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The main result of this paper, Theorem 1.5, gives explicit formulae for the kernels of the ergodic decomposition measures for infinite Pickrell measures on the space of infinite complex matrices. The kernels are obtained as the scaling limits of Christoffel-Uvarov deformations of Jacobi orthogonal polynomial ensembles.

We consider in this work stochastic differential equation (SDE) model for particles in contact with a heat bath when the memory effects are non-negligible. As a result of the fluctuation-dissipation theorem, the differential equations driven by fractional Brownian noise to model memory effects should be paired with Caputo derivatives and based on this we consider fractional stochastic differential equations (FSDEs), which should be understood in an integral form. We establish the existence of strong solutions for such equations. In the linear forcing regime, we compute the solutions explicitly and analyze the asymptotic behavior, through which we verify that satisfying fluctuation-dissipation indeed leads to the correct physical behavior. We further discuss possible extensions to nonlinear forcing regime, while leave the rigorous analysis for future works.

In this paper, we study the shape of the min-max minimal hypersurface produced by Almgren-Pitts in F. Almgren, and J. Pitts, corresponding to the fundamental class of a Riemannian manifold $(M^{n+1}, g)$of positive Ricci curvature with 2 ≤ n ≤ 6. We characterize the Morse index, volume and multiplicity of this min-max hypersurface. In particular, we show that the min-max hypersurface is either orientable and of index one, or is a double cover of a non-orientable minimal hypersurface with least area among all closed embedded minimal hypersurfaces.