CHARACTERIZATION OF SUBORDINATE SYMMETRIC MARKOV PROCESSES

In this paper, we consider subordinate symmetric Markov processes that correspond to non-killing Dirichlet forms enjoying heat kernel estimates on a metric measure space with the volume doubling property.We obtain estimates of the jump kernel of the subordinate process and establish equivalent conditions for the jump kernel following the work of Liu and Murugan (ALEA Lat.Am.J. Probab.Math.Stat.20(2) (2023), 1271-1281).In particular, we clarify the scale of the jump kernel, which is different from the diffusion type.This result is applicable to non-subordinate processes by the transferring method, which uses the stability of Dirichlet forms. IntroductionIn recent years, there has been a significant amount of research on jump processes.Since the generators of Markov processes with discontinuous paths are non-local operators, they have been studied from the areas of both probability and partial differential equations.One useful approach to studying jump processes is the time change method via subordinators, called subordination.A typical example of a subordinate process is the 𝛼-stable process on ℝ d , which is obtained from Brownian motion by a specific subordinator.Concerning applications, subordinate Brownian motions on ℝ d or on a bounded domain have been investigated, and significant number of properties, such as heat kernel estimates and Harnack inequalities, have been revealed [20][21][22][23]26].Subordination is applicable not only to Brownian motion but also to Markov processes on metric measure spaces by using the theory of Dirichlet forms.To be precise, let (M, d, 𝜇) be a metric measure space, that is, (M, d) is a locally compact separable metric space and 𝜇 is a full support Radon measure on M. Ôkura [28] established a formula for the Dirichlet forms for subordinate Markov processes.Research on jump processes especially on fractals (see, for example, the Sierpinski gasket in Figure 1 and the Sierpinski carpet in Figure 2) has been developed since the beginning of this century; see, for instance, Chen and Kumagai [10].A typical example is the '𝛼-stable process' whose Dirichlet form is given by ℰ( f , f ) = ∫ M×M ( f (x)f (y)) 2 J(x, y)𝜇(dx)𝜇(dy), J(x, y) = C V (x, d(x, y))d(x, y) 𝛼 , (1.1) 2020