SINGLE POINT PENALIZATION FOR SYMMETRIC LEVY PROCESSES

We consider a one-dimensional symmetric Levy process ξ(t), t ≥ 0, that has local time, which we denote by L(t, x), and construct the operator A + μ δ(x − a), μ > 0, where A is the generator of ξ(t), and δ(x − a) is the Dirac delta function at a ∈ ℝ. We show that the constructed operator is the generator of (U_t)_{t ≥ 0} – C₀-semigroup on L₂(ℝ), which is given by (U_t f)(x) = E f (x − ξ(t)) e^{μ L(t,x−a)}, f ∈ L₂(ℝ) ∩ C_b(ℝ), and prove the Feynman–Kac formula for the delta function-type potentials. Furthermore, we construct a family of penalized distributions {Q_T,x^μ}_{T ≥ 0} of form Q_T,x^μ = e^{μ L(T,x−a)} / Ee^{μ L(T,x−a)} P_T,x, where P_T,x is the measure of the process ξ(t), t ≤ T. We show that this family weakly converges to a Feller process as T → ∞, study the Feynman–Kac semigroup generated by this Feller process and prove a limit theorem for the distribution of ξ(T) under Q_T,x^.

stochastic processes, Levy processes, local time, Feynman–Kac formula, Feynman–Kac semigroup, penalization