Stochastic bifurcation models
It is studied an ordinary differential equation controlled by a stochastic process. The results are encompassing the existence and uniqueness of solutions for the following ordinary differential equations \[ \frac{-dx_t}{dt}= \begin{cases} \beta_1 &\text{if }x_t< B_t,\\ \beta_2 &\text{if }x_t> B_t,\;t\in \mathbb{R},\;x(t_0)= x_0. \end{cases} \] There are proved: 1) If \(B_t\) is a typical Brownian motion path, then there will be a unique Lipschitz solution; 2) if \(\beta_1< 0\) and \(\beta_2> 0\), then a bifurcation situation appears provided \(B_t\) is a Brownian motion. There are given seven theorems including local time, time and direction of bifurcation. The results are illustrated by several examples.
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