We prove that, for a two-dimensional Riemannian manifold, the Ricci flow is obtained by a Wiener process.

Existing definitions for optimal orbit determination are not satisfactory. Any useful definition must explicitly address questions relating to sequential processing, linearization, performance function and its extremalization, state estimate structure completeness, use of physics in applicable stochastic processes, and criteria for validation. Such a definition is presented.

Suppose we are given a multi-dimensional Itô process, which can be regarded as a model for an underlying asset price together with related stochastic processes, e.g., volatility. The drift and diffusion terms for this Itô process are permitted to be arbitrary adapted processes. We construct a weak solution to a diffusion-type equation that matches the distribution of the Itô process at each fix...

Let N(T, d, δ) denote the covering number, i.e., for every δ > 0, let N(T, d, δ) denote the minimal number of d-balls of radius δ required to cover T . The supremum of a stochastic process Xt defined on T , supt∈T Xt can be quantified in terms of N(T, d, δ) (see (Talagrand, 2005, Chapter 1) for instance) under various assumptions on the process Xt. Department of Mathematical Sciences, Universit...

A new, wide class of relativistic stochastic processes is introduced. All relativistic processes considered so far in the literature are members of this class. For each process, the stochastic equations of motion are obtained in an arbitrary Lorentz frame. The associated Kolmogorov equation is also derived for the first time, in both (3 + 1)and manifestly covariant forms.

We consider a stochastic process (Xt)t≥0 that grows linearly in time and experiences collapses at times governed by a Poisson process with rate λ. The collapses are modeled by multiplying the process level by a random variable supported on [0, 1). For the hitting time defined as τy = inf{t > 0|Xt = y} we derive power series for the Laplace transform and all moments. We further discuss the asymp...

Many real phenomena may be modelled as locally finite unions of d-dimensional time dependent random closed sets in R, described by birth-and-growth stochastic processes, so that their mean volume and surface densities, as well as the so called mean extended volume and surface densities, may be studied in terms of relevant quantities characterizing the process. We extend here known results in th...

We propose a variety of models of random walk, discrete in space and time, suitable for simulating stable random variables of arbitrary index α (0 < α ≤ 2), in the symmetric case. We show that by properly scaled transition to vanishing space and time steps our random walk models converge to the corresponding continuous Markovian stochastic processes, that we refer to as Lévy-Feller diffusion pr...

We shall deal only with processes which evolve at discrete instances of time. Typically, the time index can be k0, k0 + 1, . . . , N , with k0 and N both finite, or it can be the nonnegative integers Z+ = 0, 1, . . ., or it can be all the integers Z. Let T be such an index set, which can be finite or countably infinite. Assume also that there is an underlying probability space (Ω,S, P ) with re...

Many applications require stochastic processes specified on twoor higherdimensional domains; spatial or spatial-temporal modelling, for example. In these applications it is attractive, for conceptual simplicity and computational tractability, to propose a covariance function that is separable; e.g. the product of a covariance function in space and one in time. This paper presents a representati...