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

2010/01/01 by Li Ma, Wei Sun, Ma, Li +1
Mathematics · #31C25 #60J45 #60J57 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:31C25 #msc:60J45 #msc:60J57

paper · pdf · doi:10.48550/arxiv.1001.0203

arxiv created 2010/01/05 · arxiv updated 2010/01/14

Abstract

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(X_t)-u(X₀)=M^u_t+N^u_t. In this paper, we investigate the strong continuity of the generalized Feynman-Kac semigroup defined by P^u_tf(x)=E_x[e^N^u_tf(X_t)]. Let Q^u(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 (Q^u,D(E)_b) is lower semi-bounded if and only if there exists a constant α₀≥ 0 such that ‖P^u_t‖₂≤ e^{α₀ t} for every t>0. If one of these conditions holds, then (P^u_t)_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.

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