Given the generator for the pure density function diffusion on the partition

where

for initial data . It is possible to define and find a Lyapunov function related to boundedness and non-explosion of the density function diffusion(refs). A function is a Lyapunov function if

1. .

2

3. Given the generator for the diffusion, constants such that

Given the generator for the matter density diffusion for all suppose there exists a Lyapunov function then the SDE does not explode and the density diffusion is finite and bounded for all

Proof:

For , Define the hitting times

Now Ito’s formula is applied to

+

Since and are continuous and bounded on a ball of radius , the local martingale is a martingale. For the generator G of the pure diffusion

Since and are continuous and bounded on a ball os radius , the local martingale is a martingale. For the generator G of the pure diffusion

+

Taking the expectation this reduces to

=

+

From Gronwall’s inequality it follows that

Letting and observing that does not explode before comoving time

<

But

=

Now

so that

taking the limit or gives

or and so is always less than

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