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.