Stochastic processes

Brownian Motion #

$$ \begin{align*} u(x,t+\tau)dx=\int_{-\infty}^{\infty}u(x+\Delta,t)\phi(\Delta)d \Delta dx \end{align*} $$

Langevin’s model #

$$ \begin{align*} F_{f}=-6\pi \eta rv\\ \\ m \ddot x=F_{f}+F_{r} \end{align*} $$

RC circuit #

In laplace domain, V(s) output voltage and W(s) input voltage $$ \begin{align*} V(s) &= \frac{1}{1+RCs}W(s) \end{align*} $$

Dynamic model of car #

$$ \begin{align*} \frac{d^{2}x_{1}}{dt^{2}}=w_{1}(t)\\ \frac{d^{2}x_{2}}{dt^{2}}=w_{2}(t) \end{align*} $$ convert it to matrix form of first order system.

Noisy Pendulum #

$$ \begin{align*} \ddot \theta &= -g \sin(\theta)+w(t)\\ w(t): \text{noise} \end{align*} $$

convert it to matrix form

Spring #

$$ \begin{align*} \ddot x+ \gamma \dot x + v^{2}x=w(t)\

\end{align*} $$

convert to matrix form

White noise process #

w(t)$\in R^{s}$ is a random function with the properties

• $w(t_{1}),w(t_{2})$ are independent
• $E(w(t))=0$
• $C_{w}(t,s)=E[w(t)w^{T}(s)]=\delta(t-s)Q$