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L¹-convergence and hypercontractivity of diffusion semigroups on manifolds

Feng-Yu Wang (2004)

Studia Mathematica

Let P t be the Markov semigroup generated by a weighted Laplace operator on a Riemannian manifold, with μ an invariant probability measure. If the curvature associated with the generator is bounded below, then the exponential convergence of P t in L¹(μ) implies its hypercontractivity. Consequently, under this curvature condition L¹-convergence is a property stronger than hypercontractivity but weaker than ultracontractivity. Two examples are presented to show that in general, however, L¹-convergence...

L₁-uniqueness of degenerate elliptic operators

Derek W. Robinson, Adam Sikora (2011)

Studia Mathematica

Let Ω be an open subset of d with 0 ∈ Ω. Furthermore, let H Ω = - i , j = 1 d i c i j j be a second-order partial differential operator with domain C c ( Ω ) where the coefficients c i j W l o c 1 , ( Ω ̅ ) are real, c i j = c j i and the coefficient matrix C = ( c i j ) satisfies bounds 0 < C(x) ≤ c(|x|)I for all x ∈ Ω. If 0 d s s d / 2 e - λ μ ( s ) ² < for some λ > 0 where μ ( s ) = 0 s d t c ( t ) - 1 / 2 then we establish that H Ω is L₁-unique, i.e. it has a unique L₁-extension which generates a continuous semigroup, if and only if it is Markov unique, i.e. it has a unique L₂-extension which generates a submarkovian semigroup. Moreover...

Laplace transform identities for diffusions, with applications to rebates and barrier options

Hardy Hulley, Eckhard Platen (2008)

Banach Center Publications

We start with a general time-homogeneous scalar diffusion whose state space is an interval I ⊆ ℝ. If it is started at x ∈ I, then we consider the problem of imposing upper and/or lower boundary conditions at two points a,b ∈ I, where a < x < b. Using a simple integral identity, we derive general expressions for the Laplace transform of the transition density of the process, if killing or reflecting boundaries are specified. We also obtain a number of useful expressions for the Laplace transforms...

Less than one implies zero

Felix L. Schwenninger, Hans Zwart (2015)

Studia Mathematica

In this paper we show that from an estimate of the form s u p t 0 | | C ( t ) - c o s ( a t ) I | | < 1 , we can conclude that C(t) equals cos(at)I. Here ( C ( t ) ) t 0 is a strongly continuous cosine family on a Banach space.

Linear and nonlinear abstract differential equations of high order

Veli B. Shakhmurov (2015)

Open Mathematics

The nonlocal boundary value problems for linear and nonlinear degenerate abstract differential equations of arbitrary order are studied. The equations have the variable coefficients and small parameters in principal part. The separability properties for linear problem, sharp coercive estimates for resolvent, discreetness of spectrum and completeness of root elements of the corresponding differential operator are obtained. Moreover, optimal regularity properties for nonlinear problem is established....

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