Ergodicity for stochastic neural field equations

Anna-Mariya Otsetova; Jonas M. Tölle

doi:10.48550/arXiv.2505.14012

Ergodicity for stochastic neural field equations

Research output: Contribution to journal › Article › Scientific

Abstract

We investigate the well-posedness and long-time behavior of a general continuum neural field model with Gaussian noise on possibly unbounded domains. In particular, we give conditions for the existence of invariant probability measures by restricting the solution flow to an invariant subspace with a nonlocal metric. Under the assumption of a sufficiently large decay parameter relative to the noise intensity, the growth of the connectivity kernel, and the Lipschitz regularity of the activation function, we establish exponential ergodicity and exponential mixing of the associated Markovian Feller semigroup and the uniqueness of the invariant measure with second moments.

Original language	English
Number of pages	30
Journal	arXiv.org
DOIs	https://doi.org/10.48550/arXiv.2505.14012
Publication status	Submitted - 21 May 2025
MoE publication type	B1 Non-refereed journal articles

Keywords

stochastic Amari neural field model
stochastic equations in Hilbert space
ergodic Feller semigroup
exponential ergodicity
exponential mixing
existence and uniqueness of invariant measures

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10.48550/arXiv.2505.14012

LiBERA: Stochastic interacting systems: Limiting Behavior, Evaluation, Regularity and Applications
Tölle, J. (Principal investigator), Peltola, E. (Project Member) & Viitasaari, L. (Co-PI)
01/01/2025 → 31/12/2028
Project: EU Horizon Europe MC

Cite this

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title = "Ergodicity for stochastic neural field equations",

abstract = "We investigate the well-posedness and long-time behavior of a general continuum neural field model with Gaussian noise on possibly unbounded domains. In particular, we give conditions for the existence of invariant probability measures by restricting the solution flow to an invariant subspace with a nonlocal metric. Under the assumption of a sufficiently large decay parameter relative to the noise intensity, the growth of the connectivity kernel, and the Lipschitz regularity of the activation function, we establish exponential ergodicity and exponential mixing of the associated Markovian Feller semigroup and the uniqueness of the invariant measure with second moments.",

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author = "Anna-Mariya Otsetova and T{\"o}lle, \{Jonas M.\}",

year = "2025",

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doi = "10.48550/arXiv.2505.14012",

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T1 - Ergodicity for stochastic neural field equations

AU - Otsetova, Anna-Mariya

AU - Tölle, Jonas M.

PY - 2025/5/21

Y1 - 2025/5/21

N2 - We investigate the well-posedness and long-time behavior of a general continuum neural field model with Gaussian noise on possibly unbounded domains. In particular, we give conditions for the existence of invariant probability measures by restricting the solution flow to an invariant subspace with a nonlocal metric. Under the assumption of a sufficiently large decay parameter relative to the noise intensity, the growth of the connectivity kernel, and the Lipschitz regularity of the activation function, we establish exponential ergodicity and exponential mixing of the associated Markovian Feller semigroup and the uniqueness of the invariant measure with second moments.

AB - We investigate the well-posedness and long-time behavior of a general continuum neural field model with Gaussian noise on possibly unbounded domains. In particular, we give conditions for the existence of invariant probability measures by restricting the solution flow to an invariant subspace with a nonlocal metric. Under the assumption of a sufficiently large decay parameter relative to the noise intensity, the growth of the connectivity kernel, and the Lipschitz regularity of the activation function, we establish exponential ergodicity and exponential mixing of the associated Markovian Feller semigroup and the uniqueness of the invariant measure with second moments.

KW - stochastic Amari neural field model

KW - stochastic equations in Hilbert space

KW - ergodic Feller semigroup

KW - exponential ergodicity

KW - exponential mixing

KW - existence and uniqueness of invariant measures

U2 - 10.48550/arXiv.2505.14012

DO - 10.48550/arXiv.2505.14012

M3 - Article

SN - 2331-8422

JO - arXiv.org

JF - arXiv.org

ER -

Ergodicity for stochastic neural field equations

Abstract

Keywords

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Projects

LiBERA: Stochastic interacting systems: Limiting Behavior, Evaluation, Regularity and Applications

Cite this