Probabilistic Cauchy Functional Equations

In this short note, we introduce probabilistic Cauchy functional equations, specifically, functional equations of the following form: f(X₁ + X₂) d= f(X₁) + f(X₂), where X₁ and X₂ represent two independent identically distributed real-valued random variables governed by a distribution μ having appropriate support on the real line. The symbol d= denotes equality in distribution. When μ 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.