Forward and backward propagators.

(Propagator and transitional probability) Let be a subset of $\QTR{cal}{R}^{n}$ , $X_{t}$ is a $\QTR{cal}{R}^{n}$ -valued stochastic process with the property ( Markov property ) and is a function .

We define the "transition function" $\pi$ and the two-parameter families of operators .

1. The function $\pi$ acts as follows

(Transition function)

2. We define the family of forward propagators acting on probability measures as follows

(Forward propagator)

for any set

(Measure

carried forward by the process $X_{t}$ ).

3. We define the family of backward propagators acting on Borel measurable functions as follows

(Backward propagator)

4. The transition function $\pi$ is called "homogeneous" if depends only on and . In such case we use the notations MATH

Proposition

(Basic properties of propagator 1)

1. Let $m\left( B\right) =$ Prob then Prob , .

2. Let then , .

Proof

(1). According to the formula ( Total_probability_rule ) with and $C_{\alpha}$ covering all MATH The dot marks the position where the operator acts.

(2) According to the formula ( Chain_rule ) MATH

Proposition

Let , then

Claim

1. To carry the measure from to is to carry it from $\,s$ to and then to carry it from to :

(Kolmogorov-Chapman equation)

2. . (Chain rule (compare with the formula ( Chain_rule )).

Proof

1. By definition of $P^{\ast}$ , MATH By the formula ( Total_probability_rule ) we have hence, MATH We change the order of integration (see the proposition ( Fubini theorem )) MATH and use the definition of $P^{\ast}$ : MATH MATH

2. The technique of the proposition ( Basic properties of propagator 1 )-2 applies with little changes.

Proposition

(Basic properties of propagator 2)

1. ,

2. , .

3. , .

4. , .

Proof

1. MATH

2. MATH

3. MATH

4. MATH

Note that when any of the operators acts on the transitional probability the following rules apply. The backward operator $P_{t_{0}t_{1}}$ acts on the backward space argument and extends the backward time argument $t_{1}$ backward in time. The forward *-operator acts on the forward space set and extends the forward parameter $t_{2}$ forward in time. The precise manner of the extension is natural in every case.

The next observation is on the symmetry of and $P^{\ast}$ .

Notation

We introduce the notation for the natural scalar product of function and measure: MATH

Proposition

The propagators $P_{st}$ and $P_{st}^{\ast}$ are adjoint operators with respect to the scalar product : MATH for any integrable and probability measure .

Proof

By the definitions, MATH

Notation.

Index.

Contents.