Geodesic
A counterexample to the stochastic version of the Brouwer fixed point theorem
Vestnik rossijskih universitetov. Matematika, Tome 26 (2021) no. 134, pp. 143-150

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It is shown that the stochastic counterpart of the classical fixed point theorem for continuous maps in a finite dimensional Euclidean space (“Brouwer's theorem”) is not, in general, true. This result implies, in particular, that a careful choice of invariant sets in the stochastic version of Brouwer's theorem is necessary in the theory of stochastic nonlinear operators.
Keywords: local operators, convergence in probability, fixed points. 
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     title = {A counterexample to the stochastic version of the {Brouwer} fixed point theorem},
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A. V. Ponosov. A counterexample to the stochastic version of the Brouwer fixed point theorem. Vestnik rossijskih universitetov. Matematika, Tome 26 (2021) no. 134, pp. 143-150. http://geodesic.mathdoc.fr/item/VTAMU_2021_26_134_a3/