Communications

On the volume of the intersection of two Wiener sausages

Erwin Bolthausen

The lectures will give an overview of the attempts to derive the exact $x\rightarrow\infty$ behavior of\[ P\left( \left\vert R^{\left( 1\right) }\cap R^{\left( 2\right) }\right\vert \geq x\right) \] where $R^{\left( 1\right) },R^{\left( 2\right) }$ are the ranges of two independent infinite length random walks in dimension $d\geq5,$ and on the similar question for the Wiener sausage. The conjecture is that \[ P\left( \left\vert R^{\left( 1\right) }\cap R^{\left( 2\right) }\right\vert \geq x\right) =\exp\left[ -c\left( d\right) x^{\left( d-2\right) /d}\right] , \] with a fairly explicitely known constant. Although, there is a fair amount of evidence for the conjecture, there still is no proof for it.

First lecture: The Khanin-Mazel-Sinai-Shlosman result [3] on the intersection of the ranges of two independent random walks. Detailed proof. Introduction into the results of the paper [1]. Outline of the Donser-Varadhan approach to the volume of the Wiener sausage.

Second lecture: Presentation of some of the key arguments in [1]: The large deviation result. The "Swiss cheese picture". The concentration of measure argument. Derivation of the upper bound.

Third lecture: Discussion of the analytical properties of the variational problem, including a proof of the important leakage of mass for $d\geqå5.$ Introduction into the two-path problem analyzed in [2]: The upper bound.

Forth lecture: Continuation of the analysis of the two-path problem. Solution of the analytic variational problem leading to the variational formula for $c\left( d\right) .$ Discussion of the open infinite time-horizon problem.

References

[1] Van Den Berg, M., Bolthausen, E., and Den Hollander, F.: Moderate deviations for the volume of the Wiener sausage. Ann. Math. 153, 355-406 (2001)

[2] Van Den Berg, M., Bolthausen, E., and Den Hollander, F.: On the volume of the intersection of two Wiener sausages. Ann. Math. 159, 741-783 (2004)

[3] Khanin, K.M., Mazel, A.E., Shlosman, S.B., and Sinai, Ya. G.: Loop condensation effects in the behavior of random walks, }in "The Dynkin Festschrift" (M. Freidlin, ed.), Progr. Probab. 34, Birkhäuser, Boston, 1994, pp. 167-184

Large deviations for the intersection local times of Brownian motions and random walks

Xia Chen

The sample path intersection has long been of interest to physicists and mathematicians. It presents a physically relevant model for real world phenomena such as random polymers and quantum field. On the other hand, its analysis has provided mathematical challenges. Thus studying the behavior of intersection local times and related functionals is both physically relevant and often requires a variety of new mathematical ideas. In this lecture we focus on the large deviation problems arising from this area. The self-intersection local time $$Q_n=\#\{(j,k);\hskip.1in 1\le j<k\le n\hskip.05in\hbox{and}\hskip.05in S(j)=S(k)\} $$ of the random walk $S(n)$ on $\mathbb{Z}^d$ measures the self-intersection of the random trajectory $\{S(1),\cdots, S(n)\}$; while the range defined as $$R_n=\#\{S(1),\cdots, S(n)\}.$$

The self-intersection of a single random walk is closed related to the mutual intersection between two independent and identically distributed random walks $S(n)$ and $\widetilde{S}(n)$. The relevant quantities in mutual intersection are mutual intersection local time  $$I_n=\#\{(j,k)\in [1, n];\hskip.05in S(j)=\widetilde{S}(k)\}$$ and the range intersection $$J_n=\#\Big(\{S(1),\cdots, S(n)\}\cap\{\widetilde{S}(1),\cdots,\widetilde{S}(n)\}\Big).$$

The notion of the intersection local times can be extended from random walks to Brownian motions in the "sub-critical dimensions'' (which mean different things for different type of intersections). Compared with their discrete counterparts, the Brownian intersection local times need a more technical setup. On the other hand, there are more analytical tools available  to deal with continuous case. In the spirit of invariance principle, the study in the Brownian cases turns out to be a crucial step toward the later success in the discrete cases, in addition to the importance for its own sake.
 
This lecture presents some progresses recently made on the large deviations for sample path intersections and addresses the following topics:
  1. Definition and renormalization of Brownian intersection local times.
  2. Le Gall's moment identity and method of high moment asymptotics.
  3. LDP by Feynman-Kac formula.
  4. Large deviations for $Q_n$, $R_n$, $I_n$, $J_n$ and the relations among these results.
  5. Unsolved problems.

References

Bolthausen, E. (1999). Large deviations and interacting random walks. Lecture Notes in Math. 1781 1-124.

Chen, X. Random Walk Intersections: Large Deviations and Related Topics. Math. Surv. Mono. 157, Providence 2009.

Le Gall, J-F. (1992). Some properties of planar Brownian motion. École d'Été de Probabilités de Saint-Flour XX. 1990. Lecture Notes in Math. 1527 111-235. Springer, Berlin.

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