Null Controllability of a Stochastic Parabolic Equation with a Space-Dependent Analytic Noise Coefficient
Abstract
We consider a stochastic parabolic equation on the n-dimensional flat torus 𝕋n=(ℝ/(2πℤ))n, n∈ℕ, whose only control acts in the drift. The multiplicative-noise coefficient is adapted and may depend jointly on the sample point, time, and space. Its spatial profiles are assumed to be real analytic with a uniform positive radius of analyticity, while the corresponding analytic norm is only required to be square integrable in time, uniformly along sample paths. We prove a state-only observability inequality with an L1-in-time mean-square observation norm and deduce null controllability from every measurable spatial set of positive measure by one adapted drift control. The same one-time interpolation estimate also yields approximate controllability to arbitrary square-integrable random terminal targets. The proof is based on a direct analytic energy estimate for the adjoint state, followed by propagation of smallness and a telescoping argument.
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BibTeX
@misc{lu2026null,
title = {{Null Controllability of a Stochastic Parabolic Equation with a Space-Dependent Analytic Noise Coefficient}},
author = {Qi Lü and Yu Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28937},
eprint = {2609.28937},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28937},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Null Controllability of a Stochastic Parabolic Equation with a Space-Dependent Analytic Noise Coefficient
AU - Qi Lü
AU - Yu Wang
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.28937
UR - https://arxiv.org/abs/2609.28937
AB - We consider a stochastic parabolic equation on the n-dimensional flat torus 𝕋^n=(ℝ/(2πℤ))^n, n∈ℕ, whose only control acts in the drift. The multiplicative-noise coefficient is adapted and may depend jointly on the sample point, time, and space. Its spatial profiles are assumed to be real analytic with a uniform positive radius of analyticity, while the corresponding analytic norm is only required to be square integrable in time, uniformly along sample paths. We prove a state-only observability inequality with an L^1-in-time mean-square observation norm and deduce null controllability from every measurable spatial set of positive measure by one adapted drift control. The same one-time interpolation estimate also yields approximate controllability to arbitrary square-integrable random terminal targets. The proof is based on a direct analytic energy estimate for the adjoint state, followed by propagation of smallness and a telescoping argument.
ER -