Let , and is the comoving proper time for the density function blowup. And The original nonlinear ODE defined over is given by (-) so that . In the stochastic extension, the SDE for the density function diffusion for all is with initial data with . Then for the coefficient the following hold

(1) For all , there is a such that the Lipschitz condition

holds for all and

(2) There is a such that the linear growth condition of the coefficient holds for all and , namely

There exists a strong solution for at least all such that for some

<=

To prove (1), let then then

which is

or

giving

For finite but

which holds for all finite density functions since . To prove the linear growth condition (2). Let then

which is

To prove (3), The second moment over is

=+ =

+

Using the result of (-) this reduces to

Using the Ito isometry and the Ito calculas whereby , this gives

From the growth condition for the coefficient for any such that

or using (67) and .

We then have:

<=

Recall the Gronwall inequality. Let

be continuous functions where exists then if

and then

If then

Then

Given

such that

so (-) is established. Taking then .

A stronger result is established in the following proposition. It will be useful to note the following classical results from functional analysis. Let and be conjugate exponents such that with . Let

<=

The following result gives further support to (-) in that and for all .

: Let or and

with . If the coefficient has linear growth then for some then

does not blowup for all or . The stochastic expectations are bounded from above such that for any $\exists (q,Q)>0$ such that

and to all orders

Then in the limit as

If is a solution of the SDE for all then . We consider the general case for all and then consider the specific cases for and for the mean and variance. There is a constant $q_{p}$ such that

The expectation is then

+

Applying the Buckholder-Gundy_Davis inequality, such that

*

From the conditions and $latex |\Psi[D]|^{2}=D^{4}(D-1) <= C(1+|C|)^{2}$ we have

\

From the inequalities (-)

giving

From the Gronwall inequality it follows that

The cases for follow. In the limit as

and the results are estabished.

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