Probabilistic Analysis of Distributed Fractional-Order Stochastic Systems Driven by Fractional Brownian Motion: Existence, Uniqueness, and Transportation Inequalities
Abstract
:1. Introduction
2. Preliminaries
2.1. Fractional Brownian Motion and Wick Product
2.2. Malliavin Derivative
- (H1) and on for some and .
- (H2) There exists such that for ,
- (H3) There exists such that for all , ,
3. Existence and Uniqueness
- (1)
- Structural Picard sequence:
- (2)
- Prove that the sequence converges uniformly: Then, for any , we obtain by mathematical induction that
- (3)
- Verify that the limit function is a solution: According to the Borel–Cantelli lemma, converges uniformly on [0,T] to , taking the limit for (9):
4. Transportation Inequalities for (4)
- (1)
- Construct the measure transformation.
- (2)
- Define a new noise process.
- (3)
- Rewrite the original equation.
- (4)
- Construct coupling equation.
- (5)
- (Control Wasserstein distance.
5. Example
- (1)
- When is a probability density function, define
- (2)
- When , define . Obviously, satisfies (H2), that is, for all ,
- (3)
- (When , define . Obviously, satisfies (H3), that is, for all ,
- (4)
- with .
- (5)
- is a fractional Brownian motion with .
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Xia, G.; Xu, L.; Li, Z. Probabilistic Analysis of Distributed Fractional-Order Stochastic Systems Driven by Fractional Brownian Motion: Existence, Uniqueness, and Transportation Inequalities. Symmetry 2025, 17, 650. https://doi.org/10.3390/sym17050650
Xia G, Xu L, Li Z. Probabilistic Analysis of Distributed Fractional-Order Stochastic Systems Driven by Fractional Brownian Motion: Existence, Uniqueness, and Transportation Inequalities. Symmetry. 2025; 17(5):650. https://doi.org/10.3390/sym17050650
Chicago/Turabian StyleXia, Guangyue, Liping Xu, and Zhi Li. 2025. "Probabilistic Analysis of Distributed Fractional-Order Stochastic Systems Driven by Fractional Brownian Motion: Existence, Uniqueness, and Transportation Inequalities" Symmetry 17, no. 5: 650. https://doi.org/10.3390/sym17050650
APA StyleXia, G., Xu, L., & Li, Z. (2025). Probabilistic Analysis of Distributed Fractional-Order Stochastic Systems Driven by Fractional Brownian Motion: Existence, Uniqueness, and Transportation Inequalities. Symmetry, 17(5), 650. https://doi.org/10.3390/sym17050650