It is argued that the Markovian Brownian motion of a particle immersed in an environment does feature continuous but not differentiable paths. As far as non-Markovian effects are concerned, do such Brownian trajectories hold non-differentiable?
If they are completely random, they couldn't be differenciable, isn't it? Then the differenciable (or not) characteristics of the brownian movements should be related to the quantum indeterminacy. But I'm not sure that brownian motion is an emergent fenomena of quantum: it involves the macroscopic world (molecules, liquid, solids), so perhaps we must focus in other direction to determine its motion caractheristics. Once the brownian impulse has started in the quantum scale, it begins to be a classical movement in the macroscopic scale. Perhaps the only real quantum factor in the brownian motion is the time of starting impulses, not the motion itself.
Discrete stochastic processes are of course mathematical idealizations of real physical fluctuation phenomena, but within this framework stochastic trajectories (whether Markovian or non-Markovian) are usually not differentiable. This is somewhat strange question to ask - do you have a specific application in mind?
As far as non-Markovian effects on Brownian motion are concerned, I have shown that the Brownian paths are indeed differentiable functions. This upshot has been recently published in the Physica A. The abstract of this paper is the following:
"Non-Markovian effects on the Brownian movement of a free particle in the presence as well as in the absence of inertial force are investigated under the framework of generalized Fokker–Planck equations (Rayleigh and Smoluchowski equations). More specifically, it is predicted that non-Markovian features can diminish the values of both the root mean square displacement and the root mean square momentum, thereby assuring the mathematical property of analyticity of such physically observable quantities for all times t ≥ 0. Accordingly, the physical concept of non-Markovian Brownian trajectory turns out to be mathematically well defined by differentiable functions for all t ≥ 0. Another consequence of the non-Markovicity property is that the Langevin stochastic equations underlying the Fokker–Planck equations should be interpreted as genuine differential equations and not as integral equations according to a determined interpretation rule (Doob’s rule, for instance)."
See please my paper:
http://www.researchgate.net/publication/220004561_Non-Markovian_effects_on_the_Brownian_motion_of_a_free_particle_Physica_A_390_(2011)_30953107
Article Non-Markovian effects on the Brownian motion of a free parti...
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