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Strong convergence rates of tamed exponential Euler schemes for superlinear hyperbolic SPDEs

Katharina Klioba

arXiv:2609.08872v1math.NAcs.NAmath.APmath.FAmath.PR

TL;DR

Superlinear nonlinearities make convergence analysis difficult for hyperbolic SPDEs, where parabolic smoothing is absent and explicit schemes may lose moment stability. The paper extends a stochastic Kato framework to tamed exponential Euler schemes for locally Lipschitz nonlinearities. It establishes pathwise uniform strong convergence rates up to 1/2, alongside well-posedness and stability results.

  • Problem

    Convergence results for non-parabolic SPDEs have largely treated globally Lipschitz nonlinearities, while superlinear growth can undermine moment stability of explicit schemes.

    Source: §1 · p. 2

  • Method

    The paper extends stochastic Kato analysis to locally Lipschitz, polynomial-growth nonlinearities and studies nonlinearities-stopped and more general tamed exponential Euler schemes.

    Source: §1.1 · p. 2, §1.2 · p. 3, §1.3 · p. 5

  • Results

    Pathwise uniform strong convergence rates up to 1/2 are established for nonlinearities-stopped and general tamed schemes, with optimal-rate applications to several hyperbolic SPDEs.

    Source: §1.2 · pp. 3–4

  • Takeaways & Limitations

    The analysis provides local and global well-posedness and uniform a priori estimates for the mild solution, as well as convergence and stability results for its approximation.

    Source: §1.3 · p. 5

    Rates above 1/2 are outside the first-order Wiener-increment setting discussed; higher rates would require additional Brownian information, such as iterated stochastic integrals.

    Source: §8 · p. 41

Abstract

from arXiv · show

In this paper, we prove pathwise uniform convergence at rates up to $1/2$ for tamed exponential Euler schemes for semilinear hyperbolic stochastic evolution equations with superlinearly growing nonlinearities and multiplicative noise. We take the term hyperbolic to mean that the leading operator generates a contractive $C_0$-semigroup but no parabolic smoothing occurs. Under local Lipschitz, polynomial growth, coercivity, and monotonicity conditions on the nonlinearities, we establish pathwise uniform strong error estimates of the form \begin{equation*} \Big(\mathbb{E}\max_{0\le j \le N} \|U(t_j)-U^j\|_X^p\Big)^{1/p} \lesssim \sqrt{k} \end{equation*} on a Hilbert space $X$ for $p\in [2,\infty)$. Here, $U$ is the mild solution and $U^j$ is the tamed exponential Euler approximation at time $t_j=jk$ with step size $k>0$. This extends previous convergence results for non-parabolic SPDEs from globally to locally Lipschitz nonlinearities, allowing both drift and diffusion to grow polynomially. In a stochastic Kato framework, we further establish local and global well-posedness as well as uniform a priori estimates for the mild solution and its approximation. Applications to nonlinear stochastic transport, Airy, wave-type, and dissipatively damped nonlinear Schr\"odinger equations are included, covering different nonlinearities-stopped and fractionally tamed schemes. For the Klein-Gordon equation with cubic velocity damping, this complements previous results obtained for additive noise.

1. Introduction

The paper addresses the joint challenges of superlinear nonlinearities and absent parabolic smoothing in hyperbolic SPDEs. It extends stochastic Kato analysis to locally Lipschitz, polynomial-growth nonlinearities and proves convergence, well-posedness, and stability results for tamed exponential Euler schemes.

1. Introduction

  • Earlier convergence analyses focused largely on globally Lipschitz nonlinearities, while superlinear explicit schemes can lose moment stability; this work treats both challenges together.The stochastic Kato framework is extended to locally Lipschitz nonlinearities with polynomial growth.Source: §1 · p. 2, §1.1 · p. 2

1.1. Setting and contributions.

  • The contribution includes pathwise-uniform rates, alternative tamings, local and global well-posedness, stability estimates, and optimal rates for selected superlinear-noise equations.The paper focuses on temporal discretization; implementable SPDE schemes also require a separate spatial discretization.Source: §1.1 · p. 3

1.2. Main results.

  • Pathwise uniform strong convergence rates up to 1/2 are established for nonlinearities-stopped and more general tamed schemes under coercivity and monotonicity conditions.The bounds are for Lp maximal errors with p in [2, ∞); the general taming class includes fractional taming and growth-dependent stopping.Source: §1.1 · p. 2, §1.2 · pp. 3–4
  • The analysis includes optimal-rate applications to stochastic transport, dissipatively damped nonlinear Schrödinger, and velocity-damped Klein–Gordon equations, plus an Airy application with rate 1/3.The Airy rate reflects the coercivity available in Y = H1; the paper notes that weakened coercivity and higher-order tamed schemes remain possible directions.Source: §1.2 · pp. 4–5

1.3. Method of proof.

  • The proof controls stopping and local discretization errors, then uses monotonicity and stochastic Gronwall estimates to bound the remaining solution error.A priori estimates for the mild solution follow from coercivity and stochastic Gronwall; general tamings require preservation of coercivity, step-size-dependent growth bounds, and consistency.Source: §1.3 · p. 5
  • Taming and coercivity yield scheme moment bounds, while semigroup contractivity supports pointwise stability and stronger coercivity controls pathwise-uniform stability.Local well-posedness is established by reducing to a globally Lipschitz case using projections and handling explosion times.Source: §1.3 · p. 5

2. Preliminaries

This section fixes the Hilbert-space and time-grid notation and collects semigroup, stochastic-integration, and stochastic Gronwall tools used in the analysis. The semigroup framework exploits contractivity and interpolation to control time increments without parabolic smoothing.

2. Preliminaries

  • The paper works with separable Hilbert spaces X and Y, a uniform grid t_j = jk, and a C0-semigroup; Y embeds continuously into X.The notation also fixes a cylindrical Brownian motion and the L2(H, X) space of Hilbert–Schmidt operators.Source: Notation · p. 6

2.1. Semigroups and interpolation.

  • Contractivity on X and Y, together with embeddings into interpolation spaces, yields bounds on semigroup differences between times.For fractional α, the stated embedding is into D_A(α, ∞); for α = 1, it is into D(A).Source: §2.1 · p. 6

2.2. Stochastic integration.

  • The stochastic-integration tools include maximal bounds for stochastic convolutions and a localized BDG inequality for convergence of stochastic integrals.The section also records cylindrical Brownian motion conventions and the relation to Q-Wiener processes.Source: §2.2 · p. 7

2.3. A stochastic Gronwall lemma.

  • The stochastic Gronwall inequality accommodates an additional martingale term and gives maximal estimates with an arbitrarily small moment loss for λ approximately 1.Its hypotheses include a continuous adapted nonnegative process and increasing adapted terms.Source: §2.3 · pp. 7–8

3. Local well-posedness

The section constructs local solutions for equations with locally Lipschitz nonlinearities in a two-space stochastic Kato setting. Projection-based localization reduces the problem to globally Lipschitz equations, while uniqueness and a blow-up criterion yield a unique maximal local solution.

3. Local well-posedness

  • The stochastic Kato setting assumes a contractive semigroup on X and Y, continuous embedding Y into X, and locally Lipschitz, locally bounded nonlinearities.The local well-posedness argument proceeds by projection, uniqueness, a blow-up criterion, and a limiting construction.Source: §3 · p. 8

3.1. Reduction to the globally Lipschitz setting.

  • Projections onto bounded Y-balls reduce the local problem to globally Lipschitz equations, whose solutions agree with the original equation up to the localization stopping times.The projection maps are uniformly 1-Lipschitz, and the truncations produce globally Lipschitz nonlinearities.Source: §3 · p. 8, §3.1 · p. 9

3.2. Local uniqueness.

  • Local solutions with the same initial value agree almost surely until the earlier of their stopping times.The proof applies a contraction argument to stopped solutions on successive short intervals.Source: §3.2 · pp. 10–11

3.3. Blow-up criterion and local well-posedness.

  • Under Assumption 3.1, there exists a unique maximal local solution, and its stopping time is an explosion time.The blow-up criterion compares lifetimes: if one solution explodes, another with the same initial value cannot stop earlier; two explosion times coincide.Source: §3.3 · pp. 11–12

4. Global well-posedness

An Itô-type inequality for mild solutions supports global existence and pathwise uniform a priori estimates under an additional coercivity condition. The resulting theorem gives a global solution in Y with a uniform-in-time moment bound.

4. Global well-posedness

  • An Itô-type inequality is established for mild solutions because they need not be semimartingales, and contractivity yields an upper bound independent of the generator.The proof uses Yosida approximations to obtain the inequality.Source: §4 · pp. 13, 16
  • The coercivity-based a priori estimate rules out finite-time explosion, yielding global well-posedness and the estimate on the closed interval [0, T].The proof first obtains the estimate by localization and a stochastic Gronwall argument, then extends the solution continuously to T.Source: §4 · pp. 16, 18
  • Under the stated q-coercivity condition and Assumption 3.1, the unique maximal local solution is global and has a uniform-in-time Y-norm moment bound.The theorem assumes q in [2, ∞) and an initial value in Lq(Ω; Y).Source: §4 · p. 16

5. The nonlinearities-stopped exponential Euler scheme

The scheme advances the solution on a uniform time grid using the semigroup and nonlinearities evaluated at the previous grid point. It stops those nonlinearities when their combined growth exceeds a step-size-dependent threshold.

5. The nonlinearities-stopped exponential Euler scheme

  • Explicit exponential Euler schemes may diverge for superlinear nonlinearities, so this scheme stops the nonlinearities when their growth exceeds a threshold.The stopping approach adapts a scheme from the parabolic setting to hyperbolic equations; the paper also discusses other tamings.Source: §5 · p. 18
  • At each step, the update combines semigroup propagation, a drift increment, and a diffusion increment driven by the Brownian-motion increment.Both nonlinearities are evaluated at the current grid value and multiplied by an indicator that is one when their combined norms are at most k^-θ.Source: §5 · p. 18
  • The continuous extension of the scheme agrees with the grid values, while the piecewise constant extension generally differs between grid points.The semilinear integrated counterparts evaluate nonlinearities at the last grid point inside the convolutions.Source: §5 · pp. 18–19

6. Stability of the scheme

The section establishes stability bounds for the stopped scheme and its semilinear integrated counterparts, progressing from pointwise to pathwise uniform stability.

6. Stability of the scheme

  • Stopping limits nonlinear growth so the stochastic-integral terms remain controlled when the step size satisfies θ≤1/4.Source: §6 · p. 19

6.1. Pointwise stability of the scheme.

  • For θ∈(0, 1/4], q-coercivity and semigroup contractivity give pointwise Y-stability; stopping alone suffices without local Lipschitz continuity or local Y-boundedness.Source: §6.1 · p. 20

6.2. Pointwise stability of the semilinear integrated counterparts.

  • The semilinear integrated counterparts are pointwise stable under polynomial growth on Y, with corresponding coercivity and initial-data integrability assumptions.Source: §6.2 · p. 22

6.3. Pathwise uniform stability.

  • A slightly stronger (q+ε)-coercivity condition yields pathwise uniform stability, with bounds independent of k, N, and θ.A two-step argument controls a critical term with stronger coercivity, then applies a BDG-type maximal inequality to handle the time-grid supremum, including superlinear noise.Source: §6 · p. 20, §6.3 · pp. 23–24

7. Pathwise uniform convergence rates

This section establishes pathwise uniform convergence rates up to 1/2 in time for the nonlinearities-stopped exponential Euler scheme under the stated assumptions. Its right-continuous piecewise constant interpolation achieves the same rate on the full time interval up to a square-root logarithmic correction.

7. Pathwise uniform convergence rates

  • The nonlinearities-stopped exponential Euler scheme converges pathwise uniformly at rates up to 1/2 in time under monotonicity in X and the stated assumptions.For the piecewise constant interpolation over the full time interval, the rate holds up to a square-root logarithmic correction.Source: §7 · p. 26

7.2.2. Error between the mild solution and the semilinear integrated counterparts

  • The mild-solution error is controlled using an Itô-type inequality, monotonicity, local Lipschitz continuity, and a stochastic Gronwall argument.The resulting bound depends on the first error component and increments over small time intervals; higher-moment estimates close the overall estimate.Source: §7 · p. 26, §7.2.2 · pp. 30–31

7.1. Assumptions.

  • The analysis assumes local Lipschitz continuity, polynomial growth, coercivity, and monotonicity; the assumptions also yield scheme stability and, with stronger coercivity, global well-posedness and pathwise uniform solution estimates.The convergence-rate argument specifically imposes the monotonicity condition.Source: §7.1 · p. 27

7.2.1. Error between the scheme and the semilinear integrated counterparts

  • The pathwise uniform error between the scheme and its semilinear integrated counterparts converges at rate α.The estimate is obtained for the continuous extension of the scheme and yields the grid-point estimate.Source: §7.2 · p. 27, §7.2.1 · p. 28

7.2.3. Error between the scheme and the solution.

  • The nonlinearities-stopped exponential Euler scheme converges pathwise uniformly to the solution at rate α ∈ (0, 1/2] under the stated assumptions.The error-bound constant is independent of the step size, number of steps, and initial value.Source: §7.2.3 · p. 33
  • Larger α can lower the required moment parameter r, allowing larger noise coefficients and initial values with fewer moments.Higher Lipschitz and growth parameters ν and ρ make the coercivity assumption more restrictive, but affect the noise coefficient rather than admissible growth order.Source: §7.2.3 · p. 34
  • Extension to 2-smooth Banach spaces remains challenging because projections onto Y-balls need not be X-Lipschitz, requiring a different well-posedness approach.The convergence extension would also require γ-radonifying operators and coercivity and monotonicity conditions formulated with dual pairings.Source: §7.2.3 · p. 34

7.3. Error estimate for an interpolation to the full time interval.

  • The right-continuous piecewise constant interpolation achieves the same pathwise uniform rate on [0, T] up to a square-root logarithmic correction.The proof relies on a logarithmic square-function estimate for stochastic integrals.Source: §7.3 · p. 34

8. Tamed exponential Euler schemes

This section extends stability and convergence analysis from stopped nonlinearities to broader tamed exponential Euler schemes, including optimized stopping parameters and continuous fractional taming. It describes convergence rates under specified growth conditions and notes that, with multiplicative noise, first-order Wiener increments limit rates to 1/2; extending higher rates to the superlinear case remains open.

Scalar tamings

  • Scalar tamings allow separate drift and diffusion growth parameters, and [0,1]-valued scalar factors preserve coercivity under the stated sufficient conditions.The nonlinearities-stopped scheme is included as a special case.Source: §8 · pp. 36, 39

Propositions 8.5–8.6

  • Tamed coercivity and growth assumptions yield pointwise stability in Y with a bound uniform in the step size k.The stability result assumes a contractive C0-semigroup and q-coercivity; the continuous extension is also stable.Source: §8 · p. 37

Theorem 8.8

  • Under the stated assumptions, tamed exponential Euler schemes converge pathwise uniformly at rate α, with the error bound’s constant independent of k, N, and u0.The theorem allows separate growth parameters for the tamed drift and diffusion.Source: §8 · p. 38

Nonlinearities-stopped scheme

  • Theorem 8.8 recovers the nonlinearities-stopped scheme’s previous convergence bound under the same regularity assumptions.For this scheme, the theorem applies with θF = θG = θ and β = ρ(1 + αθ).Source: §8 · p. 39

Corollary 8.10

  • A coercivity-preserving stopping modification yields rate α and optimizes stopping parameters for nonlinearities with different polynomial growth orders.When ρG > ρF, the optimal parameters are θF = ρF / (2(ρF + ρG)), θG = ρG / (2(ρF + ρG)), and β* = ρG + 2α(ρF + ρG).Source: §8 · pp. 39–40

Example 8.11

  • Fractional taming with a shared factor for drift and diffusion converges pathwise uniformly at rate α; with globally Lipschitz, linearly growing G, the noise need not be tamed.The respective initial-data integrability exponents are r = ρq(1 + ν + 4α) and r = ρq(2 + ν).Source: §8 · p. 41

8. Tamed exponential Euler schemes

  • Choosing a_k(x)=b_k(x)=(1+k^αR(x))^-1 results in convergence at rate α for θ_F=θ_G=α, β=2ρ, and thus r=ρq(2+ν).For higher rates in the case of (8.14), verifying the growth bounds for the taming is necessary.Source: §8 · p. 41
  • For multiplicative noise, first-order Wiener increments naturally cap rates at 1/2; higher rates require more Brownian information, such as iterated stochastic integrals in exponential Milstein schemes.Pathwise uniform rate 1 was obtained for globally Lipschitz nonlinearities [35]; combining Milstein terms with this paper’s taming for superlinear cases is plausible but beyond scope.Source: §8 · p. 41

9. Applications to hyperbolic SPDEs with polynomial nonlinearities

This section applies the exponential Euler framework to hyperbolic SPDEs, including stochastic transport, dissipatively damped nonlinear Schrödinger, Klein–Gordon, and Airy equations. It establishes pathwise convergence and, for some applications, global well-posedness for tamed or nonlinearities-stopped schemes.

9.3. The stochastic Klein–Gordon equation with nonlinear velocity damping.

  • For the wave-type Klein–Gordon equation with cubic velocity damping, nonlinearities-stopped and fractionally tamed schemes achieve pathwise convergence at the optimal rate 1/2.The noise acts multiplicatively on the velocity variable.Source: §9 · p. 41, §9.3 · p. 46

9.1. The nonlinear stochastic transport equation.

  • For small quadratic noise, stopped and fractionally tamed transport schemes converge pathwise uniformly at rate 1/2; fractional estimates allow Y = H1/2+ε.The sign of the Allen–Cahn nonlinearity helps compensate for the noise. Using different stopping indicators reduces the initial-data moment requirement from r = 15q to about 13.6q and relaxes the coercivity-related noise smallness condition.Source: §9 · p. 41, §9.1 · p. 44

9.2. The nonlinear stochastic Schr¨odinger equation with dissipative damping.

  • Under the damping and covariance assumptions, the dissipatively damped stochastic nonlinear Schrödinger equation has a unique global mild solution, and stopped and fractionally tamed schemes converge pathwise uniformly at rate 1/2.The paper also establishes the same rate for schemes that tame only the drift, using either stopping or smooth fractional taming.Source: §9.2 · p. 46

9.3. The stochastic Klein–Gordon equation with nonlinear velocity damping.

  • For the stochastic Klein–Gordon equation with cubic velocity damping, a unique global mild solution and pathwise uniform convergence hold for the tamed exponential Euler scheme.The result also holds when the drift is stopped at the stated threshold; stopping both drift and diffusion requires the stronger initial moment condition U0 ∈ L^(15q) F0(Ω; Y).Source: §9.3 · p. 47
  • The Klein–Gordon result complements additive-noise results for related damped waves and stochastic Klein–Gordon equations [20] with stronger pathwise uniform error estimates for explicit schemes.The cited related work reports mean-square temporal rate 1 in dimension 1 for and rate 1 for [20].Source: §9.3 · p. 47

9.4. The nonlinear stochastic Airy equation.

  • The Airy equation’s third-order leading operator limits the available rate to α ≤ 1/3, while attaining 1/2 for multiplicative noise would require a generalized coercivity assumption left for future work.Coercivity is verified only on H1; attempting H2 coercivity for the cubic case does not yield the needed bound.Source: §9.4 · pp. 48–49
  • For the stochastic Airy equation with a generalized Allen–Cahn nonlinearity of arbitrary odd order, both nonlinearities-stopped and drift-only-tamed schemes converge pathwise uniformly, and the equation has a unique global mild solution.The initial value is required to have the stated finite H1 moment, with Q^(1/2) ∈ L2(L2, H1).Source: §9.4 · p. 48
  • For cubic Airy drift, the required initial moment exponents are 13q for the nonlinearities-stopped scheme and 11q for the drift-only-tamed scheme; for quintic drift they are 95/3 q and 85/3 q, respectively.Higher polynomial orders require higher initial moments and, with quadratic noise, a more restrictive smallness condition.Source: §9.4 · p. 48
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