### Abstract

The goal of this paper is to supplement the large deviation principle of the Freidlin-Wentzell theory on exit problems for diffusion processes with results of classical central limit theorem type. Namely, we describe a class of situations where conditioning on exit through unlikely locations leads to a Gaussian scaling limit for the exit distribution. Our results are based on Doob’s h-transform and new asymptotic convergence gradient estimates for elliptic nonlinear equations that allow one to reduce the problem to the Levinson case. We devote an appendix to a rigorous and general discussion of h-transform.

Original language | English (US) |
---|---|

Pages (from-to) | 6487-6517 |

Number of pages | 31 |

Journal | Transactions of the American Mathematical Society |

Volume | 368 |

Issue number | 9 |

DOIs | |

State | Published - 2016 |

### Fingerprint

### Keywords

- Diffusion
- Doob’s h- transform
- Elliptic PDE
- Exit problems
- Hamilton-Jacobi-Bellman equation
- Region of strong regularity
- Scaling limit
- Small noise
- Viscosity solution

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**Scaling limits for conditional diffusion exit problems and asymptotics for nonlinear elliptic equations.** / Bakhtin, Yuri; Święch, Andrzej.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 368, no. 9, pp. 6487-6517. https://doi.org/10.1090/tran/6574

}

TY - JOUR

T1 - Scaling limits for conditional diffusion exit problems and asymptotics for nonlinear elliptic equations

AU - Bakhtin, Yuri

AU - Święch, Andrzej

PY - 2016

Y1 - 2016

N2 - The goal of this paper is to supplement the large deviation principle of the Freidlin-Wentzell theory on exit problems for diffusion processes with results of classical central limit theorem type. Namely, we describe a class of situations where conditioning on exit through unlikely locations leads to a Gaussian scaling limit for the exit distribution. Our results are based on Doob’s h-transform and new asymptotic convergence gradient estimates for elliptic nonlinear equations that allow one to reduce the problem to the Levinson case. We devote an appendix to a rigorous and general discussion of h-transform.

AB - The goal of this paper is to supplement the large deviation principle of the Freidlin-Wentzell theory on exit problems for diffusion processes with results of classical central limit theorem type. Namely, we describe a class of situations where conditioning on exit through unlikely locations leads to a Gaussian scaling limit for the exit distribution. Our results are based on Doob’s h-transform and new asymptotic convergence gradient estimates for elliptic nonlinear equations that allow one to reduce the problem to the Levinson case. We devote an appendix to a rigorous and general discussion of h-transform.

KW - Diffusion

KW - Doob’s h- transform

KW - Elliptic PDE

KW - Exit problems

KW - Hamilton-Jacobi-Bellman equation

KW - Region of strong regularity

KW - Scaling limit

KW - Small noise

KW - Viscosity solution

UR - http://www.scopus.com/inward/record.url?scp=84958787556&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84958787556&partnerID=8YFLogxK

U2 - 10.1090/tran/6574

DO - 10.1090/tran/6574

M3 - Article

AN - SCOPUS:84958787556

VL - 368

SP - 6487

EP - 6517

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 9

ER -