Milstein scheme for SPDEs · p. 2

Main result.

  • With sufficient regularity, Milstein schemes attain the optimal rate 1; rational schemes have a logarithmic correction, while exponential schemes do not.The results cover concrete equations including Schrödinger and Maxwell equations.Source: Main result · pp. 3–4
  • The analysis provides optimal Milstein convergence rates for hyperbolic SPDEs, including rational semigroup approximations and pathwise uniform estimates for p-th moments.The framework also gives error estimates on the full interval through a suitable continuous-time extension.Source: Main result · pp. 3–4

Method of proof.

  • The proof uses Gâteaux differentiability and Hölder continuity of derivatives, stochastic maximal estimates, and a semigroup-based error splitting.This avoids assuming twice Fréchet differentiability and supports pathwise uniform estimates beyond the root-mean-square case.Source: Method of proof · pp. 5–6

Overview.

  • The paper applies its theoretical results to Schrödinger, Maxwell, and transport equations and reports simulations for stochastic Schrödinger equations.The simulations are described as validating the theoretical findings.Source: Overview · p. 6

2. Preliminaries

The preliminaries establish stochastic-integration tools, differentiability notions, and semigroup-approximation conditions used in the analysis. They also fix notation and introduce a discrete Grönwall variant.

Notation.

  • The paper fixes the uniform time grid, solution and approximation notation, and conventions for probability norms and function spaces.The step size is h = T/M, and ⌊t⌋ denotes the grid point immediately before t.Source: Notation · p. 6

2.1. Stochastic integration in Hilbert spaces.

  • A cylindrical Brownian motion is characterized through centered Gaussian variables, covariance, and filtration properties; its one-dimensional case recovers standard Brownian motion.A Q-Wiener process can be represented using cylindrical Brownian motion by replacing G with GQ^1/2.Source: §2.1 · p. 7
  • The stochastic-convolution maximal inequality applies to quasi-contractive semigroups, while a logarithmic square-function estimate is an essential tool for rational Milstein convergence.For the stochastic-convolution inequality, the stated constants are B2 = 2 and Bp = 4√p for p > 2.Source: §2.1 · pp. 7–8

2.2. Gˆateaux and Fr´echet differentiability.

  • Fréchet differentiability implies Gâteaux differentiability, but the converse generally needs continuity of the Gâteaux derivative; Lipschitz continuity alone bounds that derivative uniformly.For the stated Nemytskii operator on L2, Fréchet differentiability at a point forces it to be affine linear.Source: §2.2 · pp. 8–9

2.3. Approximation of semigroups.

  • Semigroup schemes are assessed by their approximation order and contractivity; exponential Euler, implicit Euler, and Crank–Nicolson have different regularity-dependent orders.Exponential Euler has any order on X; implicit Euler has order α on D(A^2α), and Crank–Nicolson has order α on D(A^(3α/2)) when S is contractive. A discrete Grönwall variant is also introduced for subsequent estimates.Source: §2.3 · p. 9, §2.4 · p. 9

3. Setting and Assumptions

The setting is a stochastic evolution equation driven by cylindrical Brownian motion whose generator produces a contractive semigroup. The assumptions and Milstein construction combine regularity on a stronger space with Lipschitz control on the base space.

3. Setting and Assumptions

  • The equation is posed on Hilbert spaces X and Y, with a contractive semigroup and Y embedded into an interpolation or operator domain determined by α.The noise takes values in Hilbert–Schmidt operators, and the solution begins from initial data ξ.Source: §3 · p. 10
  • The assumptions use global Lipschitz continuity on X and linear growth and invariance on Y for the nonlinear drift and multiplicative noise.The derivatives and the composition G′G also have specified regularity and growth conditions on the spaces in the assumptions.Source: §3 · pp. 10, 12
  • The rational Milstein scheme advances the solution on an equidistant grid using a contractive semigroup approximation, Wiener increments, and iterated stochastic integrals.Setting R = S gives the exponential Milstein scheme.Source: §3 · pp. 11–12
  • Hölder-continuous derivatives on Y support Taylor expansions for F and G without requiring Fréchet differentiability of F on X.This distinction permits nonlinear Nemytskii operators, for which Fréchet differentiability on X can force affine linearity.Source: §3 · pp. 12–13

4. Well-posedness and Stability

This section establishes mild-solution well-posedness and stability estimates needed for convergence analysis. It proves pointwise moment bounds for the rational Milstein scheme and describes conditions for pathwise uniform stability.

4. Well-posedness and Stability

  • Under the stated assumptions, the equation has a unique mild solution with an a priori moment bound in X or Y.For Y, the theorem additionally requires invariance and linear growth conditions; the initial condition and noise must have the specified integrability.Source: §4 · p. 13
  • The rational Milstein scheme satisfies pointwise strong stability on Y, with moments bounded uniformly over the time-step index.The proof uses contractivity, linear growth, stochastic-integral estimates, and a discrete Grönwall argument.Source: §4 · p. 14
  • With additional integrability of the data and forcing terms, the scheme also has pathwise uniform stability, with a constant independent of M and h.The proof of this stronger estimate uses a dilation argument for discrete stochastic convolutions.Source: §4 · p. 15

5. Convergence Rates

This section establishes Milstein convergence estimates for rational and exponential schemes, covering grid points and continuous-time extensions. It describes the rational-scheme rate with a logarithmic correction, the exponential scheme’s rate without one, and lower-rate limitations and extensions to broader spaces and assumptions.

5.1. Main error estimates at the grid points.

  • An additional stochastic integral term yields decay of order min{α + 1/2, 1} on X, allowing rates above 1/2.Source: §5.1 · p. 16
  • The rational Milstein scheme achieves pathwise uniform strong error of order h^α up to a logarithmic correction, for α in (1/2, 1].The theorem’s bound is stated for the maximum error over grid points through T; the assumptions include A1–A4 and regularity of the initial value.Source: §5.1 · pp. 17, 24
  • For the rational scheme, estimating the discrete stochastic convolution uses a logarithmic square-function estimate, introducing the correction while exploiting the scheme’s approximation order α.Source: §5.1 · p. 18
  • Stochastic Fubini enables pathwise uniform error estimates for general p in [2, ∞), unlike the cited orthogonality argument for mean-square error [28, Sec. 6.1].Source: §5.1 · pp. 21–22
  • Exponential Milstein schemes omit the logarithmic correction because the logarithmic square-function estimate is unnecessary for terms without the semigroup.Source: §5.1 · p. 24
  • At grid points, the exponential scheme converges at rate α; for affine-linear coefficients, the rational scheme has rate α up to a logarithmic correction, while the exponential scheme has no correction.The linear-case result assumes α ∈ (0, 1] and the stated temporal Hölder regularity of the coefficients.Source: §5.1 · pp. 25–27
  • For α ≤ 1/2, Milstein is not advantageous: Euler and A-stable rational schemes without Milstein terms reach min{α, 1/2} up to rational-scheme logarithmic corrections, without derivative assumptions.These alternatives also require less computation because they avoid simulating iterated stochastic integrals [34, Thm. 6.4, Cor. 6.5].Source: §5.1 · p. 25
  • The differentiability assumption on F can be weakened to Gateaux differentiability on Y, but the corresponding relaxation is unavailable for G because G′G must be defined on all of X.The weaker F assumption preserves the same order of decay, although with larger constants.Source: §5.1 · p. 26
  • The convergence analysis extends to 2-smooth Banach spaces with Pisier’s contraction property, and pathwise uniform stability can also be established under linear growth of G′G.For Lemma 5.14, Hilbert–Schmidt operators are replaced by γ-radonifying operators [54, Thm. 4.1; 35, Sec. 3].Source: §5.1 · p. 26, §5.2 · p. 28

5.2. Error estimates on the full time interval.

  • The continuous-time extensions preserve grid-point values and satisfy full-interval error estimates, with no logarithmic correction for the exponential extension.The extension uses the scheme, Wiener increments, and iterated stochastic integrals within each time interval.Source: §5.2 · p. 28
  • Piecewise-constant extensions of the rational scheme attain only a lower, h^1/2-scale rate with a logarithmic factor, reflecting solution path regularity.For the scalar Brownian-motion equation, piecewise-affine interpolation has the same supremum-error scale because its error is a Brownian bridge [30, Chapter 5.6.B].Source: §5.2 · p. 30

6. Applications to Hyperbolic Equations

This section applies Milstein schemes to stochastic Schrödinger, Maxwell, and nonlinear transport equations, establishing convergence results under stated regularity conditions. The rates depend on the equation and scheme.

6.1. Linear stochastic Schr¨odinger equation.

  • For linear stochastic Schrödinger equations, EX-Milstein attains order 1, while IE- and CN-Milstein attain order 1 up to a logarithmic correction under stronger regularity.The stated order-1 conditions for EX-Milstein are V ∈ Hσ+2, Q1/2 ∈ L2(L2, Hσ+2), σ > d/2, and ξ ∈ L2p; for IE-/CN-Milstein they use V and Q1/2 regularity Hσ+ℓ, with ℓ = 4 and 3, respectively, and ξ ∈ L2p(Ω; Hσ+ℓ). Rational schemes require higher regularity, with ℓ = 4 for IE and ℓ = 3 for CN.Source: §6.1 · pp. 31–32

6.2. Stochastic Schr¨odinger equation with a nonlocal nonlinearity.

  • For the nonlocal nonlinear stochastic Schrödinger equation, EX-, CN-, and IE-Milstein attain rate 1, with a logarithmic correction for CN and IE but not EX.The stated rate applies in Hσ for σ > d/2 when Q1/2 ∈ L2(L2, Hσ+ℓ) and ξ ∈ L2p(Ω; Hσ+ℓ), under Assumption A7 and the scheme-specific regularity requirement ℓ ≥ 2, 3, or 4 for EX, CN, or IE, respectively.Source: §6.2 · p. 33

6.3. Stochastic Maxwell’s equations.

  • For stochastic Maxwell equations, the exponential Milstein scheme reaches convergence rates up to 1, improving on the cited non-Milstein rates of 1/2 and 1/2 up to logarithmic correction.The theorem assumes Assumption A8, initial data ξ ∈ L2p(Ω; D(A)), and Q1/2 ∈ L2(L2(O)6, H1+β(O)6) for β > 3/2.Source: §6.3 · p. 36
  • The EX-Milstein scheme converges at rate 1 for stochastic Maxwell’s equations, while CN-Milstein converges at rate 2/3 up to a logarithmic factor.The stated CN-Milstein result applies for all ξ ∈ L4p/3.Source: §6.3 · p. 36
  • For Maxwell’s equations, IE-Milstein reaches only rate 1/2 up to a logarithmic correction, while greater regularity is expected to let IE- and CN-Milstein reach rate 1 up to that correction.The IE rate matches IE without a Milstein term [34, Thm. 6.21], which the authors say should be preferred; fractional Maxwell operator domains are not explicitly characterized.Source: §6.3 · p. 37

6.4. Nonlinear stochastic transport equation.

  • For nonlinear stochastic transport, the exponential Milstein scheme converges at rate α under the theorem’s regularity assumptions.These include ξ ∈ L2αp(Ω; Hα), Q1/2 ∈ L2(L2, Hα), differentiable ϕ and ψ with bounded (2α−1)-Hölder continuous derivatives, Lipschitz ψ′ψ, and ϕ(0) = 0.Source: §6.4 · p. 37
  • For nonlinear transport, IE-Milstein and CN-Milstein rates are limited to at most 1/2 and 2/3, respectively; this limitation does not apply in the linear case.The composition estimate is available only for s ∈ [0, 1], while IE and CN approximate the shift semigroup to orders α/2 and 2α/3, respectively.Source: §6.4 · p. 39

7. Numerical Simulations of the Stochastic Schr¨odinger Equation

The simulations compare exponential and rational Milstein schemes across linear and nonlinear stochastic Schrödinger examples, measuring errors against a fine-step reference. EXM and CNM show advantages in the covered cases, while the Nemytskii example lies outside the stated growth assumptions.

7. Numerical Simulations of the Stochastic Schrödinger Equation

  • The simulations cover potential, nonlocal, and Nemytskii nonlinearities; scheme errors are measured against a reference computed with exponential Euler at step size 2^-16.The study compares EXM, CNM, and IEM with exponential Euler and rational schemes across five step sizes; it also notes that iterated-integral simulation costs can differ, motivating effective order measured against computational cost.Source: §7 · pp. 39–40

7.1. A linear example.

  • Observed EXM and CNM rates were close to 1 and 2/3, respectively, while IEM and schemes without Milstein terms remained near 1/2.These results are for the linear equation with a potential.Source: §7.1 · p. 40

7.2. A nonlinear example.

  • For the nonlocal nonlinearity, EXM and CNM show a clear advantage, and numerical rates closely match expected rates.Figure 1(b) shows the errors; Table 2 presents expected and numerical rates for this case and the potential case.Source: §7.2 · p. 40, Figure 1 · p. 41, Table 2 · p. 41

7.3. A further nonlinear example.

  • In the Nemytskii example, EXM and CNM numerically converge above 1/2 despite expected rates limited to 1/2 under the applicable setting.Error estimates under weaker growth conditions remain an open question.Source: §7.3 · pp. 41–42
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