In this paper, random and stochastic processes are defined on fractal curves. Fractal calculus is used to define the cumulative distribution function, probability density function, moments, variance, and correlation function of stochastic processes on fractal curves. A new framework, which is a generalization of mean square calculus, is formulated. The sequence of random variables on the fractal curve, fractal mean square continuity, mean square F α -derivative, and fractal mean square integral are discussed. The mean square solution of a fractal stochastic equation is derived and plotted to illustrate the details.

In this paper, we develop fractal calculus by defining improper fractal integrals and their convergence and divergence conditions with related tests and by providing examples. Using fractal calculus that provides a new mathematical model, we investigate the effect of fractal time on the evolution of the physical system, for example, electrical circuits. In these physical models, we change the dimension of the fractal time; as a result, the order of the fractal derivative changes; therefore, the corresponding solutions also change. We obtain several analytical solutions that are non-differentiable in the sense of ordinary calculus by means of the local fractal Laplace transformation. In addition, we perform a comparative analysis by solving the governing fractal equations in the electrical circuits and using their smooth solutions, and we also show that when $\alpha =1,$ we get the same results as in the standard version.

In this paper, random and stochastic processes are defined on fractal curves. Fractal calculus is used to define cumulative distribution function, probability density function, moments, variance and correlation function of stochastic process on fractal curve. A new framework which is a generalization of mean square calculus is formulated. Sequence of random variable on fractal curve, fractal mean square continuity, mean square $F^α$-derivative, and fractal mean square integral. The mean square solution of a fractal stochastic equation is derived and plotted in order to show the details.