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Variational principles for the exit time of non-symmetric diffusions

Lu-Jing Huang, Kyung-Youn Kim, Yong-Hua Mao

Abstract

In this paper we develop some new variational principles for the exit time of non-symmetric diffusions from a domain. As applications, we give some comparison theorems and monotonicity law between different diffusions.

Variational principles for the exit time of non-symmetric diffusions

Abstract

In this paper we develop some new variational principles for the exit time of non-symmetric diffusions from a domain. As applications, we give some comparison theorems and monotonicity law between different diffusions.

Paper Structure

This paper contains 8 sections, 10 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Applications
  3. Comparison theorems
  4. Monotonicity law
  5. Proof of Theorems \ref{['vf_ht_unb']} and \ref{['expm1']}
  6. The exit time on smooth bounded domains
  7. The exit time on arbitrary domains
  8. Ergodic diffusions

Key Result

Theorem 1.1

Let $D\subset \mathbb{R}^d$ be a domain. Assume that $L$ is the operator defined in L satisfying Assumption A on any subdomain $D'\Subset D$ and $\lambda_0( D)<0$. Then for any $\beta>0$, and In particular, if $L$ is self-adjoint with respect to the Lebesgue measure, i.e. $b=0$, then vf_htunb and vf-meanunb can be reduced to

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • Lemma 3.1
  • ...and 6 more