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.


60F15 Strong limit theorems
60H05 Stochastic integrals


Zbl 0201.49502
Full Text: DOI