Submarkovian Semigroups Research Articles

Analyticity of Submarkovian Semigroups

Let A be a generator of a submarkovian semigroup in Li{M, dp.). We investigate the domain of analyticity of exp(-L4) in LP(M, dp) . The same problem for the generator perturbed by potential is considered.

Semi-groupes sous-Markoviens engendr�s par des op�rateurs matriciels

We characterize generators of sub-Markovian semigroups onLp(Ω) by a version of Kato's inequality. This will be used to show (under precise assumptions) that the semigroup generated by a matrix operatorA=(Aij)1≦i,j≦n on (Lp(Ω))n is sub-Markovian if and only if the semigroup generated by the sum of each rowAi1+...+Ain (1≦i≦n), is sub-Markovian. The corresponding result on (C0(X))n characterizes dissipative operator matrices.

Some inequalities for sub-Markovian generators and their applications to the perturbation theory

We characterize the domain in L p {L^p} -space of generators of submarkovian semigroups in terms of the form domain in L 2 {L^2} and give the corresponding inequality. Using this inequality we obtain a criterion for the formal difference A − B A - B of such generators to be a generator of a contraction semigroup in L p {L^p} . The conditions on perturbation are expressed in terms of forms, i.e., in L 2 {L^2} -terms.

Uniformly contractive semigroups and submarkovian semigroups (I)

Let (M, ℱ,d, x) be a σ-finite measure space with a σ-field ℱ countably generated. We call a linear mapT uniformly contractive if which maps measurable functions onM to measurable functions and If a linear mapT which maps measurable functions onM to measurable functions has positivity property, namely,Tf≧0 forf≧0, we call it a submarkovian operator. In this article we prove

Semi-groupes d'opérateurs et espaces fonctionnels sur les groupes de Lie

We prove imbedding theorems in the setting of abstract symmetric sub-markovian semi-groups, under some natural dimensional hypothesis. Our results are sharp and extend classical ones. They involve Lipschitz, Besov, and Sobolev spaces, as well as Lebesgue spaces, and include a generalization of the Herz-Bernstein Theorem on the Wiener algebra. They typically apply to function spaces associated to left invariant sub-laplacians on unimodular Lie groups, thanks to estimates obtained by Varopoulos.

Natural densities of Markov transition probabilities

Let (P t ) t>0 , (P * ) t>0 be two measurable submarkovian semigroups on a measurable spaceE which are absolutely continuous and in duality with respect to a σ-finite measure μ. Then we show that there exists a unique measurable function $$p:]0,\infty [ \times E \times E \to \mathop \mathbb{R}\limits^{\_} _ +$$ satisfying (i) $$\begin{gathered} P_t f(x) = \smallint p(t,x,y)f(y)\mu (dy), \hfill \\ P_t^* f(x) = \smallint p(t,y,x)f(y)\mu (dy)(f \in E_ + ,x \in E) \hfill \\ \end{gathered} $$ (ii) $$p(s + t,x,y) = \smallint p(s,x,z)p(t,z,y)\mu (dz)(s,t > 0,x,y \in E).$$ A similar result is shown for inhomogeneous transition functions.

Compact symmetry groups and generators for sub-Markovian semigroups

We consider closed infinitesimal operators L with dense domain D in the algebra of all continuous functions, vanishing at infinity, on a given locally compact Hausdorff space X. Necessary and sufficient conditions are given for L to be the infinitesimal generator of a strongly continuous sub-Markovian semigroup on C0(X). The conditions involve: (1) commutativity with a compact group G acting continuously on X (i.e., spatial G-symmetry), (2) a tangential notion defined in terms of the G-action, and finally, (3) a dispersive estimate for L. The dispersive estimate is satisfied on a subalgebra which is a core for L, and contained in the domain D. If G is also a Lie group, we explicitly construct such a core algebra in the domain.

Application of the Sector Condition to the Classification of Submarkovian Semigroups

Application of the Sector Condition to the Classification of Submarkovian Semigroups

Application of the sector condition to the classification of sub-Markovian semigroups

Let p t {p_{t}} , t > 0 t > 0 , be a strongly continuous submarkovian semigroup on a real Hilbert space L 2 ( X , m ) {L^2}(X, m) . The measure m is assumed to be excessive and the L 2 {L^2} generator A is assumed to satisfy an estimate (the sector condition) which permits the application of Dirichlet spaces (not necessarily symmetric). Other submarkovian semigroups P t ∼ P_t^ \sim with the same local generator and cogenerator and relative to which m is again excessive are classified in terms of generators for processes which live on a suitable boundary.