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**Transience and recurrence of a Brownian path with limited local time.**
*(English)*
Zbl 1364.60095

Following the work of I. Benjamini and N. Berestycki [Ann. Inst. Henri Poincaré, Probab. Stat. 47, No. 2, 539–558 (2011; Zbl 1216.60028)], the authors study the behavior of a Brownian motion conditioned on the event that its local time at zero stays below a given increasing function \(f\) up to time \(t\). Under some mild assumptions on \(f\), it was previously proved that in the limit \(t\rightarrow \infty\) these conditional probabilities are tight and every weak limit point is transient if moreover \(I(f)=\int_1^{\infty} f(t) t^{-3/2} dt < \infty\).

Under an additional condition on the growth of \(f\), it is proved in the present paper that a limit exists and this limit is explicitly identified. When \(I(f)=\infty\), under another growth condition, the limit exists and corresponds to a recurrent process. In this case, the repulsion envelope is introduced to understand how much slower than \(f\) the local time of the process grows as a result of the conditioning.

The proofs use the fact that the right inverse of the local time of the Brownian motion is a stable subordinator of index \(1/2\). As such it enjoys the so-called “one large jump principle”. Further, it is proved that the probability of a general subordinator to stay above a given curve up to time \(t\) is the solution to a general ordinary differential equation. The explicit and precise computation of the asymptotics of this probability is performed for the inverse local time of the Brownian motion.

Under an additional condition on the growth of \(f\), it is proved in the present paper that a limit exists and this limit is explicitly identified. When \(I(f)=\infty\), under another growth condition, the limit exists and corresponds to a recurrent process. In this case, the repulsion envelope is introduced to understand how much slower than \(f\) the local time of the process grows as a result of the conditioning.

The proofs use the fact that the right inverse of the local time of the Brownian motion is a stable subordinator of index \(1/2\). As such it enjoys the so-called “one large jump principle”. Further, it is proved that the probability of a general subordinator to stay above a given curve up to time \(t\) is the solution to a general ordinary differential equation. The explicit and precise computation of the asymptotics of this probability is performed for the inverse local time of the Brownian motion.

Reviewer: Dominique Lepingle (Orléans)