Joint probability distributions for a class of non-Markovian processes

We consider joint probability distributions for the class of coupled Langevin equations introduced by Fogedby [H. C. Fogedby, Phys. Rev. E 50, 1657 (1994)]. We generalize well-known results for the single-time probability distributions to the case of $N$-time joint probability distributions. It is shown that these probability distribution functions can be obtained by an integral transform from distributions of a Markovian process. The integral kernel obeys a partial differential equation with fractional time derivatives reflecting the non-Markovian character of the process.

Figure 1

Sketch of the process $t\left(s\right)$ which relates the arclength $s$ to physical time $t$. Since the increment $\tau \left(s\right)$ of Eq. (3) is positive, the curve $t\left(s\right)$ is monotonically increasing, implying the validity of the relation (14).