Probabilistic Cauchy Functional Equations

In this short note, we introduce probabilistic Cauchy functional equations, specifically, functional equations of the following form: $$f(X_1+X_2)f(X_1)+f(X_2),$$ 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 $$ 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.