Noise-driven dynamical systems - Noise-induced Phenomena in the Environmental Sciences - page 36

Box 2.3: Direct derivation of the master equation for the processes
driven by WSN

In this box, we determine the master equation for process ( 2.57 ) driven by WSN, using
the approach by Cox and Miller ( 1965 ). We first transform generic multiplicative
Langevin equation ( 2.57 ) into an additive equation by using the transformation
z = φ φ )d φ , 1 / g ( ρ ( z ) = f [ φ ( z )] / g [ φ ( z )]. The Langevin equation for z is

d z
d t = ρ

( z ) + ξ sn . (B2.3-1)

In the infinitesimal interval d t the probability of having no jumps is (1 − λ d t ) +◦ (d t );

in this case, at time z + d t , the variable z will take the value

z ( t + d t ) = z ( t ) − z , (B2.3-2)

where

t + d t

z =− ρ [ z ( τ )]d τ = ρ [ z ( τ )]d t +◦ (d t ) , (B2.3-3)

t

where ) is a high-order infinitesimal. The probability that a jump occurs in the same

interval d t is

◦ ( ·

λ d t +◦ (d t ). In this case,

z ( t + d t ) = z ( t ) + h + z , (B2.3-4)

where h is the size of the jump, with distribution p H ( h ). As a consequence, the
probability that the process takes a value in [ z , z + d z ] at time t + d t can be expressed as

p ( z , t + d t )d z = (1 − λ d t ) p ( z + z , t )d( z + z )

d t z
0

p ( z + z , z )d( z + z )d z

+ λ t ) p H ( z − , (B2.3-5)

where the second term on the right-hand side accounts for the case in which the process
reaches the z level because of a jump induced by the noise. Now, if expression ( B2.3-3 )
for z is substituted into Eq. ( B2.3-5 ) and all terms of the order ◦ (d t ) are neglected, we

obtain

, + = − λ − ρ , − ρ

p ( z t d t )d z ) (1 d t ) p [ z ( z )d t t ]d[ z ( z )d t ]

d t z
0

p [ z − ρ ( z )d t z )d[ z − ρ ( z )d t ]d z

+ λ , −

t ] p H ( z

d t ) p ( z t ) 1 ( z )d t

( z )d t ∂
∂

∂
∂

= (1 − λ , t ) − ρ

z p ( z

, −

z ρ

d t d z z
0

p ( z , z )d z .

+ λ t ) p H ( z − (B2.3-6)

Finally, dividing by d z , subtracting p ( z , t ) from both sides, dividing by d t , and taking

the limit as d t → 0, we obtain (see Rodriguez-Iturbe et al. , 1999b ) the master equation:

z

∂ p ( z , t )

=− ∂
∂

p ( z , z )d z .

z [ p ( z

, t ) ρ ( z )] − λ p ( z , t ) + λ t ) p H ( z − (B2.3-7)

∂ t

0