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Random series and discrete path integral methods: The Lévy-Ciesielski implementation
1Department of Chemistry, Brown University, Providence, Rhode Island 02912, USA. Cristian_Predescu@brown.edu
Summary
This study connects discrete and random series path integral methods for numerical analysis. Both approaches achieve similar convergence rates, O(1/n^2), for the Feynman-Kaç formula.
Area of Science:
- Numerical analysis
- Computational mathematics
- Stochastic processes
Background:
- Path integral methods are crucial for solving the Feynman-Kaç formula.
- Discrete and series representations are primary numerical techniques.
- Understanding their relationship is key for computational efficiency.
Purpose of the Study:
- To analyze the relationship between discrete and series path integral methods.
- To interpret discrete methods via Feynman-Kaç formula discretization.
- To connect Lévy-Ciesielski representation methods to discrete path integrals.
Main Methods:
- Direct discretization of the Feynman-Kaç formula.
- Analysis of Lévy-Ciesielski representation (primitive, partial averaging, reweighted).
- Establishing connections between series subsequences and discrete path integrals.
Main Results:
- A direct connection is established between discrete and random series path integral methods.
- Subsequences of Lévy-Ciesielski methods are shown to be discrete path integrals.
- Sharp estimates for convergence rates of series methods are derived.
Conclusions:
- The study bridges discrete and random series path integral approaches.
- Both methods demonstrate comparable convergence rates, O(1/n^2).
- This provides a unified understanding for numerical applications of the Feynman-Kaç formula.