Search ResultsShowing 1-20 of 20
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arXiv:2404.02401 (Published 2024-04-03)
Transformations and quadratic forms on Wiener spaces
Categories: math.PRTwo-way relationships between transformations and quadratic forms on Wiener spaces are investigated with the help of change of variables formulas on Wiener spaces. Further the evaluation of Laplace transforms of quadratic forms via Riccati or linear second order ODEs will be shown.
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arXiv:2203.09345 (Published 2022-03-17)
Operator Lie Algebras of Rotations and Transformations in White Noise
The infinitesimal generator of a one-parameter subgroup of the infinite dimensional rotation group associated with the complex Gelfand triple $ (E) \subset L^2(E^*, \mu) \subset (E)^* $ is of the form $$ R_\kappa = \int_{T\times T} \kappa(s,t) (a_s^* a_t - a_t^* a_s) ds dt $$ where $\kappa \in E \otimes E^*$ is a skew-symmetric distribution. Hence $R_\kappa$ is twice the conservation operator associated with a skew-symmetric operator $S$. The Lie algebra containing $R_\kappa$, identity operator, annihilation operator, creation operator, number operator, (generalized) Gross Laplacian is discussed. We show that this Lie algebra is associated with the orbit of the skew-symmetric operator $S$.
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arXiv:1909.11217 (Published 2019-09-24)
On Transformations of Markov Chains and Poisson Boundary
Categories: math.PRA discrete-time Markov chain can be transformed into a new Markov chain by looking at its states along iterations of an almost surely finite stopping time. By the optional stopping theorem, any bounded harmonic function with respect to the transition function of the original chain is harmonic with respect to the transition function of the transformed chain. The reverse inclusion is in general not true. Our main result provides a sufficient condition on the stopping time which guarantees that the space of bounded harmonic functions for the transformed chain embeds in the space of bounded harmonic sequences for the original chain. We also obtain a similar result on positive unbounded harmonic functions, under some additional conditions. Our work was motivated by and is analogous to Forghani-Kaimanovich, the well-studied case when the Markov chain is a random walk on a discrete group.
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arXiv:1909.08832 (Published 2019-09-19)
Generalised Krein-Feller operators and Liouville Brownian motion via transformations of measure spaces
Comments: 13 pages, 4 figuresWe consider generalised Kre\u{\i}n-Feller operators $\Delta_{\nu, \mu} $ with respect to compactly supported Borel probability measures $\mu$ and $\nu$ under the natural restrictions $\mathrm{supp}(\nu)\subset\mathrm{supp}(\mu)$ and $\mu$ atomless. We show that the solutions of the eigenvalue problem for $\Delta_{\nu, \mu} $ can be transferred to the corresponding problem for the classical Kre\u{\i}n-Feller operator $\Delta_{\nu, \Lambda}=\partial_{\mu}\partial_{x}$ with respect to the Lebesgue measure $\Lambda$ via an isometric isomorphism of the underlying Banach spaces. In this way we reprove the spectral asymptotic on the eigenvalue counting function obtained by Freiberg. Additionally, we investigate infinitesimal generators of generalised Liouville Brownian motions associated to generalised Kre\u{\i}n-Feller operator $\Delta_{\nu, \mu}$ under von Neumann boundary condition. Extending the measure $\mu$ and $\nu$ to the real line allows us to determine its walk dimension.
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arXiv:1811.06062 (Published 2018-11-01)
Central limit theorems with a rate of convergence for sequences of transformations
Using Stein's method, we prove an abstract result that yields multivariate central limit theorems with a rate of convergence for time-dependent dynamical systems. As examples we study a model of expanding circle maps and a quasistatic model. In both models we prove multivariate central limit theorems with a rate of convergence.
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arXiv:1810.10760 (Published 2018-10-25)
Quenched normal approximation for random sequences of transformations
We study random compositions of transformations having certain uniform fiberwise properties and prove bounds which in combination with other results yield a quenched central limit theorem equipped with a convergence rate, also in the multivariate case, assuming fiberwise centering. For the most part we work with non-stationary randomness and non-invariant, non-product measures. Independently, we believe our work sheds light on the mechanisms that make quenched central limit theorems work, by dissecting the problem into three separate parts.
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arXiv:1610.05636 (Published 2016-10-18)
On the probability of hitting the boundary for Brownian motions on the SABR plane
Comments: 11 pages. arXiv admin note: substantial text overlap with arXiv:1502.03254Categories: math.PRStarting from the hyperbolic Brownian motion as a time-changed Brownian motion, we explore a set of probabilistic models--related to the SABR model in mathematical finance--which can be obtained by geometry-preserving transformations, and show how to translate the properties of the hyperbolic Brownian motion (density, probability mass, drift) to each particular model. Our main result is an explicit expression for the probability of any of these models hitting the boundary of their domains, the proof of which relies on the properties of the aforementioned transformations as well as time-change methods.
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arXiv:1501.05506 (Published 2015-01-22)
Transformations of polynomial ensembles
Comments: 17 pagesA polynomial ensemble is a probability density function for the position of $n$ real particles of the form $\frac{1}{Z_n} \, \prod_{j<k} (x_k-x_j) \, \det \left[ f_k (x_j) \right]_{j,k=1}^n$, for certain functions $f_1, \ldots, f_n$. Such ensembles appear frequently as the joint eigenvalue density of random matrices. We present a number of transformations that preserve the structure of a polynomial ensemble. These transformations include the restriction of a Hermitian matrix by removing one row and one column, a rank-one modification of a Hermitian matrix, and the extension of a Hermitian matrix by adding an extra row and column with complex Gaussians. A special case of the latter result gives an elementary approach to the joint eigenvalue density of a GUE matrix.
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Transformations of Wiener Measure and Orthogonal Expansions
Categories: math.PRIn this paper we study the structure of square integrable functionals measurable with respect to coalescing stochastic flows. The case of $L^2$ space generated by the process $\eta(\cdot)=w(\min(\tau,\cdot)),$ where $w$ is a Brownian motion and $\tau$ is the first moment when $w$ hits the given continuous function $g$ is considered. We present a new construction of multiple stochastic integrals with respect to the process $\eta.$ Our approach is based on the change of measure technique. The analogue of the It\^o-Wiener expansion for the space $L^2(\eta)$ is constructed.
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On the vaguelet and Riesz properties of L^2-unbounded transformations of orthogonal wavelet bases
In this work, we prove that certain L^2-unbounded transformations of orthogonal wavelet bases generate vaguelets. The L^2-unbounded functions involved in the transformations are assumed to be quasi-homogeneous at high frequencies. We provide natural examples of functions which are not quasi-homogeneous and for which the resulting transformations are not vaguelets. We also address the related question of whether the considered family of functions is a Riesz basis in L^2(R). The Riesz property could be deduced directly from the results available in the literature or, as we outline, by using the vaguelet property in the context of this work. The considered families of functions arise in wavelet-based decompositions of stochastic processes with uncorrelated coefficients.
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arXiv:1209.4314 (Published 2012-09-19)
Transformations of random walks on groups via Markov stopping times
Categories: math.PRWe describe a new construction of a family of measures on a group with the same Poisson boundary. Our approach is based on applying Markov stopping times to an extension of the original random walk.
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arXiv:1112.3687 (Published 2011-12-15)
Stochastic symmetries and transformations of stochastic differential equations
Categories: math.PRIn this article, we introduce the notion of stochastic symmetry of a differential equation. It consists in a stochastic flow that acts over a solution of a differential equation and produces another solution of the same equation. In the ordinary case, we give necessary conditions in order to obtain such symmetries. These conditions involve the infinitesimal generator of the flow and the coefficients of the equation. Moreover, we show how to obtain necessary conditions in order to find an application that transforms a stochastic differential equation that one would like to solve into a target equation that one previously know how to solve.
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arXiv:1002.4796 (Published 2010-02-25)
Transformations of one-dimensional Gibbs measures with infinite range interaction
Journal: Markov Proc. Rel. Fields, 16, pp. 737-752, 2010Keywords: one-dimensional gibbs measures, infinite range interaction, transformations, study single-site stochastic, uniqueness regimeTags: journal articleWe study single-site stochastic and deterministic transforma- tions of one-dimensional Gibbs measures in the uniqueness regime with infinite-range interactions. We prove conservation of Gibbsianness and give quantitative estimates on the decay of the transformed potential. As examples, we consider exponentially decaying potentials, and potentials decaying as a power-law.
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arXiv:0903.1005 (Published 2009-03-05)
Transformations des lois multivariées avec queues régulières
Comments: 11 pagesCategories: math.PRLet $X$ be a random vector in $\rd$ with a regularly varying tail. We consider two transformations $\|X\|f(\frac{X}{\|X\|})$, $f: \sd\to\sd$, and $Xf(\frac{X}{\|X\|})$, $f: \sd\to \mathbb{R}_+$. Some sufficient conditions for preserving the property of regularity of the tail for this kind of transformations are given.
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arXiv:0803.1413 (Published 2008-03-10)
On some transformations of bilateral birth-and-death processes with applications
Comments: 4 pages; appeared in: The 17th Symposium of Information Theory and Its Applications (SITA '94), Hiroshima, Japan, December 6-9, 1994, pp. 739-742Categories: math.PRSubjects: 60J80A method yielding simple relationships among bilateral birth-and-death processes is outlined. This allows one to relate birth and death rates of two processes in such a way that their transition probabilities, first-passage-time densities and ultimate crossing probabilities are mutually related by some product-form expressions.
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Transformations of Lévy Processes
Comments: Revised version. Proof shortened and streamlined considerablyJournal: Commun. Stoch. Anal. 4 (553-577) 2010Keywords: lévy processes, hermitian linear functional vanishing, generator process, transformations, levy processTags: journal articleA L\'evy process on a *-bialgebra is given by its generator, a conditionally positive hermitian linear functional vanishing at the unit element. A *-algebra homomorphism k from a *-bialgebra C to a *-bialgebra B with the property that k respects the counits maps generators on B to generators on C. A tranformation between the corrresponding two L\'evy processes is given by forming infinitesimal convolution products. This general result is applied to various situations, e.g., to a *-bialgebra and its associated primitive tensor *-bialgebra (called "generator process") as well as its associated group-like *-bialgebra (called Weyl-*-bialgebra). It follows that a L\'evy process on a *-bialgebra can be realized on Bose Fock space as the infinitesimal convolution product of its generator process such that the vacuum vector is cyclic for the L\e'vy process. Moreover, we obtain convolution approximations of the Az\'ema martingale by the Wiener process and vice versa.
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arXiv:0710.1596 (Published 2007-10-08)
Transformations of Markov Processes and Classification Scheme for Solvable Driftless Diffusions
Comments: 31 pagesSubjects: 60J60We propose a new classification scheme for diffusion processes for which the backward Kolmogorov equation is solvable in analytically closed form by reduction to hypergeometric equations of the Gaussian or confluent type. The construction makes use of transformations of diffusion processes to eliminate the drift which combine a measure change given by Doob's h-transform and a diffeomorphism. Such transformations have the important property of preserving analytic solvability of the process: the transition probability density for the driftless process can be expressed through the transition probability density of original process. We also make use of tools from the theory of ordinary differential equations such as Liouville transformations, canonical forms and Bose invariants. Beside recognizing all analytically solvable diffusion process known in the previous literature fall into this scheme and we also discover rich new families of analytically solvable processes.
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arXiv:0707.0538 (Published 2007-07-04)
Transformations of infinitely divisible distributions via improper stochastic integrals
Comments: 44 pagesCategories: math.PRLet $X^{(\mu)}(ds)$ be an $\mathbb{R}^d$-valued homogeneous independently scattered random measure over $\mathbb{R}$ having $\mu$ as the distribution of $X^{(\mu)}((t,t+1])$. Let $f(s)$ be a nonrandom measurable function on an open interval $(a,b)$ where $-\infty\leqslant a<b\leqslant\infty$. The improper stochastic integral $\int_{a+}^{b-} f(s)X^{(\mu)}(ds)$ is studied. Its distribution $\Phi_f(\mu)$ defines a mapping from $\mu$ to an infinitely divisible distribution on $\mathbb{R}^d$. Three modifications (compensated, essential, and symmetrized) and absolute definability are considered. After their domains are characterized, necessary and sufficient conditions for the domains to be very large (or very small) in various senses are given. The concept of the dual in the class of purely non-Gaussian infinitely divisible distributions on $\mathbb{R}^d$ is introduced and employed in studying some examples. The $\tau$-measure $\tau$ of function $f$ is introduced and whether $\tau$ determines $\Phi_f$ is discussed. Related transformations of L\'evy measures are also studied.
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arXiv:math/0601243 (Published 2006-01-11)
On some transformations between positive self--similar Markov processes
Categories: math.PRA path decomposition at the infimum for positive self-similar Markov processes (pssMp) is obtained. Next, several aspects of the conditioning to hit 0 of a pssMp are studied. Associated to a given a pssMp $X,$ that never hits 0, we construct a pssMp $X^{\downarrow}$ that hits 0 in a finite time. The latter can be viewed as $X$ conditioned to hit 0 in a finite time and we prove that this conditioning is determined by the pre-minimum part of $X.$ Finally, we provide a method for conditioning a pssMp that hits 0 by a jump to do it continuously.
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Perte d'information dans les transformations du jeu de pile ou face
Comments: Published at http://dx.doi.org/10.1214/009117906000000124 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)Journal: Annals of Probability 2006, Vol. 34, No. 4, 1550-1588Categories: math.PRSubjects: 60J05Keywords: pile ou face, perte dinformation dans, transformations, entre ces filtrations lorsquelles sont, filtration naturelleTags: journal articleSoit $(\epsilon_n)_{n\in\mathbf{Z}}$ un jeu de pile ou face, c'est-\`{a}-dire une suite de variables al\'{e}atoires ind\'{e}pendantes de loi $(\delta_{-1}+\delta_1)/2$, et $(H_n)_{n\in\mathbf{Z}}$ un processus \`{a} valeurs dans $\{-1,1\}$, pr\'{e}visible dans la filtration naturelle de $(\epsilon_n)_{n\in\mathbf{Z}}$. Alors $(H_n\epsilon_n)_{n\in \mathbf{Z}}$ est encore un jeu de pile ou face, dont la filtration naturelle est contenue dans celle de $(\epsilon_n)_{n\in\mathbf{Z}}$. Le but de l'article est d'obtenir des conditions pour que ces filtrations soient \'{e}gales et de d\'{e}crire l'\'{e}cart entre ces filtrations lorsqu'elles sont diff\'{e}rentes. Nous nous int\'{e}ressons plus particuli\`{e}rement au cas des transformations homog\`{e}nes, o\`{u} le processus $(H_n\epsilon_n)_{n\in\mathbf{Z}}$ est une fonctionnelle de $(\epsilon_n)_{n\in\mathbf{Z}}$ qui commute avec les translations. Nous \'{e}tudions de fa\c{c}on approfondie les transformations homog\`{e}nes de longueur finie, o\`{u} $H_n$ est de la forme $\phi(\epsilon_{n-d},...,\epsilon_{n-1})$ avec $d\in\mathbf {N}$ et $\phi:\{-1;1\}^d\to\{-1;1\}$ fix\'{e}s.