MATHEMATICS
Single point penalization for symmetric Levy processes
T. Abildaevab a St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, Saint Petersburg, Russian Federation
b Saint Petersburg State University
Abstract:
Weconsider a one-dimensional symmetric Levy process
$\xi(t)$,
$t\geq0$, that has local time, which we denote by
$L(t,x)$, and construct the operator
$\mathcal A+\mu\delta(x-a)$,
$\mu>0$, where
$\mathcal A$ is the generator of
$\xi(t)$, and and
$\delta(x-a)$ is the Dirac delta function at
$a\in\mathbb R$. We show that the constructed operator is the generator of
$\{U_{t\geq0}\}$,
$C_0$-semigroup on
$L_2(\mathbb R)$, which is given by
$$
(U_tf)(x)=\mathbf E f(x-\xi(t))e^{\mu L(t,x-a)},
\quad f\in L_2(\mathbb R)\cap C_b(\mathbb R),
$$
and prove the Feynman–Kac formula for the delta function-type potentials. Furthermore, we construct a family of penalized distributions
$\{\mathbf Q^\mu_{T,x}\}_{T\geq0}$ of form
$$
\mathbf Q^\mu_{T,x}=\frac{e^{\mu L(T,x-a)}}
{\mathbf E e^{\mu L(T,x-a)}}
\mathbf P_{T,x},
$$
where
$\mathbf P_{T,x}$ is the measure of the process
$\xi(t)$,
$t\leq T$. We show that this family weakly converges to a Feller process as
$T\to\infty$, study the Feynman–Kac semigroup generated by this Feller process and prove a limit theorem for the distribution of
$\xi(T)$ under
$\mathbf Q^\mu_{T,x}$.
Keywords:
stochastic processes, Levy processes, local time, Feynman–Kac formula, Feynman–Kac semigroup, penalization.
UDC:
519.214.6 Presented: I. A. IbragimovReceived: 24.07.2025
Revised: 19.08.2025
Accepted: 19.08.2025
DOI:
10.7868/S3034504925050068
