Operator exponentials
Boundary conditions
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Operator exponentials in biology
Adam Bobrowski
Polish Academy of Sciences
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Neurotransmiters
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
The most important function in mathematics
W. Rudin (Real and Complex Analysis):
„The exponential function is the most important function in
mathematics. It is defined by:
ez =
∞
X
zn
n=0
n!
,
where z is a complex number.”
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
(1)
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
The most important function in mathematics
W. Rudin (Real and Complex Analysis):
„The exponential function is the most important function in
mathematics. It is defined by:
ez =
∞
X
zn
n=0
n!
,
where z is a complex number.”
We will think of
R+ := [0, ∞) 3 t 7→ etz ,
or rather of
R+ 3 t 7→ etA ,
Adam Bobrowski
A - operator.
CIMPA, Muizenberg, 22.07.2013
(1)
Operator exponentials
Boundary conditions
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Neurotransmiters
Outline
Operator exponentials (one-parameter semigroups of
operators).
Boundary conditions for semigroups:
Wright-Fisher model of population genetics,
McKendrick model of an age-structured population.
Lord Kelvin’s method of images.
Modelling neurotransmitters.
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Operator
A function having functions as arguments and values.
Examples:
f →
7 f 0,
f →
7 f 00 ,
f 7→ f 2 ,
f 7→ g , g (x) = f (x + a), x ∈ R, a − given,
(f (1), . . . , f (n)) 7→ (f (1), . . . , f (n))An×n .
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
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Neurotransmiters
Operator
A function having functions as arguments and values.
Examples:
f →
7 f 0,
f →
7 f 00 ,
f 7→ f 2 ,
f 7→ g , g (x) = f (x + a), x ∈ R, a − given,
(f (1), . . . , f (n)) 7→ (f (1), . . . , f (n))An×n .
Linear operator:
A(αf + βg ) = αAf + βAg ,
f , g ∈ D(A), α, β − scalars .
Above - only one operator is not linear (note: no parentheses).
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
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Continuous operator:
Arguments and values in a Banach space B.
Banach space (continuous functions, p-integrable
functions, etc.):
linear structure,
compatible norm,
completness (Cauchy sequences converge).
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Neurotransmiters
Operator exponentials
Boundary conditions
Pause
Continuous operator:
Arguments and values in a Banach space B.
Banach space (continuous functions, p-integrable
functions, etc.):
linear structure,
compatible norm,
completness (Cauchy sequences converge).
Continuity: A : B → B (above: three operators are
continuous)
kfn − f k → 0 ⇒ kAfn − Af k.
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Neurotransmiters
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Continuous operator:
Arguments and values in a Banach space B.
Banach space (continuous functions, p-integrable
functions, etc.):
linear structure,
compatible norm,
completness (Cauchy sequences converge).
Continuity: A : B → B (above: three operators are
continuous)
kfn − f k → 0 ⇒ kAfn − Af k.
For linear operators continuity equivalent to boundedness:
∃M>0 ∀f ∈B
kAf k ¬ Mkf k.
Smallest M denoted kAk; the space L(B) of bounded
linear operators on B is a Banach space.
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
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Neurotransmiters
Operator exponential for a bounded operator
Bounded A given. How to define etA ?
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Operator exponential for a bounded operator
Bounded A given. How to define etA ?
The key to (??) - absolute convergence.
In a Banach space an absolutely convergent series
converges:
k
N
X
fk k ¬
k=n
Adam Bobrowski
N
X
kfk k.
k=n
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
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Neurotransmiters
Operator exponential for a bounded operator
Bounded A given. How to define etA ?
The key to (??) - absolute convergence.
In a Banach space an absolutely convergent series
converges:
k
N
X
fk k ¬
k=n
We define
etA :=
kfk k.
k=n
∞ n n
X
t A
n=0
N
X
n!
t ∈ R.
,
n
n
n
n
t A ∞ |t| kAk
(This is possible since ∞
, by
n=0 n! ¬
n=0
n!
kABk ¬ kAk kBk). In a complex space, we can in fact
define ezA .
P
Adam Bobrowski
P
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
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Neurotransmiters
Proporties and examples
Good properties: if A and B commute (AB = BA), then
eA+B = eA eB = eB eA .
In particular,
e(t+s)A = etA esA ,
e−tA = (etA )−1 ,
Adam Bobrowski
et(A−I ) = e−t etA .
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Proporties and examples
Good properties: if A and B commute (AB = BA), then
eA+B = eA eB = eB eA .
In particular,
e−tA = (etA )−1 ,
e(t+s)A = etA esA ,
et(A−I ) = e−t etA .
Example 1. a ­ 0, b ­ 0, a + b > 0.
!
−a a
A=
,
b −b
!
e
tA
1
b + ae−(a+b)t a − ae−(a+b)t
=
.
a + b b − be−(a+b)t a + be−(a+b)t
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
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Neurotransmiters
Example 2.
B = C [0, ∞], Af (x) = a[f (x + 1) − f (x)], x ­ 0, a > 0 given.
etA ?
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
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Neurotransmiters
Example 2.
B = C [0, ∞], Af (x) = a[f (x + 1) − f (x)], x ­ 0, a > 0 given.
etA f (x) = e−at et(A+aI ) f (x)
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Example 2.
B = C [0, ∞], Af (x) = a[f (x + 1) − f (x)], x ­ 0, a > 0 given.
etA f (x) = e−at et(A+aI ) f (x) =
∞
X
n=0
Adam Bobrowski
e−at
an t n
f (x + n)
n!
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Example 2.
B = C [0, ∞], Af (x) = a[f (x + 1) − f (x)], x ­ 0, a > 0 given.
etA f (x) = e−at et(A+aI ) f (x) =
∞
X
n=0
e−at
an t n
f (x + n)
n!
= E f (x + N(t)).
etA , t ­ 0 describes Poisson process.
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Example 3.
B = C [0, ∞], Af = f 0 , D(A) = C 1 [0, ∞]. A – unbounded.
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
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Neurotransmiters
Example 3.
B = C [0, ∞], Af = f 0 , D(A) = C 1 [0, ∞]. A – unbounded.
fn (x) = e−nx , kfn k = 1, kAfn k = n.
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Example 3.
B = C [0, ∞], Af = f 0 , D(A) = C 1 [0, ∞]. A – unbounded. For
f ∈ C ∞ [0, ∞]:
?
etA f (x) =
∞ n n
X
t A
n=0
n!
f (x)
= f (x) + tf 0 (x) +
t 2 00
t3
f (x) + f 000 (x) + ...
2
3!
=
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Example 3.
B = C [0, ∞], Af = f 0 , D(A) = C 1 [0, ∞]. A – unbounded. For
f ∈ C ∞ [0, ∞]:
?
etA f (x) =
∞ n n
X
t A
n=0
n!
f (x)
t 2 00
t3
f (x) + f 000 (x) + ...
2
3!
(provided f is analytic).
= f (x) + tf 0 (x) +
= f (x + t),
Deterministic movement to the right (shifts to the left).
e−tA undefined - shifts etA do not have inverses.
Differential operators of 1st order – deterministic
movements
etA = limn→∞ (I − tA
)−n .
n
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Where does the semigroup property come from?
What is the real reason for:
etA esA = e(s+t)A ,
Adam Bobrowski
s, t ­ 0?
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Where does the semigroup property come from?
What is the real reason for:
etA esA = e(s+t)A ,
s, t ­ 0?
A – bounded, f ∈ B, u(t) = etA f :
u 0 (t) = AetA f
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Where does the semigroup property come from?
What is the real reason for:
etA esA = e(s+t)A ,
s, t ­ 0?
A – bounded, f ∈ B, u(t) = etA f :
u 0 (t) = AetA f = Au(t), t ­ 0
u(0) = f .
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Where does the semigroup property come from?
What is the real reason for:
etA esA = e(s+t)A ,
s, t ­ 0?
A – bounded, f ∈ B, u(t) = etA f :
Trajectory t 7→ etA f
is a solution to the differential equation
u 0 (t) = Au(t), t ­ 0
u(0) = f ,
and this solution is unique.
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Where does the semigroup property come from?
(cont.)
Fix s ­ 0 and f ∈ B. Consider v (t) = e(t+s)A f = u(t + s).
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Where does the semigroup property come from?
(cont.)
Fix s ­ 0 and f ∈ B. Consider v (t) = e(t+s)A f = u(t + s). We
have
v 0 (t) = u 0 (t + s) = Au(t + s) = Av (t), t ­ 0.
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Where does the semigroup property come from?
(cont.)
Fix s ­ 0 and f ∈ B. Consider v (t) = e(t+s)A f = u(t + s). We
have
v 0 (t) = u 0 (t + s) = Au(t + s) = Av (t), t ­ 0.
Since v (0) = u(s) = esA f , uniqueness of solutions forces
e(t+s)A f = v (t) = etA v (0) = etA esA f .
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Semigroup property reflects uniqueness of solutions
If A is a (not necessarily bounded) operator such that the
differential equation (in a Banach space)
du(t)
= Au(t), t ­ 0,
dt
u(0) = f
has exactly one solution uf for f in a dense set, and if this
solution depends continuously on f , then the formula
etA f = uf (t)
defines an exponential function for A (a semigroup of
operators).
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Back to Example 3
A partial differential equation:
∂u(t, x)
∂u(t, x)
=
, t ­ 0,
u(0, x) = f (x), x ∈ [0, ∞),
∂t
∂x
has exactly one solution for f ∈ C 1 [0, ∞], given by
u(t, x) = f (x + t).
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Back to Example 3
A partial differential equation:
∂u(t, x)
∂u(t, x)
=
, t ­ 0,
u(0, x) = f (x), x ∈ [0, ∞),
∂t
∂x
has exactly one solution for f ∈ C 1 [0, ∞], given by
u(t, x) = f (x + t).
This equation is identical to the ordinary differential equation
du(t)
= Au(t), t ­ 0, u(0) = f ∈ C 1 [0, ∞],
dt
d
in the Banach space C [0, ∞], where A = dx
. This gives
d
et dx f (x) = f (x + t).
(Note that for f 6∈ C 1 [0, ∞] - generalized solution.)
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
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Neurotransmiters
Example 4
The PDE:
∂u(t, x)
1 ∂ 2 u(t, x)
, t ­ 0,
=
∂t
2 ∂x 2
u(0, x) = f (x), x ∈ R,
has the unique solution for f ∈ C 2 [−∞, ∞] given by
1 Z ∞ − y2
u(t, x) = √
e 2t f (x + y ) dy .
2πt −∞
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Example 4
The PDE:
∂u(t, x)
1 ∂ 2 u(t, x)
, t ­ 0,
=
∂t
2 ∂x 2
u(0, x) = f (x), x ∈ R,
has the unique solution for f ∈ C 2 [−∞, ∞] given by
1 Z ∞ − y2
u(t, x) = √
e 2t f (x + y ) dy .
2πt −∞
Hence, arguing as above,
t 12
e
d2
dx 2
f (x) = √
1 Z ∞ − y2
e 2t f (x + y ) dy = E f (x + w (t));
2πt −∞
this is the Brownian motion semigroup.
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Summary
one operator (usually ubounded) - entire dynamics,
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Neurotransmiters
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Summary
one operator (usually ubounded) - entire dynamics,
for continuous functions a > 0 and b
Af (x) = a(x)f 00 (x) + b(x)f 0 (x) + integral operator, x ∈ R
‘composition’ of three processes:
diffusion with variance depending on position (x),
deterministic movement along trajectories of the ODE
x 0 (t) = b(x(t)), and
jumps (for example, Poisson process),
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Summary
one operator (usually ubounded) - entire dynamics,
for continuous functions a > 0 and b
Af (x) = a(x)f 00 (x) + b(x)f 0 (x) + integral operator, x ∈ R
‘composition’ of three processes:
diffusion with variance depending on position (x),
deterministic movement along trajectories of the ODE
x 0 (t) = b(x(t)), and
jumps (for example, Poisson process),
however, ‘operator’ = ‘map’ + ‘domain’!
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Classical boundary conditions
Heat equation on the right half-axis:
∂u(t, x)
1 ∂ 2 u(t, x)
,
=
∂t
2 ∂x 2
x ­ 0, t ­ 0, u(0, x) = f (x).
2
d
2
0
A = 12 dx
2 , D(A) = {f ∈ C [0, ∞] : f (0) = 0} Neuman
b.c. (reflection),
D(A) = {f ∈ C 2 [0, ∞] : f (0) = 0} Dirichlet b.c. (killed
Brownian motion),
D(A) = {f ∈ C 2 [0, ∞] : f 0 (0) = γf (0)} Robin b.c.
(elastic B.m.) γ = const. > 0,
boundary condition ←→ domain of A.
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Classical boundary conditions
Heat equation on the right half-axis:
∂u(t, x)
1 ∂ 2 u(t, x)
,
=
∂t
2 ∂x 2
x ­ 0, t ­ 0, u(0, x) = f (x).
2
d
2
0
A = 12 dx
2 , D(A) = {f ∈ C [0, ∞] : f (0) = 0} Neuman
b.c. (reflection),
D(A) = {f ∈ C 2 [0, ∞] : f (0) = 0} Dirichlet b.c. (killed
Brownian motion),
D(A) = {f ∈ C 2 [0, ∞] : f 0 (0) = γf (0)} Robin b.c.
(elastic B.m.) γ = const. > 0,
boundary condition ←→ domain of A.
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
W. Feller on R. A. Fisher and S. Wright
The theory of evolution provides examples of stochastic
processes which have not yet been treated systematically.
Existing methods are ... due to R. A. Fisher and S. Wright.
They have ... with great ingenuity and admirable
resourcefulness ... discovered ... facts of the general theory
stochastic processes.
Essential part of Wright’s theory is equivalent to assuming a
certain diffusion equation for gene frequency ...
This diffusion equation ... is of a singular type and lead to new
types of boundary conditions.
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Wright-Fisher model
Pause
Neurotransmiters
Operator exponentials
Boundary conditions
Wright-Fisher model
Pause
Neurotransmiters
Operator exponentials
Boundary conditions
Pause
Wright-Fisher model
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Neurotransmiters
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Diffusion approximation of the Wright-Fisher
model
Af (x) = a(x)f 00 (x) + b(x)f 0 (x), x ∈ (0, 1).
At the boundary, the process stops Af (0) = Af (1) = 0.
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Wright-Fisher model with mutations
Process may come back from the boundary!
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Neurotransmiters
Operator exponentials
Boundary conditions
Pause
Classification of boundary points (Feller)
boundary type
regular
exit
entrance
natural
accessible?
Y
Y
N
N
Adam Bobrowski
absorbing?
N
Y
N
Y
CIMPA, Muizenberg, 22.07.2013
Neurotransmiters
Operator exponentials
Boundary conditions
Boundary conditions for
A=
00
Pause
d2
dx 2
at 0 (regular!)
d2
w C [0, ∞],
dx 2
0
af (0) − bf (0) + cf (0) − d
Z
R+
∗
f dµ = 0,
a, b, c, d ­ 0, c ­ d, µ probabilistic measure
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Neurotransmiters
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Elementary return Brownian motion
1
Af = f 00 ,
2
af 00 (0) + cf (0) − d
x
t
Z
R+
∗
f dµ = 0.
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Elementary return Brownian motion
1
Af = f 00 ,
2
af 00 (0) + cf (0) − d
x
τ
c
P(τ > t) = e − a t
t
Z
R+
∗
f dµ = 0.
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Elementary return Brownian motion
1
Af = f 00 ,
2
af 00 (0) + cf (0) − d
x
µ
p=
d
c
τ
c
P(τ > t) = e − a t
t
Z
R+
∗
f dµ = 0.
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Elementary return Brownian motion
1
Af = f 00 ,
2
af 00 (0) + cf (0) − d
Z
R+
∗
f dµ = 0.
x
µ
p=
d
c
τ
c
P(τ > t) = e − a t
Adam Bobrowski
p =1−
d
c
- particle ‘dies’
t
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Elastic boundary
Feller 1953: ‘description very desirable’; P. Lévy w 1948
Z
1
bf 0 (0) = cf (0) − d
f dµ.
Af = f 00 ,
2
R+
∗
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Neurotransmiters
Operator exponentials
Boundary conditions
Pause
Elastic boundary
Feller 1953: ‘description very desirable’; P. Lévy w 1948
Z
1
bf 0 (0) = cf (0) − d
f dµ.
Af = f 00 ,
2
R+
∗
x
t
Neurotransmiters
Operator exponentials
Boundary conditions
Pause
Elastic boundary
Feller 1953: ‘description very desirable’; P. Lévy w 1948
Z
1
bf 0 (0) = cf (0) − d
f dµ.
Af = f 00 ,
2
R+
∗
x
τ
c +
P(τ > t) = e − b t
t
Neurotransmiters
Operator exponentials
Boundary conditions
Pause
Elastic boundary
Feller 1953: ‘description very desirable’; P. Lévy w 1948
Z
1
bf 0 (0) = cf (0) − d
f dµ.
Af = f 00 ,
2
R+
∗
x
µ
τ
c +
P(τ > t) = e − b t
t
Neurotransmiters
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Elastic boundary
Feller 1953: ‘description very desirable’; P. Lévy w 1948
Z
1
bf 0 (0) = cf (0) − d
f dµ.
Af = f 00 ,
2
R+
∗
x
µ
p =1−
τ
c +
P(τ > t) = e − b t
Adam Bobrowski
d
c
- particle ‘dies’
t
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
McKendrick model
B = L1 (0, ∞). Non-negative φ ∈ B – population density:
Z d
φ(a) da − number of individuals with age ∈ [c, d].
c
‘a’ = ‘age’.
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
McKendrick model
B = L1 (0, ∞). Non-negative φ ∈ B – population density:
Z d
φ(a) da − number of individuals with age ∈ [c, d].
c
‘a’ = ‘age’.
∂φ(t, a)
∂φ(t, a)
=−
− µ(a)φ(t, a).
∂t
∂a
Derivative in L1 , not pointwise.
Aφ = −
∂
φ − µφ
∂a
D(A) = {abs. cont. functions with φ(0) =
Z ∞
b(a)φ(a) da.}
0
Adam Bobrowski
(2)
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Lord Kelvin’s method of images
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Neurotransmiters
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Feller’s construction of reflected Brownian motion
f ∈ C [0, ∞]
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Feller’s construction of reflected Brownian motion
Ef ∈ C [−∞, ∞]
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Feller’s construction of reflected Brownian motion
etA Ef ∈ C [−∞, ∞]
1 Z ∞ − y2
tA
e 2t g (x + y ) dy
e g (x) = √
2πt −∞
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Feller’s construction of reflected Brownian motion
etAr f = RetA Ef
Lord Kelvin’s method of images
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Same for any boundary condition (A.B. 2010)
Find ‘distorted’ images.
(Nearly) explicit formulae!
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
(Nearly) arbitrary b.c.
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Neurotransmiters
Operator exponentials
Boundary conditions
Pause
(Nearly) arbitrary b.c.
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Neurotransmiters
Operator exponentials
Boundary conditions
Pause
(Nearly) arbitrary b.c.
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Neurotransmiters
Operator exponentials
Boundary conditions
Pause
(Nearly) arbitrary b.c.
etAB f = RetA EB f .
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Neurotransmiters
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Convergence of semigroups
etA1
etA2
Adam Bobrowski
et?
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
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Neurotransmiters
A toy example
C [0, 1], Af = f 0 ,
f ∈ D(A) ⇐⇒ f ∈ C 1 [0, 1], f 0 (1) = a[f (0) − f (1)].
jump intensity = a
x =0
−→
v =1
x =1
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
A toy example
C [0, 1], Af = f 0 ,
f ∈ D(A) ⇐⇒ f ∈ C 1 [0, 1], f 0 (1) = a[f (0) − f (1)].
jump intensity = a
x =0
−→
v =1
x =1 a→∞
Adam Bobrowski
0=1
v =1
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Let’s continue to play
C [0, 1], Af = f 0 ,
f ∈ D(A) ⇐⇒ f ∈ C 1 [0, 1], f 0 (1) = a[pf (0)+(1−p)f (0.5)−f (1)].
jump intensity = a
x =0
−→
v =1
x =1
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Let’s continue to play
C [0, 1], Af = f 0 ,
f ∈ D(A) ⇐⇒ f ∈ C 1 [0, 1], f 0 (1) = a[pf (0)+(1−p)f (0.5)−f (1)].
jump intensity = a
x =0
−→
v =1
x = 1a → ∞
Adam Bobrowski
v =1
v =1
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
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Neurotransmiters
ODE model (Aristizabal and Glavinovič, 2004)
R3
Synthesis
R2
R1
C3
E
R0
C2
C1
Stimulus
Large pool


U10
 0
U2 
U30
Small pool

− R01C1 −

Immediately available pool


0
U1
  

0
= A U2  + 
,
1
(E
−
U
)
U3
3
R3 C3

A=

1
R1 C2
1
R1 C 1
0
1
R1 C1
− R21C2 −
1
R2 C3
0
1
R1 C2


1
R2 C 2  .
− R21C3
Ui – voltage on the ith capacitor (neurotransmitter’s density in the
ith pool), Ci -capacity, Ri1Cj –replenishment rate.
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
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Neurotransmiters
PDE model (Bielecki and Kalita, 2008)
σ1
σ3
immediately available
σ2
small pool
large pool
A = σ4
Brownian motion in 3D
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Connection?
How to do that?
ρ0 = σ4ρ + production,
↓?


U10
 0
U2 
U30
− R 1C
0 1
A=

U1

 
= A U2  + production,
U3
− R 1C
1 1
1
R1 C 2
0
1
R1 C1
− R 1C − R 1C
2 2
1 2
1
R2 C3
0
1
R2 C 2
− R 1C
2 3
!
.
Probabilistically: how to approximate a Markov chain by a
diffusion?
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
A connection
fast diffusion in pools,
transmission rates - very low,
appropriate ‘tunning’ of the rates of diffusion and
transmission −→ communication of ‘lumped’ points as
states of a Markov chains.
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Details (1):
O
0
r3
r3
r2
r2
r1
1D model
spherical
symmetry
pools = 3
intervals
separated by
semi-permeable
membranes
r1
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Details (2): transmission conditions
Af = σf 00 ,
σ(x) = σi , x ∈ (ri+1 , ri ), i = 1, 2, 3.
C [0, r3 ]×C [r3 , r2 ]×C [r2 , r1 ].
f 0 (0) = 0
f 0 (r3 −) = k32 [f (r3 +) − f (r3 −)]
f (x)
f 0 (r3 +) = k23 [f (r3 +) − f (r3 −)]
f 0 (r2 −) = k21 [f (r2 +) − f (r2 −)]
r3
r2
r1 x
f 0 (r2 +) = k12 [f (r2 +) − f (r2 −)]
f 0 (r1 −) = −kf (r1 −).
k, kij - transmission coefficients.
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Transmission conditions
t
0
r3
Adam Bobrowski
r2
r1 x
CIMPA, Muizenberg, 22.07.2013
Neurotransmiters
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Details (3): approximating processes
An f = nσf 00 ,
σ(x) = σi , x ∈ (ri+1 , ri ), i = 1, 2, 3.
C [0, r3 ]×C [r3 , r2 ]×C [r2 , r1 ].
f 0 (0) = 0
f 0 (r3 −) = n−1 k32 [f (r3 +) − f (r3 −)]
f (x)
f 0 (r3 +) = n−1 k23 [f (r3 +) − f (r3 −)]
f 0 (r2 −) = n−1 k21 [f (r2 +) − f (r2 −)]
r3
r2
r1 x
f 0 (r2 +) = n−1 k12 [f (r2 +) − f (r2 −)]
f 0 (r1 −) = −n−1 kf (r1 −).
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Probabilistic intuition
t
n→∞
0
r3
r2
large
pool
small
pool
r1 x
Parameters’s tunning delicate!
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
imm.
available
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Probabilistic intuition in analytic terms
C [α, β], Bf = f 00 for f ∈ C 2 [α, β] such that f 0 (α) = f 0 (β) =
0 (reflection). Then
ntB
lim e
n→∞
1 Zβ
f =
f,
β−α α
Adam Bobrowski
t > 0, f ∈ C [α, β].
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Probabilistic intuition in analytic terms
C [α, β], Bf = f 00 for f ∈ C 2 [α, β] such that f 0 (α) = f 0 (β) =
0 (reflection). Then
ntB
lim e
n→∞
1 Zβ
f =
f,
β−α α
t > 0, f ∈ C [α, β].
(Big volumes on such convergence!)
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
The limit
lim etAn f = etA Pf , t > 0,
n→∞
Pf = ((r1 − r2 )−1
Z r1
r2
f , (r2 − r3 )−1
Z r2
r3
0
0
0
−k10
− k12
k12
0

0
0
0
0 
k21
−k21 − k23 k23
A=
,
0
0
0
k32
−k32

f , r3−1
Z r3
f ),
0

Adam Bobrowski
kij0 =
CIMPA, Muizenberg, 22.07.2013
σi kij
,
|Ωi |
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
The limit
lim etAn f = etA Pf , t > 0,
n→∞
Pf = ((r1 − r2 )−1
Z r1
r2
f , (r2 − r3 )−1
Z r2
r3
0
0
0
−k10
− k12
k12
0

0
0
0
0 
k21
−k21 − k23 k23
A=
,
0
0
0
k32
−k32

 0
U1
 0
U2 
U30
f , r3−1
Z r3
f ),
0

− R01C1 −

=

1
R1 C 2
1
R1 C1
0
Adam Bobrowski
1
R1 C1
− R21C2 −
1
R2 C3
kij0 =
0
1
R1 C2
σi kij
,
|Ωi |


U1

1 
R2 C2  U2  .
1
U3
R2 C3
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Coefficients’s interpretation:
Coefficients:
kij0 =
σi kij
.
|Ωi |
Replenishment rate:
directly proportional to transmission coefficients,
directly proportional to diffusion coefficients,
inversely proportional to intervals’ lenghts,
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Neurotransmiters
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
A generalization
A
A
D
C
B
F
E
B
C
E
F
G
D
G
fast diffusion on each edge,
low permeability of the membranes at vertexes,
linear graph: edges ‘lumped’ to vertexes, vertexes
‘extended’ to edges,
Markov chain on vertexes of the linear graph (= edges of
the original graph).
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013
Operator exponentials
Boundary conditions
Pause
Neurotransmiters
Thank you for your attention.
Adam Bobrowski
CIMPA, Muizenberg, 22.07.2013