No abstract

No abstract

We consider Funaki's model of a random string taking values in R d . It is specified by the following stochastic PDE,whereẆ =Ẇ (x, t) is two-parameter white noise, also taking values in R d . We find the dimensions in which the string hits points, and in which it has double points of various types. We also study the question of recurrence and transience.

We consider reaction-diffusion equations of KPP type in one spatial dimension, perturbed by a Fisher-Wright white noise, under the assumption of uniqueness in distribution. Examples include the randomly perturbed Fisher-KPP equationswhereẆ =Ẇ (t, x) is a space-time white noise.

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.