article published Martin Kolb author 48880 Mladen Savov author 96 department In this note we investigate the behaviour of Brownian motion conditioned on a growth constraint of its local time which has been previously investigated by Berestycki and Benjamini. For a class of non-decreasing positive functions f(t);t>0, we consider the Wiener measure under the condition that the Brownian local time is dominated by the function f up to time T. In the case where f(t)/t3/2 is integrable we describe the limiting process as T goes to infinity. Moreover, we prove two conjectures in [BB10] in the case for a class of functions f, for which f(t)/t3/2 just fails to be integrable. Our methodology is more general as it relies on the study of the asymptotic of the probability of subordinators to stay above a given curve. Immediately or with adaptations one can study questions like the Brownian motioned conditioned on a growth constraint of its local time at the maximum or more generally a Levy process conditioned on a growth constraint of its local time at the maximum or at zero. We discuss briefly the former. Institute of Mathematical Statistics2016 eng http://dx.doi.org/10.1214/15-AOP1069 446 M. Kolb, M. Savov, The Annals of Probability 44 (2016). Kolb, Martin, and Mladen Savov. “Transience and Recurrence of a Brownian Path with Limited Local Time.” <i>The Annals of Probability</i> 44, no. 6 (2016). <a href="http://dx.doi.org/10.1214/15-AOP1069">http://dx.doi.org/10.1214/15-AOP1069</a>. Kolb, M., &#38; Savov, M. (2016). Transience and recurrence of a Brownian path with limited local time. <i>The Annals of Probability</i>, <i>44</i>(6). <a href="http://dx.doi.org/10.1214/15-AOP1069">http://dx.doi.org/10.1214/15-AOP1069</a> M. Kolb and M. Savov, “Transience and recurrence of a Brownian path with limited local time,” <i>The Annals of Probability</i>, vol. 44, no. 6, 2016, doi: <a href="http://dx.doi.org/10.1214/15-AOP1069">http://dx.doi.org/10.1214/15-AOP1069</a>. Kolb M, Savov M. Transience and recurrence of a Brownian path with limited local time. <i>The Annals of Probability</i>. 2016;44(6). doi:<a href="http://dx.doi.org/10.1214/15-AOP1069">http://dx.doi.org/10.1214/15-AOP1069</a> @article{Kolb_Savov_2016, title={Transience and recurrence of a Brownian path with limited local time}, volume={44}, DOI={<a href="http://dx.doi.org/10.1214/15-AOP1069">http://dx.doi.org/10.1214/15-AOP1069</a>}, number={6}, journal={The Annals of Probability}, publisher={Institute of Mathematical Statistics}, author={Kolb, Martin and Savov, Mladen}, year={2016} } Kolb, Martin, and Mladen Savov. “Transience and Recurrence of a Brownian Path with Limited Local Time.” <i>The Annals of Probability</i>, vol. 44, no. 6, Institute of Mathematical Statistics, 2016, doi:<a href="http://dx.doi.org/10.1214/15-AOP1069">http://dx.doi.org/10.1214/15-AOP1069</a>. 333572022-09-14T04:22:23Z2022-09-14T04:22:26Z