Journal of the Mathematical Society of Japan

Local time penalizations with various clocks for one-dimensional diffusions

Christophe PROFETA, Kouji YANO, and Yuko YANO

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We study some limit theorems for the law of a generalized one-dimensional diffusion weighted and normalized by a non-negative function of the local time evaluated at a parametrized family of random times (which we will call a clock). As the clock tends to infinity, we show that the initial process converges towards a new penalized process, which generally depends on the chosen clock. However, unlike with deterministic clocks, no specific assumptions are needed on the resolvent of the diffusion. We then give a path interpretation of these penalized processes via some universal $\sigma$-finite measures.


The first and the second authors were supported by JSPS-MAEDI Sakura program. The second author was supported by MEXT KAKENHI grant 26800058, 24540390 and 15H03624. The third author was supported by MEXT KAKENHI grant 23740073.

Article information

J. Math. Soc. Japan, Volume 71, Number 1 (2019), 203-233.

Received: 30 August 2016
Revised: 13 August 2017
First available in Project Euclid: 5 November 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 60J60: Diffusion processes [See also 58J65] 60G44: Martingales with continuous parameter

penalization one-dimensional diffusion local time excursion measure


PROFETA, Christophe; YANO, Kouji; YANO, Yuko. Local time penalizations with various clocks for one-dimensional diffusions. J. Math. Soc. Japan 71 (2019), no. 1, 203--233. doi:10.2969/jmsj/75947594.

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