## Survival asymptotics for Brownian motion in a Poisson field of decaying traps.(English)Zbl 0793.60086

Summary: Let $$W(t)$$ be the Wiener sausage in $$\mathbb{R}^ d$$, that is, the $$a$$- neighborhood for some $$a>0$$ of the path of Brownian motion up to time $$t$$. It is shown that integrals of the type $$\int^ t_ 0\nu(s)d| W(s)|$$, with $$t\to\nu(t)$$ nonincreasing and $$\nu(t)\sim\nu t^{- \gamma}$$, $$t\to\infty$$, have a large deviation behavior similar to that of $$| W(t)|$$ established by M. D. Donsker and S. R. S. Varadhan [Commun. Pure Appl. Math. 28, 525-565 (1975; Zbl 0333.60077)]. Such a result gives information about the survival asymptotics for Brownian motion in a Poisson field of spherical traps of radius $$a$$ when the traps decay independently with lifetime distribution $$\nu(t)/\nu(0)$$. There are two critical phenomena: (i) in $$d\geq 3$$ the exponent of the tail of the survival probability has a cross-over at $$\gamma=2/d$$; (ii) in $$d\geq 1$$ the survival strategy changes at time $$s=[\gamma/(1+\gamma)]t$$, provided $$\gamma<1/2$$, $$d=1$$, respectively, $$\gamma<2/d$$, $$d\geq 2$$.

### MSC:

 60J55 Local time and additive functionals 60G17 Sample path properties 60G57 Random measures

Zbl 0333.60077
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