274 Modeling and Inverse Problems in the Presence of Uncertainty
the driv ing process being a diffusion process, a nd the ca se with the driving
process being a continuous time Markov chain.
A diffusion driving process For the given joint probability density func-
tion ˜ϕ
X,Y
, we can obtain the probability density function of X(t)
p(t, x) =
Z
Ω
y
˜ϕ
X,Y
(t, x, y)dy, (7.125)
where Ω
y
denotes the set of all possible values for y. Equation (7 .125) com-
bined with Equation (7.118) for ˜ϕ
X,Y
(with appropriate boundary and initial
conditions) can be used to describe the population density in a population
ecology system where the growth rates g of the species are affected by a driv-
ing diffusion process. This could be due to the climate variability, which is
often de ...