# A New Approach to Fractional Kinetic Evolutions

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## Abstract

**:**

## 1. Introduction

**Remark**

**1.**

**Remark**

**2.**

## 2. Preliminaries: General Kinetic Equations for Birth-and-Death Processes with Migration

**Remark**

**3.**

**Remark**

**4.**

## 3. CTRW Modeling of Interacting Particle Systems

## 4. Main Results

**Lemma**

**1.**

**Proposition**

**1.**

**Remark**

**5.**

**Theorem**

**1.**

## 5. Proof of Lemma 1

## 6. Proof of Proposition 1

## 7. Proof of Theorem 1

## 8. Extension to Binary and $\mathit{k}$-ary Interaction

**Remark**

**6.**

**Proposition**

**2.**

## 9. Example: Fractional Smoluchovski Coagulation Evolution

## 10. Example: Fractional Boltzmann Collisions Evolution

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. CTRW Approximation of Stable Processes

**Proposition**

**A1.**

## Appendix B. Time-Nonhomogeneous Stable-like Processes

**Remark**

**A1.**

## Appendix C. Stationary Problems and Dynkin’s Martingales

## References

- Kolokoltsov, V.N. Nonlinear Markov Processes and Kinetic Equations; Cambridge Tracts in Mathematics; Cambridge University Press: Cambridge, UK; New York, NY, USA, 2010; Volume 182. [Google Scholar]
- Zaslavsky, G.M. Fractional kinetic equation for Hamiltonian chaos. Phys. D Nonlinear Phenom.
**1994**, 76, 110–122. [Google Scholar] [CrossRef] - Caputo, M. Linear models of dissipation whose Q is almost frequency independent—II. Geophys. J. Intern.
**1967**, 13, 529–539. [Google Scholar] [CrossRef] - Dzherbashian, M.M.; Nersesian, A.B. Fractional derivatives and the Cauchy problem for differential equations of fractional order. Fract. Calc. Appl. Anal.
**2020**, 23, 1810–1836. [Google Scholar] [CrossRef] - Kochubei, A.N.; Kondratiev, Y. Fractional kinetic hierarchies and intermittency. Kinet. Relat. Models
**2017**, 10, 725–740. [Google Scholar] [CrossRef] - Kolokoltsov, V.N.; Malafeyev, O.A. Many Agent Games in Socio-Economic Systems: Corruption, Inspection, Coalition Building, Network Growth, Security; Springer Series in Operations Research and Financial Engineering; Springer Nature: Cham, Switzerland, 2019. [Google Scholar]
- Baleanu, D.; Diethelm, K.; Scalas, E.; Trujillo, J.J. Fractional Calculus: Models and Numerical Methods, 2nd ed.; Series on Complexity, Nonlinearity and Chaos; World Scientific Publishing: Singapore, 2017; Volume 5. [Google Scholar]
- Kiryakova, V. Generalized Fractional Calculus and Applications; Pitman Research Notes in Mathematics Series; Longman Scientific: Harlow, UK; John Wiley and Sons: New York, NY, USA, 1994; Volume 301. [Google Scholar]
- Podlubny, I. Fractional Differential Equations, An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications; Mathematics in Science and Engineering; Academic Press, Inc.: San Diego, CA, USA, 1999; Volume 198. [Google Scholar]
- Diethelm, K. The Analysis of Fractional Differential Equations; Lecture Notes in Mathematics; Springer: Berlin/Heidelberg, Germany, 2010. [Google Scholar]
- Kallenberg, O. Foundations of Modern Probability, 2nd ed.; Springer: Berlin/Heidelberg, Germany, 2002. [Google Scholar]
- Kolokoltsov, V.N. Markov Processes, Semigroups and Generators; DeGruyter Studies in Mathematics; DeGruyter: Berlin, Germany, 2011; Volume 38. [Google Scholar]
- Norris, J. Cluster Coagulation. Comm. Math. Phys.
**2000**, 209, 407–435. [Google Scholar] [CrossRef] - Gnedenko, B.V.; Korolev, V.Y. Random Summation: Limit Theorems and Applications; CRC Press: Boca Raton, FL, USA, 1996. [Google Scholar]
- Meerschaert, M.M.; Scheffler, H.-P. Limit Distributions for Sums of Independent Random Vectors; Wiley Series in Probability and Statistics; John Wiley and Son: Hoboken, NJ, USA, 2001. [Google Scholar]
- Kolokoltsov, V.N. Differential Equations on Measures and Functional Spaces; Birkhäuser Advanced Texts Basler Lehrbücher; Birkhäuser: Cham, Switzerland, 2019. [Google Scholar]

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**MDPI and ACS Style**

Kolokoltsov, V.N.; Troeva, M.
A New Approach to Fractional Kinetic Evolutions. *Fractal Fract.* **2022**, *6*, 49.
https://doi.org/10.3390/fractalfract6020049

**AMA Style**

Kolokoltsov VN, Troeva M.
A New Approach to Fractional Kinetic Evolutions. *Fractal and Fractional*. 2022; 6(2):49.
https://doi.org/10.3390/fractalfract6020049

**Chicago/Turabian Style**

Kolokoltsov, Vassili N., and Marianna Troeva.
2022. "A New Approach to Fractional Kinetic Evolutions" *Fractal and Fractional* 6, no. 2: 49.
https://doi.org/10.3390/fractalfract6020049