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We first prove that the realization $A_{\mathrm {min}}$ of $A:={\operatorname {\mathrm {div}}}(Q\nabla )-V$ in $L^2({\mathbb {R}}^d)$ with unbounded coefficients generates a symmetric sub-Markovian and ultracontractive semigroup on $L^2({\mathbb {R}}^d)$ which coincides on $L^2({\mathbb {R}}^d)\cap C_b({\mathbb {R}}^d)$ with the minimal semigroup generated by a realization of $A$ on $C_b({\mathbb {R}}^d)$. Moreover, using time-dependent Lyapunov functions, we prove pointwise upper bounds for the heat kernel of $A$ and deduce some spectral properties of $A_{\min }$ in the case of polynomially and exponentially growing diffusion and potential coefficients.

We present a method for estimating the edge of a two-dimensionalbounded set, given a finite random set of points drawn from the interior.The estimator is based both on a Parzen-Rosenblatt kernel and extreme values of point processes. We give conditionsfor various kinds of convergence and asymptotic normality.We propose a method of reducing the negative bias and edge effects,illustrated by some simulations.

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General kernel estimates of Schrödinger-type operators with unbounded diffusion terms

Extreme values and kernel estimates of point processes boundaries

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2 results

General kernel estimates of Schrödinger-type operators with unbounded diffusion terms

Extreme values and kernel estimates of point processes boundaries