On the generalized Feynman-Kac transformation for nearly symmetric Markov processes

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Suppose X is a right process which is associated with a non-symmetric Dirichlet form (E,D(E)) on L²(E;m). For u∈ D(E), we have Fukushima's decomposition: u(Xₜ)-u(X₀)=Mᵘₜ+Nᵘₜ. In this paper, we investigate the strong continuity of the generalized Feynman-Kac semigroup defined by Pᵘₜf(x)=Eₓ[eⁿᵘₜf(Xₜ)]. Let Qᵘ(f,g)=E(f,g)+E(u,fg) for f,g∈ D(E)b. Denote by J₁ the dissymmetric part of the jumping measure J of (E,D(E)). Under the assumption that J₁ is finite, we show that (Qᵘ,D(E)b) is lower semi-bounded if and only if there exists a constant α₀≥ 0 such that Pᵘₜ₂≤ eα⁰ ᵗ for every t>0. If one of these conditions holds, then (Pᵘₜ)ₜ≥₀ is strongly continuous on L²(E;m). If X is equipped with a differential structure, then this result also holds without assuming that J₁ is finite.