Abstract:
We give a survey of the basic statistical ideas underlying the definition of entropy in information theory and their connections with the entropy in the theory of dynamical systems and in statistical mechanics.

Abstract:
Let $\Omega$ be a connected open subset of $\Ri^d$. We analyze $L_1$-uniqueness of real second-order partial differential operators $H=-\sum^d_{k,l=1}\partial_k\,c_{kl}\,\partial_l$ and $K=H+\sum^d_{k=1}c_k\,\partial_k+c_0$ on $\Omega$ where $c_{kl}=c_{lk}\in W^{1,\infty}_{\rm loc}( \Omega), c_k\in L_{\infty,{\rm loc}}(\Omega)$, $c_0\in L_{2,{\rm loc}}(\Omega)$ and $C(x)=(c_{kl}(x))>0$ for all $x\in\Omega$. Boundedness properties of the coefficients are expressed indirectly in terms of the balls $B(r)$ associated with the Riemannian metric $C^{-1}$ and their Lebesgue measure $|B(r)|$. \noindent\hspace{10mm}First we establish that if the balls $B(r)$ are bounded, the T\"acklind condition $\int^\infty_Rdr\,r(\log|B(r)|)^{-1}=\infty$ is satisfied for all large $R$ and $H$ is Markov unique then $H$ is $L_1$-unique. If, in addition, $C(x)\geq \kappa\, (c^{T}\!\otimes\, c)(x)$ for some $\kappa>0$ and almost all $x\in\Omega$, $\divv c\in L_{\infty,{\rm loc}}(\Omega)$ is upper semi-bounded and $c_0$ is lower semi-bounded then $K$ is also $L_1$-unique. \noindent\hspace{10mm}Secondly, if the $c_{kl}$ extend continuously to functions which are locally bounded on $\partial\Omega$ and if the balls $B(r)$ are bounded we characterize Markov uniqueness of $H$ in terms of local capacity estimates and boundary capacity estimates. For example, $H$ is Markov unique if and only if for each bounded subset $A$ of $\overline\Omega$ there exist $\eta_n \in C_c^\infty(\Omega)$ satisfying $\lim_{n\to\infty} \|\one_A\Gamma(\eta_n)\|_1 = 0$, where $\Gamma(\eta_n)=\sum^d_{k,l=1}c_{kl}\,(\partial_k\eta_n)\,(\partial_l\eta_n)$, and $\lim_{n\to\infty}\|\one_A (\one_\Omega-\eta_n )\, \varphi\|_2 = 0$ for each $\varphi \in L_2(\Omega)$ or if and only if $\capp(\partial\Omega)=0$.

Abstract:
Let $h$ be a quadratic form with domain $W_0^{1,2}(\Ri^d)$ given by \[ h(\varphi)=\sum^d_{i,j=1}(\partial_i\varphi,c_{ij}\,\partial_j\varphi) \] where $c_{ij}=c_{ji}$ are real-valued, locally bounded, measurable functions and $C=(c_{ij})\geq 0 $. If $C$ is strongly elliptic, i.e.\ if there exist $\lambda, \mu>0$ such that $\lambda\,I\geq C\geq \mu \,I>0$, then $h$ is closable, the closure determines a positive self-adjoint operator $H$ on $L_2(\Ri^d)$ which generates a submarkovian semigroup $S$ with a positive distributional kernel~$K$ and the kernel satisfies Gaussian upper and lower bounds. Moreover, $S$ is conservative, i.e.\ $S_t\one=\one$ for all $t>0$. Our aim is to examine converse statements. First we establish that $C$ is strongly elliptic if and only if $h$ is closable, the semigroup $S$ is conservative and $K$ satisfies Gaussian bounds. Secondly, we prove that if the coefficients are such that a Tikhonov growth condition is satisfied then $S$ is conservative. Thus in this case strong ellipticity of $C$ is equivalent to closability of $h$ together with Gaussian bounds on $K$. Finally we consider coefficients $c_{ij}\in W^{1,\infty}_{\rm loc}(\Ri^d)$. It follows that $h$ is closable and a growth condition of the T\"acklind type is sufficient to establish the equivalence of strong ellipticity of $C$ and Gaussian bounds on $K$.

Abstract:
Let $\Omega$ be an open subset of $\Ri^d$ and $H_\Omega=-\sum^d_{i,j=1}\partial_i c_{ij} \partial_j$ a second-order partial differential operator on $L_2(\Omega)$ with domain $C_c^\infty(\Omega)$ where the coefficients $c_{ij}\in W^{1,\infty}(\Omega)$ are real symmetric and $C=(c_{ij})$ is a strictly positive-definite matrix over $\Omega$. In particular, $H_\Omega$ is locally strongly elliptic. We analyze the submarkovian extensions of $H_\Omega$, i.e. the self-adjoint extensions which generate submarkovian semigroups. Our main result establishes that $H_\Omega$ is Markov unique, i.e. it has a unique submarkovian extension, if and only if $\capp_\Omega(\partial\Omega)=0$ where $\capp_\Omega(\partial\Omega)$ is the capacity of the boundary of $\Omega$ measured with respect to $H_\Omega$. The second main result establishes that Markov uniqueness of $H_\Omega$ is equivalent to the semigroup generated by the Friedrichs extension of $H_\Omega$ being conservative.

Abstract:
Let $\Omega$ be an open subset of $\Ri^d$ with $0\in \Omega$. Further let $H_\Omega=-\sum^d_{i,j=1}\partial_i\,c_{ij}\,\partial_j$ be a second-order partial differential operator with domain $C_c^\infty(\Omega)$ where the coefficients $c_{ij}\in W^{1,\infty}_{\rm loc}(\bar\Omega)$ are real, $c_{ij}=c_{ji}$ and the coefficient matrix $C=(c_{ij})$ satisfies bounds $00$ where $\mu(s)=\int^s_0dt\,c(t)^{-1/2}$ then we establish that $H_\Omega$ is $L_1$-unique, i.e.\ it has a unique $L_1$-extension which generates a continuous semigroup, if and only if it is Markov unique, i.e.\ it has a unique $L_2$-extension which generates a submarkovian semigroup. Moreover these uniqueness conditions are equivalent with the capacity of the boundary of $\Omega$, measured with respect to $H_\Omega$, being zero. We also demonstrate that the capacity depends on two gross features, the Hausdorff dimension of subsets $A$ of the boundary the set and the order of degeneracy of $H_\Omega$ at $A$.

Abstract:
Let $S=\{S_t\}_{t\geq0}$ be the submarkovian semigroup on $L_2(\Ri^d)$ generated by a self-adjoint, second-order, divergence-form, elliptic operator $H$ with Lipschitz continuous coefficients $c_{ij}$. Further let $\Omega$ be an open subset of $\Ri^d$. Under the assumption that $C_c^\infty(\Ri^d)$ is a core for $H$ we prove that $S$ leaves $L_2(\Omega)$ invariant if, and only if, it is invariant under the flows generated by the vector fields $Y_i=\sum^d_{j=1}c_{ij}\partial_j$.

Abstract:
Let $H$ be the symmetric second-order differential operator on $L_2(\Ri)$ with domain $C_c^\infty(\Ri)$ and action $H\varphi=-(c \varphi')'$ where $ c\in W^{1,2}_{\rm loc}(\Ri)$ is a real function which is strictly positive on $\Ri\backslash\{0\}$ but with $c(0)=0$. We give a complete characterization of the self-adjoint extensions and the submarkovian extensions of $H$. In particular if $\nu=\nu_+\vee\nu_-$ where $\nu_\pm(x)=\pm\int^{\pm 1}_{\pm x} c^{-1}$ then $H$ has a unique self-adjoint extension if and only if $\nu\not\in L_2(0,1)$ and a unique submarkovian extension if and only if $\nu\not\in L_\infty(0,1)$. In both cases the corresponding semigroup leaves $L_2(0,\infty)$ and $L_2(-\infty,0)$ invariant. In addition we prove that for a general non-negative $ c\in W^{1,\infty}_{\rm loc}(\Ri)$ the corresponding operator $H$ has a unique submarkovian extension.

Abstract:
We analyze degenerate, second-order, elliptic operators $H$ in divergence form on $L_2({\bf R}^{n}\times{\bf R}^{m})$. We assume the coefficients are real symmetric and $a_1H_\delta\geq H\geq a_2H_\delta$ for some $a_1,a_2>0$ where \[ H_\delta=-\nabla_{x_1} c_{\delta_1, \delta'_1}(x_1) \nabla_{x_1}-c_{\delta_2, \delta'_2}(x_1) \nabla_{x_2}^2 . \] Here $x_1\in{\bf R}^n$, $x_2\in{\bf R}^m$ and $c_{\delta_i, \delta'_i}$ are positive measurable functions such that $c_{\delta_i, \delta'_i}(x)$ behaves like $|x|^{\delta_i}$ as $x\to0$ and $|x|^{\delta_i'}$ as $x\to\infty$ with $\delta_1,\delta_1'\in[0,1>$ and $\delta_2,\delta_2'\geq0$. Our principal results state that the submarkovian semigroup $S_t=e^{-tH}$ is conservative and its kernel $K_t$ satisfies bounds \[ 0\leq K_t(x ;y)\leq a (|B(x ;t^{1/2})| |B(y ;t^{1/2})|)^{-1/2} \] where $|B(x ;r)|$ denotes the volume of the ball $B(x ;r)$ centred at $x$ with radius $r$ measured with respect to the Riemannian distance associated with $H$. The proofs depend on detailed subelliptic estimations on $H$, a precise characterization of the Riemannian distance and the corresponding volumes and wave equation techniques which exploit the finite speed of propagation. We discuss further implications of these bounds and give explicit examples that show the kernel is not necessarily strictly positive, nor continuous.

Abstract:
Let $S=\{S_t\}_{t\geq0}$ be the semigroup generated on $L_2(\Ri^d)$ by a self-adjoint, second-order, divergence-form, elliptic operator $H$ with Lipschitz continuous coefficients. Further let $\Omega$ be an open subset of $\Ri^d$ with Lipschitz continuous boundary $\partial\Omega$. We prove that $S$ leaves $L_2(\Omega)$ invariant if, and only if, the capacity of the boundary with respect to $H$ is zero or if, and only if, the energy flux across the boundary is zero. The global result is based on an analogous local result.

Abstract:
We analyze degenerate, second-order, elliptic operators $H$ in divergence form on $L_2(\Ri^{n}\times\Ri^{m})$. We assume the coefficients are real symmetric and $a_1H_\delta\geq H\geq a_2H_\delta$ for some $a_1,a_2>0$ where \[ H_\delta=-{\nabla}_{x_1}\cdot(c_{\delta_1, \delta'_1}(x_1)\,\nabla_{x_1})-c_{\delta_2, \delta'_2}(x_1)\,\nabla_{x_2}^2 \;. \] Here $x_1\in\Ri^n$, $x_2\in\Ri^m$ and $c_{\delta_i, \delta'_i}$ are positive measurable functions such that $c_{\delta_i, \delta'_i}(x)$ behaves like $|x|^{\delta_i}$ as $x\to0$ and $|x|^{\delta_i'}$ as $x\to\infty$ with $\delta_1,\delta_1'\in[0,1\rangle$ and $\delta_2,\delta_2'\geq0$. Our principal results state that the submarkovian semigroup $S_t=e^{-tH}$ is conservative and its kernel $K_t$ satisfies bounds \[ 0\leq K_t(x\,;y)\leq a\,(|B(x\,;t^{1/2})|\,|B(y\,;t^{1/2})|)^{-1/2} \] where $|B(x\,;r)|$ denotes the volume of the ball $B(x\,;r)$ centred at $x$ with radius $r$ measured with respect to the Riemannian distance associated with $H$. The proofs depend on detailed subelliptic estimations on $H$, a precise characterization of the Riemannian distance and the corresponding volumes and wave equation techniques which exploit the finite speed of propagation. We discuss further implications of these bounds and give explicit examples that show the kernel is not necessarily strictly positive, nor continuous.