On a class of measures on configuration spaces

In this thesis we have explored a new class of measures $\nu_{theta}$ on configuration spaces $\Gamma_X$ (of countable subsets of Euclidean space $X=\mbb{R}^d$), obtained as a push-forward of ``lattice'' Gibbs measure $\theta$ on $X^{\mbb{Z}^d}$. For these measures, we have proved the finiteness of the first and second moments and the integration by parts formula. It has also been proved that the generator of the Dirichlet form of $\nu_{\theta}$ satisfies log-Sobolev inequality, which is not typical for measures on configuration spaces. Stochastic dynamics of a particle in random environment distributed according to the measure $\nu_{\theta}$, is presented as an example of possible application of this construction. We consider a toy model of a market, where this stochastic dynamics represents the volatility process of certain European derivative security. We have derived the ``Black-Scholes type" pricing partial differential equation for this derivative security.