We introduce and study conditional differential equations, a kind of random differential equations. We give necessary and sufficient conditions for the existence of a solution of such an equation. We apply our main result to a Malthus type model.

Let $A=\mathrm{d}/\mathrm{d}\theta$ denote the generator of the rotation group in the space $C\left(\Gamma \right)$, where $\Gamma$ denotes the unit circle. We show that the stochastic Cauchy problem $$\mathrm{d}U\left(t\right)=AU\left(t\right)+f\mathrm{d}{b}_t,U\left(0\right)=0,\left(1\right)$$ where $b$ is a standard Brownian motion and $f\in C\left(\Gamma \right)$ is fixed, has a weak solution if and only if the stochastic convolution process $t\mapsto {(f*b)}_t$ has a continuous modification, and that in this situation the weak solution has a continuous modification. In combination with a recent result of Brzeźniak, Peszat and Zabczyk it follows that (1) fails to have a weak solution for all...

The problem of continuous dependence for inverses of fundamental matrices in the case when uniform convergence is violated is presented here.