The classical Ornstein-Uhlenbeck diffusion neuronal model is generalized by inclusion of a time-dependent input whose strength exponentially decreases in time. The behavior of the membrane potential is consequently seen to be modeled by a process whose mean and covariance classify, it as Gaussian-Markov. The effect of the input on the neuron's firing characteristics is investigated by comparing...

We consider the nonlinear filtering model with Ornstein-Uhlenbeck process as noise and obtain an analogue of the Bayes’ formula for the filter. For this we need to consider a modified model, where the instaneteneous effect h(Xt) of the signal in the usual model is replaced by ξ t = α ∫ t (t− 1 α )∨0 h(Xu) du, (where α is a large parameter). This means that there is a lingering effect of the sig...

This paper uses linear programming to numerically evaluate the Laplace transform of the exit time distribution and the resolvent of the moments of various Markov processes in bounded regions. The linear programming formulation is developed from a martingale characterization of the processes and the use of occupation measures. The LP approach naturally provides both upper and lower bounds on the...

The Ornstein-Uhlenbeck semigroup combines Gaussian diffusion with the flow of a linear vector field. In infinite dimensional settings there can be non-Gaussian invariant measures. This gives a context for one version of the renormalization group. The adjoint of the OrnsteinUhlenbeck semigroup with respect to an invariant measure need not be an Ornstein-Uhlenbeck semigroup. This adjoint is the a...

Using positive semidefinite processes of Ornstein-Uhlenbeck type a multivariate Ornstein-Uhlenbeck (OU) type stochastic volatility model is introduced. We derive many important statistical and probabilistic properties, e.g. the complete second order structure and a state-space representation. Noteworthy, many of our results are shown to be valid for the more general class of multivariate stocha...

Multivariate Lévy-driven mixed moving average (MMA) processes of the type Xt = ∫ ∫ f(A, t − s)Λ(dA, ds) cover a wide range of well known and extensively used processes such as Ornstein-Uhlenbeck processes, superpositions of Ornstein-Uhlenbeck (supOU) processes, (fractionally integrated) CARMA processes and increments of fractional Lévy processes. In this paper, we introduce multivariate MMA pro...

We propose a generalization of the Ornstein-Uhlenbeck process in 1+1 dimensions which is the product of a temporal Ornstein-Uhlenbeck process with a spatial one and has exponentially decaying autocorrelation. The generalized Langevin equation of the process, the corresponding Fokker-Planck equation, and a discrete integral algorithm for numerical simulation are given. The process is an alternat...

We prove that the self-avoiding walk has Ornstein-Zernike decay and some related properties for all noncritical temperatures at which the model is defined.