The probabilistic point of view on the generalized fractional partial differential equations

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The probabilistic point of view on the generalized fractional partial differential equations. / Kolokoltsov, Vassili N.

в: Fractional Calculus and Applied Analysis, Том 22, № 3, 26.06.2019, стр. 543-600.

Результаты исследований: Научные публикации в периодических изданиях › Обзорная статья › Рецензирование

BibTeX

@article{ef5e70d298c74f238234b7a029ca21ed,

title = "The probabilistic point of view on the generalized fractional partial differential equations",

abstract = "This paper aims at unifying and clarifying the recent advances in the analysis of the fractional and generalized fractional Partial Differential Equations of Caputo and Riemann-Liouville type arising essentially from the probabilistic point of view. This point of view leads to the path integral representation for the solutions of these equations, which is seen to be stable with respect to the initial data and key parameters and is directly amenable to numeric calculations (Monte-Carlo simulation). In many cases these solutions can be compactly presented via the wide class of operator-valued analytic functions of the Mittag-Leffler type, which are proved to be expressed as the Laplace transforms of the exit times of monotone Markov processes.",

keywords = "cardiac imaging, cardiac troponin, crossreactivity, skeletal muscle",

author = "Kolokoltsov, {Vassili N.}",

year = "2019",

month = jun,

day = "26",

doi = "10.1515/fca-2019-0033",

language = "English",

volume = "22",

pages = "543--600",

journal = "Fractional Calculus and Applied Analysis",

issn = "1311-0454",

publisher = "De Gruyter",

number = "3",

}

RIS

TY - JOUR

T1 - The probabilistic point of view on the generalized fractional partial differential equations

AU - Kolokoltsov, Vassili N.

PY - 2019/6/26

Y1 - 2019/6/26

N2 - This paper aims at unifying and clarifying the recent advances in the analysis of the fractional and generalized fractional Partial Differential Equations of Caputo and Riemann-Liouville type arising essentially from the probabilistic point of view. This point of view leads to the path integral representation for the solutions of these equations, which is seen to be stable with respect to the initial data and key parameters and is directly amenable to numeric calculations (Monte-Carlo simulation). In many cases these solutions can be compactly presented via the wide class of operator-valued analytic functions of the Mittag-Leffler type, which are proved to be expressed as the Laplace transforms of the exit times of monotone Markov processes.

AB - This paper aims at unifying and clarifying the recent advances in the analysis of the fractional and generalized fractional Partial Differential Equations of Caputo and Riemann-Liouville type arising essentially from the probabilistic point of view. This point of view leads to the path integral representation for the solutions of these equations, which is seen to be stable with respect to the initial data and key parameters and is directly amenable to numeric calculations (Monte-Carlo simulation). In many cases these solutions can be compactly presented via the wide class of operator-valued analytic functions of the Mittag-Leffler type, which are proved to be expressed as the Laplace transforms of the exit times of monotone Markov processes.

KW - cardiac imaging

KW - cardiac troponin

KW - crossreactivity

KW - skeletal muscle

UR - http://www.scopus.com/inward/record.url?scp=85070281264&partnerID=8YFLogxK

U2 - 10.1515/fca-2019-0033

DO - 10.1515/fca-2019-0033

M3 - Review article

AN - SCOPUS:85070281264

VL - 22

SP - 543

EP - 600

JO - Fractional Calculus and Applied Analysis

JF - Fractional Calculus and Applied Analysis

SN - 1311-0454

IS - 3

ER -

ID: 51530048