Let X(t) be a diffusion process satisfying the stochastic differential equation dX(t) = a(X(t))dW(t) + b(X(t))dt. We analyse the asymptotic behaviour of p(t) = ProbX(t) ≥ 0 as t → ∞ and construct an equation such that $limsu{p}_{t\to \infty }{t}^{-1}{\int }_0^{t}p\left(s\right)ds=1$ and $limin{f}_{t\to \infty }{t}^{-1}{\int }_0^{t}p\left(s\right)ds=0$.

Strassen’s functional form of the law of the iterated logarithm is formulated for partial sums of random variables with values in a strict inductive limit of Frechet spaces of Hilbert space type. The proof depends on obtaining Berry-Essen estimates for Hilbert space valued random variables.

Existence of strong and weak solutions to stochastic inclusions $x_t-x_s\in {\int }_s^t{F}_{\tau }\left(x_{\tau }\right)d\tau +{\int }_s^t{G}_{\tau }\left(x_{\tau }\right)d{w}_{\tau }+{\int }_s^t{\int }_{{ℝ}^n}{H}_{\tau ,z}\left(x_{\tau }\right)q(d\tau ,dz)$ and $x_t-x_s\in {\int }_s^t{F}_{\tau }\left(x_{\tau }\right)d\tau +{\int }_s^t{G}_{\tau }\left(x_{\tau }\right)d{w}_{\tau }+{\int }_s^t{\int }_{\left|z\right|\le 1}{H}_{\tau ,z}\left(x_{\tau }\right)q(d\tau ,dz)+{\int }_s^t{\int }_{\left|z\right|>1}{H}_{\tau ,z}\left(x_{\tau }\right)p(d\tau ,dz)$, where p and q are certain random measures, is considered.

An integral Markov operator $P$ appearing in biomathematics is investigated. This operator acts on the space of probabilistic Borel measures. Let $\mu$ and $\nu$ be probabilistic Borel measures. Sufficient conditions for weak and strong convergence of the sequence $(P^n\mu -P^n\nu )$ to $0$ are given.

We generalize the results of Komlós, Major and Tusnády concerning the strong approximation of partial sums of independent and identically distributed random variables with a finite r-th moment to the case when the parameter set is two-dimensional. The most striking result is that the rates of convergence are exactly the same as in the one-dimensional case.