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The virial theorem is established in the framework of resolution-scalerelativity for stochastic dynamics characterized by a diffusion constant D. Itonly relies on a simple time average just like the classical virial theorem,while the quantum mechanical virial theorem involves the expectation values ofthe observables. Nevertheless, by the emergence of a quantum-like potentialterm, the resolution-scale relativity virial theorem also encompasses quantummechanical dynamics under the identification hbar <--> 2mD. This provides anillustration of the scale relativistic approach to the foundation of standardquantum mechanics. Furthermore, it is pointed out that, if the resolution-scalerelativity principle is implemented in macroscopic systems that are complexand/or chaotic, then the application of the classical virial theorem in theanalysis of the dynamics of astrophysical systems neglects the contributionfrom a resolution-scale relativistic quantum-like potential. It is shown thatthis quantum-like potential could account for some fraction of the dark matterhypothesis.

The virial theorem for non-differentiable dynamical paths in resolution-scale relativity