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Now showing items 1-10 of 84

#### On minimal parabolic functions and time-homogenous parabolic h-transforms

(American Mathematical Society, 1999-03-29)

Does a minimal harmonic function h remain minimal when it is viewed as a parabolic function? The question is answered for a class of long thin
semi-infinite tubes D [is an element of the subset of real numbers to the power of] d of variable width and minimal harmonic functions h corresponding to the boundary point of D "at ...

#### On non-increase of Brownian motion

(Institute of Mathematical Statistics, 1990-07)

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

#### Stochastic bifurcation models

(Institute of Mathematical Statistics, 1999-01)

We study an ordinary differential equation controlled by a stochastic process. We present results on existence and uniqueness of solutions, on associated local times (Trotter and Ray-Knight theorems), and on time and direction of bifurcation. A relationship with Lipschitz approximations to Brownian paths is also discussed.

#### An asymptotically 4-stable process

(CRC Press, 1995)

An asymptotically 4-stable process is constructed. The model identifies the 4-stable process with a sequence of processes converging in a very weak sense. It is proved that the 4-th variation of the process is a linear function of time and its quadratic variation may be identified with a Brownian motion.

#### Variation of iterated Brownian motion

(American Mathematical Society, 1994)

In this paper, we study higher order variations of iterated Brownian motion (IBM) with view towards possible applications to the construction of the stochastic integral with respect to IBM. We prove that the 4-th variation of IBM is a deterministic linear function. This clearly means that the quadratic variation is infinite ...

#### 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 counterexample to the "hot spots" conjecture

(Princeton University and Institute for Advanced Study, 1999-01)

We construct a counterexample to the "hot spots" conjecture; there exists a bounded connected
planar domain (with two holes) such that the second eigenvalue of the Laplacian in that domain with
Neumann boundary conditions is simple and such that the corresponding eigenfunction attains its strict
maximum at an interior point ...

#### The level sets of iterated Brownian motion

(Springer-Verlag, 1995)

We show that the Hausdorff dimension of every level set of iterated Brownian motion is equal to 3/4.

#### Sets avoided by Brownian motion

(Institute of Mathematical Statistics, 1998-04)

A fixed two-dimensional projection of a three-dimensional Brownian motion is almost surely neighborhood recurrent; is this simultaneously true of all the two-dimensional projections with probability one? Equivalently: three-dimensional Brownian motion hits any infinite cylinder with probability one; does it hit all cylinders? ...

#### A boundary Harnack principle in twisted Hölder domains

(Annals of Mathematics, 1991-09)

The boundary Harnack principle for the ratio of positive harmonic functions is shown to hold in twisted Hölder domains of order [alpha] for [alpha is an element of the set](1/2, 1]. For each [alpha is an element of the set] (0, 1/2), there exists a twisted Hölder domain of order [alpha] for which the boundary Harnack principle ...