Summary :This thesis is devoted to the study of some properties (like existence, uniqueness or ergodicity) of the solutions of stochastic differential equations related to Statistical Mechanics; they are modelization of random dynamics of infinite particle systems on the lattice Zd.We first state the ergodicity of infinite-dimensional brownian diffusions when the drift is the gradient of a Hamilton function, denoted by h. Thanks to their Harris recurrence property, these processes are ergodic, that is the law of the dynamical system converges for infinite time to an equilibrium state which does not depend of the initial conditions. Moreover this limit distribution is identified as the Gibbs measure associated to the Hamiltonien h. In the next chapter, we study the ergodicity of gradient diffusions but now in an infinite-dimensional situation, i.e when the configuration space is RZd. Usually, they are no more recurrent, then we have to state conditions on the Hamilton function h in order to obtain the convergence to equilibrium. Since the limit distribution is the unique invariant measure for the dynamics, we deduce that the set of regular Gibbs measures associated to h contains at most one element (in other words, this is no phase transition). We illustrate our results with some examples. In the last chapter, we consider the class of infinite-dimensional Ginzburg-Landau diffusions. These processes modelize at each site of lattice Zd the sum of the currents of magnetization with neighbour sites. We prove existence and uniqueness of these conservative diffusions under regularity conditions of the coefficients. Then, we present a family of Gibbs measures (indexed by harmonic sequences on Zd) which are invariant for a typical examples of Ginzburg-Landau dynamics. |