34
4 Stochastic Processes
with dx = σ ξ
√
dτ , where we assume that the distributions of jumps ξ is described
by the Gaussian from (4.8). In the next step we expand ψ(x, t) up to second order
in the spatial variable x and find
ψ(x, t + dτ ) =
∞
−∞
ψ(x, t) −
∂ψ
∂ x
σ ξ
√
dτ +
1
2
∂
2
ψ
∂ x 2 σ
2
ξ
2 dτ
φ(ξ )dξ . (4.14)
Since the Gaussian φ(ξ ) is normalized, has mean zero and unit rms, only the first
and third term in the square bracket remain after performing the integral over dξ.
Equating the resulting equation with the left-hand side of (4.12), we finally obtain
∂ψ
∂t
=
σ
2
2
∂
2
ψ
∂ x 2 ,
(4.15)
which is referred to as the the diffusion equation.
A special solution G(x, t) of this linear partial differential equation can be found
by introducing its Fourier-transform ˜
G(k, t) with respect to the spatial variable x
G(x, t) =
1
2π
∞
−∞
e
−ikx ˜
G(k, t)dk
(4.16)
and inserting it in (4.15), which leads to
∂ ˜
G
∂t
= −
σ
2
2
k
2 ˜
G
or
d ˜
G
˜
G
= −
σ
2 k
2
2
dt ,
(4.17)
where we separated the variables. This equation is easily integrated and we obtain
˜
G(k, t) = ˜
G 0 e
−σ
2 k
2 t/2 with an integration constant ˜
G 0 . Using (4.16) to calculate the
solution G(x, t) we find
G(x, t) =
˜
G 0
2π
∞
−∞
e
−ikx−σ
2 k
2 t/2 dk =
˜
G 0
√
2πσ 2 t
e
−x
2 /2σ
2 t
,
(4.18)
where we solved the integral by completing the square in the exponent and then using
∞
−∞ e
−ay
2 dy =
√
π/a.
This solution G(x, t) has the well-known behavior of a spreading-out Gaussian
already used in Sect. 4.1 for (4.1). Note also that G(x, t) is a special solution that
vanishes at x → ±∞. Moreover, considering the solution in the limit t → 0, we find
that the Gaussian becomes increasingly more pointed and approaches Dirac’s deltafunction in the limit. It therefore mimics the spreading of, for example, heat injected at
a single point. If we were to inject equal amounts of heat at two separate points x 1 and
x 2 at the same time, we could model it as the spreading of two fundamental solutions,
4 Stochastic Processes
with dx = σ ξ
√
dτ , where we assume that the distributions of jumps ξ is described
by the Gaussian from (4.8). In the next step we expand ψ(x, t) up to second order
in the spatial variable x and find
ψ(x, t + dτ ) =
∞
−∞
ψ(x, t) −
∂ψ
∂ x
σ ξ
√
dτ +
1
2
∂
2
ψ
∂ x 2 σ
2
ξ
2 dτ
φ(ξ )dξ . (4.14)
Since the Gaussian φ(ξ ) is normalized, has mean zero and unit rms, only the first
and third term in the square bracket remain after performing the integral over dξ.
Equating the resulting equation with the left-hand side of (4.12), we finally obtain
∂ψ
∂t
=
σ
2
2
∂
2
ψ
∂ x 2 ,
(4.15)
which is referred to as the the diffusion equation.
A special solution G(x, t) of this linear partial differential equation can be found
by introducing its Fourier-transform ˜
G(k, t) with respect to the spatial variable x
G(x, t) =
1
2π
∞
−∞
e
−ikx ˜
G(k, t)dk
(4.16)
and inserting it in (4.15), which leads to
∂ ˜
G
∂t
= −
σ
2
2
k
2 ˜
G
or
d ˜
G
˜
G
= −
σ
2 k
2
2
dt ,
(4.17)
where we separated the variables. This equation is easily integrated and we obtain
˜
G(k, t) = ˜
G 0 e
−σ
2 k
2 t/2 with an integration constant ˜
G 0 . Using (4.16) to calculate the
solution G(x, t) we find
G(x, t) =
˜
G 0
2π
∞
−∞
e
−ikx−σ
2 k
2 t/2 dk =
˜
G 0
√
2πσ 2 t
e
−x
2 /2σ
2 t
,
(4.18)
where we solved the integral by completing the square in the exponent and then using
∞
−∞ e
−ay
2 dy =
√
π/a.
This solution G(x, t) has the well-known behavior of a spreading-out Gaussian
already used in Sect. 4.1 for (4.1). Note also that G(x, t) is a special solution that
vanishes at x → ±∞. Moreover, considering the solution in the limit t → 0, we find
that the Gaussian becomes increasingly more pointed and approaches Dirac’s deltafunction in the limit. It therefore mimics the spreading of, for example, heat injected at
a single point. If we were to inject equal amounts of heat at two separate points x 1 and
x 2 at the same time, we could model it as the spreading of two fundamental solutions,