Asymptotic Properties Of The Additive Functionals

Let(ε,D(ε)) be a regular symmetric Dirichlet form on L²(E, m) which is associated with a symmetric Hunt process X=(Ω,F, (F_t)_t≥0, (X_t)_t≥0, (P_x)_x∈E). Suppose p∈D(ε) and (?) is a quasi-continuous version ofρ, then by the classical Fukushima decomposition we have(?)(X_t)-(?)(X₀)=M_t^ρ+N_t^ρP_x-a.s. q.e x∈E,where M_t^ρand N_t^ρare the martingale additive functional and continuous additive functional of zero energy respectively.In this paper we mainly study the limitation (?)(l/t)log E_x(e^N_tρ). necessary of bounded variation process, we couldn't use the results in [45] directly. Considering the generality of the process, neither could we use the methods in [44, 50] directly. Here, we change the unbounded variation process into bounded variation process by Girsanov transformation, and focus on the properties of the process after transformation. In chapter two, firstly we get that if the symmetric Hunt process satisfies some conditions, thenwhere (Q,D(ε)_b) is defined asQ(u,v)=ε(u,v)+ε(uv,ρ)D(ε)_b=D(ε)∩L^∞(E,m).Secondly, we give some examples about diffusion process andαstable like process to demonstrate the theorem above. Our results extend the results of [44, 50]. In chapter three and chapter four we discuss the properties of Brownian motion and diffusion process after Girsanov transformation respectively, here by a new method which adapt to more general process we get some new results which supplement the results of [45]. The difficulties in this paper are whether the transition density function exists after Girsanov transformation and if it does exist then whether it could be dominated by some function. According to me, though there are a large amount of documents about the quadratic forms after Girsanov transformation, there is few papers about the transition density function after the transformation. In this paper we do some work about it and get some properties of the transition density function.