The escape probability is a deterministic concept that quantifies some aspects of stochastic dynamics. This issue has been investigated previously for dynamical systems driven by Gaussian Brownian motions. The present work considers escape probabilities for dynamical systems driven by non-Gaussian Lévy motions, especially symmetric α-stable Lévy motions. The escape probabilities are characterized as solutions of the Balayage-Dirichlet problems of certain partial differential-integral equations. Differences between escape probabilities for dynamical systems driven by Gaussian and non-Gaussian noises are highlighted. In certain special cases, analytic results for escape probabilities are given.
|Original language||English (US)|
|Title of host publication||Malliavin Calculus and Stochastic Analysis|
|Subtitle of host publication||A Festschrift in Honor of David Nualart|
|Publisher||Springer New York LLC|
|Number of pages||22|
|State||Published - 2013|
|Name||Springer Proceedings in Mathematics and Statistics|
Bibliographical noteFunding Information:
We have benefited from our previous collaboration with Ting Gao, Xiaofan Li, and Renming Song. We thank Ming Liao, Renming Song, and Zhen–Qing Chen for helpful discussions. This work was done while Huijie Qiao was visiting the Institute for Pure and Applied Mathematics (IPAM), Los Angeles. This work is partially supported by the NSF of China (No. 11001051 and No. 11028102) and the NSF grant DMS-1025422.
- Balayage-Dirichlet problem
- Discontinuous stochastic dynamical systems
- Escape probability
- Lévy processes
- Non-Gaussian noise
- Nonlocal differential equation