A Liouville-Type Problem in Partial Differential Equations

1972 ◽  
Vol 23 (2) ◽  
pp. 187-188
Author(s):  
Kuang-Ho Chen
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Type Problem ◽  
Liouville Type ◽  
Partial Differential

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

2019 ◽  
Vol 22 (3) ◽  
pp. 543-600 ◽  
Author(s):  
Vassili N. Kolokoltsov
Keyword(s):  
Monte Carlo Simulation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Point Of View ◽  
Fractional Partial Differential Equations ◽  
Exit Times ◽  
Liouville Type ◽  
Partial Differential ◽  
Path Integral Representation ◽  
Probabilistic Point

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.

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