Roger Tribe

Stochastic p.d.e.'s arising from the long range contact and long range voter processes

SummaryA long range contact process and a long range voter process are scaled so that the distance between sites decreases and the number of neighbors of each site increases. The approximate densities of occupied sites, under suitable tine scaling, converge to continuous space time densities which solve stochastic p.d.e.s. For the contact process the limiting equation is the Kolmogorov-Petrovskii-Piscuinov equation driven by branching white noise. For the voter process the limiting equation is the heat equation driven by Fisher-Wright white noise.

A Feynman-Kac-type formula for the deterministic and stochastic wave equations and other p.d.e.’s

We establish a probabilistic representation for a wide class of linear deterministic p.d.e.s with potential term, including the wave equation in spatial dimensions 1 to 3. Our representation applies to the heat equation, where it is related to the classical Feynman-Kac formula, as well as to the telegraph and beam equations. If the potential is a (random) spatially homogeneous Gaussian noise, then this formula leads to an expression for the moments of the solution.

A phase transition for a stochastic PDE related to the contact process

SummaryWe consider the one-dimensional heat equation, with a semilinear term and with a nonlinear white noise term. R. Durrett conjectured that this equation arises as a weak limit of the contact process with longrange interactions. We show that our equation possesses a phase transition. To be more precise, we assume that the initial function is nonnegative with bounded total mass. If a certain parameter in the equation is small enough, then the solution dies out to 0 in finite time, with probability 1. If this parameter is large enough, then the solution has a positive probability of never dying out to 0. This result answers a question of Durett.

Multi-Scaling of the n -Point Density Function for Coalescing Brownian Motions

This paper gives a derivation for the large time asymptotics of the n-point density function of a system of coalescing Brownian motions on R.

Parameter Estimates and Exact Variations for Stochastic Heat Equations Driven by Space-Time White Noise

Abstract In this article we calculate the exact quadratic variation in space and quartic variation in time for the solutions to a one dimensional stochastic heat equation driven by a multiplicative space-time white noise. We use the knowledge of exact variations to estimate the drift parameter appearing in the equation.

On the Large Time Asymptotics¶of Decaying Burgers Turbulence

Abstract: The decay of Burgers turbulence with compactly supported Gaussian “white noise” initial conditions is studied in the limit of vanishing viscosity and large time. Probability distribution functions and moments for both velocities and velocity differences are computed exactly, together with the “time-like” structure functions Tn(t,τ)≡< (u(t+τ) -u(t))n>.The analysis of the answers reveals both well known features of Burgers turbulence, such as the presence of dissipative anomaly, the extreme anomalous scaling of the velocity structure functions and self similarity of the statistics of the velocity field, and new features such as the extreme anomalous scaling of the “time-like” structure functions and the non-existence of a global inertial scale due to multiscaling of the Burgers velocity field.We also observe that all the results can be recovered using the one point probability distribution function of the shock strength and discuss the implications of this fact for Burgers turbulence in general.

Large time behavior of interface solutions to the heat equation with Fisher-Wright white noise

SummaryThe one-dimensional heat equation driven by Fisher-Wright white noise is studied. From initial conditions with compact support, solutions retain this compact support and die out in finite time. There exist interface solutions which change from the value 1 to the value 0 in a finite region. The motion of the interface location is shown to approach that of a Brownian motion under rescaling. Solutions with a finite number of interfaces are approximated by a system of annihilating Brownian motions.

Pfaffian Formulae for One Dimensional Coalescing and Annihilating Systems

The paper considers instantly coalescing, or instantly annihilating, systems of one-dimensional Brownian particles on the real line. Under maximal entrance laws, the distribution of the particles at a fixed time is shown to be Pfaffian point processes closely related to the Pfaffian point process describing one dimensional distribution of real eigenvalues in the real Ginibre ensemble of random matrices. As an application, an exact large time asymptotic for the

A travelling wave solution to the kolmogorov equation with noise

n

What is the probability that a large random matrix has no real eigenvalues

-point density function for coalescing particles is derived.