Assume the local time is $L(t,y)$ and we know $P_x(L(t,y) \in d\tau)$ where $P_x$ denotes the probability measure for a stochastic process starts at $x$. Can we then derive the transition density function $p(t,x,y)$ (from x to y after time t)? Or what is the relationship between $P_x(L(t,y) \in d\tau)$ and transition densitity function?
Intuitively, I feels like $\int^\infty_0 P_x(L(t,y) \in d\tau)d\tau$ has some direct connection with $p(t,x,y)$. Any kind of adivce is appreciated!
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