The role of noise in solutions to SDE

Uvarova L.A., Merzlyakov V.D.

MSTU "STANKIN", Vadkovskiy per., 3a, Moscow, 127055, Russian Federation, +7 (499) 973-30-76

There are many application areas where it is important that the trajectories of the strong numerical approximations be close to the strong solution of an SDE. These direct simulations of the trajectories can provide considerable insight into the qualitative behavior and dynamics of the stochastic differential equation.

In this paper we use the phase-locked loop model, Duffing equation, Fitzugh-Nagumo model and the Ermak-McCammon Brownian dynamics algorithm. These equations allow us to evaluate the role of noise in different mathematical models and application areas.

The solutions to these equations were tested with different instances of noise used in each case. Then, for each solution were found the equilibrium states, which allowed for an easier evaluation of the effect the change of noise has on the solution.

The results can also be used in evaluation of phase transitions under in reaction to outside forces.

One of the results of this paper is the confirmation of the hypothesis of the effect of noise on the processes of genetic regulation. The results allow that even the smallest changes and fluctuations in the thermal forces can result in irreversible changes to the forming DNA chain which, in turn, might cause changes or damage to the working process of the organism's systems.

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