Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

On Some Boundary Property of Dirichlet Operator Family for the Second Order Ordinary Differential Equation

Author(s):

B. P. Harlamov

Institute of Problems of Mechanical Engineering, RAS, Saint-Petersburg, Russia

b.p.harlamov@gmail.com

Abstract:

An ordinary differential equation of the second order is considered. Coefficients of the equation are assumed to provide a unique Dirichlet problem solution on every finite interval. A consistency condition with respect to convolution of operators Dirichlet family of this equation is defined. The Dirichlet problem solutions representing in a kernel form are investigated. In one-dimensional case such a sub-probability kernel is reduced to two functions (left and right) corresponding to two boundaries of the interval. The left (right) boundary of the initial interval is said to be unattainable if the the left (right) sub-probability is equal to zero. In terms of the initial differential equation some sufficient conditions are derived for the left (right) boundary to be unattainable. One can control these conditions by a constructive method. As interpretation of obtain results unattainable boundaries of a diffusion semi-Markov process are demonstrated

Keywords

• diffusion semi-Markov process
• Dirichlet problem
• integral equation
• ordinary differential equation of the second order
• sub-probability kernel

References:

1. Hilbert, D., Trudinger N. C. Elliptical differential equation with partial derivatives of the second order. M. Nauka, 1989
2. Dynkin, E. B. Markov processes. M. : FM, 1963
3. Harlamov, B. P. Continuous semi-Markov processes. SPb. : Nauka, 2001
4. Sobolev, S. L. Equations of mathematical physics. M. : Tekhizdat, 1954
5. Fihtengolts, G. M. Course of differential and integral calculation. Fizmatgiz, 1959
6. Kamke, E. Reference book on ordinary differential equations. M. : Nauka, 1971
7. Harlamov, B. P. On unattainable boundary of diffusion process interval range: semi-Markov approach. Zapiski nauchnykh seminarov POMI, v. 466, 2017, 313-330
8. Gihman, I. I., Skorokhod , A. V. Stochastic differential equations. Kyev: Naukova Dumka, 1968
9. Cherny, A. S., Engelbert, H. -J. Singular Stochastic Differential Equations. Springer, Berlin-Heidelberg-New York, 2005

Full text (pdf)