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Now showing items 1-7 of 7

#### A representation of local time for Lipschitz surfaces

(Springer-Verlag GmbH, 1990)

Suppose that D [is an element of the set of Real numbers to the power of n], n [is greater than or equal to] 2, is a Lipschitz domain and let N[subscript]t(r) be the number of excursions of Brownian motion inside D with diameter greater than r which started before time t. Then rN[subscript]t(r) converges as r --> 0 to a ...

#### Non-intersection exponents for Brownian paths. Part I: Existence and an invariance principle

(Springer-Verlag GmbH, 1990)

Let X and Y be independent three-dimensional Brownian motions, X(0) = (0; 0; 0), Y (0) = (1; 0; 0) and let p [subscript]r = P(X[0; r] [intersected with] Y [0; r] = [empty set]. Then the "non-
intersection exponent" [from] lim [subscript]r [to infinity] -log p [subscript]r / log r exists and is equal to a similar "non-intersection ...

#### Percolation dimension of fractals

(Academic Press (Elsevier), 1990-01)

"Percolation dimension" is introduced in this note. It characterizes certain fractals and its definition is based on the Hausdorff dimension. It is shown that percolation dimension and "boundary dimension" are in a sense independent from the Hausdorff dimension and, therefore, provide an additional tool for classification of ...

#### On non-increase of Brownian motion

(Institute of Mathematical Statistics, 1990-07)

A new proof of the non-increase of Brownian paths is given.

#### Non-intersection exponents for Brownian paths. Part II: Estimations and applications to a random fractal.

(Institute of Mathematical Statistics, 1990-07)

Let X and Y be independent two-dimensional Brownian motions, X(0) = (0; 0); Y(0) = ([epsilon]; 0), and let p([epsilon]) = P(X[0; 1] [intersected with] Y [0; 1] = [empty set], q([epsilon]) = {Y [0; 1] does not contain a closed loop around 0}. Asymptotic estimates (when [epsilon] --> 0) of p([epsilon]); q([epsilon]),
and some ...

#### A probabilistic proof of the boundary Harnack principle

(Birkhauser Boston, Inc., 1990)

The main purpose of this paper is to give a probabilistic proof of Theorem 1.1, one using elementary properties of Brownian motion. We also obtain the fact that the Martin boundary equals the Euclidean boundary as an easy corollary of Theorem 1.1. The boundary Harnack principle may be viewed as a Harnack inequality for ...

#### Minimal Fine Derivatives and Brownian Excursions

(Nagoya University, 1990-09)

Let f be an analytic function defined on D [is a subset of] [complex numbers] C. If [the derivative of the function f at the point x] has a limit
when [the set] x [into the set] z [is an element of the set partial derivative] D in the minimal fine topology then the limit will be called a minimal fine derivative. Several ...