On Inversions and Doob h-Transforms of Linear Diffusions

Alili, Larbi; Graczyk, Piotr; Żak, Tomasz

doi:10.1007/978-3-319-18585-9_6

Cited by 3 publications

(15 citation statements)

References 29 publications

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“…Remark Theorem (iii) and Proposition (ii) give another “operator‐like” proof of Theorem when X is a diffusion and for continuous X ‐harmonic functions, see also Remark 7 in .…”

Section: Inversion Property and Kelvin Transform Of X‐harmonic Functionsmentioning

confidence: 95%

“…We furthermore assume that X is irreducible, on E , in the sense that, starting from anywhere in

\overset{˚}{S} ∖ {0}

, the process can reach with positive probability any nonempty open subset of E . This is a multidimensional generalization of the situation considered in , where the authors constructed the dual of a one dimensional regular diffusion living on a compact interval

[l, r]

and killed upon exiting the interval.…”

Section: Inversion Property and Kelvin Transform Of X‐harmonic Functionsmentioning

confidence: 99%

“…The IP with the spherical inversion for isotropic stable processes in

{double-struckR}^{n}

was proved in . The continuous case in dimension 1 was studied in . The spherical inversions of self‐similar Markov processes under a reversibility condition have been studied in , and, in the particular case of 1‐dimensional stable processes in .…”

Section: Inversion Property and Kelvin Transform Of X‐harmonic Functionsmentioning

confidence: 99%

“…According to , the process Y has the Inversion Property, with characteristics

(I_{0}, h_{0}, v_{0})

that can be determined by [, Theorem 1]. It is natural to conjecture that the Hyperbolic Brownian Motion X has IP with characteristics

(I, h, v_{0})

, where

I false(x false) = σ tanh \frac{I_{0} false(r false)}{2} and h false(x false) = h_{0} false(r false) .

When

n = 3

, by [, Section 5.2], we have

I_{0} false(r false) = \frac{1}{2} ln coth r

h_{0} false(r false) = coth r - 1

and

v_{0} false(r false) = 2 cosh r sinh r

. If the Hyperbolic Brownian Motion

X_{t}

had IP with the involution I and the excessive function h , then, by Theorem and Proposition , if

L f = 0

then

L (h f \circ I) = 0

.…”

Section: Applicationsmentioning

confidence: 99%

“…In the pointwise recurrent case

α > n = 1

one must consider the process

X_{t}^{0}

killed at 0. In the recent papers , inversions involving dual processes were studied for diffusions on

double-struckR

and for self‐similar Markov processes on

{double-struckR}^{n}

n \geq 1

.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Space and time inversions of stochastic processes and Kelvin transform

Alili

Chaumont

Graczyk

et al. 2018

Mathematische Nachrichten

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Let X be a standard Markov process. We prove that a space inversion property of X implies the existence of a Kelvin transform of X‐harmonic, excessive and operator‐harmonic functions and that the inversion property is inherited by Doob h‐transforms. We determine new classes of processes having space inversion properties amongst transient processes satisfying the time inversion property. For these processes, some explicit inversions, which are often not the spherical ones, and excessive functions are given explicitly. We treat in details the examples of free scaled power Bessel processes, non‐colliding Bessel particles, Wishart processes, Gaussian Ensemble and Dyson Brownian Motion.

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“…Remark Theorem (iii) and Proposition (ii) give another “operator‐like” proof of Theorem when X is a diffusion and for continuous X ‐harmonic functions, see also Remark 7 in .…”

Section: Inversion Property and Kelvin Transform Of X‐harmonic Functionsmentioning

confidence: 95%

“…We furthermore assume that X is irreducible, on E , in the sense that, starting from anywhere in

\overset{˚}{S} ∖ {0}

[l, r]

and killed upon exiting the interval.…”

Section: Inversion Property and Kelvin Transform Of X‐harmonic Functionsmentioning

confidence: 99%

“…The IP with the spherical inversion for isotropic stable processes in

{double-struckR}^{n}

Section: Inversion Property and Kelvin Transform Of X‐harmonic Functionsmentioning

confidence: 99%

“…According to , the process Y has the Inversion Property, with characteristics

(I_{0}, h_{0}, v_{0})

that can be determined by [, Theorem 1]. It is natural to conjecture that the Hyperbolic Brownian Motion X has IP with characteristics

(I, h, v_{0})

, where

I false(x false) = σ tanh \frac{I_{0} false(r false)}{2} and h false(x false) = h_{0} false(r false) .

When

n = 3

, by [, Section 5.2], we have

I_{0} false(r false) = \frac{1}{2} ln coth r

h_{0} false(r false) = coth r - 1

and

v_{0} false(r false) = 2 cosh r sinh r

. If the Hyperbolic Brownian Motion

X_{t}

had IP with the involution I and the excessive function h , then, by Theorem and Proposition , if

L f = 0

then

L (h f \circ I) = 0

.…”

Section: Applicationsmentioning

confidence: 99%

“…In the pointwise recurrent case

α > n = 1

one must consider the process

X_{t}^{0}

killed at 0. In the recent papers , inversions involving dual processes were studied for diffusions on

double-struckR

and for self‐similar Markov processes on

{double-struckR}^{n}

n \geq 1

.…”

Section: Introductionmentioning

confidence: 99%

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Space and time inversions of stochastic processes and Kelvin transform

Alili

Chaumont

Graczyk

et al. 2018

Mathematische Nachrichten

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Inversion, duality and Doob $h$-transforms for self-similar Markov processes

Alili¹,

Chaumont²,

Graczyk³

et al. 2017

Electron. J. Probab.

Self Cite

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We show that any R d \{0}-valued self-similar Markov process X, with index α > 0 can be represented as a path transformation of some Markov additive process (MAP) (θ, ξ) in S d−1 × R. This result extends the well known Lamperti transformation. Let us denote by X the self-similar Markov process which is obtained from the MAP (θ, −ξ) through this extended Lamperti transformation. Then we prove that X is in weak duality with X, with respect to the measure π(x/ x ) x α−d dx, if and only if (θ, ξ) is reversible with respect to the measure π(ds)dx, where π(ds) is some σ-finite measure on S d−1 and dx is the Lebesgue measure on R. Besides, the dual process X has the same law as the inversionThese results allow us to obtain excessive functions for some classes of self-similar Markov processes such as stable Lévy processes.

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Independent factorization of the last zero arcsine law for Bessel processes with drift

Panzo

2021

Electron. Commun. Probab.

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We show that the last zero before time t of a recurrent Bessel process with drift starting at 0 has the same distribution as the product of a right-censored exponential random variable and an independent beta random variable. This extends a recent result of Schulte-Geers and Stadje [19] from Brownian motion with drift to recurrent Bessel processes with drift. We give two proofs, one of which is intuitive, direct, and avoids heavy computations. For this we develop a novel additive decomposition for the square of a Bessel process with drift that may be of independent interest.

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On Inversions and Doob h-Transforms of Linear Diffusions

Cited by 3 publications

References 29 publications

Space and time inversions of stochastic processes and Kelvin transform

Space and time inversions of stochastic processes and Kelvin transform

Inversion, duality and Doob $h$-transforms for self-similar Markov processes

Independent factorization of the last zero arcsine law for Bessel processes with drift

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