No free lunch for continuity of stochastic convolutions
Abstract
We construct compact, exponentially stable C0-semigroups S on Hilbert spaces X such that, for every T>0, the stochastic convolution $U_g(t)=\int_0^t S(t-s)g(s)\,\mathrm{d}β_s $ is unbounded on [0,T] with positive probability for some predictable g∈ L∞(Ω;L2(0,T;X)), where β is a real Brownian motion. One example is even an analytic semigroup. For the other, the negative generator A has sectorial angle π/2 and a bounded H∞-calculus. Our main tool is a necessary condition: for exponentially stable semigroups, an L2-maximal estimate on ℝ+ forces a lower square-function estimate for the negative generator. We deduce this implication from a new identity involving the stochastic convolution, the subordinated Poisson semigroup, and a stopped Brownian motion. Combining this condition with Schauder multipliers on a conditional trigonometric basis yields counterexamples to the maximal estimate. An extrapolation argument then produces the integrands with unbounded stochastic convolutions. Finally, an energy-adapted chaining argument gives continuity and maximal estimates for arbitrary C0-semigroups under the stronger condition g∈ L2([0,T]; L∞(Ω;X)).
Related
- No Free Lunch in Flow Surrogates under Time-Varying Boundary Conditions: A Two-Regime Study
- There Is No Free Lunch in RLVR: Every pass@1 Gain Has a pass@k Price
- Inequalities for rank-two permanents and finite free convolutions
- Zeros and Exponential Profiles of Polynomials I: Limit Distributions, Finite Free Convolutions and Repeated Differentiation
- Joint Uniform Continuity of Reduced Free Products for strong convergence
BibTeX
@misc{veraar2026free,
title = {{No free lunch for continuity of stochastic convolutions}},
author = {Mark Veraar and Joris van Winden},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12688},
eprint = {2609.12688},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12688},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - No free lunch for continuity of stochastic convolutions
AU - Mark Veraar
AU - Joris van Winden
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12688
UR - https://arxiv.org/abs/2609.12688
AB - We construct compact, exponentially stable C_0-semigroups S on Hilbert spaces X such that, for every T>0, the stochastic convolution $U_g(t)=\int_0^t S(t-s)g(s)\,\mathrm{d}β_s $ is unbounded on [0,T] with positive probability for some predictable g∈ L^∞(Ω;L^2(0,T;X)), where β is a real Brownian motion. One example is even an analytic semigroup. For the other, the negative generator A has sectorial angle π/2 and a bounded H^∞-calculus. Our main tool is a necessary condition: for exponentially stable semigroups, an L^2-maximal estimate on ℝ_+ forces a lower square-function estimate for the negative generator. We deduce this implication from a new identity involving the stochastic convolution, the subordinated Poisson semigroup, and a stopped Brownian motion. Combining this condition with Schauder multipliers on a conditional trigonometric basis yields counterexamples to the maximal estimate. An extrapolation argument then produces the integrands with unbounded stochastic convolutions. Finally, an energy-adapted chaining argument gives continuity and maximal estimates for arbitrary C_0-semigroups under the stronger condition g∈ L^2([0,T]; L^∞(Ω;X)).
ER -