## Journal of the Mathematical Society of Japan

### Local time penalizations with various clocks for one-dimensional diffusions

#### Abstract

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.

#### Note

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

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

Dates
Revised: 13 August 2017
First available in Project Euclid: 5 November 2018

https://projecteuclid.org/euclid.jmsj/1541408432

Digital Object Identifier
doi:10.2969/jmsj/75947594

Mathematical Reviews number (MathSciNet)
MR3909919

Zentralblatt MATH identifier
07056562

#### Citation

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. https://projecteuclid.org/euclid.jmsj/1541408432

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