Collatz conjecture: The summary of stochastic analysis for collatz sequences

I have applied Brownian motin model with costant drift to collatz sequences, and derived some fornula.

We thnk about the collatz sequences of natural number $n$ and stopping times $T_s$ .

"Stopping times" means operation times the collatz sequences arrives to $1$ .

Definition:

$\log$ :natural logarithm

$s=\frac{2}{\log3}\sim1.82$

$v=\alpha\frac{\log(\frac{4}{3})}{\log3}\sim0.262\alpha$

$\alpha$ is " $1$ " in stochastic model, but almost " $1.015$ " from simulation result.

The probability density function of stopping times for "n" is

$p_{n}(T_{s})=\frac{2\sqrt{3+v}}{\sqrt{2\pi(2T_{s}+(s\log(n)))^3)}}exp(-\frac{(3s\log(n)-2vT_{s})^2}{2(3+v)(2T_{s}+slog(n))})$

The expected value and variance of stopping times are

$E[T_{s}]=\frac{3s\log(n)}{2v}$

$V[T_{s}]=\frac{(3+v)^2slog(n)}{4v^3}$

Additionaly,

$\frac{1}{n}\sum_{k=1}^{n}E_n[T_{s}]\sim\frac{3s}{2v}(\log(n)-1)$

The maximum value of the collatz sequences for numbers included in $[1,n]$ is

$M\sim n^{1+\frac{1}{2vs}}$

The maximum stopping times for numbers included in $[1,n]$ is

$T_M\sim\gamma s\log(n)-\frac{3\gamma}{2\beta}\log(\gamma)-\frac{\gamma}{2\beta}log(s\log(n))$ eq(1)

Here,

$\gamma=\frac{3}{v}+\frac{3+v}{sv^2}-\frac{1}{2}$

$\beta=\frac{v^2}{3+v}\gamma$

About the comparison result of this model and actual caliculation, please see previous articles.

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