We characterize the linear and harmonizable fractional stable motions as the self-similar stable processes with stationary increments whose left-equivalent (or right-equivalent) stationary processes are moving averages and harmonizable respectively.

We develop Tanaka-like evolution equations describing the local time Lxt of certain measure valued super processes. For example, if Xt, t [greater-or-equal, slanted] 0, is a planar super Brownian motion and [lambda] 0 then L = < G[lambda], X0> - <G[lambda], X> + [lambda] [integral operator]t0 <G[lambda], Xs>ds + [integral operator]t0<G[lambda], X[Delta] (ds)>,...</g[lambda],></g[lambda],></g[lambda],></g[lambda],>

We define the Burgers superprocess to be the solution of the stochastic partial differential equation where t=0, , and W is space-time white noise. Taking [gamma]=0 gives the classic Burgers equation, an important, non-linear, partial differential equation. Taking [lambda]=0 gives the...

We study the uniform convergence of the quadratic variation of Gaussian processes, taken over large families of curves in the parameter space. A simple application of our main result shows that the quadratic variation of the Brownian sheet along all rays issuing from a point in [0, 1]2 converges...

For a two-dimensional, homogeneous, Gaussian random field X(t) and compact, convex S [subset of] R2 we show that as u -- [infinity] the set Au = {t [set membership, variant] S : X(t) [greater-or-equal, slanted] u} possesses, with probabilityapproaching one, components that are approximately...

Our primary aim is to "build" versions of generalised Gaussian processes from simple, elementary components in such a way that as many as possible of the esoteric properties of these elusive objects become intuitive. For generalised Gaussian processes, or fields, indexed by smooth functions or...

For a d-dimensional random field X(t) define the occupation measure corresponding to the level [alpha] by the Lebesgue measure of that portion of the unit cube over which X(t)[greater-or-equal, slanted][alpha]. Denoting this by M[X, [alpha]], it is shown that for sample continuous Gaussian...