The parabolic Harnack inequality for the time dependent Ginzburg-Landau type SPDE and its application

Hiroshi Kawabi

doi:10.1007/s11118-004-6456-4

The parabolic Harnack inequality for the time dependent Ginzburg-Landau type SPDE and its application

Hiroshi Kawabi

Research output: Contribution to journal › Article › peer-review

3 Citations (Scopus)

Abstract

The main purpose of this paper is to establish the parabolic Harnack inequality for the transition semigroup associated with the time dependent Ginzburg-Landau type stochastic partial differential equation (=SPDE, in abbreviation). In view of quantum field theory, this dynamics is called a P(φ)₁-time evolution. We prove the main result by adopting a stochastic approach which is different from Bakry-Emery's Γ₂- method. As an application of our result, we study some estimates on the transition probability for our dynamics. We also discuss the Varadhan type asymptotics.

Original language	English
Pages (from-to)	61-84
Number of pages	24
Journal	Potential Analysis
Volume	22
Issue number	1
DOIs	https://doi.org/10.1007/s11118-004-6456-4
Publication status	Published - 2005 Feb
Externally published	Yes

Keywords

Gradient estimate
Parabolic Harnack inequality
SPDE
Varadhan type small time asymptotics

ASJC Scopus subject areas

Analysis

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10.1007/s11118-004-6456-4

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abstract = "The main purpose of this paper is to establish the parabolic Harnack inequality for the transition semigroup associated with the time dependent Ginzburg-Landau type stochastic partial differential equation (=SPDE, in abbreviation). In view of quantum field theory, this dynamics is called a P(φ)1-time evolution. We prove the main result by adopting a stochastic approach which is different from Bakry-Emery's Γ2- method. As an application of our result, we study some estimates on the transition probability for our dynamics. We also discuss the Varadhan type asymptotics.",

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AB - The main purpose of this paper is to establish the parabolic Harnack inequality for the transition semigroup associated with the time dependent Ginzburg-Landau type stochastic partial differential equation (=SPDE, in abbreviation). In view of quantum field theory, this dynamics is called a P(φ)1-time evolution. We prove the main result by adopting a stochastic approach which is different from Bakry-Emery's Γ2- method. As an application of our result, we study some estimates on the transition probability for our dynamics. We also discuss the Varadhan type asymptotics.

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KW - Parabolic Harnack inequality

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KW - Varadhan type small time asymptotics

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The parabolic Harnack inequality for the time dependent Ginzburg-Landau type SPDE and its application

Abstract

Keywords

ASJC Scopus subject areas

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