2. The process has independent increments; that is, for arbitrary 
Ene Se a ee Eh emsinernements 
ib Z n 
x(t) - x(t 3), x(t,_j) - x(t,_5),---sx(t,) - x(t,), x(t) 
are independent.* 
3. The distribution of x(t) - x(s) for arbitrary t and s depends 
only on t-s. In this case, the process is said to have stationary 
increments. 
4. The transition probabilities are Gaussian. In the one- 
dimensional case, the transition probability density is 
oe ++ Ne, wiht, O) = E exp - we /2At 
4 7 V2TAt 
5. w(0) = O with probability one, and E[w(t)] = 0 for all t>0O. 
Sample functions of a Wiener process have interesting properties. 
They can be continuous functions but are nowhere differentiable. Their 
paths are of infinite length. Yet it is for just such perturbations 
that (4.2) will be solved. 
If w in (4.2) had bounded variation, the solution could be written 
in terms of the transport matrix (x, t) of the linear system 
y= Any, (4.3) 
The solution of (4.2) would be 
t 
Ed (i) sO CEO) eect | O(t, tT) d w(t) (4.4) 
0 
where the value of x at t = 0 is the random variable c. The expectation 
of c is m and its covariance matrix is [.. 
*Independent random variables are defined on page 7 of Masi.” 
47 
