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.

MSC:

 60J55 Local time and additive functionals 60J65 Brownian motion 60G17 Sample path properties

Zbl 1216.60028
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