Consider an infinite dimensional diffusion process process on , where is the circle, defined by the action of its generator on ) local functions as $Lf\left(\eta \right)={\sum }_{i\in {𝐙}^d}\left(\frac{1}{2}{a}_i\frac{{\partial }^{2}f}{\partial {\eta }_i^2}+b_i\frac{\partial f}{\partial {\eta }_i}\right)$. Assume that the coefficients, and are smooth, bounded, finite range with uniformly bounded second order partial derivatives, that is only a function of ${\eta }_i$ and that ${inf}_{i,\eta }{a}_i\left(\eta \right)>0$. Suppose is an invariant product measure. Then, if is the Lebesgue measure or if ,...

Let be a non-negative function of class C from ${}^2$ to $$, which vanishes exactly at two points and . Let (, ) be the set of functions of a real variable which tend to at -∞ and to at +∞ and whose one dimensional energy $$E_1\left(v\right)={\int }_{\left[W\left(v\right)+{\left|v^{\text{'}}\right|}^2/2\right]}x$$ is finite. Assume that there exist two isolated minimizers and of the energy over (, ). Under a mild coercivity condition on the potential and a generic spectral condition on the linearization...