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.
"Local time penalizations with various clocks for one-dimensional diffusions." J. Math. Soc. Japan 71 (1) 203 - 233, January, 2019. https://doi.org/10.2969/jmsj/75947594