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#### 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.

#### Intersection local time for points of infinite multiplicity

(Institute of Mathematical Statistics, 1994-04)

For each a [is an element of the set] (0, 1/2), there exists a random measure [beta] [subscript] a which is supported on the set of points where two-dimensional Brownian motion spends a units of local time. The measure [beta] [subscript] a is carried by a set which has Hausdorff dimension equal to 2−a. A Palm measure ...

#### Local time flow related to skew Brownian motion

(Institute of Mathematical Statistics, 2001-10)

We define a local time flow of skew Brownian motions, i.e., a family of solutions to the stochastic differential equation defining the skew Brownian motion, starting from different points but driven by the same Brownian motion. We prove several results on distributional and path properties of the flow. Our main result is a ...

#### Variably skewed Brownian motion

(Institute of Mathematical Statistics, 2000-03-01)

Given a standard Brownian motion B, we show that the equation X [subscript] t = x [subscript] 0 + B [subscript] t + [beta](L [to the power of X] [subscript] t ); t [is greater than or equal to] 0 ; has a unique strong solution X. Here L [to the power of X] is the symmetric local time of X at 0, and [beta] is a given differentiable ...

#### Iterated law of iterated logarithm

(Institute of Mathematical Statistics, 1995-10)

Suppose [epsilon] [is a member of the set] [0, 1) and let theta [subscipt epsilon] (t) = (1 − [epsilon]) [square root of] (2tln [subscript] 2 t). Let L [to the power of epsilon] [subscript] t denote the amount of local time spent by Brownian motion on the curve [theta subscript epsilon] (s) before time t. If [epsilon] > 0 ...

#### The supremum of Brownian local times on Holder curves

(Elsevier, 2001-11)

For f : [maps the set] [0, 1] [into the set of real numbers] R, we consider L ([to the power of] f [subscript] t), the local time of spacetime
Brownian motion on the curve f. Let S [subscript alpha] be the class of all functions whose Holder norm of order [alpha] is less than or equal to 1. We show that the supremum of L ...

#### Uniqueness for reflecting Brownian motion in lip domains

(Elsevier, 2005-03)

A lip domain is a Lipschitz domain where the Lipschitz constant is strictly less than one. We prove strong existence and pathwise uniqueness for the solution X = {X [subscript] t, t [is less than or equal to] 0} to the Skorokhod equation dX [subscript] t = dW [subscript] t + n(X [subscript] t)dL [subscript] t, in planar lip ...