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Free Jacobi Process

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  • N. Demni

    (Université de Paris VI)

Abstract

In this paper, we define and study two parameters dependent free processes (λ,θ) called free Jacobi, obtained as the limit of its matrix counterpart when the size of the matrix goes to infinity. The main result we derive is a free SDE analogous to that satisfied in the matrix setting, derived under injectivity assumptions. Once we did, we examine a particular case for which the spectral measure is explicit and does not depend on time (stationary). This allows us to determine easily the parameters range ensuring our injectivity requirements so that our result applies. Then, we show that under an additional condition of invertibility at time t=0, this range extends to the general setting. To proceed, we set a recurrence formula for the moments of the process via free stochastic calculus.

Suggested Citation

  • N. Demni, 2008. "Free Jacobi Process," Journal of Theoretical Probability, Springer, vol. 21(1), pages 118-143, March.
    • Handle: RePEc:spr:jotpro:v:21:y:2008:i:1:d:10.1007_s10959-007-0110-1
    DOI: 10.1007/s10959-007-0110-1
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    References listed on IDEAS

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    1. Silverstein, J. W. & Choi, S. I., 1995. "Analysis of the Limiting Spectral Distribution of Large Dimensional Random Matrices," Journal of Multivariate Analysis, Elsevier, vol. 54(2), pages 295-309, August.
    2. M. Capitaine & C. Donati-Martin, 2005. "Free Wishart Processes," Journal of Theoretical Probability, Springer, vol. 18(2), pages 413-438, April.
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