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1
Feynman formulae for Feller semigroups
Ya.A. Butko, R.L. Schilling, O.G. Smolyanov
1 Introduction
Feller processes are a particular kind of continuous-time Markov pro cesses,
which generalize the class of stochastic processes with stationary and inde-
pendent increments or L´evy processes. A sto chastic process (ξ
t
)
t≥0
in R
d
is called Feller process if it generates a strongly continuous positivity pre-
serving contraction semigroup (T
t
)
t≥0
on the space C
∞
(R
d
) of continuous
functions vanishing at infinity (i.e. Feller semigroup): T
t
f(q) = E
q
[f(ξ
t
)]
for any f ∈ C
∞
(R
d
). Note that diffusion processes in R
d
also belong to the
class of Feller processes.
It is well known that (under a mild richness condition on the domain)
the infinitesimal generator A of a Feller semigroup is a pseudo-differential
operator (ΨDO, for short), i.e. an operator of the form
Af(q) = H(q, D)f(q) = (2 π)
−d
Z
R
d
Z
R
d
exp{i(q −x)p}H(q, p)f(x) dx dp,
f ∈ C
∞
c
(R
d
), with the symbol H : R
d
× R
d
→ C, (q, p) 7→ H(q, p). In a
similar way, each operator T
t
can be represented as a pseudo-differential
operator λ
t
(q, D) with symbol λ
t
(q, p) = E
q
£
e
i(ξ
t
−q)p
¤
. It is known that
H(q, p) = lim
t→0
λ
t
(q,p)−1
t
, see e.g. [9]. If (ξ
t
)
t≥0
is a L´evy process, we have
H(q, p) = H(p) and λ
t
(q, p) = e
tH(p)
— this is due to the fact that the
generator is an operator with constant (i.e. independent of the state-space
variable q) “coefficients”. In the general case, there is no such straightfor-
ward connection between the symbols of the semigroup and its generator
and this gives rise to several interesting problems: which symbols H(q, p)
do lead to Feller processes and, if so, how can we represent and/or approx-
imate the symbol λ
t
(q, p). The existence problem has been discussed at
length in a series of papers, see [7, 9] and the literature given there, and we
want now to concentrate on the problem how to represent the semigroup
resp. its symbol if the (symbol of the ) generator is known.
Consider the evolution equation
∂f
∂t
(t, q) = H(q, D)f(t, q), where
H(q, D) is the generator of some Feller process (ξ
t
)
t≥0
. The solution of the
Cauchy problem for this equation with the initial data f(0, q) = f
0
(q) can
be obtained by the Feynman-Kac formula f(t, q) ≡ (T
t
f
0
)(q) = E
q
[f
0
(ξ
t
)].
Here the expectation E
q
[f
0
(ξ
t
)] is a functional (path) integral over the
2
set of paths of the process (ξ
t
)
t≥0
with respect to the measure gener-
ated by this process. If (ξ
t
)
t≥0
is a diffusion process, then E
q
[f
0
(ξ
t
)] =
R
C([0,t],R
d
)
f
0
(ξ
t
)µ(dξ), where µ is a Gaussian measure, corresponding to this
process; in particular, the Wiener measure corresponds to the process of
Brownian motion.
The heuristic notion of a path integral has been introduced by R. Feyn-
man, see [4], to represent a solution of the Schr¨odinger equation with a
potential. Feynman has obtained the solution of this equation as a limit
of some finite dimensional integrals (actually, these integrals range over
Cartesian products of the configuration space of the system, described by
this equation), then, this limit has been interpreted as an integral over
a set of paths of the system. This kind of path integrals are called now
Feynman path integrals with respect to a Feynman pseudomeasure on the
set of paths in the configuration space.
The classical Feynman-Kac formula, representing the solution of the
Cauchy problem for the heat equation by a functional integral with re-
spect to the Wiener measure can also be obtained applying Feynman’s
construction. Here the functional integral is a limit of finite dimensional
integrals containing Gaussian exponents which are transition densities of
the process of Brownian motion. However, in most cases the transition
densities of Feller processes cannot be expressed by elementary functions
and, hence, in order to compute functional integrals in Feynman-Kac for-
mulae (and to simulate stochastic processes) we need to approximate these
densities (or functional integrals themselves) somehow. This give rise to
Feynman formulae.
A Feynman formula is the representation of the solution of an initial
(and boundary value) problem for an evolution equation as the limit of
finite dimensional integrals of some elementary functions, when the di-
mension of integrals tends to infinity.
1
Obviously, the finite dimensional
integrals in a Feynman formula, obtained for some problem, give approxi-
mations for a functional integral in the Feynman-Kac formula representing
the solution of the same problem. And these approximations can be used
for direct calculations and simulations.
The notion of a Feynman formula has been introduced in [12] and the
method to obtain Feynman formulae for evolutionary equations has been
developed in a series of papers [10]–[12], [1], [2]. This method is based
on Chernoff’s theorem. By the Chernoff theorem a strongly continuous
1
In the case of an evolution equation on an infinite dimensional space, Feynman formulae are the limit
of infinite dimensional integrals over finite Cartesian products of the infinite dimensional configuration
or the phase space of the system described by this equation.
3
semigroup (T
t
)
t≥0
on a Banach space can be represented as a strong limit:
T
t
= lim
n→∞
[F (t/n)]
n
where F (t) is an operator-valued function satisfy-
ing certain conditions. This equality is called a Feynman formula for the
semigroup (T
t
)
t≥0
. We call this Feynman formula a Lagrangian Feynman
formula, if the F (t), t > 0, are integral operators with elementary ker-
nels; if the F (t) are ΨDOs, we speak of Hamiltonian Feynman formulae.
In particular, we obtain Hamiltonian Feynman formulae for a semigroup
T
t
≡ e
tH(q,D)
generated by a ΨDO H(q, D) with symbol H(q, p) if
e
tH(q,D)
= strong- lim
n→∞
£
e
t
n
H
(q, D)
¤
n
,
where e
t
n
H
(q, D) is the ΨDO with symbol e
t
n
H(q,p)
. Note that, in general
e
t
n
H
(q, D) is not a semigroup and that λ
t
(q, p) 6= e
tH(q,p)
. Our terminology
is inspired by the fact that the Lagrangian Feynman formula gives approx-
imations to a functional integral over a set of paths in the configuration
space of a system (whose evolution is described by the semigroup (T
t
)
t≥0
),
while the Hamiltonian Feynman formula corresponds to a functional inte-
gral over a set of paths in the phase space of the same system.
In this note the Hamiltonian Feynman formula has been proved for a
certain class of Feller semigroups. The generators of these semigroups are
ΨDOs whose symbols H(q, p) are continuous functions of a variable (q, p)
such that −H(q, p) are smooth and negative definite with respect to the
variable p for each fixed q and bounded with respect to the variable q for
each fixed p. Also the Lagrangian Feynman formula has been obtained for
the Feller semigroup associated to a Cauchy type process with a symbol
H(q, p) = −a(q)|p| where the function a(·) is continuous, positive and
bounded. Note, that Lagrangian Feynman formula for diffusion processes
has been obtained in the paper [1].
2 Notations and preliminaries
Let C
∞
c
(R) be a set of infinitely differentiable on R functions with compact
supports and S(R) be the Swartz space of tempered functions. Let us also
consider a space C
∞
(R) of all continuous functions vanishing at infinity.
It is a Banach space with the norm ||f||
∞
= sup
x∈R
|f(x)|.
We call ψ continuous negative definite function if ψ : R
d
→ C is con-
tinuous and for any choice of k ∈ N and vectors ξ
1
, ..., ξ
k
∈ R
d
the matrix
¡
ψ(ξ
i
) + ψ(ξ
j
) − ψ(ξ
i
− ξ
j
)
¢
i,j=1,...,k
is positive Hermitian. The character-
istic exponent of a L´evy process is a continuous negative definite function
4
and vice versa, to every continuous negative definite function ψ satisfying
ψ(0) = 0 exists a L´evy process X
t
such that E[e
iξX
t
] = e
−tψ(ξ)
.
Let (T
t
)
t≥0
be a strongly continuous contraction semigroup on C
∞
(R)
which is positivity preserving (i.e. 0 ≤ u implies 0 ≤ T
t
u). Then (T
t
)
t≥0
is
called Feller semigroup.
If X, X
1
, X
2
are Banach spaces, then L(X
1
, X
2
) denotes the space of
continuous linear mappings from X
1
to X
2
with strong operator topology,
L(X) = L(X, X), k · k denotes the operator norm on L(X) and Id the
identity operator in X. If D(T ) ⊂ X is a linear subspace and T : D(T ) →
X is linear (operator), then D( T ) denotes the domain of T .
The derivative at the origin of a function F : [0, ε) → L(X), ε > 0, is a
linear mapping F
0
(0) : D(F
0
(0)) → X such that
F
0
(0)g := lim
t&0
t
−1
(F (t)g − F (0)g),
where D(F
0
(0)) is the vector space of all elements g ∈ X for which the
above limit exists.
In the sequel we use the following version of Chernoff theorem (cf. [3],
[12]).
Theorem 2.1 (Chernoff theorem) Let X be a Banach space, F :
[0, ∞) → L(X) be a (strongly) continuous mapping such that F (0) = Id
and kF (t)k ≤ e
at
for some a ∈ [0, ∞) and all t ≥ 0. Let D be a linear sub-
space of D(F
0
(0)) such that the restriction of the operator F
0
(0) to this sub-
space is closable. Let (L, D(L)) be this closure. If (L, D(L)) is the genera-
tor of a strongly continuous semigroup (T
t
)
t≥0
, then for any t
0
> 0 the se-
quence (F (t/n))
n
)
n∈N
converges to (T
t
)
t≥0
as n → ∞ in the strong operator
topology, uniformly with respect to t ∈ [0, t
0
], i.e., T
t
= lim
n→∞
(F (t/n))
n
locally uniformly in L(X).
A family of operators (F (t))
t≥0
is called Chernoff equivalent to the semi-
group (T
t
)
t≥0
if this family satisfies the assertions of the Chernoff theorem
with respect to this semigroup, i.e. by the Chernoff theorem locally uni-
formly in L(X)
T
t
= lim
n→∞
(F (t/n))
n
(1)
and the equality ( 1) is called Feynman formula for the semigroup (T
t
)
t≥0
.
5
3 Hamiltonian Feynman formula for Feller semi-
groups
Let ψ : R × R → C be a continuous function such that for each x ∈ R
a mapping p 7→ ψ(x, p) is negative definite, i.e. p 7→ e
−tψ(x,p)
is positive
definite for all t > 0. Let also ψ(x, ·) ∈ C
2
(R) for each x ∈ R. Let for each
p ∈ R the following estimates hold (the symbols ψ
0
1
and ψ
00
2
denote respec-
tively the first and the second derivative of ψ w.r.t. its second variable):
sup
x∈R
|ψ(x, p)| ≤ f
0
(p), (2)
sup
x∈R
|ψ
0
2
(x, p)| ≤ f
1
(p), (3)
sup
x∈R
|ψ
00
2
(x, p)| ≤ f
2
(p), (4)
where functions f
0
, f
1
and f
2
are continuous on R and have at most a
polynomial growth at infinity.
Let us consider a ΨDO H(x, D) with a symbol H(x, p) = −ψ(x, p), i.e.
for each ϕ ∈ C
∞
c
(R) we have
H(x, D)ϕ(x) =
−1
2π
Z
R
Z
R
exp{ip(x − q)}ψ(x, p)ϕ(q)dqdp.
Assumtion 3.1 (i) We assume that the function H(x, p) satisfies suffi-
cient conditions for H(x, D) is closable and its closure is a generator of a
strongly continuous semigroup on C
∞
(R).
(ii) We assume also that the set C
∞
c
(R) of test functions is a core for
this generator.
Remark 3.2 The conditions on the function H(x, p) to fulfill item (i) of
the Assumption 3.1 can be found for example in Vol. 2 of [7](Theo. 2.6.4,
2.6.9, 2.7.9, 2.7.16, 2.7.19, 2.8.1 e.t.c.) and in [9]. See also examples at the
end of this section. The item (ii) of the Assumption 3.1 holds for example
for generators of L´evy processes, see [8, Theo. 31.5].
Let F (t) be a ΨDO with a symbol e
tH
(x, D), i.e. for each ϕ ∈ C
∞
c
(R)
F (t)ϕ(x) =
1
2π
Z
R
Z
R
exp{ip(x − q)}exp{−tψ(x, p)}ϕ(q)dqdp.
6
Theorem 3.3 Under assumption (3.1) a family (F (t))
t≥0
is Chernoff
equivalent to a strongly continuous semigroup (T
t
)
t≥0
, generated by a clo-
sure of a ΨDO H(x, D) with a symbol H(x, p) = −ψ(x, p), and the Hamil-
tonian Feynman formula T
t
= lim
n→∞
£
F (
t
n
)
¤
n
is valid in L(C
∞
(R)), locally
uniformly with respect to t ≥ 0.
To prove the Theorem it is sufficient to show that for each ϕ ∈ C
∞
c
(R)
the function F (t)ϕ belongs to C
∞
(R), for each t > 0 a mapping F (t) can
be extended to a contraction F (t) : C
∞
(R) → C
∞
(R), and that for all
ϕ ∈ C
∞
c
(R), x ∈ R the equality F
0
(0)ϕ(x) = H(x, D)ϕ(x) holds. Then
one needs only to apply the Chernoff theorem.
Remark 3.4 For any ϕ ∈ C
∞
c
(R) and any q
0
∈ R the Hamiltonian Feyn-
man formula in the Theorem 3.3 has the following view:
(T
t
ϕ)(q
0
) = lim
n→∞
1
(2π)
n
Z
R
2n
exp{i
n
X
k=1
p
k
(q
k−1
− q
k
)}×
× exp{−
t
n
n
X
k=1
ψ(q
k−1
, p
k
)}ϕ(q
n
)dq
1
dp
1
...dq
n
dp
n
. (5)
Remark 3.5 If in Assumption 3.1 (i) we require the existence of not just a
strongly continuous but a Feller semigroup, then the obtained Hamiltonian
Feynman formula gives approximations to the Feynman–Kac formula for
the corresponding Feller process.
Example 3.6 Let us consider a symbol H
1
(q, p) = −a(q)p
2
, where a(·) ∈
C
∞
(R) is a positive and bounded on R function. Then H
1
(q, D) generates a
Feller semigroup (T
1
t
)
t≥0
related to the process of diffusion with a variable
diffusion coefficient. By the Hamiltonian Feynman formula (5) for any
ϕ ∈ C
∞
c
(R) and any q
0
∈ R we have:
(T
1
t
ϕ)(q
0
) = lim
n→∞
1
(2π)
n
Z
R
2n
exp{i
n
X
k=1
p
k
(q
k−1
− q
k
)}×
× exp{−
t
n
n
X
k=1
a(q
k−1
)p
2
k
}ϕ(q
n
)dq
1
dp
1
...dq
n
dp
n
.
Example 3.7 Let us consider a symbol H
2
(q, p) =
p
p
2
+ m
2
(q) − m(q),
where m(·) is a strictly positive, smooth and bounded on R function. If
additionally the function m(·) is such that the Assumption 3.1 holds (it is
7
so for ex. if m ≡ const), then the following Hamiltonian Feynman formula
is valid for the corresponding semigroup (T
2
t
)
t≥0
:
(T
2
t
ϕ)(q
0
) = lim
n→∞
1
(2π)
n
Z
R
2n
exp{i
n
X
k=1
p
k
(q
k−1
− q
k
)}×
× exp{−
t
n
n
X
k=1
q
p
2
k
+ m
2
(q
k−1
) − m(q
k−1
)}ϕ(q
n
)dq
1
dp
1
...dq
n
dp
n
,
where ϕ ∈ C
∞
c
(R) and q
0
∈ R. The operator H
2
(q, D) can b e considered
as a Hamiltonian of a free relativistic (quasi)particle with a variable mass
(cf. [5], [6]).
4 Lagrangian Feynman formula for the Feller semi-
group associated to a Cauchy type process with a
variable coefficient
Let a function a : R → R be continuous and for some A, B > 0 the
inequalities A ≤ a(x) ≤ B hold for all x ∈ R. Let us define an opera-
tor H
0
(·, D) ≡ −a
√
−∆ as a pseudo-differential operator with a symbol
H
0
(q, p) = −a(q)|p|, i.e. for each ϕ in C
∞
c
(R)
H
0
(x, D)ϕ(x) = −
a(x)
2π
Z
R
exp{ixp}|p|
Z
R
exp{−ipq}ϕ(q)dqdp.
Since the operator H
0
(·, D) is a composition of the operator −
√
∆, gener-
ating the Cauchy process, and the operator of multiplication on a bounded
continuous function a(·), then H
0
(·, D) has the same domain as −
√
∆ and
the set C
∞
c
(R) is a core for H
0
(·, D).
Assumtion 4.1 We assume that the coefficient a(·) satisfies sufficient
conditions for the existence of a strongly continuous semigroup (T
t
)
t≥0
on
the space C
∞
(R) with the generator H
0
(·, D).
The assumption holds for example in the case when a is a Lipschitz
continuous function (see [9]). In the case a(x) ≡ 1 the operator H
0
(·, D) is
a generator of the Feller semigroup, associated with the standard Cauchy
process.
Let us introduce a family of operators (F (t))
t≥0
on the space C
∞
(R)
such that F (0) = Id and for any t > 0 and any ϕ ∈ C
∞
(R)
F (t)ϕ(x) =
1
π
Z
R
a(x)t
(a(x)t)
2
+ y
2
ϕ(x − y)dy.
8
Theorem 4.2 Let the Assumption 4.1 holds. Then the family (F (t))
t≥0
is Chernoff equivalent to the semigroup (T
t
)
t≥0
, generated by the operator
H
0
(·, D) and for any ϕ ∈ C
∞
(R) the Lagrangian Feynman formula holds:
T
t
ϕ(x) =
= lim
n→∞
Z
R
n
p(t/n, x, x
1
)p(t/n, x
1
, x
2
)...p(t/n, x
n−1
, x
n
)ϕ(x
n
)dx
1
...dx
n
,
where p(t, x, y) =
1
π
a(x)t
(a(x)t)
2
+(x−y)
2
.
To prove the Theorem it is sufficient to show that the family (F
t
)
t≥0
is
strongly continuous, ||F (t)|| = 1 for any t ≥ 0, for any ϕ ∈ C
∞
c
(R) and
any x ∈ R the equality F
0
(0)ϕ(x) = H
0
(x, D)ϕ(x) holds. Then one needs
only to apply the Chernoff theorem.
Acknowledgements. This work has been supported by the Grant of
the President of Russian Federation (MK-943.2010.1.) and by the Erasmus
Mundus External Window Cooperation Program (EM ECW-L04 TUD 08-
31).
References
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Yana A. Butko,
Department of Fundamental Sciences,
Bauman Moscow State Technical University,
105005, 2nd Baumanskaya str., 5, Moscow, Russia.
Email: yanabutko@yandex.ru,
Oleg G. Smolyanov,
Department of Mechanics and Mathematics,
Lomonosov Moscow State University,
119992, Vorob’evy gory, 1, Moscow, Russia.
Email: Smolyanov@yandex.ru,
Ren´e L. Schilling
Institute of Mathematical Stochastic,
Dresden Technical University,
01062, Dresden, Germany.
Email: rene.schilling@tu-dresden.de
