arXiv subjects

Anna Srapionyan

Publications and source records attributed to Anna Srapionyan.

3 recordsLinked to original sources

arXiv2021

On non-local ergodic Jacobi semigroups: spectral theory, convergence-to-equilibrium and contractivity

In this paper, we introduce and study non-local Jacobi operators, which generalize the classical (local) Jacobi operators. We show that these operators extend to generators of ergodic Markov semigroups with unique invariant probability measures and study their spectral and convergence properties. In particular, we derive a series expansion of the semigroup in terms of explicitly defined polynomials, which generalize the classical Jacobi orthogonal polynomials. In addition, we give a complete characterization of the spectrum of the non-self-adjoint generator and semigroup. We show that the variance decay of the semigroup is hypocoercive with explicit constants, which provides a natural generalization of the spectral gap estimate. After a random warm-up time, the semigroup also decays exponentially in entropy and is both hypercontractive and ultracontractive. Our proofs hinge on the development of commutation identities, known as intertwining relations, between local and non-local Jacobi operators and semigroups, with the local objects serving as reference points for transferring properties from the local to the non-local case.

math.PR↗

arXiv2019

Spectral projections correlation structure for short-to-long range dependent processes

Let $\mathbf{X}=(\mathbf{X}_t)_{t \geq 0}$ be a stochastic process issued from $x \in \mathbb R$ that admits a marginal stationary measure $ν$, i.e. $ν\mathbf{P}_t f = νf$ for all $t \geq 0$, where $\mathbf{P}_t f(x)= \mathbb{E}_x[f(\mathbf{X}_t)]$. In this paper, we introduce the (resp. biorthogonal) spectral projections correlation functions which are expressed in terms of projections into the eigenspaces of $\mathbf{P}_t$ (resp. and of its adjoint in the weighted Hilbert space $L^2(ν)$). We obtain closed-form expressions involving eigenvalues, the condition number and/or the angle between the projections in the following different situations: when $\mathbf{X}=X$ with $X=(X_t)_{t \geq 0}$ is a Markov process, $\mathbf{X}$ is the subordination of $X$ in the sense of Bochner, and $\mathbf X$ is a non-Markovian process which is obtained by time-changing $X$ with an inverse of a subordinator. It turns out that these spectral projections correlation functions have different expressions with respect to these classes of processes which enables to identify substantial and deep properties about their dynamics. This interesting fact can be used to design original statistical tests to make inferences, for example, about the path properties of the process (presence of jumps), distance from symmetry (self-adjoint or non-self-adjoint) and short-to-long-range dependence. To reveal the usefulness of our results, we apply them to a class of non-self-adjoint Markov semigroups studied in Patie and Savov [28], and then time-change by subordinators and their inverses.

math.PR↗

arXiv2016

Riesz bases generated by the spectra of Sturm-Liouville problems

Let $\{λ_n^2\}_{n = 0}^\infty$ be the spectra of a Sturm-Liouville problem on $[0,π]$. We investigate the question: Do the systems $\{\cos(λ_n x)\}_{n= 0}^\infty$ or $\{\sin(λ_n x)\}_{n = 0}^\infty$ form Riesz bases in ${L^2}[0,π]$? The answer is almost always positive.

math.SP↗