We show that a stochastic (Markov) operator S acting on a Schatten class C-1 satisfies the Noether condition (i.e. S' (A) = A and S' (A(2)) = A(2), where A is an element of C-infinity is a Hermitian and bounded operator on a fixed separable and complex Hilbert space (H, <.,.>)), if and only if S(E-A(G)XEA(G)) = E-A (G)S(X)E-A (G) for any state X is an element of C-1 and all Borel sets G subset of R, where E-A (G) denotes the orthogonal projection coming from the spectral resolution A = integral(sigma(A)) zE(A)(dz). Similar results are obtained for stochastic one-parameter continuous semigroups.
In this paper it is shown that irregular boundary points for p-harmonic functions as well as for quasiminimizers can be divided into semiregular and strongly irregular points with vastly different boundary behaviour. This division is emphasized by a large number of characterizations of semiregular points. The results hold in complete metric spaces equipped with a doubling measure supporting a Poincaré inequality. They also apply to Cheeger p-harmonic functions and in the Euclidean setting to -harmonic functions, with the usual assumptions on .
We use sphericalization to study the Dirichlet problem, Perron solutions and boundary regularity for p-harmonic functions on unbounded sets in Ahlfors regular metric spaces. Boundary regularity for the point at infinity is given special attention. In particular, we allow for several "approach directions" towards infinity and take into account the massiveness of their complements. In 2005, Llorente-Manfredi-Wu showed that the p-harmonic measure on the upper half space R-+(n), n amp;gt;= 2, is not subadditive on null sets when p not equal 2. Using their result and spherical inversion, we create similar bounded examples in the unit ball B subset of R-n showing that the n-harmonic measure is not subadditive on null sets when n amp;gt;= 3, and neither are the p-harmonic measures in B generated by certain weights depending on p not equal 2 and n amp;gt;= 2. (C) 2019 Elsevier Inc. All rights reserved.
In the setting of metric measure spaces equipped with a doubling measure supporting a weak p-Poincaré inequality with 1 ≤ p < ∞, we show that any uniform domain Ω is an extension domain for the Newtonian space N1, p (Ω) and that Ω, together with the metric and the measure inherited from X, supports a weak p-Poincaré inequality. For p > 1, we obtain a near characterization of N1, p-extension domains with local estimates for the extension operator. © 2006 Elsevier Inc. All rights reserved.
We consider an oblique derivative problem in a wedge for non-divergence parabolic equations with time-discontinuous coefficients. We obtain weighted coercive estimates of solutions in anisotropic Sobolev spaces. (C) 2015 Elsevier Inc. All rights reserved.
This paper considers to the equation integral(S) U(Q)/vertical bar P - Q vertical bar(N-1) dS(Q) = F(P), P is an element of S, where the surface S is the graph of a Lipschitz function phi on R-N, which has a small Lipschitz constant. The integral on the left-hand side is the single layer potential corresponding to the Laplacian in RN+1. Let Lambda(r) be the Lipschitz constant of phi on the ball centered at the origin with radius 2r. Our analysis is carried out in local L-p-spaces and local Sobolev spaces, where 1 less than p less than infinity, and results are presented in terms of Lambda. Estimates of solutions to the equation are provided, which can be used to obtain knowledge about the behavior of the solutions near a point on the surface. These estimates are given in terms of seminorms. Solutions are also shown to be unique if they are subject to certain growth conditions. Local estimates are provided and some applications are supplied.
We consider the Neumann problem for the heat equation perturbed by a dissipation term au, where a is a function of space and time variables, small in some integral sense, and u is the temperature. We derive a two term asymptotic representation for the solution for large time which can be used, in particular, to study boundedness and stability properties of the solution in the case when the leading term of the asymptotic expansion does not allow to do this analysis.
In this paper, for a finite subset A subset of {2,3, center dot center dot center dot }, we introduce the notion of longest block function L-n (x,A) for the Luroth expansion of x epsilon [0,1) with respect to A and consider the asymptotic behavior of L-n (x,A) as n tends to infinity. We also obtain the Hausdorff dimensions of the level sets and exceptional set arising from the longest block function. (c) 2017 Elsevier Inc. All rights reserved.
We consider two types of multilinear pseudodifferential operators. First, we prove the boundedness of multilinear pseudodifferential operators with symbols which are only measurable in the spatial variables in Lebesgue spaces. These results generalise earlier work of the present authors concerning linear pseudo-pseudodifferential operators. Secondly, we investigate the boundedness of bilinear pseudodifferential operators with symbols in the Hormander S-p,delta(m) classes. These results are new in the case p less than 1, that is, outwith the scope of multilinear Calderon-Zygmund theory.
Let mu(epsilon) be the probability measures on D[0,T] of suitable Markov processes {xi(epsilon)(t)}0 amp;lt;= t amp;lt;= T (possibly with small jumps) depending on a small parameter epsilon amp;gt;0, where D[0,T] denotes the space of all functions on [0, T] which are right continuous with left limits. In this paper we investigate asymptotic expansions for the Laplace transforms integral(D[0,T]) exp{epsilon F-1(x)}mu(epsilon)(dx) as epsilon -amp;gt; 0 for smooth functionals F on D[0,T]. This study not only recovers several well-known results, but more importantly provides new expansions for jump Markov processes. Besides several standard tools such as exponential change of measures and Taylors expansions, the novelty of the proof is to implement the expectation asymptotic expansions on normal deviations which were recently derived in [13]. (c) 2017 Elsevier Inc. All rights reserved.
In this paper, we consider a family of Markov bridges with jumps constructed from truncated stable processes. These Markov bridges depend on a small parameter h greater than 0, and have fixed initial and terminal positions. We propose a new method to prove a large deviation principle for this family of bridges based on compact level sets, change of measures, duality and various global and local estimates of transition densities for truncated stable processes.
In this paper we consider a family of generalized Brownian bridges with a small noise, which was used by Brennan and Schwartz [3] to model the arbitrage profit in stock index futures in the absence of transaction costs. More precisely, we study the large deviation principle of these generalized Brownian bridges as the noise becomes infinitesimal. (C) 2015 Elsevier Inc. All rights reserved.