Conditional maximal distributions of processes related to higher-order heat-type equations

Beghin, Luisa; Hochberg, Kenneth J.; Orsingher, Enzo

doi:10.1016/s0304-4149(99)00074-5

Cited by 26 publications

(29 citation statements)

References 5 publications

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“…When N is an odd integer, the integral of p is not absolutely convergent and then similar estimates may not be obtained; this makes the study of X very much harder in this case. Nevertheless, we have found, formally at least, remarkable formulas which agree with those of Beghin et al [2,3] in the case N = 3. They obtained them by using a Feynman-Kac approach and solving differential equations.…”

Section: Introductionsupporting

confidence: 90%

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First Hitting Time and Place, Monopoles and Multipoles for Pseudo-Processes Driven by the Equation $\partial u/\partial t=\pm\partial^N u/\partial x^N$

Lachal

2007

Electron. J. Probab.

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Consider the high-order heat-type equation ∂u/∂t = ±∂ N u/∂x N for an integer N > 2 and introduce the related Markov pseudo-process (X(t)) t 0 . In this paper, we study several functionals related to (X(t)) t 0 : the maximum M (t) and minimum m(t) up to time t; the hitting times τ + a and τ − a of the half lines (a, +∞) and (−∞, a) respectively. We provide explicit expressions for the distributions of the vectors (X(t), M (t)) and (X(t), m(t)), as well as those of the vectors (τ + a , X(τ + a )) and (τ − a , X(τ − a )).AMS 2000 subject classifications: primary 60G20; secondary 60J25. Key words and phrases: pseudo-process, joint distribution of the process and its maximum/minimum, first hitting time and place, Multipoles, Spitzer identity.where κ N = (−1) 1+N/2 if N is even and κ N = ±1 if N is odd. Let p(t; z) be the fundamental solution of Eq. (1.1) and put p(t; x, y) = p(t; x − y).The function p is characterized by its Fourier transform +∞ −∞ e iuξ p(t; ξ) dξ = e κ N t(−iu) N .( 1.2) With Eq. (1.1) one associates a Markov pseudo-process (X(t)) t 0 defined on the real line and governed by a signed measure P, which is not a probability measure, according to the usual rules of ordinary stochastic processes: P x {X(t) ∈ dy} = p(t; x, y) dy

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Section: Introductionsupporting

confidence: 90%

“…Hochberg [8], Beghin et al [2,3], Lachal [12] explicitly derived the distribution of M (t) and that of m(t) (with possible conditioning on some values of X(t));…”

Section: Introductionmentioning

confidence: 99%

First Hitting Time and Place, Monopoles and Multipoles for Pseudo-Processes Driven by the Equation $\partial u/\partial t=\pm\partial^N u/\partial x^N$

Lachal

2007

Electron. J. Probab.

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“…1 and . We consider the pseudorandom walk ( ) ∈N related to a family of real parameters { , ∈ {− , .…”

Section: G ( ) =̃(mentioning

confidence: 99%

“…For instance, let us quote the works of Beghin et al [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] and the references therein. We observe that (5) and (7) are closely related to the continuous -iterated Laplacian d 2 /d 2 .…”

Section: G ( ) =̃(mentioning

confidence: 99%

From Pseudorandom Walk to Pseudo-Brownian Motion: First Exit Time from a One-Sided or a Two-Sided Interval

Lachal

2014

International Journal of Stochastic Analysis

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Let be a positive integer, a positive constant and ( ) ≥1 be a sequence of independent identically distributed pseudorandom variables. We assume that the 's take their values in the discrete set {− , − + 1, . . . , − 1, } and that their common pseudodistribution is characterized by the (positive or negative) real numbers P{ = } = 0 + (−1) −1 ( 2 + ) for any ∈ {− , − + 1, . . . , − 1, }. Let us finally introduce ( ) ≥0 the associated pseudorandom walk defined on Z by 0 = 0 and = ∑ =1 for ≥ 1. In this paper, we exhibit some properties of ( ) ≥0 . In particular, we explicitly determine the pseudodistribution of the first overshooting time of a given threshold for ( ) ≥0 as well as that of the first exit time from a bounded interval. Next, with an appropriate normalization, we pass from the pseudorandom walk to the pseudo-Brownian motion driven by the high-order heat-type equation / = (−1) −1 2 / 2 . We retrieve the corresponding pseudodistribution of the first overshooting time of a threshold for the pseudo-Brownian motion (Lachal, 2007). In the same way, we get the pseudodistribution of the first exit time from a bounded interval for the pseudo-Brownian motion which is a new result for this pseudoprocess.

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“…On pourra consulter avec profit [1] et [2] pour des résultats relatifs au cas N = 3 qui ont été obtenus par une autre approche (reposant sur des équations différentielles). Bien que ce cas ne rentre pas dans le contexte présent puisque ρ = +∞, il est à noter que nos résultats concordent exactement, formellement, avec ceux de [2] lorsque N = 3.…”

Section: Exemple : Le Cas « Biharmonique » N =unclassified

Lois conjointes du processus et de son maximum, des premier instant et position d'atteinte d'une demi-droite pour le pseudo-processus régi par l'équation ∂∂t=±∂N∂xN
Lachal
¹

2006
Comptes Rendus. Mathématique
7
0
0
0
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Conditional maximal distributions of processes related to higher-order heat-type equations

Cited by 26 publications

References 5 publications

First Hitting Time and Place, Monopoles and Multipoles for Pseudo-Processes Driven by the Equation $\partial u/\partial t=\pm\partial^N u/\partial x^N$

First Hitting Time and Place, Monopoles and Multipoles for Pseudo-Processes Driven by the Equation $\partial u/\partial t=\pm\partial^N u/\partial x^N$

From Pseudorandom Walk to Pseudo-Brownian Motion: First Exit Time from a One-Sided or a Two-Sided Interval

Lois conjointes du processus et de son maximum, des premier instant et position d'atteinte d'une demi-droite pour le pseudo-processus régi par l'équation ∂∂t=±∂N∂xN
Lachal
¹

2006
Comptes Rendus. Mathématique
7
0
0
0
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Conditional maximal distributions of processes related to higher-order heat-type equations

Cited by 26 publications

References 5 publications

First Hitting Time and Place, Monopoles and Multipoles for Pseudo-Processes Driven by the Equation $\partial u/\partial t=\pm\partial^N u/\partial x^N$

First Hitting Time and Place, Monopoles and Multipoles for Pseudo-Processes Driven by the Equation $\partial u/\partial t=\pm\partial^N u/\partial x^N$

From Pseudorandom Walk to Pseudo-Brownian Motion: First Exit Time from a One-Sided or a Two-Sided Interval

Lois conjointes du processus et de son maximum, des premier instant et position d'atteinte d'une demi-droite pour le pseudo-processus régi par l'équation ∂∂t=±∂N∂xNLachal1 2006Comptes Rendus. Mathématique7000View full textAdd to dashboardCite

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Lois conjointes du processus et de son maximum, des premier instant et position d'atteinte d'une demi-droite pour le pseudo-processus régi par l'équation ∂∂t=±∂N∂xN
Lachal
¹

2006
Comptes Rendus. Mathématique
7
0
0
0
View full text Add to dashboard Cite