## Search

Now showing items 1-10 of 13

#### Shy Couplings

(2005)

A pair (X; Y) of Markov processes is called a Markov coupling if X and Y have the same transition probabilities and (X;Y) is a Markov process. We say that a coupling is "shy" if there exists a (random) [Epsilon] > 0 such that dist(X [subscript] t; Y [subscript] t) > [Epsilon] for all t [is greater than or equal to] 0. We ...

#### Traps for Reflected Brownian Motion

(Springer-Verlag GmbH, 2005-08-16)

Consider an open set D [is an element of the set] R [The set of Real Numbers] [superscript]d, d [is greater than or equal to] 2, and a closed ball B [is a proper subset of] D. Let E[superscript]xT[subscript]B denote the expectation of the hitting time of B for reflected Brownian motion in D starting from x [is an element of ...

#### The Heat Equation and Reflected Brownian Motion in Time-Dependent Domains

(Institute of Mathematical Statistics, 2004-01)

The paper is concerned with reflecting Brownian motion (RBM) in domains with deterministic moving boundaries, also known as "non-cylindrical domains," and its connections with partial differential equations. Construction is given for reflecting Brownian motion in C3-smooth time-dependent domains in the n-dimensional Euclidean ...

#### Comparison of potential theoretic properties of rough domains

(2005)

We discuss the relationships between the notion of intrinsic ultracontractivity, parabolic Harnack principle, compactness of the 1-resolvent of the Neumann Laplacian, and non-trap property for Euclidean domains with finite Lebesgue measure. In particular, we give an answer to an open problem raised by Davies and Simon in 1984 ...

#### Synchronous couplings of reflected Brownian motions in smooth domains

(2005)

For every bounded planar domain D with a smooth boundary, we define a "Lyapunov exponent" [Lambda](D) using a fairly explicit formula. We consider two reflected Brownian motions in D, driven by the same Brownian motion (i.e., a "synchronous coupling"). If [Lambda] (D) > 0 then the distance between the two Brownian particles ...

#### Weak convergence of reflecting Brownian motions

(Institute of Mathematical Statistics, 1998-05-23)

We will show that if a sequence of domains D [subscript] k increases to a domain D then the reflected Brownian motions in D [subscript] k's converge to the reflected Brownian motion in D, under mild technical assumptions. Our theorem follows easily from known results and is perhaps known as a "folk law" among the specialists ...

#### 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 ...

#### Stochastic differential equations driven by stable processes for which pathwise uniqueness fails

(North-Holland (Elsevier), 2004-05)

Let Z [subscript] t be a one-dimensional symmetric stable process of order [alpha] with [alpha is an element of the set] (0, 2) and consider the stochastic differential equation
dX [subscript] t = [omega] (X [subscript] t−)dZ [subscript]t.
For [beta] < 1 [divided by alpha] ^ 1, we show there exists a function that is ...

#### Coalescence of synchronous couplings

(Springer-Verlag GmbH, 2002-08)

We consider a pair of reflected Brownian motions in a Lipschitz planar domain starting from different points but driven by the same Brownian motion. First we construct such a pair of processes in a certain weak sense, since it is not known whether a strong solution to the Skorohod equation in Lipschitz domains exists. Then ...

#### 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 ...