Multivariate Markov processes for stochastic systems with delays: application to the stochastic Gompertz model with...

Area of Science:

• Mathematical Physics
• Stochastic Analysis
• Dynamical Systems

Background:

• Stochastic delay differential equations (SDDEs) are crucial for modeling systems with time lags.
• Existing methods for analyzing SDDEs can be complex and computationally intensive.
• A unified framework for analyzing SDDEs using Markov diffusion processes is needed.

Purpose of the Study:

• To develop a method for describing stochastic processes with delays in terms of Markov diffusion processes.
• To derive multivariate Langevin and Fokker-Planck equations for SDDEs.
• To apply the developed framework to a population growth model (Gompertz model).

Main Methods:

• Method of steps applied to stochastic processes with delays.
• Derivation of multivariate Langevin and Fokker-Planck equations.
• Consideration of Ito and Stratonovich calculus.
• Analysis of natural, periodic, and reflective boundary conditions.

Main Results:

• Successfully described stochastic processes with delays using Markov diffusion processes.
• Derived generalized Langevin and Fokker-Planck equations for SDDEs.
• Recovered the generalized delay Fokker-Planck equation.
• Demonstrated the applicability to the Gompertz model with delay and multiplicative white noise.

Conclusions:

• The method of steps provides a powerful framework for analyzing SDDEs.
• The derived equations offer a new tool for studying delayed stochastic systems.
• The application to the Gompertz model highlights the practical relevance of the approach.