The paper describes an interpretation of R. Nevanlinna's theory on the distribution of values taken by a meromorphic function in terms of probability theory. A meromorphic function transforms Brownian paths in its domain into Brownian paths on the Riemann sphere which run up to a stopping time T. The expected value of this stopping time gives the Nevanlinna characteristic of the function. A later section relates the effect of killing the Brownian motion in the domain at an exponential rate to the order of meromorphic functions. Finally the methods of Burkholder, Gundy and Silverstein are used to study the distribution of the stopping time T and to compare it with a maximal function.