907 research outputs found

### Hitting properties of a random string

We consider Funaki's model of a random string taking values in R^d. It is
specified by the following stochastic PDE, du = u_{xx} + W, where W=W(x,t) is
two-parameter white noise, also taking values in R^d. We study hitting
properties, double points, and recurrence. The main difficulty is that the
process has the Markov property in time, but not in space. We find: (1) The
string hits points if d<6. (2) For fixed t, there are points x,y such that
u(t,x)=u(t,y) iff d < 4. (3) There exist points t,x,y such that u(t,x)=u(t,y)
iff d < 8. (4) There exist points s,t,x,y such that u(t,x)=u(s,y) iff d < 12.
(5) The string is recurrent iff d < 7

### Solutions of semilinear wave equation via stochastic cascades

We introduce a probabilistic representation for solutions of quasilinear wave
equation with analytic nonlinearities. We use stochastic cascades to prove
existence and uniqueness of the solution

### Super-Brownian motion with extra birth at one point

A super-Brownian motion in two and three dimensions is constructed where
"particles" give birth at a higher rate, if they approach the origin. Via a
log-Laplace approach, the construction is based on Albeverio et al. (1995) who
calculated the fundamental solutions of the heat equation with one-point
potential in dimensions less than four

### Regularity of the density for the stochastic heat equation

We study the smoothness of the density of a semilinear heat equation with
multiplicative spacetime white noise. Using Malliavin calculus, we reduce the
problem to a question of negative moments of solutions of a linear heat
equation with multiplicative white noise. Then we settle this question by
proving that solutions to the linear equation have negative moments of all
orders

### Some non-linear s.p.d.e.'s that are second order in time

We extend Walsh's theory of martingale measures in order to deal with
hyperbolic stochastic partial differential equations that are second order in
time, such as the wave equation and the beam equation, and driven by spatially
homogeneous Gaussian noise. For such equations, the fundamental solution can be
a distribution in the sense of Schwartz, which appears as an integrand in the
reformulation of the s.p.d.e. as a stochastic integral equation. Our approach
provides an alternative to the Hilbert space integrals of Hilbert-Schmidt
operators. We give several examples, including the beam equation and the wave
equation, with nonlinear multiplicative noise terms

### Multiple points of the Brownian sheet in critical dimensions

It is well known that an $N$-parameter $d$-dimensional Brownian sheet has no
$k$-multiple points when $(k-1)d>2kN$, and does have such points when
$(k-1)d<2kN$. We complete the study of the existence of $k$-multiple points by
showing that in the critical cases where $(k-1)d=2kN$, there are a.s. no
$k$-multiple points.Comment: Published at http://dx.doi.org/10.1214/14-AOP912 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org

### Some properties for superprocess under a stochastic flow

For a superprocess under a stochastic flow, we prove that it has a density
with respect to the Lebesgue measure for d=1 and is singular for d>1. For d=1,
a stochastic partial differential equation is derived for the density. The
regularity of the solution is then proved by using Krylov's L_p-theory for
linear SPDE. A snake representation for this superprocess is established. As
applications of this representation, we prove the compact support property for
general d and singularity of the process when d>1

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