## The strong law of large numbers for a Brownian polymer.(English)Zbl 0873.60014

The main purpose of this paper is to prove the asymptotic relation $$\lim_{t\to \infty}(X(t)/t)=c$$ a.s. for some $$c>0$$, where $$X$$ satisfies (pointwise) the equation $X(t)=W(t)+\int_0^tds\int_0^s f(X(s)-X(u))du.$ Here $$W$$ denotes a standard Wiener process and the function $$f$$ is required to be nonnegative, Lipschitz continuous with compact support, and bounded away from $$0$$ for certain arguments. A proof is given for the case $$\text{supp}(f)=[-1,1]$$ and $$|f|_\infty\leq 1$$. The main argument of the proof is based on the corresponding asymptotic behaviour of a “truncated”/“tail” version of $$X$$, say $$X^T$$, defined pointwise via $X^T(t)=W(t)+\int_0^t ds\int_{(s-T)\vee 0}f(X^T(s)-X^T(u))du.$ For large values of $$T$$, $$T$$ fixed, Durrett’s observation that $$\lim_{t\to\infty}(X^T(t)/t)=c^T$$ a.s. is shown first for some positive constant $$c^T$$ depending on $$T$$. This proof is based on Markov chains and the Harris-recurrent property. In order to validate the main result for $$X$$ from the corresponding result for $$X^T$$, a series of lemmata is provided next. Frequently the L. E. Dubins and D. A. Freedman version of the Borel-Cantelli lemma [in: Proc. 5th Berkeley Symp. Math. Stat. Probab., Univ. Calif. 1965/66, 2, Part 1, 183-214 (1967; Zbl 0201.49502)] is invoked. These auxiliary proofs as well as the remainder of the paper are very technical.

### MSC:

 60F15 Strong limit theorems 60H05 Stochastic integrals

Zbl 0201.49502
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