Let $ (X_t) _ {t in[0,1]} $ to be a continuous Markov process with a state space $ mathbb R $, and with a local time process $ ( ell (x)) _ {x in mathbb R} $:

$$ int_0 ^ 1f (X_t) ~ dt = int_ mathbb R ell (x) f (x) ~ dx. $$

We can set the local time of self-intersection of $ X $ as

$$ gamma: = int_ mathbb R ell (x) ^ 2 ~ dx, $$

that we can represent intuitively as

$$ gamma = int_ {[0,1]^ 2} delta (X_t-X_s) ~ dsdt, $$

that is to say the number of times that the path of $ X $ intersects with himself.

Question.Are there any results regarding the exponential moments finished for the local intersection time ofgeneralunidimensional Markov process, that is to say

$$ mathbb E[e^{alphagamma}]0 $$

or

$$ mathbb E[e^{alphagamma}]

begin {cases}

< infty & text {if} alphaalpha_0

end {cases} $$

for some critical exponents $ alpha_0> $ 0.

For example, it is well known that $ X $ is the Brownian 1D motion, so there are exponential moments of all orders, and if $ X $ is the 2D Brownian motion, then the exponential moments explode after a critical exponent.

Before attempting to extend these results, I wonder if there are more general assertions of this type in the literature for other 1D processes, such as (but not limited to) Brownian bridges, reflecting the Brownian movements (at one or two limits). Brownian periodic movements (on intervals), etc.