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CTRW approximations for fractional equations with variable order. Chapter in a bookстатья
- Автор: Kolokoltsov V.N.
- Сборник: The IMA Volumes in Mathematics and its Applications
- Том: 165
- Глава в коллективной монографии
- Год издания: 2023
- Издательство: Springer Nature
- Местоположение издательства: Switzerland
- Первая страница: 3
- Последняя страница: 26
- Номер статьи: 1
- DOI: 10.1007/978-3-031-34089-5
- Аннотация: The standard diffusion processes are known to be obtained as the limits of appropriate random walks. These prelimiting random walks can be quite different, however. The diffusion coefficient can be made responsible for the size of jumps or for the intensity of jumps. The diffusion limit does not feel the difference. The situation changes if we model jump-type approximations via CTRW with non-exponential waiting times. If we make the diffusion coefficient responsible for the size of jumps and take waiting times from the domain of attraction of an alpha α-stable law with a constant intensity alpha α, then the standard scaling would lead in the limit of small jumps and large intensities to the most standard fractional diffusion equation. However, if we choose the CTRW approximations with fixed jump sizes and use the diffusion coefficient to distinguish intensities at different points, then we obtain in the limit the equations with variable position-dependent fractional derivatives. In this chapter, we build rigorously these approximations and prove their convergence to the corresponding fractional equations for the cases of multidimensional diffusions and more general Feller processes.
- Добавил в систему: Колокольцов Василий Никитич