A stochastic Schauder-Tychonoff type theorem and its applications

Erika Hausenblas; Ankit Kumar; Jonas M. Tölle

doi:10.48550/arXiv.2602.17285

A stochastic Schauder-Tychonoff type theorem and its applications

Erika Hausenblas
, Ankit Kumar^*
, Jonas M. Tölle

^*Corresponding author for this work

Montanuniversität Leoben

Research output: Contribution to journal › Article › Scientific

Abstract

One standard way to prove existence for deterministic, highly nonlinear PDEs is to use the Schauder-Tychonoff fixed-point theorem. In what follows, we introduce and verify a stochastic variant of the Schauder-Tychonoff theorem. We apply our existence result to nonlinear stochastic diffusion equations with non-Lipschitz perturbations.

Original language	English
Number of pages	58
Journal	arXiv.org
DOIs	https://doi.org/10.48550/arXiv.2602.17285
Publication status	Submitted - 18 Feb 2026
MoE publication type	B1 Non-refereed journal articles

Keywords

Stochastic Schauder-Tychonoﬀ-type theorem
pattern formation in ecology
nonlinear stochastic partial diﬀerential equation
flows in porous media
pathwise uniqueness
multiplica- tive Wiener noise
nonlinear gradient noise

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10.48550/arXiv.2602.17285Licence: Other

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

The Stochastic Klausmeier System and A Stochastic Schauder-Tychonoff Type Theorem
Hausenblas, E. & Tölle, J. M., 1 Aug 2024, In: Potential Analysis. 61, 2, p. 185-246
Research output: Contribution to journal › Article › Scientific › peer-review

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title = "A stochastic Schauder-Tychonoff type theorem and its applications",

abstract = "One standard way to prove existence for deterministic, highly nonlinear PDEs is to use the Schauder-Tychonoff fixed-point theorem. In what follows, we introduce and verify a stochastic variant of the Schauder-Tychonoff theorem. We apply our existence result to nonlinear stochastic diffusion equations with non-Lipschitz perturbations.",

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author = "Erika Hausenblas and Ankit Kumar and T{\"o}lle, \{Jonas M.\}",

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AU - Kumar, Ankit

AU - Tölle, Jonas M.

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Y1 - 2026/2/18

N2 - One standard way to prove existence for deterministic, highly nonlinear PDEs is to use the Schauder-Tychonoff fixed-point theorem. In what follows, we introduce and verify a stochastic variant of the Schauder-Tychonoff theorem. We apply our existence result to nonlinear stochastic diffusion equations with non-Lipschitz perturbations.

AB - One standard way to prove existence for deterministic, highly nonlinear PDEs is to use the Schauder-Tychonoff fixed-point theorem. In what follows, we introduce and verify a stochastic variant of the Schauder-Tychonoff theorem. We apply our existence result to nonlinear stochastic diffusion equations with non-Lipschitz perturbations.

KW - Stochastic Schauder-Tychonoﬀ-type theorem

KW - pattern formation in ecology

KW - nonlinear stochastic partial diﬀerential equation

KW - flows in porous media

KW - pathwise uniqueness

KW - multiplica- tive Wiener noise

KW - nonlinear gradient noise

U2 - 10.48550/arXiv.2602.17285

DO - 10.48550/arXiv.2602.17285

M3 - Article

SN - 2331-8422

JO - arXiv.org

JF - arXiv.org

ER -

A stochastic Schauder-Tychonoff type theorem and its applications

Abstract

Keywords

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Projects

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

Research output

The Stochastic Klausmeier System and A Stochastic Schauder-Tychonoff Type Theorem

Cite this