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 ...

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 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 ...

There exists a planar domain with piecewise smooth boundary and one hole such that the second eigenfunction for the Laplacian with Neumann boundary conditions attains its maximum and minimum inside the domain.

We consider two species of particles performing random walks in a domain in [Real numbers] [superscript] d with reflecting boundary conditions, which annihilate on contact. In addition there is a conservation law so that the total number of particles of each type is preserved: When the two particles of different species ...

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 ...

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 ...

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.

In order to activate a gene in a DNA molecule a specific protein (transcription factor) has to bind to the promoter of the gene. We formulate and partially answer the following question: how much time does a transcription factor, which activates a given gene, need in order to find this gene inside the nucleus of a cell? The ...

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 ...