Department of Radiation and Radiobiological Research,

Joint Institute for Nuclear Research,

141980 Dubna,

RUSSIA

E-mail: altaisky@mx.iki.rssi.ru

**Scale-dependent functions and stochastic quantization**

**Abstract:**

We consider quantum and random fields on the affine group $G:x'=ax+b$,
$a>0$, $x,b\in {\mathbb R}^d$, consisting of dilations and translation
of Euclidean space. The fields $\phi_a(b,\cdot)$ are constructed using
the continuous wavelet transform. The explicit dependence of the correlation
properties of the fields $\phi$ on the scale $a$ is shown to be useful
for the construction of divergence-free field theory. An examples
of the scalar field theory and stochastic quantization of

gauge theories are presented.