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 ...

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.

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 ...

Let B be a Borel subset of R [to the power of] d and let p(t, x, y) be the transition densities of Brownian motion killed on leaving B. Fix x and y in B. If p(t, x, y) is positive for one t, it is positive for every value of t. Some related results are given.

Let X [subscript] t be a Brownian motion and let S(c) be the set of reals r [is greather than or equal to] 0 such that |X ([subscript] r+t) − X [subscript] r| [is less than or equal to] c [square root of] t, 0 [is less than or equal to] t [is less than or equal to] h, for some h = h(r) > 0. It is known that S(c) is empty if ...

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 ...

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 ...

Let B be a Borel subset of R [to the power of] d with finite volume. We give an eigenvalue expansion for the transition densities of Brownian motion killed on exiting B. Let A [subscript] 1 be the time spent by Brownian motion in a closed cone with vertex 0 until time one. We show that lim [subscript] u [approaching] 0 log P ...

For [function] f [maps the set]: [0, 1] [into the set] [Real numbers], we consider L [superscript] f [subscript] t , the local time of spacetime Brownian motion on the curve f. Let S [subscript][sigma] be the class of all functions whose Hölder norm of order [sigma] is less than or equal to 1. We show that the supremum of L ...

We study the solution to the Robin boundary problem for the Laplacian in a Euclidean domain. We present some families of fractal domains where the infimum is greater than 0, and some other families of domains where it is equal to 0. We also give a new result on "trap domains" defined in [BCM], i.e., domains where reflecting ...