A probabilistic approach to a differential-difference equation arising in analytic number theory

Abstract:

The differential-difference equation \[ \begin {array}{*{20}{c}} {tv’(t) + v(t - 1) = 0,} \hfill & {t > 1,} \hfill \\ {v(t) = 0,} \hfill & {t < 0,} \hfill \\ {v(t) = {\operatorname {constant}},} \hfill & {0 \leqq t \leqq 1,} \hfill \\ \end {array} \] can be solved by the Monte-Carlo method, for the initial condition $v(t) = {e^{ - \gamma }},0 \leqq t \leqq 1$, where the $v(t)$ represent the probability density of a random variable: \[ t = \lim \limits _{n \to \infty } \sum \limits _{i = 1}^n {\prod \limits _{j = 1}^i {{x_j},} } \] where the ${x_j}$ are independent and uniformly distributed on (0, 1).