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Reflected (Degenerate) Diffusions and Stationary Measures

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XIII Symposium on Probability and Stochastic Processes

Part of the book series: Progress in Probability ((PRPR,volume 75))

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Abstract

These notes were written with the occasion of the XIII Symposium on Probability and Stochastic Processes at UNAM. We will introduce general reflected diffusions with instantaneous reflection when hitting the boundary. Two main tools for studying these processes are presented: the submartingale problem, and stochastic differential equations. We will see how these two complement each other. In the last sections, we will see in detail two processes to which this theory applies nicely, and uniqueness of a stationary distribution holds for them, despite the fact they are degenerate.

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Notes

  1. 1.

    These notes were written for a four-lecture mini-course at the XIII Symposium on Probability and Stochastic Processes, held at the Faculty of Sciences, Universidad Nacional Autónoma de México, from December 4–8, 2017.

  2. 2.

    This is a good exercise in stochastic calculus. I recommend to follow it closely, and fill the minor gaps.

  3. 3.

    We must give credit for this compilation of applications to the authors of [28].

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Acknowledgements

I am very grateful for the opportunity to speak at the XIII Symposium of Probability and Stochastic Processes. We thank the organizers at UNAM for the invitation, and for hosting an amazing conference.

We thank the support from Proyecto FONDECYT 11160591, and NĂşcleo Milenio NC130062.

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Authors and Affiliations

  1. Departamento de Matematica, Universidad Andres Bello, Santiago, Chile

    Mauricio Duarte

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  1. Mauricio Duarte
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Correspondence to Mauricio Duarte .

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Editors and Affiliations

  1. Departamento de Matemáticas, Universidad Nacional Autónoma México, México, Ciudad de México, Mexico

    Sergio I. LĂłpez

  2. CIMAT, Guanajuato, Mexico

    VĂ­ctor M. Rivero

  3. Facultad de Ciencias, Universidad Autónoma de Sinaloa, Culiacán, Sinaloa, Mexico

    Alfonso Rocha-Arteaga

  4. Departamento de Probabilidad, Universidad Nacional AutĂłnoma MĂ©xico, MĂ©xico, Ciudad de MĂ©xico, Mexico

    Arno Siri-JĂ©gousse

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Duarte, M. (2020). Reflected (Degenerate) Diffusions and Stationary Measures. In: López, S.I., Rivero, V.M., Rocha-Arteaga, A., Siri-Jégousse, A. (eds) XIII Symposium on Probability and Stochastic Processes. Progress in Probability, vol 75. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-57513-7_1

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