📂Stochastic Differential Equations

Definition of Dynamic Noise

Definition ¹

Consider a dynamical system, in particular a system represented by differential equations, given as a deterministic system $\dot{\mathbf{x}} = \mathbf{f} \left( \mathbf{x} \right)$. When a random vector $\mathbf{w}(t)$ is added to the right-hand side, this system becomes a non-deterministic stochastic process, and this $\mathbf{w}(t)$ is called dynamical noise.

Explanation

$$ \begin{align*} {{dx} \over {dt}} =& - \sigma x + \sigma y + w_{x} (t) \\ {{dy} \over {dt}} =& - xz + \rho x - y + w_{y} (t) \\ {{dz} \over {dt}} =& xy - \beta z + w_{z} (t) \end{align*} $$ For example, a system in which dynamical noise is added to the Lorenz attractor can be expressed as above.

From a researcher’s point of view, the most distinct difference from uniformly adding white noise to a trajectory $\mathbf{x} (t)$ is that when a solver is used, the noise is applied on the fly. Literally, since the amount of change itself acts as noise, the result is continuous but drifts away from the original governing equation.