In one way or another, the extension of the standard Brownian motion process {B¡: t e [0,oo)} to a (Gaussian) random field {Bt: t € R+} involves a proof of the positive semi-definiteness of the kernel used to generalize p(s, 1) = cov(Bs,B¡) = s A t to multidimensional time. Simple direct analytical proofs are provided here for the cases of (i) the Levy multiparameter Brownian motion, (ii) the C...

We will introduce a class of (m-times) integrated Brownian motions. The exact asymptotic expansions for the L2 small ball probabilities will be discussed for members of this class, of which the usual m-times integrated Brownian motion is an example. Another example will be what we call an Euler-integrated Brownian motion. We will also find very sharp estimates for the asymptotics of the eigenva...

We construct a martingale which has the same marginals as the arithmetic average of geometric Brownian motion. This provides a short proof of the recent result due to P. Carr et al [5] that the arithmetic average of geometric Brownian motion is increasing in the convex order. The Brownian sheet plays an essential role in the construction. Our method may also be applied when the Brownian motion ...

in this paper, unsteady two phase simulation of nanofluid flow and heat transfer between moving parallel plates, in presence of the magnetic field is studied. the significant effects of thermophoresis and brownian motion have been contained in the model of nanofluid flow. the three governing equations are solved simultaneously via galerkin method. comparison with other works indicates that this...

The " knotting " properties of Brownian motion are investigated. Because the path of Brownian motion in 3-space intersects itself, the topological definition of a knot does not apply. A modified concept is defined; implication in a knot-tube. It is shown that 3-space Brownian motion is implicated in infinitely many disjoint knot-tubes in every time interval. As a corollary every segment of the ...

Let X be a drifted fractional Brownian motion with Hurst index H > 1/2. We prove that there exists a fractional backward representation of X , i.e. the time reversed process is a drifted fractional Brownian motion, which continuously extends the one obtained in the theory of time reversal of Brownian diffusions when H = 1/2. We then apply our result to stochastic differential equations driven b...

We introduce and study the random non-compact metric space called the Brownian plane, which is obtained as the scaling limit of the uniform infinite planar quadrangulation. Alternatively, the Brownian plane is identified as the Gromov-Hausdorff tangent cone in distribution of the Brownian map at its root vertex, and it also arises as the scaling limit of uniformly distributed (finite) planar qu...