A pr 1 99 8 Brownian Sheet Images and Bessel – Riesz Capacity
نویسنده
چکیده
We show that the image of a 2–dimensional set under d–dimensional, 2– parameter Brownian sheet can have positive Lebesgue measure if and only if the set in question has positive (d/2)–dimensional Bessel–Riesz capacity. Our methods solve a problem of J.-P. Kahane. Consider two independent d–dimensional Brownian motions X X(t) ; t 0 and Y Y (t) ; t 0. Let E 1 and E 2 denote two disjoint compact subsets of [0, ∞[. where Leb d denotes d–dimensional Lebesgue measure. Define additive Brownian motion Z Z(s, t) ; s, t 0 by, Z(s, t) X(s) + Y (t). Consequently, self–intersection problems for a single Brownian motion naturally translate themselves to problems about the Cartesian product E 1 × E 2 and its image under the (2,d)–random field Z; we follow [1] for notation on (N, d) fields. The goal of this paper is to provide an analytical condition on E 1 × E 2 which is equivalent to (1.1). This solves a problem of J.-P. Kahane. We will actually be concerned with a more intricate problem involving the Brownian sheet. The aforementioned problem is a simple consequence of the methods employed in this paper. To explain our results, we begin with notation and definitions which we will use throughout the paper. Any s ∈ R k is written coordinatewise as s = (s (1) , · · · , s (k)). We will use the sup norm. That is, for all integers k and all x ∈ R k , |x| max 1 i k |x (i) |. Typographically, we shall single out the special case when s ∈ [0, ∞[ 2. In this case, we write s, |s|, etc. for s, |s|, etc.; s will denote 2–dimensional time and we wish to emphasize its temporal nature by emboldening it.
منابع مشابه
Brownian Sheet Images and Bessel–riesz Capacity
We show that the image of a 2–dimensional set under d–dimensional, 2–parameter Brownian sheet can have positive Lebesgue measure if and only if the set in question has positive (d/2)–dimensional Bessel–Riesz capacity. Our methods solve a problem of J.-P. Kahane.
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