arXiv subjects

Carl Mueller

Publications and source records attributed to Carl Mueller.

45 records · Page 3Linked to original sources

arXiv2006

On the large scale behavior of super-Brownian motion in three dimensions with a single point source

In a recent work, Fleischmann and Mueller (2004) showed the existence of a super-Brownian motion in R^d, d=2,3, with extra birth at the origin. Their construction made use of an analytical approach based on the fundamental solution of the heat equation with a one point potential worked out by Albeverio et al. (1995). The present note addresses two properties of this measure-valued process in the three-dimensional case, namely the scaling of the process and the large scale behavior of its mean.

math.PR↗

arXiv2006

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.

math.PR↗

arXiv2005

The heat equation with multiplicative stable Lévy noise

We study the heat equation with a random potential term. The potential is a one-sided stable noise, with positive jumps, which does not depend on time. To avoid singularities, we define the equation in terms of a construction similar to the Skorokhod integral or Wick product. We give a criterion for existence based on the dimension of the space variable, and the parameter p of the stable noise. Our arguments are different for p<1 and p>1.

math.PR↗

arXiv2002

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.

math.PR↗

arXiv2002

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.

math.PR↗

arXiv2002

A Singular Parabolic Anderson Model

We give a new example of a measure-valued process without a density, which arises from a stochastic partial differential equation with a multiplicative noise term. This process has some unusual properties. We work with the heat equation with a random potential: u_t=Delta u+kuF. Here k>0 is a small number, and x lies in d-dimensional Euclidean space with d>2. F is a Gaussian noise which is uncorrelated in time, and whose spatial covariance equals |x-y|^(-2). The exponent 2 is critical in the following sense. For exponents less than 2, the equation has function-valued solutions, and for exponents higher than 2, we do not expect solutions to exist. This model is closely related to the parabolic Anderson model; we expect solutions to be small, except for a collection of high peaks. This phenomenon is called intermittency, and is reflected in the singular nature of our process. Solutions exist as singular measures, under suitable assumptions on the initial conditions and for sufficiently small k. We investigate various properties of the solutions, such as dimension of the support and long-time behavior. As opposed to the super-Brownian motion, which satisfies a similar equation, our process does not have compact support, nor does it die out in finite time. We use such tools as scaling, self-duality and moment formulae.

math.PR↗

arXiv2002

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.

math.PR↗

arXiv1999

The Critical Parameter for the Heat Equation with a Noise Term to Blow Up in Finite Time

Consider the stochastic partial differential equation u_t=u_{xx}+u^gamma dot{W}, where x in [0,J], dot{W}=dot{W}(t,x) is 2-parameter white noise, and we assume that the initial function u(0,x) is nonnegative and not identically 0. We impose Dirichlet boundary conditions on u. We say that u blows up in finite time, with positive probability, if there is a finite random time T such that P(\lim_{t->T}sup_x u(t,x)=infty)>0. It was known that if gamma<3/2, then with probability 1, u does not blow up in finite time. It was also known that there is a positive probability of finite time blow-up for gamma sufficiently large. In this paper, we show that if gamma>3/2, then there is a positive probability that u blows up in finite time.

math.PR↗

arXiv1998

Finite time extinction of super-Brownian motions with catalysts

Consider a catalytic super-Brownian motion $X=X^Γ$ with finite variance branching. Here `catalytic' means that branching of the reactant $X$ is only possible in the presence of some catalyst. Our intrinsic example of a catalyst is a stable random measure $Γ$ on $R$ of index $0< gamma <1$. Consequently, here the catalyst is located in a countable dense subset of $R$. Starting with a finite reactant mass $X_0$ supported by a compact set, $X$ is shown to die in finite time. Our probabilistic argument uses the idea of good and bad historical paths of reactant `particles' during time periods $[T_{n},T_{n+1})$. Good paths have a significant collision local time with the catalyst, and extinction can be shown by individual time change according to the collision local time and a comparison with Feller's branching diffusion. On the other hand, the remaining bad paths are shown to have a small expected mass at time $T_{n+1}$ which can be controlled by the hitting probability of point catalysts and the collision local time spent on them.

math.PR↗