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We consider the random vector $u(t,\underline{x})=(u(t,x_1),\cdots ,u(t,x_d))$, where $t\>0,x_1,\cdots ,x_d$ are distinct points of ${ℝ}^2$ and $u$ denotes the stochastic process solution to a stochastic wave equation driven by a noise white in time and correlated in space. In a recent paper by Millet and Sanz–Solé [10], sufficient conditions are given ensuring existence and smoothness of density for $u(t,\underline{x})$. We study here the positivity of such density. Using techniques developped in [1] (see also [9]) based on Analysis on an abstract Wiener space, we characterize the set of...

We consider the random vector $u(t,\underline{x})=(u(t,x_1),\cdots ,u(t,x_d))$, where are distinct points of ${ℝ}^2$ and denotes the stochastic process solution to a stochastic wave equation driven by a noise white in time and correlated in space. In a recent paper by Millet and Sanz–Solé [10], sufficient conditions are given ensuring existence and smoothness of density for $u(t,\underline{x})$. We study here the positivity of such density. Using techniques developped in [1] (see also [9]) based on Analysis on an abstract Wiener space, we characterize the set of...

We consider a Wright-Fisher diffusion whose current state cannot be observed directly. Instead, at times < < ..., the observations are such that, given the process , the random variables () are independent and the conditional distribution of only depends on . When this conditional distribution has a specific form, we prove that the model ((), 1) is a computable filter in the sense that all distributions involved in filtering, prediction...