Nonlinear effects in white-noise driven spatial diffusion: General analytical results and probabilities of exceeding threshold

We consider a general nonlinear diffusion, typified by those deriving from Fitzhugh-Nagumo or Hindmarsh-Rose models of nerve-cell dynamics, perturbed also by 2-parameter white noise. In order to investigate the effects of the nonlinearity, we find for general boundary conditions the mean to order ?² and the four-point covariance to order ?³. The derivations involve multiple stochastic integrals in the plane. The mean and variance of the state variable are thus obtained and may be used to estimate the probabilities that a threshold value is exceeded as a function of space and time. A numerical example is given for a space-time white-noise driven diffusion with a cubic nonlinearity. From the asymptotic form of the covariance the spectral density of the process can also be obtained.