  According to the theory of functional inequalities, a non-symmetric Markov
semigroup has better properties than the corresponding symmetric one. For
instance, there exist non-symmetric Markov semigroups which are
hypercontractive (and thus converge exponentially in both $L^2$ and entropy),
but the symmetric ones are even not ergodic. In this paper, we aim to search
for reasonable conditions to ensure that a non-symmetric Markov semigroup and
its symmetrization share the properties of exponential convergence, uniform
integrability, hypercontractivity, and supercontractivity. Since in the
symmetric case these properties are precisely characterized by functional
inequalities of the Dirichlet form, the key point of the study is to prove
these inequalities for non-symmetric Markov processes. SDEs driven by Brownian
motion or L\'evy jump process are investigated.
