Previous theorems established that there is zero probability of a diffusion blowup such that for all where

and is the blowup time for the unperturbed system. From the perspective of consistency, the same conclusion should also follow from a Fokker-Planck equation, or Kolmogorov’s second equation, which is essentially an equation for conservation of probability. For a pure diffusion with non drift, the FP equation will reduce to a diffusion equation for the underlying probability density function.

Let be a generic diffusion satisfying athe nonlinear SDE

with and

some initial data with . A pure drift-free diffusion is then

The corresponding nonlinear Fokker-Planck equation for a probability density is (refs)

Taking the Ito interpretation

The Fokker-Planck equation now becomes (ref)

If then this is a nonlinear diffusion equation for

The general solution of a nonlinear Fokker-Planck equation cannot usually be found but a stationary solution is possible in the infinite-time relaxation limit such that

so that

where G is the generator of the diffusion as in (-). The stationary solution at for the probability density is

where $ latex C$ is a constant. The normalised solution is

so that

Integrating over the probability density gives

The probability that at is then

For a pure diffusion with the solution reduces to

T

Technical details of the FP equation are found in (refs.)

Given the SDE defined for all with initial matter density the Fokker-Planck diffusion equation now becomes (ref)

For a stationary solution in the limit as

=

The probability that the diffusion is finite at is then

This result can also be computed explicitly. Given the SDE defined for all

with initial matter density the Fokker-Planck diffusion equation is as before(ref)

For a stationary solution in the limit as

The stationary solution for the Fokker-Planck probability density is

The probability that the diffusion is finite at is now given by (-) as

Performing the integral i (-)

=

The first term vanishes at since so that (-) reduces to

Hence, the diffusion remains finite in the infinite-time relaxation limit.

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