The paper deals with several questions of the diffusion approximation. The goal of this paper is to create the general method of reducting the dimension of the model with the aid of the diffusion approximation. Especially, two dimensional random variables are approximated by one-dimensional diffusion process by replacing one of its coordinates by a certain characteristic, e.g. by its stationary expectation. The suggested method is used for several different systems. For instance, the method is applicable...

On a Lie group NA that is a split extension of a nilpotent Lie group N by a one-parameter group of automorphisms A, the heat semigroup ${\mu }_t$ generated by a second order subelliptic left-invariant operator ${\sum }_{j=0}^m{Y}_j+Y$ is considered. Under natural conditions there is a $\mu {̌}_t$-invariant measure m on N, i.e. $\mu {̌}_t*m=m$. Precise asymptotics of m at infinity is given for a large class of operators with Y₀,...,Yₘ generating the Lie algebra of S.

We study the asymptotic behaviour of solutions of a transport equation. We give some sufficient conditions for the complete mixing property of the Markov semigroup generated by this equation.

We study the asymptotic behaviour of the Markov semigroup generated by an integro-partial differential equation. We give new sufficient conditions for asymptotic stability of this semigroup.

Let -ℒ be the generator of a Lévy semigroup on L¹(ℝⁿ) and f: ℝ → ℝⁿ be a nonlinearity. We study the large time asymptotic behavior of solutions of the nonlocal and nonlinear equations uₜ + ℒu + ∇·f(u) = 0, analyzing their $L^p$-decay and two terms of their asymptotics. These equations appear as models of physical phenomena that involve anomalous diffusions such as Lévy flights.

We consider a diffusion process $X_t$ smoothed with (small) sampling parameter $\epsilon$. As in Berzin, León and Ortega (2001), we consider a kernel estimate ${\widehat{\alpha }}_{\epsilon }$ with window $h\left(\epsilon \right)$ of a function $\alpha$ of its variance. In order to exhibit global tests of hypothesis, we derive here central limit theorems for the $L^p$ deviations such as$$\frac{1}{\sqrt{h}}{\left(\frac{h}{\epsilon }\right)}^{\frac{p}{2}}\left({∥{\widehat{\alpha }}_{\epsilon }-\alpha ∥}_p^p-𝔼{∥{\widehat{\alpha }}_{\epsilon }-\alpha ∥}_p^p\right).$$

We consider a diffusion process Xt smoothed with (small) sampling parameter ε. As in Berzin, León and Ortega (2001), we consider a kernel estimate ${\widehat{\alpha }}_{\epsilon }$ with window h(ε) of a function α of its variance. In order to exhibit global tests of hypothesis, we derive here central limit theorems for the Lp deviations such as $$\frac{1}{\sqrt{h}}{\left(\frac{h}{\epsilon }\right)}^{\frac{p}{2}}\left({∥{\widehat{\alpha }}_{\epsilon }-\alpha ∥}_p^p-\text{I}\text{E}{∥{\widehat{\alpha }}_{\epsilon }-\alpha ∥}_p^p\right).$$