  In this short note, we introduce probabilistic Cauchy functional equations,
specifically, functional equations of the following form: $$
  f(X_1 + X_2) \stackrel{d}{=} f(X_1) + f(X_2),
  $$ where $X_1$ and $X_2$ represent two independent identically distributed
real-valued random variables governed by a distribution $\mu$ having
appropriate support on the real line. The symbol $\stackrel{d}{=}$ denotes
equality in distribution. When $\mu$ follows an exponential distribution, we
provide sufficient (regularity) conditions on the function $f$ to ensure that
the unique measurable solution to the above equation is solely linear.
Furthermore, we present some partial results in the general case, establishing
a connection to integrated Cauchy functional equations.
