Date of Completion


Embargo Period



Penalization; Phase transition; Random scaling; Bessel process; Brownian meander; Brownian co-meander; Pseudo-Brownian bridge

Major Advisor

Iddo Ben-Ari

Associate Advisor

Fabrice Baudoin

Associate Advisor

Maria Gordina

Associate Advisor

Alexander Teplyaev

Field of Study



Doctor of Philosophy

Open Access

Campus Access


We study a scaled version of a two-parameter Brownian penalization model introduced by Roynette-Vallois-Yor. The original model penalizes Brownian motion with drift h by a weight process involving the running maximum of the Brownian motion and a parameter v. It was shown there that the resulting penalized process exhibits three distinct phases corresponding to different regions of the (v,h)-plane. In this paper, we investigate the effect of penalizing the Brownian motion concurrently with scaling and identify the limit process. This extends a result of Roynette-Yor to the whole parameter plane and reveals two additional "critical" phases occurring at the boundaries between the parameter regions. One of these novel phases is Brownian motion conditioned to end at its maximum, a process we call the Brownian ascent. We then relate the Brownian ascent to some well-known Brownian path fragments and to a random scaling transformation of Brownian motion recently studied by Rosenbaum-Yor.