Leave $ (X_t) _ {t in[0,1]$ be a continuous Markov process with state space in $ 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 define the local time of auto-intersection of $ X $ as

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

that we can intuitively represent as

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

that is, the number of times the path of $ X $ it crosses itself

Question.Are there results with respect to the finite exponential moments for the auto-intersection of the local time ofgeneralOne-dimensional Markov processes, that is,

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

or

$$ mathbb E[e^{alphagamma}]

begin {cases}

< infty & text {if} alphaalpha_0

end {cases} $$

by some critical exponent $ alpha_0> 0 $.

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

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