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Exponential, and not Gaussian, decay of probability density functions wasstudied by Laplace in the context of his analysis of errors. Such Laplacepropagators for the diffusive motion of single particles in disordered mediawere recently observed in numerous experimental systems. What will happen tothis universality when an external driving force is applied? Using theubiquitous continuous time random walk with bias, and the Crooks relation inconjunction with large deviations theory, we derive two properties of thepositional probability density function $P_F(x,t)$ that hold for a widespectrum of random walk models: (I) Universal asymmetric exponential decay of$P_F(X,t)$ for large $|X|$, and (II) Existence of a time transformation thatfor large $|X|$ allows to express $P_F(X,t)$ in terms of the propagator of theunbiased process (measured at a shorter time). These findings allow us toestablish how the symmetric exponential-like tails, measured in many unbiasedprocesses, will transform into asymmetric Laplace tails when an external forceis applied.

Exponential Tails and Asymmetry Relations for the Spread of Biased Random Walks