The paper is devoted to the existence of perpetual integral functionals ${\int }_0^{\mathrm{\infty }}f(X(t))\mathrm{d}t$for several classes of d-dimensional of stochastic processes $X(t)$. The method is very simple: we establish the conditions supplying that these functionals have a finite expectation. Examples of these classes include d-dimensional fractional Brownian motion having coordinates with the same Hurst index H, for which existence is established under the assumption $d>1/H$. In particular, perpetual integral functionals exist for d-dimensional Brownian motion with d>2, compound Poisson process, Markov processes admitting densities of transitional probabilities. In the case of Brownian motion and fractional Brownian motion we establish that the perpetual integral functionals are not a constant a.s. if $f\ne 0$.

该论文致力于永久积分泛函的存在性 ${\int }_0^{\mathrm{\infty }}F(X(吨))\mathrm{d}吨$对于随机过程的几类d维$X(吨)$. 该方法非常简单：我们建立提供这些泛函具有有限期望的条件。这些类别的示例包括具有相同 Hurst 指数H 的坐标的d维分数布朗运动，其存在性在假设下成立$d>1/H$. 特别是，对于存在永久性的积分函d与维布朗运动d > 2，化合物泊松过程，马尔可夫过程承认过渡概率的密度。在布朗运动和分数布朗运动的情况下，我们确定永积分泛函不是常数，好像$F\ne 0$.