A Markov chain with continuous state space has a transition exponential density function: $$p(x_{t},x_{t+1})=\frac{1}{x_{t}}exp(-\frac{x_{t+1}}{x_{t}})$$ i.e. the realized value in period t is the mean value of the next period exponential distribution. By standard procedure, if I want to get the stationary distribution of this Markov chain, I should solve the equation: $$\phi(y)=\int_{0}^{\infty}p(x,y)\phi(x)dx$$ where p(x,y) is the transition density function defined as above and $\phi()$ is the objective stationary distribution. But is there explicit solution for this integration equation?

Furthermore,in the markov chain with continuous state space,what transition density has corresponding explicit stationary distribution in long run?