Abstract
We show the result that is stated in the title of the paper, which has consequences about decomposition of Brownian loop-soup clusters in two dimensions.
Highlights
Main result of the present paper and strategy of proofWhen x and y are two distinct boundary points of the unit disk U, let us denote byPx,y the natural probability measure on Brownian excursions from x to y in U
It is easy to see that the image of Px,y under time-reversal is Py,x. This leads to the definition of an unoriented excursion which is obtained from an oriented excursion by forgetting its orientation
Suppose that we are given a point process C = ({xi, yi})i∈I of unordered pairs of distinct points on ∂U
Summary
When x and y are two distinct boundary points of the unit disk U, let us denote by. Px,y the natural probability measure on Brownian excursions from x to y in U (that can be for instance defined as the limit when z → x of the law of Brownian motion started from z ∈ U and conditioned to exit U at y, see [5, 4]). Let us first briefly survey some relevant features from earlier papers (for more references, see [12]) in order to state the main consequence that we draw from Proposition 1.1: There exists a natural conformally invariant measure μ on unrooted Brownian loops in the unit disk introduced in [5], and for each positive c, when one samples a Poisson point process of such loops with intensity exactly cμ, one obtains the so-called Brownian loop-soup with intensity c (see again [5]). Möbius transformations – one important feature of that result is the value of the constant; here this is the constant such that when restricted to end-points on the half circle, the outer boundary of the Poisson point process of excursions with this intensity does create a restriction sample of exponent 1/4, see [16]) Combining this with Proposition 1.1 implies immediately the following fact: Corollary 1.3.
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