A Directed Continuous Time Random Walk Model with Jump Length Depending on Waiting Time

In continuum one-dimensional space, a coupled directed continuous time random walk model is proposed, where the random walker jumps toward one direction and the waiting time between jumps affects the subsequent jump. In the proposed model, the Laplace-Laplace transform of the probability density function $P(x,t)$ of finding the walker at position $x$ at time $t$ is completely determined by the Laplace transform of the probability density function $\phi (t)$ of the waiting time. In terms of the probability density function of the waiting time in the Laplace domain, the limit distribution of the random process and the corresponding evolving equations are derived.