## IMS Collections (2008)

### BibTeX

@MISC{Bhamidi08imscollections,

author = {Shankar Bhamidi and Peter Ralph},

title = {IMS Collections},

year = {2008}

}

### OpenURL

### Abstract

Brownian motion on disconnected sets, basic hypergeometric functions, and some continued fractions of Ramanujan

### Citations

807 | Probability: Theory and Examples
- Durrett
- 1996
(Show Context)
Citation Context ... with this rate of growth, for example, the standard log-normal distribution has k th moment e k2 2 , as does the discrete measure which assigns mass proportional to e −k2 2 at the points ek , k ∈ Z, =-=[12]-=-, 2.3e. From Proposition 5.1 we would expect informally that the moments of ξt should converge to those of a Brownian motion at time t as q ↓ 1. Recall that cq = q−1 (q − 1) 2 (1 + q) and observe that... |

374 |
Quantum Groups
- Kassel
- 1995
(Show Context)
Citation Context ...logue of the positive integer n, through to the very deep, such as certain quantum groups (which are not actually groups in the usual sense) being the “q-deformations” of appropriate classical groups =-=[9, 35, 40]-=-. For a very readable introduction to q-calculus see [36], and for its relation with q-series, see the tutorial [45], or the more extensive books [30] or [2]. What we need for our purposes is given in... |

193 |
Diffusion processes and their sample paths
- Itô, McKean
- 1974
(Show Context)
Citation Context ...bounded below is similar.44 S. Bhamidi, S. N. Evans, R. Peled and P. Ralph The process ξ is constructed as a time-change of standard Brownian motion, a class of processes described in Itô and McKean =-=[34]-=-, and that has been studied variously as “gap diffusions” [44], “quasidiffusions” [7, 46, 47, 48, 49, 50, 55], and (one-dimensional) “generalized diffusions” [65, 66, 67, 68]. The process ξ that we st... |

106 |
Lectures on quantum groups
- Jantzen
- 1996
(Show Context)
Citation Context ...logue of the positive integer n, through to the very deep, such as certain quantum groups (which are not actually groups in the usual sense) being the “q-deformations” of appropriate classical groups =-=[9, 35, 40]-=-. For a very readable introduction to q-calculus see [36], and for its relation with q-series, see the tutorial [45], or the more extensive books [30] or [2]. What we need for our purposes is given in... |

77 |
Quantum Calculus
- Kac, Cheung
- 2002
(Show Context)
Citation Context ...as certain quantum groups (which are not actually groups in the usual sense) being the “q-deformations” of appropriate classical groups [9, 35, 40]. For a very readable introduction to q-calculus see =-=[36]-=-, and for its relation with q-series, see the tutorial [45], or the more extensive books [30] or [2]. What we need for our purposes is given in Section 11. The interplay between q-calculus (that is, q... |

37 |
The classification of birth and death processes
- KARLIN, J
- 1957
(Show Context)
Citation Context ...l-known spectral representation of transition probabilities of a unilateral birth-and-death process, for which the appropriate eigenfunctions are a family of orthogonal polynomials (see, for example, =-=[37, 38, 39, 62]-=-). If we kill X at q−n for some n ∈ Z to obtain a process on {q−n+1, q−n+2 , . . .}, then the corresponding unilateral birth-and-death process has a specialization of the associated continuous dual q-... |

35 |
A guide to quantum groups. Cambridge Univ
- Chari, Pressley
- 1994
(Show Context)
Citation Context ...logue of the positive integer n, through to the very deep, such as certain quantum groups (which are not actually groups in the usual sense) being the “q-deformations” of appropriate classical groups =-=[9, 35, 40]-=-. For a very readable introduction to q-calculus see [36], and for its relation with q-series, see the tutorial [45], or the more extensive books [30] or [2]. What we need for our purposes is given in... |

33 |
Basic hypergeometric series, 2nd ed
- Gasper, Rahman
- 2004
(Show Context)
Citation Context ...ormations” of appropriate classical groups [9, 35, 40]. For a very readable introduction to q-calculus see [36], and for its relation with q-series, see the tutorial [45], or the more extensive books =-=[30]-=- or [2]. What we need for our purposes is given in Section 11. The interplay between q-calculus (that is, q-difference operators, q-integration, and q-difference equations), q-series (particularly bas... |

29 | A Markovian analysis of additive-increase multiplicative-decrease algorithms
- Dumas, Guillemin, et al.
(Show Context)
Citation Context ...o note that the same functional seems to have arisen in a number of applied probability settings as well, for example, in genetics [10] and in transmission control protocols on communication networks =-=[11]-=-. In [42] the Euler and Heine distributions, q-analogues of the Poisson distribution, are studied: distributional properties are derived and some statistical applications (such as fitting these distri... |

28 |
Dynamic Equations on Time Scales
- Bohner, Peterson
- 2001
(Show Context)
Citation Context ...fference equations when T = Z (as well as the somewhat less familiar theory of q-differences and qdifference equations when T is {q k : k ∈ Z} for some q > 1). The time scale calculus is described in =-=[6]-=-, where there is also discussion of the application of time scale dynamic equations to systems that evolve via a mixture of discrete and continuous mechanisms. Our first aim in this paper is to invest... |

24 |
Characterization of the Lévy measures of inverse local times of gap diffusion
- Knight
- 1981
(Show Context)
Citation Context ... and P. Ralph The process ξ is constructed as a time-change of standard Brownian motion, a class of processes described in Itô and McKean [34], and that has been studied variously as “gap diffusions” =-=[44]-=-, “quasidiffusions” [7, 46, 47, 48, 49, 50, 55], and (one-dimensional) “generalized diffusions” [65, 66, 67, 68]. The process ξ that we study is a quasidiffusion, so results on quasidiffusions apply i... |

22 | Random matrix theory over finite fields
- Fulman
- 2002
(Show Context)
Citation Context ... point is expressed via a q-analogue of the exponential function. Connections between q-series and random matrices over a finite field (resp. over a local field other than R or C) are investigated in =-=[27, 28, 29]-=- (resp. [1, 14]). 2. Brownian motion on a general unbounded closed subset of R 2.1. Existence Let T be a closed subset of R that is unbounded above and below (that is, bilaterally unbounded). We now s... |

22 | Contiguous relations, basic hypergeometric functions and orthogonal polynomials. II: Associated big q-Jacobi polynomials
- Gupta, Ismail, et al.
(Show Context)
Citation Context ...ractions of this form are listed in Ramanujan’s “lost” notebook (see the discussion in [5]), and evaluations for various ranges of the parameters (although not all the values we need) can be found in =-=[5, 32, 33]-=- (although in the last several parameter restrictions are omitted). Theorem 6.1. (i) The Laplace transform of the time to go from 1 to q−1 for both X and ˆ X is H ↓ q 0 (λ) = λ An alternative expressi... |

18 |
Linear growth, birth and death processes
- Karlin, McGregor
- 1958
(Show Context)
Citation Context ...l-known spectral representation of transition probabilities of a unilateral birth-and-death process, for which the appropriate eigenfunctions are a family of orthogonal polynomials (see, for example, =-=[37, 38, 39, 62]-=-). If we kill X at q−n for some n ∈ Z to obtain a process on {q−n+1, q−n+2 , . . .}, then the corresponding unilateral birth-and-death process has a specialization of the associated continuous dual q-... |

17 |
Special Functions. Cambridge Univ
- Askey, Roy
- 1999
(Show Context)
Citation Context ...s” of appropriate classical groups [9, 35, 40]. For a very readable introduction to q-calculus see [36], and for its relation with q-series, see the tutorial [45], or the more extensive books [30] or =-=[2]-=-. What we need for our purposes is given in Section 11. The interplay between q-calculus (that is, q-difference operators, q-integration, and q-difference equations), q-series (particularly basic hype... |

15 |
Poissonian exponential functionals, q-series, q-integrals, and the moment problem for log-normal distributions
- Bertoin, Biane, et al.
- 2004
(Show Context)
Citation Context ...egration, and q-difference equations), q-series (particularly basic hypergeometric functions), and probability has been explored in a number of settings both theoretical and applied. The recent paper =-=[4]-=- studies the connection between q-calculus and the exponential functional of a Poisson process Iq := ∫ ∞ 0 q Nt dt, q < 1, where Nt is the simple homogeneous Poisson counting process on the real line.... |

13 | The formal theory of birth-and-death processes, lattice path combinatorics, and continued fractions, Adv
- Flajolet, Guillemin
(Show Context)
Citation Context ... See Section 12 for some relevant background and notation for continued fractions. The connection between birth-and-death processes and continued fractions has already been explored, for instance, in =-=[23, 31]-=-. The role of continued fractions in this setting is to pick out the correct solutions of the (generalized) Sturm-Liouville equations [63], whose relationship to quasidiffusions in general is well-lai... |

12 |
Many server queueing processes with Poisson input and exponential service times
- Karlin, McGregor
- 1958
(Show Context)
Citation Context ...l-known spectral representation of transition probabilities of a unilateral birth-and-death process, for which the appropriate eigenfunctions are a family of orthogonal polynomials (see, for example, =-=[37, 38, 39, 62]-=-). If we kill X at q−n for some n ∈ Z to obtain a process on {q−n+1, q−n+2 , . . .}, then the corresponding unilateral birth-and-death process has a specialization of the associated continuous dual q-... |

9 |
The general diffusion operator and positivity preserving semi-groups in one dimension
- Feller
- 1954
(Show Context)
Citation Context ...asidiffusion that has all of R as its state space, but spends all its time in Q . These processes (with killing and appropriate boundary conditions) were shown by Löbus [54], extending work by Feller =-=[15, 18, 20]-=- to be the only Markov processes taking values in R whose generators are in some sense local, and satisfy a certain maximum principle. Various authors [47, 48, 51] have given beautiful spectral repres... |

9 | The Rogers–Ramanujan identities, the finite general linear groups
- Fulman
- 2000
(Show Context)
Citation Context ... point is expressed via a q-analogue of the exponential function. Connections between q-series and random matrices over a finite field (resp. over a local field other than R or C) are investigated in =-=[27, 28, 29]-=- (resp. [1, 14]). 2. Brownian motion on a general unbounded closed subset of R 2.1. Existence Let T be a closed subset of R that is unbounded above and below (that is, bilaterally unbounded). We now s... |

9 | A probabilistic proof of the Rogers–Ramanujan identities
- Fulman
(Show Context)
Citation Context ... point is expressed via a q-analogue of the exponential function. Connections between q-series and random matrices over a finite field (resp. over a local field other than R or C) are investigated in =-=[27, 28, 29]-=- (resp. [1, 14]). 2. Brownian motion on a general unbounded closed subset of R 2.1. Existence Let T be a closed subset of R that is unbounded above and below (that is, bilaterally unbounded). We now s... |

8 |
Generalized second order differential operators and their lateral conditions
- FELLER
- 1957
(Show Context)
Citation Context ...asidiffusion that has all of R as its state space, but spends all its time in Q . These processes (with killing and appropriate boundary conditions) were shown by Löbus [54], extending work by Feller =-=[15, 18, 20]-=- to be the only Markov processes taking values in R whose generators are in some sense local, and satisfy a certain maximum principle. Various authors [47, 48, 51] have given beautiful spectral repres... |

8 |
Some asymptotic properties of the transition densities of one–dimensional diffusions
- Küchler
- 1980
(Show Context)
Citation Context ...s ξ is constructed as a time-change of standard Brownian motion, a class of processes described in Itô and McKean [34], and that has been studied variously as “gap diffusions” [44], “quasidiffusions” =-=[7, 46, 47, 48, 49, 50, 55]-=-, and (one-dimensional) “generalized diffusions” [65, 66, 67, 68]. The process ξ that we study is a quasidiffusion, so results on quasidiffusions apply in this context – but it is a distinguished quas... |

7 |
A stochastic model of fragment formation when DNA replicates
- Cowan, Chiu
- 1994
(Show Context)
Citation Context ...tribution of Iq using q-calculus is given in [3]. It is interesting to note that the same functional seems to have arisen in a number of applied probability settings as well, for example, in genetics =-=[10]-=- and in transmission control protocols on communication networks [11]. In [42] the Euler and Heine distributions, q-analogues of the Poisson distribution, are studied: distributional properties are de... |

6 |
On the expansion of a continued fraction
- Hirschhorn
(Show Context)
Citation Context ...ractions of this form are listed in Ramanujan’s “lost” notebook (see the discussion in [5]), and evaluations for various ranges of the parameters (although not all the values we need) can be found in =-=[5, 32, 33]-=- (although in the last several parameter restrictions are omitted). Theorem 6.1. (i) The Laplace transform of the time to go from 1 to q−1 for both X and ˆ X is H ↓ q 0 (λ) = λ An alternative expressi... |

6 |
On spectral measures of strings and excursions of quasidiffusions
- Küchler, Salminen
- 1989
(Show Context)
Citation Context ...Löbus [54], extending work by Feller [15, 18, 20] to be the only Markov processes taking values in R whose generators are in some sense local, and satisfy a certain maximum principle. Various authors =-=[47, 48, 51]-=- have given beautiful spectral representations of quasidiffusions using Kreĭn’s theory of strings [13, 19, 41]. It is natural to ask about further properties of the Brownian motion on T. In the presen... |

5 |
Spectral asymptotics of generalized measure geometric Laplacians on Cantor like sets
- Freiberg
(Show Context)
Citation Context ... relation to diffusion processes in Itô and McKean [34], §§5.1-5.3, and were studied further by Feller [15, 16, 17, 18, 19, 20, 21] (where the support of µ is connected), Löbus [54, 55], and Freiberg =-=[24, 25, 26]-=- (where µ is atomless). 2.2. Uniqueness We next establish a uniqueness result that complements the existence result of Proposition 2.1. We will apply the following result, which is a slight variant of... |

5 |
Excursions of birth and death processes, orthogonal polynomials, and continued fractions
- Guillemin, Pinchon
- 1999
(Show Context)
Citation Context ... See Section 12 for some relevant background and notation for continued fractions. The connection between birth-and-death processes and continued fractions has already been explored, for instance, in =-=[23, 31]-=-. The role of continued fractions in this setting is to pick out the correct solutions of the (generalized) Sturm-Liouville equations [63], whose relationship to quasidiffusions in general is well-lai... |

5 |
Heine-Euler extensions of the Poisson distribution
- Kemp
- 1992
(Show Context)
Citation Context ...at the same functional seems to have arisen in a number of applied probability settings as well, for example, in genetics [10] and in transmission control protocols on communication networks [11]. In =-=[42]-=- the Euler and Heine distributions, q-analogues of the Poisson distribution, are studied: distributional properties are derived and some statistical applications (such as fitting these distributions t... |

5 |
Absorption sampling and the absorption distribution
- Kemp
- 1998
(Show Context)
Citation Context ...ng time strategies when drilling for oil and studies of parasite distributions, see the references in [42]. The q-analogue of the Pascal distribution has also been studied in the applied context, see =-=[43]-=-. The properties of q-analogues of various classical discrete distributions are also surveyed in [52]. Both Iq and the Euler distribution appear in Section 6, where they come together to form the dist... |

4 |
On a generalized gamma convolution related to the q-calculus
- Berg
- 2005
(Show Context)
Citation Context ...on process Iq := ∫ ∞ 0 q Nt dt, q < 1, where Nt is the simple homogeneous Poisson counting process on the real line. A purely analytic treatment of the distribution of Iq using q-calculus is given in =-=[3]-=-. It is interesting to note that the same functional seems to have arisen in a number of applied probability settings as well, for example, in genetics [10] and in transmission control protocols on co... |

4 |
Analytical properties of measure geometric Krein-Feller-operators on the real line
- Freiberg
- 2003
(Show Context)
Citation Context ... relation to diffusion processes in Itô and McKean [34], §§5.1-5.3, and were studied further by Feller [15, 16, 17, 18, 19, 20, 21] (where the support of µ is connected), Löbus [54, 55], and Freiberg =-=[24, 25, 26]-=- (where µ is atomless). 2.2. Uniqueness We next establish a uniqueness result that complements the existence result of Proposition 2.1. We will apply the following result, which is a slight variant of... |

4 |
q-special functions, a tutorial. In Representations of Lie groups and quantum groups
- Koornwinder
- 1994
(Show Context)
Citation Context ... the usual sense) being the “q-deformations” of appropriate classical groups [9, 35, 40]. For a very readable introduction to q-calculus see [36], and for its relation with q-series, see the tutorial =-=[45]-=-, or the more extensive books [30] or [2]. What we need for our purposes is given in Section 11. The interplay between q-calculus (that is, q-difference operators, q-integration, and q-difference equa... |

3 |
The determinant of random power series matrices over finite fields
- Abdel-Ghaffar
(Show Context)
Citation Context ...via a q-analogue of the exponential function. Connections between q-series and random matrices over a finite field (resp. over a local field other than R or C) are investigated in [27, 28, 29] (resp. =-=[1, 14]-=-). 2. Brownian motion on a general unbounded closed subset of R 2.1. Existence Let T be a closed subset of R that is unbounded above and below (that is, bilaterally unbounded). We now show existence o... |

3 | Elementary divisors and determinants of random matrices over a local field, Stochastic Process
- Evans
(Show Context)
Citation Context ...via a q-analogue of the exponential function. Connections between q-series and random matrices over a finite field (resp. over a local field other than R or C) are investigated in [27, 28, 29] (resp. =-=[1, 14]-=-). 2. Brownian motion on a general unbounded closed subset of R 2.1. Existence Let T be a closed subset of R that is unbounded above and below (that is, bilaterally unbounded). We now show existence o... |

3 |
Differential operators with the positive maximum property
- FELLER
- 1959
(Show Context)
Citation Context ...asidiffusion that has all of R as its state space, but spends all its time in Q . These processes (with killing and appropriate boundary conditions) were shown by Löbus [54], extending work by Feller =-=[15, 18, 20]-=- to be the only Markov processes taking values in R whose generators are in some sense local, and satisfy a certain maximum principle. Various authors [47, 48, 51] have given beautiful spectral repres... |

3 |
On sojourn times, excursions and spectral measures connected with quasidiffusions
- Küchler
- 1986
(Show Context)
Citation Context ...s ξ is constructed as a time-change of standard Brownian motion, a class of processes described in Itô and McKean [34], and that has been studied variously as “gap diffusions” [44], “quasidiffusions” =-=[7, 46, 47, 48, 49, 50, 55]-=-, and (one-dimensional) “generalized diffusions” [65, 66, 67, 68]. The process ξ that we study is a quasidiffusion, so results on quasidiffusions apply in this context – but it is a distinguished quas... |

2 |
On some continued fraction identities of Srinivasa Ramanujan
- Bhargava, Adiga
- 1984
(Show Context)
Citation Context ...1 + q + λq4 , − ... 1 q 1 + q + λ − 1 + q + λq−2 q − 1 + q + λq−4 . − ... Closed-form expressions for continued fractions of this form are listed in Ramanujan’s “lost” notebook (see the discussion in =-=[5]-=-), and evaluations for various ranges of the parameters (although not all the values we need) can be found in [5, 32, 33] (although in the last several parameter restrictions are omitted). Theorem 6.1... |

2 |
The semimartingale decomposition of one-dimensional quasidiffusions with natural scale. Stochastic Process
- Burkhardt, Küchler
- 1987
(Show Context)
Citation Context ...s ξ is constructed as a time-change of standard Brownian motion, a class of processes described in Itô and McKean [34], and that has been studied variously as “gap diffusions” [44], “quasidiffusions” =-=[7, 46, 47, 48, 49, 50, 55]-=-, and (one-dimensional) “generalized diffusions” [65, 66, 67, 68]. The process ξ that we study is a quasidiffusion, so results on quasidiffusions apply in this context – but it is a distinguished quas... |

2 |
A fundamental property of Markov processes with an application to equivalence under time changes
- Chacon, Jamison
- 1979
(Show Context)
Citation Context ...ext establish a uniqueness result that complements the existence result of Proposition 2.1. We will apply the following result, which is a slight variant of Corollary 3.5 of [64] extending results of =-=[8]-=-. We make the assumption that the Markov process X is a right process and that the process Y is defined on a space satisfying the usual conditions to avoid listing Walsh’s assumptions. We also state t... |

2 |
On the intrinsic form for second order differential operators
- Feller
(Show Context)
Citation Context ...erators are in some sense local, and satisfy a certain maximum principle. Various authors [47, 48, 51] have given beautiful spectral representations of quasidiffusions using Kreĭn’s theory of strings =-=[13, 19, 41]-=-. It is natural to ask about further properties of the Brownian motion on T. In the present paper we pursue this matter in a particularly nice special case, when T = Tq := {±q k : k ∈ Z} ∪ {0} for som... |

2 |
Dirichlet forms on fractal subsets of the real line
- Freiberg
(Show Context)
Citation Context ... relation to diffusion processes in Itô and McKean [34], §§5.1-5.3, and were studied further by Feller [15, 16, 17, 18, 19, 20, 21] (where the support of µ is connected), Löbus [54, 55], and Freiberg =-=[24, 25, 26]-=- (where µ is atomless). 2.2. Uniqueness We next establish a uniqueness result that complements the existence result of Proposition 2.1. We will apply the following result, which is a slight variant of... |

2 |
The spectral theory of a string
- Kats
- 1994
(Show Context)
Citation Context ...erators are in some sense local, and satisfy a certain maximum principle. Various authors [47, 48, 51] have given beautiful spectral representations of quasidiffusions using Kreĭn’s theory of strings =-=[13, 19, 41]-=-. It is natural to ask about further properties of the Brownian motion on T. In the present paper we pursue this matter in a particularly nice special case, when T = Tq := {±q k : k ∈ Z} ∪ {0} for som... |

1 |
On differential operators and boundary conditions
- Feller
- 1955
(Show Context)
Citation Context ...t of µ. Such a generator has the natural interpretation 1 d d 2 dµ dx . These operators appear in relation to diffusion processes in Itô and McKean [34], §§5.1-5.3, and were studied further by Feller =-=[15, 16, 17, 18, 19, 20, 21]-=- (where the support of µ is connected), Löbus [54, 55], and Freiberg [24, 25, 26] (where µ is atomless). 2.2. Uniqueness We next establish a uniqueness result that complements the existence result of ... |

1 |
On generalized Sturm-Liouville operators
- Feller
- 1956
(Show Context)
Citation Context ...t of µ. Such a generator has the natural interpretation 1 d d 2 dµ dx . These operators appear in relation to diffusion processes in Itô and McKean [34], §§5.1-5.3, and were studied further by Feller =-=[15, 16, 17, 18, 19, 20, 21]-=- (where the support of µ is connected), Löbus [54, 55], and Freiberg [24, 25, 26] (where µ is atomless). 2.2. Uniqueness We next establish a uniqueness result that complements the existence result of ... |

1 |
On the equation of the vibrating string
- Feller
- 1959
(Show Context)
Citation Context ...t of µ. Such a generator has the natural interpretation 1 d d 2 dµ dx . These operators appear in relation to diffusion processes in Itô and McKean [34], §§5.1-5.3, and were studied further by Feller =-=[15, 16, 17, 18, 19, 20, 21]-=- (where the support of µ is connected), Löbus [54, 55], and Freiberg [24, 25, 26] (where µ is atomless). 2.2. Uniqueness We next establish a uniqueness result that complements the existence result of ... |

1 |
A diffusion equivalent to a countable Markov chain
- Feller, McKean
- 1956
(Show Context)
Citation Context ...a distinguished quasidiffusion among the many possible quasidiffusions taking values in T. Quasidiffusions can exhibit behavior considerably different from that of ξ – for instance, Feller and McKean =-=[22]-=- described a quasidiffusion that has all of R as its state space, but spends all its time in Q . These processes (with killing and appropriate boundary conditions) were shown by Löbus [54], extending ... |

1 |
Quasidiffusions, sojourn times and spectral measures
- Küchler
- 1985
(Show Context)
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1 |
On Itô’s excursion law, local times and spectral measures for quasidiffusions
- Küchler
- 1987
(Show Context)
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1 |
A limit theorem for the excursion of quasidiffusions straddling t
- Küchler
- 1989
(Show Context)
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