2 Free Jacobi Process
نویسنده
چکیده
In this paper, we define and study free Jacobi processes of parameters λ > 0 and 0 < θ ≤ 1, as the limit of the complex version of the matrix Jacobi process already defined by Y. Doumerc. In the first part, we focus on the stationary case for which we compute the law (that does not depend on time) and derive, for λ ∈]0, 1] and 1/θ ≥ λ + 1 a free SDE analogous to the classical one. In the second part, we generalize this result under an additional condition. To proceed, we set a recurrence formula for the moments of the process using free stochastic calculus. This will also be used to compute the p. d. e. satisfied by the Cauchy transform of the free Jacobi's law.
منابع مشابه
un 2 00 6 FREE JACOBI PROCESS
In this paper, we define and study free Jacobi processes of parameters λ > 0 and 0 < θ ≤ 1, as the limit of the complex version of the matrix Jacobi process already defined by Y. Doumerc. In the first part, we focus on the stationary case for which we compute the law (that does not depend on time) and derive, for λ ∈]0, 1] and 1/θ ≥ λ + 1 a free SDE analogous to the classical one. In the second...
متن کاملFree Martingale polynomials for stationary Jacobi processes
We generalize a previous result concerning free martingale polynomials for the stationary free Jacobi process of parameters λ ∈]0.1], θ = 1/2. Hopelessly, apart from the case λ = 1, the polynomials we derive are no longer orthogonal with respect to the spectral measure. As a matter of fact, we use the multiplicative renormalization method to write down its corresponding orthogonal polynomials a...
متن کاملMicroscopic derivations of several Hamilton–Jacobi equations in infinite dimensions, and large deviation of stochastic systems
We consider Hamilton–Jacobi equations which characterize optimal controlled partial differential equations of the following types: the Allen–Cahn equation, the Cahn–Hilliard equation, a nonlinear Fokker–Planck equation, and aVlasov–Fokker–Planck equation. In each of the examples, the optimal control problem and its associated cost functional can be derived as limit from a microscopically define...
متن کاملOn a Spectral Property of Jacobi Matrices
Let J be a Jacobi matrix with elements bk on the main diagonal and elements ak on the auxiliary ones. We suppose that J is a compact perturbation of the free Jacobi matrix. In this case the essential spectrum of J coincides with [−2, 2], and its discrete spectrum is a union of two sequences {xj }, x + j > 2, x − j < −2, tending to ±2. We denote sequences {ak+1 − ak} and {ak+1 + ak−1 − 2ak} by ∂...
متن کاملThe Gauged O ( 3 ) Sigma Model : Schrödinger Representation and Hamilton - Jacobi Formulation
We first study a free particle on an (n − 1)-sphere in an extended phase space, where the originally second-class Hamiltonian and constraints are now in strong involution. This allows for a Schrödinger representation and a Hamilton-Jacobi formulation of the model. We thereby obtain the free particle energy spectrum corresponding to that of a rigid rotator. We extend these considerations to a mo...
متن کامل