The first-passage time, or hitting time, of a stochastic process 
to a set 
is the random variable
 |
(1)
|
For a discrete-time process, the infimum is taken over the nonnegative integers.
Under the usual adaptedness assumptions, 
is a stopping time.
The distribution function 
and survival function 
of 
satisfy
 |  |  |
(2)
|
 |  |  |
(3)
|
If the distribution has a probability density function 
, then
 |
(4)
|
and the mean first-passage time can be written as
![E[tau_A]=int_0^inftyS(t)dt,](/images/equations/First-PassageTime/NumberedEquation3.svg) |
(5)
|
with the value possibly infinite.
For example, let 
,
where 
is a standard Brownian motion, and let 
. The density of the first-passage time to 
is
 |
(6)
|
This formula holds for 
. Although the process reaches 
with probability one, this
distribution has infinite expectation
value. Introducing stochastic resetting
can make the corresponding mean first-passage time finite.
