Carl Mueller

On the support of solutions to the heat equation with noise

Let be 2-parameter white noise. Let satisfy and suppose that is bounded, nonnegative, with compact support and not identically 0. We show that with probability for all . This complements results of Iscoe and Shiga, who show that for y has compact support

A minicourse on stochastic partial differential equations

A Primer on Stochastic Partial Differential Equations.- The Stochastic Wave Equation.- Application of Malliavin Calculus to Stochastic Partial Differential Equations.- Some Tools and Results for Parabolic Stochastic Partial Differential Equations.- Sample Path Properties of Anisotropic Gaussian Random Fields.

Interacting particles, the stochastic Fisher–Kolmogorov–Petrovsky–Piscounov equation, and duality

The stochastic Fisher–Kolmogorov–Petrovsky–Piscunov equation is∂tU(x,t)=D∂xxU+γU(1−U)+eU(1−U)η(x,t)for 0⩽U⩽1 where η(x,t) is a Gaussian white noise process in space and time. Here D, γ and e are parameters and the equation is interpreted as the continuum limit of a spatially discretized set of Ito equations. Solutions of this stochastic partial differential equation have an exact connection to the A⇌A+A reaction–diffusion system at appropriate values of the rate coefficients and particles’ diffusion constant. This relationship is called “duality” by the probabilists; it is not via some hydrodynamic description of the interacting particle system. In this paper we present a complete derivation of the duality relationship and use it to deduce some properties of solutions to the stochastic Fisher–Kolmogorov–Petrovsky–Piscunov equation.

The heat equation with Lévy noise

We prove short-time existence for parabolic equations with Levy noise of the form where is nonnegative Levy noise of index is the power of the Laplacian, , and is a continuous nonnegative function. is a bounded open domain in . A sufficient condition for short time existence is While we cannot prove uniqueness, we show that the solution we construct is minimal among all solutions.

Hypercontractivity for the heat semigroup for ultraspherical polynomials and on the n-sphere

A logarithmic Sobolev inequality with respect to the probability measure Aλ(1 − x2)λ − (12) dx on [−1, 1] is proved for all λ > −12. From this inequality sharp hypercontractive estimates are derived for the heat semigroups for ultraspherical polynomials and on the n-sphere.

The compact support property for solutions to the heat equation with noise

SummaryWe consider all solutions of a martingale problem associated with the stochastic pde

Long time existence for the heat equation with a noise term

u_t = \tfrac{1}{2}u_{xx} + u^\gamma \dot W

Hitting properties of parabolic s.p.d.e.’s with reflection

and show thatu(t,·) has compact support for allt≧0 ifu(0,·) does and if γ<1. This extends a result of T. Shiga who derived this compact support property for γ≦1/2 and complements a result of C. Mueller who proved this property fails if γ≧1.