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This paper aims to study the asymptotic behaviour of the fundamental solutions (heat kernels) of non-local (partial and pseudo differential) equations with fractional operators in time and space. In particular, we obtain exact asymptotic formulas for the heat kernels of time-changed Brownian motions and Cauchy processes. As an application, we obtain exact asymptotic formulas for the fundamental solutions to the n-dimensional fractional heat equations in both time and space

This paper aims to study the asymptotic behaviour of the fundamental solutions (heat kernels) of non-local (partial and pseudo differential) equations with fractional operators in time and space. In particular, we obtain exact asymptotic formulas for the heat kernels of time-changed Brownian motions and Cauchy processes. As an application, we obtain exact asymptotic formulas for the fundamental solutions to the n-dimensional fractional heat equations in both time and space

AbstractWe are interested in the rate of convergence of a subordinate Markov process to its invariant measure. Given a subordinator and the corresponding Bernstein function (Laplace exponent), we characterize the convergence rate of the subordinate Markov process; the key ingredients are the rate of convergence of the original process and the (inverse of the) Bernstein function. At a technical level, the crucial point is to bound three types of moment (subexponential, algebraic, and logarithmic) for subordinators as time ttends to ∞. We also discuss some concrete models and we show that subordination can dramatically change the speed of convergence to equilibrium.

AbstractThe question of extending Lp-sub-Markovian semigroups to the spaces Lq, q> p, and the interpolation of Lp-sub-Markovian semigroups with Feller semigroups is investigated. The structure of generators of Lp-sub-Markovian semigroups is studied. Subordination in the sense of Bochner is used to discuss the construction of refinements of Lp-sub-Markovian semigroups. The rôle played by some function spaces which are domains of definition for Lp-generators is pointed out. The problem of regularising powers of generators as well as some perturbation results are discussed.

Abstract.: Under mild regularity assumptions on its domain the infinitesimal generator of a Feller process is known to be a pseudo-differential operator. We give a simple condition on the symbol of the generator in order to characterize the smoothness of the sample paths of real-valued Feller processes in terms of Besov spaces . Our result extends previous papers on the paths of Gaussian, symmetric -stable [6], [20], and Lvy processes [11].