M. Cranston

Gradient estimates on manifolds using coupling

We use a coupling method to give gradient estimates for solutions to (12 Δ + Z)u = 0 on a manifold. The size of the gradient depends on a lower bound on the Ricci curvature of the manifold and bounds on the vector field Z.

Maximal height scaling of kinetically growing surfaces.

The scaling properties of the maximal height of a growing self-affine surface with a lateral extent L are considered. In the late-time regime its value measured relative to the evolving average height scales like the roughness: h*(L) approximately L alpha. For large values its distribution obeys logP(h*(L)) approximately (-)A(h*(L)/L(alpha))(a). In the early-time regime where the roughness grows as t(beta), we find h*(L) approximately t(beta)[lnL-(beta/alpha)lnt+C](1/b), where either b = a or b is the corresponding exponent of the velocity distribution. These properties are derived from scaling and extreme-value arguments. They are corroborated by numerical simulations and supported by exact results for surfaces in 1D with the asymptotic behavior of a Brownian path.

Lifetime of conditioned Brownian motion in Lipschitz domains

SummaryIf D∋-ℝd, d≧3, is bounded and has Lipschitz boundary then the expected lifetime of any Brownian h-path process in D is finite.

COUPLING AND HARMONIC FUNCTIONS IN THE CASE OF CONTINUOUS TIME MARKOV PROCESSES

Consider two transient Markov processes (Xvt)t[epsilon]R·, (X[mu]t)t[epsilon]R· with the same transition semigroup and initial distributions v and [mu]. The probability spaces supporting the processes each are also assumed to support an exponentially distributed random variable independent of the process. We show that there exist (randomized) stopping times S for (Xvt), T for (X[mu]t) with common final distribution, L(XvSS

Noncoalescence for the Skorohod equation in a convex domain of ℝ2

SummaryGiven a convex domain of ℝ2, we show that a.s the paths of two solutions of the Skorohod equations driven by the same Brownian motion but starting at different points do not meet at the same time.

Conditional Brownian Motion, Whitney Squares and the Conditional Gauge Theorem

Let (X, P x ) be Brownian motion killed at τ D = inf {t > 0: X t ∉ D}, D a domain in ℝ2 and (X, P z x ) this motion conditioned on X τD = z. For Kato class potentials q we show \( E_x^x\left[ {\exp \left\{ { - \int\limits_0^{{\tau D}} {q\left( {{X_s}} \right)ds} } \right\}} \right] \)is bounded from zero and infinity with little or no assumption on the smoothness of the boundary.

Martin boundaries of sectorial domains

LetD be a domain inR2 whose complement is contained in a pair of rays leaving the origin. That is,D contains two sectors whose base angles sum to 2π. We use balayage to give an integral test that determines if the origin splits into exactly two minimal Martin boundary points, one approached through each sector. This test is related to other integral tests due to Benedicks and Chevallier, the former in the special case of a Denjoy domain. We then generalise our test, replacing the pair of rays by an arbitrary number.

On large deviation regimes for random media models

The focus of this article is on the different behavior of large deviations of random subadditive functionals above the mean versus large deviations below the mean in two random media models. We consider the point-to-point first passage percolation time a(n) on Z(d) and a last passage percolation time Z(n). For these functionals, we have lim(n ->infinity) a(n)/n = v and lim(n ->infinity) Z(n)/n = mu. Typically, the large deviations for such functionals exhibits a strong asymmetry, large deviations above the limiting value are radically different from large deviations below this quantity. We develop robust techniques to quantify and explain the differences.

The radial part of Brownian motion II. Its life and times on the cut locus

SummaryThis paper is a sequel to Kendall (1987), which explained how the Itô formula for the radial part of Brownian motionX on a Riemannian manifold can be extended to hold for all time including those times a whichX visits the cut locus. This extension consists of the subtraction of a correction term, a continuous predictable non-decreasing processL which changes only whenX visits the cut locus. In this paper we derive a representation onL in terms of measures of local time ofX on the cut locus. In analytic terms we compute an expression for the singular part of the Laplacian of the Riemannian distance function. The work uses a relationship of the Riemannian distance function to convexity, first described by Wu (1979) and applied to radial parts of Γ-martingales in Kendall (1993).

Limit laws for sums of products of exponentials ofiid random variables

AbstractWe study limit laws for sums of products of exponentials of nonnegative,iid random variables {Vij}, namely