Many results in the theory of Gaussian processes rely on the eigenstructure of the covariance operator. However, eigenproblems are notoriously hard to solve explicitly and closed form solutions are known only in a limited number of cases. In this paper we set up a framework for the spectral analysis of the fractional type covariance operators, corresponding to an important family of processes, which includes the fractional Brownian motion and its noise. We obtain accurate asymptotic approximations for the eigenvalues and the eigenfunctions. Our results provide a key to several problems, whose solution is long known in the standard Brownian case, but was missing in the more general fractional setting. This includes computation of the exact limits of L 2 -small ball probabilities and asymptotic analysis of singularly perturbed integral equations, arising in mathematical physics and applied probability.

For a class of non-uniformly ergodic Markov chains (X n ) satisfying exponential or polynomial beta-mixing, under observations (Y n ) subject to an IID noise with a positive density, it is shown that wrong initial data is forgotten in the mean total variation topology, with a certain exponential or polynomial rate. Mathematics Subject Classification (2000)60G35 · 62M20 · 93E11 · 93E15

This paper presents a new approach to the analysis of mixed processeswhere Bt is a Brownian motion and Gt is an independent centered Gaussian process. We obtain a new canonical innovation representation of X, using linear filtering theory. When the kernelhas a weak singularity on the diagonal, our results generalize the classical innovation formulas beyond the square integrable setting. For kernels with stronger singularity, our approach is applicable to processes with additional "fractional" structure, including the mixed fractional Brownian motion from mathematical finance. We show how previously-known measure equivalence relations and semimartingale properties follow from our canonical representation in a unified way, and complement them with new formulas for Radon-Nikodym densities.

In this paper we solve the basic fractional analogue of the classical linear-quadratic Gaussian regulator problem in continuous time. For a completely observable controlled linear system driven by a fractional Brownian motion, we describe explicitely the optimal control policy which minimizes a quadratic performance criterion.

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