Consider an infinite dimensional diffusion process process on $T^{{𝐙}^d}$, where $T$ is the circle, defined by the action of its generator $L$ on $C^2\left(T^{{𝐙}^d}\right)$ 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, $a_i$ and $b_i$ are smooth, bounded, finite range with uniformly bounded second order partial derivatives, that $a_i$ is only a function of ${\eta }_i$ and that ${inf}_{i,\eta }{a}_i\left(\eta \right)\>0$. Suppose $\nu$ is an invariant product measure. Then, if $\nu$ is the Lebesgue measure or if $d=1,2$, it is the unique invariant measure. Furthermore, if $\nu$ is translation invariant,...

We deal with the boundary value problem $$\begin{array}{ccccccc}\hfill -\Delta u\left(x\right)& ={\lambda }_{1}u\left(x\right)+g\left(\nabla u\left(x\right)\right)+h\left(x\right),\hfill & & x\in \Omega u\left(x\right)\hfill & \hfill =0,& & \hfill x\in \partial \Omega \end{array}$$ where $\Omega \subset {ℝ}^N$ is an smooth bounded domain, ${\lambda }_1$ is the first eigenvalue of the Laplace operator with homogeneous Dirichlet boundary conditions on $\Omega$, $h\in L^{max\{2,N/2\}}\left(\Omega \right)$ and $g:{ℝ}^N⟶ℝ$ is bounded and continuous. Bifurcation theory is used as the right framework to show the existence of solution provided that $g$ satisfies certain conditions on the origin and at infinity.

We consider an estimate of the mode $\theta$ of a multivariate probability density $f$ with support in ${ℝ}^d$ using a kernel estimate $f_n$ drawn from a sample $X_1,\cdots ,X_n$. The estimate ${\theta }_n$ is defined as any $x$ in $\{X_1,\cdots ,X_n\}$ such that $f_n\left(x\right)={max}_{i=1,\cdots ,n}{f}_n\left(X_i\right)$. It is shown that ${\theta }_n$ behaves asymptotically as any maximizer ${\widehat{\theta }}_n$ of $f_n$. More precisely, we prove that for any sequence ${\left(r_n\right)}_{n\ge 1}$ of positive real numbers such that $r_n\to \infty$ and $r_n^{d}logn/n\to 0$, one has $r_n\parallel {\theta }_n-{\widehat{\theta }}_n\parallel \to 0$ in probability. The asymptotic normality of ${\theta }_n$ follows without further work.

The existence of decaying positive solutions in ${ℝ}_{+}$ of the equations $\left(E_{\lambda }\right)$ and $\left(E_{\lambda }^1\right)$ displayed below is considered. From the existence of such solutions for the subhomogeneous cases (i.e. $t^{1-p}F(r,tU,t|U^{\text{'}}\left|\right)\searrow 0$ as $t\nearrow \infty$), a super-sub-solutions method (see § 2.2) enables us to obtain existence theorems for more general cases.