Schickentanz, Dominic Tobias Raphael (2024)
Brownian Motion with Limited Occupation Times.
Technische Universität Darmstadt
doi: 10.26083/tuprints-00026733
Dissertation, Erstveröffentlichung, Verlagsversion

Kurzbeschreibung (Abstract)

In this thesis, we condition a Brownian motion on spending limited time outside an interval. More precisely, we bound the occupation time outside the interval by a deterministic constant. This is accomplished by conditioning on approximations of the event in question and passing to the weak limit.

We start our analysis with the case of an unbounded interval, where we describe the resulting process in terms of a path decomposition. In particular, we exactly determine the distributions of the total occupation time outside and the last entrance time into the interval. Additionally, we provide limiting theorems for the mentioned quantities as the starting point tends to ∞ or -∞, respectively.

If the interval is bounded, we focus on starting points inside. In this setting, we prove that the resulting process does not leave the interval at all, but satisfies the very same SDE as a Brownian motion which is conditioned to stay inside the interval. This result is a very rare extreme example of entropic repulsion. On our way, we explicitly determine the exact asymptotic behavior of the probability that a Brownian motion spends limited time outside the interval during the first T time units, as T→∞.

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