We show that to every stochastic process X one can associate a unique collection
(
Φ
,
Φ
+
,
T
(
t
)
,
E
(
U
)
,
p
∗
)
(\Phi ,{\Phi _ + },T(t),E(U),{p^\ast })
consisting of a linear space
Φ
\Phi
, on which is defined a linear functional
p
∗
{p^ \ast }
, together with a convex subset
Φ
+
{\Phi _ + }
which is invariant under the semigroup of operators
T
(
t
)
T(t)
and the resolution of the identity
E
(
U
)
E(U)
. The joint distributions of X, there being one version for each
ϕ
∈
Φ
+
\phi \in {\Phi _ + }
, are then given by
\[
P
ϕ
(
X
(
t
1
)
∈
U
1
,
⋯
,
X
(
t
1
+
⋯
+
t
n
)
∈
U
n
)
=
p
∗
E
(
U
n
)
T
(
t
n
)
⋯
E
(
U
1
)
T
(
t
1
)
ϕ
.
{P_\phi }(X({t_1}) \in {U_1}, \cdots ,X({t_1} + \cdots + {t_n}) \in {U_n}) = {p^ \ast }E({U_n})T({t_n}) \cdots E({U_1})T({t_1})\phi .
\]
To each
ϕ
\phi
contained in the extreme points
Φ
+
+
{\Phi _{ + + }}
of
Φ
+
{\Phi _ + }
and each time t we find a probability measure
P
t
∗
(
ϕ
,
⋅
)
P_t^ \ast (\phi , \cdot )
on
Φ
+
+
{\Phi _{ + + }}
such that
T
(
t
)
ϕ
=
∫
Φ
+
+
ψ
P
t
∗
(
ϕ
,
d
ψ
)
T(t)\phi = {\smallint _{{\Phi _{ + + }}}}\psi P_t^ \ast (\phi ,d\psi )
.
P
t
∗
P_t^ \ast
is the transition probability function of a temporally homogeneous Markov process
X
∗
{X^ \ast }
on
Φ
+
+
{\Phi _{ + + }}
for which there exists a function f such that
X
=
f
(
X
∗
)
X = f({X^ \ast })
. We show that in a certain sense
X
∗
{X^ \ast }
is the smallest of all Markov processes Y for which there exists a function g with
X
=
g
(
Y
)
X = g(Y)
. We then apply these results to a class of stochastic process in which future and past are independent given the present and the conditional distribution, on the past, of a collection of random variables in the future.