Correlation of independent random processes
No. Quoting Wikipedia's article Independence (probability theory):


  If $X$ and $Y$ are independent random variables,
  then the expectation operator $\operatorname{E}$ has the property
  
  $$\operatorname{E}[X Y] = \operatorname{E}[X]\operatorname{E}[Y].$$


Consider your $X(t_1)$ and $Y(t_2)$ as $X$ and $Y$ in this answer. If both $\operatorname{E}[X] \ne 0$ and $\operatorname{E}[X] \ne 0,$ then the product  $\operatorname{E}[X]\operatorname{E}[Y]$ will be non-zero.