Typical absolute continuity for classes of dynamically defined measures

Lower bounds for the dynamically defined measures

The dynamically defined measure (DDM) Φ arising from a finite measure φ0 on an initial σ-algebra on a set X and an invertible map acting on the latter is considered. Several lower bounds for it are obtained and sufficient conditions for its positivity are deduced under the general assumption that there exists an invariant measure Λ such that Λ ≪ φ0. In particular, DDMs arising from the Hellinge...

On the Absolute Continuity of a Class of Invariant Measures

Let X be a compact connected subset of Rd, let Sj , j = 1, ...,N , be contractive self-conformal maps on a neighborhood of X, and let {pj(x)}j=1 be a family of positive continuous functions on X. We consider the probability measure μ that satisfies the eigen-equation

Absolute Continuity and Singularity of Probability Measures in Functional Spaces

Absolute Continuity between the Wiener and Stationary Gaussian Measures

It is known that the entropy distance between two Gaussian measures is finite if, and only if, they are absolutely continuous with respect to one another. Shepp [5] characterized the correlations corresponding to stationary Gaussian measures that are absolutely continuous with respect to the Wiener measure. By analyzing the entropy distance, we show that one of his conditions, involving the spe...

Absolute continuity for some one-dimensional processes

We introduce an elementary method for proving the absolute continuity of the time marginals of one-dimensional processes. It is based on a comparison between the Fourier transform of such time marginals with those of the one-step Euler approximation of the underlying process. We obtain some absolute continuity results for stochastic differential equations with Hölder continuous coefficients. Fu...