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Stochastic Homogenization of Hamilton–Jacobi Equations on a Junction

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Abstract

We consider the specified stochastic homogenization of first order evolutive Hamilton–Jacobi equations on a very simple junction, i.e. the real line with a junction at the origin. Far from the origin, we assume that the considered hamiltonian is closed to given stationary ergodic hamiltonians (which are different on the left and on the right). Near the origin, there is a perturbation zone which allows are to pass from one hamiltonian to the other. The main result of this paper is a stochastic homogenization as the length of the transition zone goes to zero. More precisely, at the limit we get two deterministic right and left hamiltonians with a deterministic junction condition at the origin. The main difficulty and novelty of the paper come from the fact that the hamiltonian is not stationary ergodic. To the best of our knowledge, this is the first specified stochastic homogenization result. This work is motivated by traffic flow applications.

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Acknowledgements

The authors would like to thank P. Cardaliaguet and P. Calka for fruitful discussions in the preparation of this work. This project was co-financed by the European Union with the European regional development fund (ERDF,18P03390/18E01750/18P02733) and by the Normandie Regional Council via the M2SiNUM project and by ANR MFG (ANR-16-CE40-0015-01).

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Authors and Affiliations

  1. Normandie Univ, INSA de Rouen Normandie, LMI (EA 3226 - FR CNRS 3335), 76000, Rouen, France

    Rim Fayad & Nicolas Forcadel

  2. Normandie Univ, 685 Avenue de l’Université, 76801, St Etienne du Rouvray Cedex, France

    Rim Fayad & Nicolas Forcadel

  3. Mathematics Department Hadath, Lebanese University, Faculty of Sciences-I, Beyrouth, Lebanon

    Hassan Ibrahim

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  1. Rim Fayad
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  2. Nicolas Forcadel
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  3. Hassan Ibrahim
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Correspondence to Nicolas Forcadel.

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Communicated by P.-L. Lions.

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Fayad, R., Forcadel, N. & Ibrahim, H. Stochastic Homogenization of Hamilton–Jacobi Equations on a Junction. Arch Rational Mech Anal 243, 1223–1267 (2022). https://doi.org/10.1007/s00205-021-01737-1

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