Sojourns and extremes of a diffusion process on a fixed interval

1982 ◽  
Vol 14 (4) ◽  
pp. 811-832 ◽  
Author(s):  
Simeon M. Berman
Keyword(s):  
Diffusion Process ◽  
Lebesgue Measure ◽  
Stationary Distribution ◽  
Real Line ◽  
Diffusion Coefficients ◽  
Limit Theorems ◽  
Random Variable ◽  
Fixed Interval ◽  
The Real ◽  
And Diffusion

Let X(t), , be an Ito diffusion process on the real line. For u > 0 and t > 0, let Lt(u) be the Lebesgue measure of the set . Limit theorems are obtained for (i) the distribution of Lt(u) for u → ∞and fixed t, and (ii) the tail of the distribution of the random variable max[0, t]X(s). The conditions on the process are stated in terms of the drift and diffusion coefficients. These conditions imply the existence of a stationary distribution for the process.

Sojourns and extremes of a diffusion process on a fixed interval

1982 ◽  
Vol 14 (04) ◽  
pp. 811-832 ◽  
Author(s):  
Simeon M. Berman
Keyword(s):  
Diffusion Process ◽  
Lebesgue Measure ◽  
Stationary Distribution ◽  
Real Line ◽  
Diffusion Coefficients ◽  
Limit Theorems ◽  
Random Variable ◽  
The Real ◽  
Simple Limit ◽  
And Diffusion

Let X(t), , be an Ito diffusion process on the real line. For u > 0 and t > 0, let Lt (u) be the Lebesgue measure of the set . Limit theorems are obtained for (i) the distribution of Lt (u) for u → ∞and fixed t, and (ii) the tail of the distribution of the random variable max[0, t] X(s). The conditions on the process are stated in terms of the drift and diffusion coefficients. These conditions imply the existence of a stationary distribution for the process.

Multifractal formalism for the inverse of random weak Gibbs measures

2019 ◽  
Vol 20 (04) ◽  
pp. 2050024
Author(s):  
Zhihui Yuan
Keyword(s):  
Probability Measure ◽  
Lebesgue Measure ◽  
Real Line ◽  
Gibbs Measures ◽  
Borel Probability Measure ◽  
Cantor Set ◽  
Multifractal Formalism ◽  
The Real ◽  
Random Dynamics ◽  
Borel Probability

Any Borel probability measure supported on a Cantor set included in [Formula: see text] and of zero Lebesgue measure on the real line possesses a discrete inverse measure. We study the validity of the multifractal formalism for the inverse measures of random weak Gibbs measures. The study requires, in particular, to develop in this context of random dynamics a suitable version of the results known for heterogeneous ubiquity associated with deterministic Gibbs measures.

Lebesque measure zero subsets of the real line and an infinite game

1996 ◽  
Vol 61 (1) ◽  
pp. 246-249 ◽  
Author(s):  
Marion Scheepers
Keyword(s):  
Lebesgue Measure ◽  
Real Line ◽  
Measure Zero ◽  
Cardinal Number ◽  
Minimal Cardinality ◽  
Combinatorial Characterization ◽  
The Real ◽  
Lebesque Measure ◽  
The Ideal

Let denote the ideal of Lebesgue measure zero subsets of the real line. Then add() denotes the minimal cardinality of a subset of whose union is not an element of . In [1] Bartoszynski gave an elegant combinatorial characterization of add(), namely: add() is the least cardinal number κ for which the following assertion fails:For every family of at mostκ functions from ω to ω there is a function F from ω to the finite subsets of ω such that:1. For each m, F(m) has at most m + 1 elements, and2. for each f inthere are only finitely many m such that f(m) is not an element of F(m).The symbol A(κ) will denote the assertion above about κ. In the course of his proof, Bartoszynski also shows that the cardinality restriction in 1 is not sharp. Indeed, let (Rm: m < ω) be any sequence of integers such that for each m Rm, ≤ Rm+1, and such that limm→∞Rm = ∞. Then the truth of the assertion A(κ) is preserved if in 1 we say instead that1′. For each m, F(m) has at most Rm elements.We shall use this observation later on. We now define three more statements, denoted B(κ), C(κ) and D(κ), about cardinal number κ.

scholarly journals Joint distribution of a Lévy process and its running supremum

2018 ◽  
Vol 55 (2) ◽  
pp. 488-512 ◽  
Author(s):  
Laure Coutin ◽  
Monique Pontier ◽  
Waly Ngom
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Weak Solution ◽  
Diffusion Process ◽  
Lebesgue Measure ◽  
Joint Distribution ◽  
Random Variable ◽  
Jump Diffusion ◽  
Marginal Density ◽  
The Law

Abstract Let X be a jump-diffusion process and X* its running supremum. In this paper we first show that for any t > 0, the law of the pair (X*t, Xt) has a density with respect to the Lebesgue measure. This allows us to show that for any t > 0, the law of the pair formed by the random variable Xt and the running supremum X*t of X at time t can be characterized as a weak solution of a partial differential equation concerning the distribution of the pair (X*t, Xt). Then we obtain an expression of the marginal density of X*t for all t > 0.

scholarly journals Empirical and Theoretical Characterization of the Diffusion Process of Different Gadolinium-Based Nanoparticles within the Brain Tissue after Ultrasound-Induced Permeabilization of the Blood-Brain Barrier

2019 ◽  
Vol 2019 ◽  
pp. 1-13 ◽  
Author(s):  
Allegra Conti ◽  
Rémi Magnin ◽  
Matthieu Gerstenmayer ◽  
Nicolas Tsapis ◽  
Erik Dumont ◽  
...  
Keyword(s):  
Diffusion Process ◽  
Brain Tissue ◽  
Blood Brain Barrier ◽  
Diffusion Coefficients ◽  
Brain Barrier ◽  
Apparent Diffusion Coefficients ◽  
Blood Brain ◽  
The Brain ◽  
And Diffusion

Low-intensity focused ultrasound (FUS), combined with microbubbles, is able to locally, and noninvasively, open the blood-brain barrier (BBB), allowing nanoparticles to enter the brain. We present here a study on the diffusion process of gadolinium-based MRI contrast agents within the brain extracellular space after ultrasound-induced BBB permeabilization. Three compounds were tested (MultiHance, Gadovist, and Dotarem). We characterized their diffusion through in vivo experimental tests supported by theoretical models. Specifically, by estimation of the free diffusion coefficients from in vitro studies and of apparent diffusion coefficients from in vivo experiments, we have assessed tortuosity in the right striatum of 9 Sprague Dawley rats through a model correctly describing both vascular permeability as a function of time and diffusion processes occurring in the brain tissue. This model takes into account acoustic pressure, particle size, blood pharmacokinetics, and diffusion rates. Our model is able to fully predict the result of a FUS-induced BBB opening experiment at long space and time scales. Recovered values of tortuosity are in agreement with the literature and demonstrate that our improved model allows us to assess that the chosen permeabilization protocol preserves the integrity of the brain tissue.

scholarly journals Order and Metric Compatible Symbolic Sequence Processing

2016 ◽  
Author(s):  
Daniel J Greenhoe
Keyword(s):  
Stochastic Process ◽  
Sequence Analysis ◽  
Linear Space ◽  
Metric Space ◽  
Real Line ◽  
Distance Geometry ◽  
Weighted Graph ◽  
Random Variable ◽  
The Real ◽  
Sequence Processing

A traditional random variable X is a function that maps from a stochastic process to the real line (X,<=,d,+,.), where R is the set of real numbers, <= is the standard linear order relation on R, d(x,y)=|x-y| is the usual metric on R, and (R, +, .) is the standard field on R. Greenhoe(2015b) has demonstrated that this definition of random variable is often a poor choice for computing statistics when the stochastic process that X maps from has structure that is dissimilar to that of the real line. Greenhoe(2015b) has further proposed an alternative statistical system, that rather than mapping a stochastic process to the real line, instead maps to a weighted graph that has order and metric geometry structures similar to that of the underlying stochastic process. In particular, ideally the structure X maps from and the structure X maps to are, with respect to each other, both isomorphic and isometric.Mapping to a weighted graph is useful for analysis of a single random variable.for example the expectation EX of X can be defined simply as the center of its weighted graph. However, the mapping has limitations with regards to a sequence of random variables in performing sequence analysis (using for example Fourier analysis or wavelet analysis), in performing sequence processing (using for example FIR filtering or IIR filtering), in making diagnostic measurements (using a post-transform metric space), or in making goptimalh decisions (based on gdistanceh measurements in a metric space or more generally a distance space). Rather than mapping to a weighted graph, this paper proposes instead mapping to an ordered distance linear space Y=(R^n,<=,d,+,.,R,+,x), where (R,+,x) is a field, + is the vector addition operator on R^n x R^n, and . is the scalar-vector multiplication operator on R x R^n. The linear space component of Y provides a much more convenient (as compared to the weighted graph) framework for sequence analysis and processing. The ordered set and distance space components of Y allow one to preserve the order structure and distance geometry inherent in the underlying stochastic process, which in turn likely provides a less distorted (as compared to the real line) framework for analysis, diagnostics, and optimal decision making.

scholarly journals Order and metric geometry compatible stochastic processing

2015 ◽  
Author(s):  
Daniel J Greenhoe
Keyword(s):  
Stochastic Process ◽  
Real Line ◽  
Order Relation ◽  
Random Variable ◽  
Poor Quality ◽  
Order Structure ◽  
Metric Geometry ◽  
The Real ◽  
R And D ◽  
The Face

A traditional random variable X is a function that maps from a stochastic process to the real line. Here, "real line" refers to the structure (R,<=,|x-y|), where R is the set of real numbers, <= is the standard linear order relation on R, and d(x,y)=|x-y| is the usual metric on R. The traditional expectation value E(X) of X is then often a poor choice of a statistic when the stochastic process that X maps from is a structure other than the real line or some substructure of the real line. If the stochastic process is a structure that is not linearly ordered (including structures totally unordered) and/or has a metric space geometry very different from that induced by the usual metric, then statistics such as E(X) are often of poor quality with regards to qualitative intuition and quantitative variance (expected error) measurements. For example, the traditional expected value of a fair die is E(X)=(1/6)(1+2+...+6)=3.5. But this result has no relationship with reality or with intuition because the result implies that we expect the value of [ooo] (die face value "3") or [oooo] (dice face value "4") more than we expect the outcome of say [o] or [oo]. The fact is, that for a fair die, we would expect any pair of values equally. The reason for this is that the values of the face of a fair die are merely symbols with no order, and with no metric geometry other than the discrete metric geometry. On a fair die, [oo] is not greater or less than [o]; rather [oo] and [o] are simply symbols without order. Moreover, [o] is not "closer" to [oo] than it is to [ooo]; rather, [o], [oo], and [ooo] are simply symbols without any inherit order or metric geometry. This paper proposes an alternative statistical system, based somewhat on graph theory, that takes into account the order structure and metric geometry of the underlying stochastic process.

scholarly journals Central limit theorems for the real eigenvalues of large Gaussian random matrices

2017 ◽  
Vol 06 (01) ◽  
pp. 1750002 ◽  
Author(s):  
N. J. Simm
Keyword(s):  
Central Limit ◽  
Limit Theorems ◽  
Polynomial Function ◽  
Random Variable ◽  
Real Matrix ◽  
Two Component System ◽  
The Real ◽  
Two Component ◽  
Standard Normal ◽  
Independent Identically Distributed

Let [Formula: see text] be an [Formula: see text] real matrix whose entries are independent identically distributed standard normal random variables [Formula: see text]. The eigenvalues of such matrices are known to form a two-component system consisting of purely real and complex conjugated points. The purpose of this paper is to show that by appropriately adapting the methods of [E. Kanzieper, M. Poplavskyi, C. Timm, R. Tribe and O. Zaboronski, Annals of Applied Probability 26(5) (2016) 2733–2753], we can prove a central limit theorem of the following form: if [Formula: see text] are the real eigenvalues of [Formula: see text], then for any even polynomial function [Formula: see text] and even [Formula: see text], we have the convergence in distribution to a normal random variable [Formula: see text] as [Formula: see text], where [Formula: see text].

scholarly journals On the Measurability of Functions with Respect to Certain Classes of Measures

2004 ◽  
Vol 11 (3) ◽  
pp. 489-494
Author(s):  
A. Kharazishvili ◽  
A. Kirtadze
Keyword(s):  
Lebesgue Measure ◽  
Real Line ◽  
The Real

Abstract The concept of measurability of real-valued functions with respect to various classes of measures is introduced and the associated notion of an absolutely nonmeasurable function is investigated. A characterization of such functions is given. Also, it is shown that functions produced by the classical Vitali partition of the real line are measurable with respect to the class of all extensions of the Lebesgue measure on this line.

scholarly journals Order and Metric Compatible Symbolic Sequence Processing

2016 ◽  
Author(s):  
Daniel J Greenhoe
Keyword(s):  
Stochastic Process ◽  
Sequence Analysis ◽  
Linear Space ◽  
Metric Space ◽  
Real Line ◽  
Distance Geometry ◽  
Weighted Graph ◽  
Random Variable ◽  
The Real ◽  
Sequence Processing

A traditional random variable X is a function that maps from a stochastic process to the real line (X,<=,d,+,.), where R is the set of real numbers, <= is the standard linear order relation on R, d(x,y)=|x-y| is the usual metric on R, and (R, +, .) is the standard field on R. Greenhoe(2015b) has demonstrated that this definition of random variable is often a poor choice for computing statistics when the stochastic process that X maps from has structure that is dissimilar to that of the real line. Greenhoe(2015b) has further proposed an alternative statistical system, that rather than mapping a stochastic process to the real line, instead maps to a weighted graph that has order and metric geometry structures similar to that of the underlying stochastic process. In particular, ideally the structure X maps from and the structure X maps to are, with respect to each other, both isomorphic and isometric.Mapping to a weighted graph is useful for analysis of a single random variable.for example the expectation EX of X can be defined simply as the center of its weighted graph. However, the mapping has limitations with regards to a sequence of random variables in performing sequence analysis (using for example Fourier analysis or wavelet analysis), in performing sequence processing (using for example FIR filtering or IIR filtering), in making diagnostic measurements (using a post-transform metric space), or in making goptimalh decisions (based on gdistanceh measurements in a metric space or more generally a distance space). Rather than mapping to a weighted graph, this paper proposes instead mapping to an ordered distance linear space Y=(R^n,<=,d,+,.,R,+,x), where (R,+,x) is a field, + is the vector addition operator on R^n x R^n, and . is the scalar-vector multiplication operator on R x R^n. The linear space component of Y provides a much more convenient (as compared to the weighted graph) framework for sequence analysis and processing. The ordered set and distance space components of Y allow one to preserve the order structure and distance geometry inherent in the underlying stochastic process, which in turn likely provides a less distorted (as compared to the real line) framework for analysis, diagnostics, and optimal decision making.

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