Expected Value of a Infinite Process
You are asking about the expected value of the leading term of a sequence. There can be no general answer given the assumptions you make, since for example

$$x_{t+1} = x_{t} + v_{t+1}, \;\;\;E(v_{t+1}) = 0,\;\;\; x_0 \;\; \text{given}$$

is a unit root process, and its expected value is equal to $E(x_{t+1})=x_0,\;\;\; \forall t$, while

$$x_{t+1} = a +x_{t} + v_{t+1}, \;\;\;E(v_{t+1}) = 0,\;\;\; x_0 \;\; \text{given}$$

is also a unit-root process, but $E(x_{t+1}) = x_0 + a\cdot(t+1)$

which is very different compared to the previous case.