Edwin A. Perkins

Brownian motion on the Sierpinski gasket

SummaryWe construct a “Brownian motion” taking values in the Sierpinski gasket, a fractal subset of ℝ2, and study its properties. This is a diffusion process characterized by local isotropy and homogeneity properties. We show, for example, that the process has a continuous symmetric transition density, pt(x,y), with respect to an appropriate Hausdorff measure and obtain estimates on pt(x,y).

Super-Brownian motion: Path properties and hitting probabilities

SummarySample path properties of super-Brownian motion including a one-sided modulus of continuity and exact Hausdorff measure function of the range and closed support are obtained. Analytic estimates for the probability of hitting balls lead to upper bounds on the Hausdorff measure of the set of k-multiple points and a sufficient condition for a set to be “polar”.

Degenerate stochastic differential equations with Hölder continuous coefficients and super-Markov chains

We consider the operator. Formula math. acting on functions in C 2 b (R d + ). We prove uniqueness of the martingale problem for this degenerate operator under suitable nonnegativity and regularity conditions on γ ij and b i . In contrast to previous work, the b i need only be nonnegative on the boundary rather than strictly positive, at the expense of the γ ij and b i being Holder continuous. Applications to super-Markov chains are given. The proof follows Stroock and Varadhans perturbation argument, but the underlying function space is now a weighted Holder space and each component of the constant coefficient process being perturbed is the square of a Bessel process.

Uniform measure results for the image of subsets under Brownian motion

SummaryThe paper obtains bounds on the Hausdorff and packing measures of the imageX(E) of a Borel setE by a transient strictly stable processXt which a.s. hold for allE and for every measure function

The Cereteli-Davis Solution to the H1-Embedding Problem and an Optimal Embedding in Brownian Motion

h_{\beta ,\gamma } (s) = s^\beta \left| {\log s} \right|^{\gamma ^ \star }

Measure-valued Markov branching processes conditioned on non-extinction

. In some cases examples are constructed to show that the bounds are sharp.

On pathwise uniqueness for stochastic heat equations with non-Lipschitz coefficients

SummaryWe show that if s(t, x) is the local time of a Brownian motion B, and Φ(t)=(2t¦log|logt∥)1/2 then Φ−m({s=0 and x real a.s., where Φ−m(E) is the Hausdorff Φ-measure of E. This solves a problem of Taylor and Wendel who proved the above equality, up to a multiplicative constant, for x=0.

On the Hausdorff dimension of the Brownian slow points

Necessary and sufficient conditions are found on a mean-zero probability, μ, for the existence of a stopping time, T, and a Brownian motion, B, such that BT has law μ and \( B_T^{*} \) is integrable. This result, due to Burgess Davis (the classical analogue was first solved by O. D. Cereteli), leads naturally to a stopping time, T, that stochastically minimizes both sups≤TBs and -infs≤TBs.