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 processes, which generalize the class of stochastic processes with stationary and independent increments or L´evy processes. A stochastic process (ξt )t≥0 in Rd is called Feller process if it generates a strongly continuous positivity preserving contraction semigroup (Tt )t≥0 on the space C∞ (Rd ) of continuous functions vanishing at infinity (i.e. Feller semigroup): Tt f (q) = Eq [f (ξt )] for any f ∈ C∞ (Rd ). Note that diffusion processes in Rd 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 Z Z −d exp{i(q − x)p}H(q, p)f (x) dx dp, Af (q) = H(q, D)f (q) = (2π) Rd

Rd

f ∈ Cc∞ (Rd ), with the symbol H : Rd × Rd → C, (q, p) 7→ H(q, p). In a similar way, each operator Tt can be represented as ¤a pseudo-differential £ operator λt (q, D) with symbol λt (q, p) = Eq ei(ξt −q)p . It is known that H(q, p) = limt→0 λt (q,p)−1 , see e.g. [9]. If (ξt )t≥0 is a L´evy process, we have t H(q, p) = H(p) and λt (q, p) = etH(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 straightforward 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 approximate the symbol λt (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 (ξt )t≥0 . The solution of the Cauchy problem for this equation with the initial data f (0, q) = f0 (q) can be obtained by the Feynman-Kac formula f (t, q) ≡ (Tt f0 )(q) = Eq [f0 (ξt )]. Here the expectation Eq [f0 (ξt )] is a functional (path) integral over the

2

set of paths of the process (ξt )t≥0 with respect to the measure generated by this process. If (ξt )t≥0 is a diffusion process, then Eq [f0 (ξt )] = R C([0,t],Rd ) f0 (ξt )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. Feynman, 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 respect 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 formulae (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 dimension of integrals tends to infinity.1 Obviously, the finite dimensional integrals in a Feynman formula, obtained for some problem, give approximations 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 (Tt )t≥0 on a Banach space can be represented as a strong limit: Tt = limn→∞ [F (t/n)]n where F (t) is an operator-valued function satisfying certain conditions. This equality is called a Feynman formula for the semigroup (Tt )t≥0 . We call this Feynman formula a Lagrangian Feynman formula, if the F (t), t > 0, are integral operators with elementary kernels; if the F (t) are ΨDOs, we speak of Hamiltonian Feynman formulae. In particular, we obtain Hamiltonian Feynman formulae for a semigroup Tt ≡ etH(q,D) generated by a ΨDO H(q, D) with symbol H(q, p) if £ t ¤n etH(q,D) = strong- lim e n H (q, D) , n→∞

t

t

where e n H (q, D) is the ΨDO with symbol e n H(q,p) . Note that, in general t e n H (q, D) is not a semigroup and that λt (q, p) 6= etH(q,p) . Our terminology is inspired by the fact that the Lagrangian Feynman formula gives approximations to a functional integral over a set of paths in the configuration space of a system (whose evolution is described by the semigroup (Tt )t≥0 ), while the Hamiltonian Feynman formula corresponds to a functional integral 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 |f (x)|. x∈R

We call ψ continuous negative definite function if ψ : Rd → C is cond ¡tinuous and for any choice ¢of k ∈ N and vectors ξ1 , ..., ξk ∈ R the matrix ψ(ξi ) + ψ(ξj ) − ψ(ξi − ξj ) i,j=1,...,k is positive Hermitian. The characteristic 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 Xt such that E[eiξXt ] = e−tψ(ξ) . Let (Tt )t≥0 be a strongly continuous contraction semigroup on C∞ (R) which is positivity preserving (i.e. 0 ≤ u implies 0 ≤ Tt u). Then (Tt )t≥0 is called Feller semigroup. If X, X1 , X2 are Banach spaces, then L(X1 , X2 ) denotes the space of continuous linear mappings from X1 to X2 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−1 (F (t)g − F (0)g), t&0

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 ≤ eat for some a ∈ [0, ∞) and all t ≥ 0. Let D be a linear subspace of D(F 0 (0)) such that the restriction of the operator F 0 (0) to this subspace is closable. Let (L, D(L)) be this closure. If (L, D(L)) is the generator of a strongly continuous semigroup (Tt )t≥0 , then for any t0 > 0 the sequence (F (t/n))n )n∈N converges to (Tt )t≥0 as n → ∞ in the strong operator topology, uniformly with respect to t ∈ [0, t0 ], i.e., Tt = limn→∞ (F (t/n))n locally uniformly in L(X). A family of operators (F (t))t≥0 is called Chernoff equivalent to the semigroup (Tt )t≥0 if this family satisfies the assertions of the Chernoff theorem with respect to this semigroup, i.e. by the Chernoff theorem locally uniformly in L(X) Tt = lim (F (t/n))n (1) n→∞

and the equality ( 1) is called Feynman formula for the semigroup (Tt )t≥0 .

5

3

Hamiltonian Feynman formula for Feller semigroups

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, ·) ∈ C2 (R) for each x ∈ R. Let for each p ∈ R the following estimates hold (the symbols ψ10 and ψ200 denote respectively the first and the second derivative of ψ w.r.t. its second variable): sup |ψ(x, p)| ≤ f0 (p),

(2)

sup |ψ20 (x, p)| ≤ f1 (p),

(3)

sup |ψ200 (x, p)| ≤ f2 (p),

(4)

x∈R

x∈R

x∈R

where functions f0 , f1 and f2 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 ∈ Cc∞ (R) we have Z Z −1 exp{ip(x − q)}ψ(x, p)ϕ(q)dqdp. H(x, D)ϕ(x) = 2π R R

Assumtion 3.1 (i) We assume that the function H(x, p) satisfies sufficient 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 Cc∞ (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 etH (x, D), i.e. for each ϕ ∈ Cc∞ (R) Z Z 1 F (t)ϕ(x) = exp{ip(x − q)} exp{−tψ(x, p)}ϕ(q)dqdp. 2π R R

6

Theorem 3.3 Under assumption (3.1) a family (F (t))t≥0 is Chernoff equivalent to a strongly continuous semigroup (Tt )t≥0 , generated by a closure of a ΨDO H(x, D) with a symbol £ H(x, ¤ p) = −ψ(x, p), and the Hamilt n tonian Feynman formula Tt = lim F ( n ) is valid in L(C∞ (R)), locally n→∞ uniformly with respect to t ≥ 0. To prove the Theorem it is sufficient to show that for each ϕ ∈ Cc∞ (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 0 ϕ ∈ C∞ c (R), x ∈ R the equality F (0)ϕ(x) = H(x, D)ϕ(x) holds. Then one needs only to apply the Chernoff theorem. Remark 3.4 For any ϕ ∈ Cc∞ (R) and any q0 ∈ R the Hamiltonian Feynman formula in the Theorem 3.3 has the following view: Z

1 (Tt ϕ)(q0 ) = lim n→∞ (2π)n

exp{i

n X

pk (qk−1 − qk )}×

k=1

R2n

n

tX ψ(qk−1 , pk )}ϕ(qn )dq1 dp1 ...dqn dpn . (5) × exp{− n k=1

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 H1 (q, p) = −a(q)p2 , where a(·) ∈ C ∞ (R) is a positive and bounded on R function. Then H1 (q, D) generates a Feller semigroup (Tt1 )t≥0 related to the process of diffusion with a variable diffusion coefficient. By the Hamiltonian Feynman formula (5) for any ϕ ∈ Cc∞ (R) and any q0 ∈ R we have: 1 (Tt1 ϕ)(q0 ) = lim n→∞ (2π)n

Z exp{i R2n

n X

pk (qk−1 − qk )}×

k=1 n

tX × exp{− a(qk−1 )p2k }ϕ(qn )dq1 dp1 ...dqn dpn . n k=1

p Example 3.7 Let us consider a symbol H2 (q, p) = p2 + m2 (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 (Tt2 )t≥0 : Z n X 1 2 (Tt ϕ)(q0 ) = lim exp{i pk (qk−1 − qk )}× n→∞ (2π)n k=1

R2n

n

tX × exp{− n

q

p2k + m2 (qk−1 ) − m(qk−1 )}ϕ(qn )dq1 dp1 ...dqn dpn ,

k=1

where ϕ ∈ Cc∞ (R) and q0 ∈ R. The operator H2 (q, D) can be considered as a Hamiltonian of a free relativistic (quasi)particle with a variable mass (cf. [5], [6]).

4

Lagrangian Feynman formula for the Feller semigroup 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 operator H0 (·, D) ≡ −a −∆ as a pseudo-differential operator with a symbol H0 (q, p) = −a(q)|p|, i.e. for each ϕ in Cc∞ (R) Z Z a(x) exp{ixp}|p| exp{−ipq}ϕ(q)dqdp. H0 (x, D)ϕ(x) = − 2π R R √ Since the operator H0 (·, D) is a composition of the operator − ∆, generating the Cauchy process, and the operator of multiplication on a bounded √ continuous function a(·), then H0 (·, D) has the same domain as − ∆ and the set Cc∞ (R) is a core for H0 (·, D). Assumtion 4.1 We assume that the coefficient a(·) satisfies sufficient conditions for the existence of a strongly continuous semigroup (Tt )t≥0 on the space C∞ (R) with the generator H0 (·, 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 H0 (·, 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) Z 1 a(x)t F (t)ϕ(x) = ϕ(x − y)dy. π (a(x)t)2 + y 2 R

8

Theorem 4.2 Let the Assumption 4.1 holds. Then the family (F (t))t≥0 is Chernoff equivalent to the semigroup (Tt )t≥0 , generated by the operator H0 (·, D) and for any ϕ ∈ C∞ (R) the Lagrangian Feynman formula holds: Tt ϕ(x) =

Z

= lim

p(t/n, x, x1 )p(t/n, x1 , x2 )...p(t/n, xn−1 , xn )ϕ(xn )dx1 ...dxn ,

n→∞ Rn

where p(t, x, y) =

a(x)t 1 π (a(x)t)2 +(x−y)2 .

To prove the Theorem it is sufficient to show that the family (Ft )t≥0 is strongly continuous, ||F (t)|| = 1 for any t ≥ 0, for any ϕ ∈ Cc∞ (R) and any x ∈ R the equality F 0 (0)ϕ(x) = H0 (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 0831).

References [1] Butko Ya., Grothaus M., Smolyanov O.G., Feynman Formula for a Class of Second-Order Parabolic Equations in a Bounded Domain. Doklady Math. 2008. V. 78. N. 1. P. 590-595. [2] Butko Ya. A., Feynman formulas and functional integrals for diffusion with drift in a domain on a manifold. Math. Notes. 2008. V. 83. N. 3-4. P. 301-316. [3] Chernoff P., Product formulas, nonlinear semigroups and addition of unbounded operators. Mem. Am. Math. Soc., 140 1974. [4] Feynman R. P., Space-time approach to nonrelativistic quantum mechanics. Rev. Mod. Phys. 20 (1948), 367–387. [5] Gadella M., Smolyanov O.G., Feynman Formulas for Particles with Position-Dependent Mass. Doklady Math. 2007. V. 77. N. 1. P. 120123.

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[6] Ichinose T., and Tamura H., Imaginary-time integral for a relativistic spinless particle in an electromagnetic field. Commun.Math.Phys., 105: 239–257, 1986. [7] Jacob N., Pseudo-differential operators and Markov processes. Vol. 1-3. Imperial College Press, 2001. [8] Sato K., Levy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press. 1999. [9] Schilling R.L. and Schnurr A., The symbol associated with the solution of a stochastic differential equation. Preprint 2009. [10] Smolyanov O.G., Weizs¨acker H.v. and Wittich O., Diffusion on compact Riemannian manifolds, and surface measures. Dokl. Math. 61 (2000), 230–234. [11] Smolyanov O.G., Weizsaecker H. v., Wittich O., Chernoff’s Theorem and Discrete Time Approximations of Brownian Motion on Manifolds. Potent. Anal. 2007. V. 26. N. 1. P. 1-29. [12] Smolyanov O.G., Tokarev A.G., Truman A., Hamiltonian Feynman path integrals via the Chernoff formula. J.Math. Phys. 2002. V.43. N. 10. P. 5161-5171. Yana A. Butko, Department of Fundamental Sciences, Bauman Moscow State Technical University, 105005, 2nd Baumanskaya str., 5, Moscow, Russia. Email: [email protected], Oleg G. Smolyanov, Department of Mechanics and Mathematics, Lomonosov Moscow State University, 119992, Vorob’evy gory, 1, Moscow, Russia. Email: [email protected], Ren´e L. Schilling Institute of Mathematical Stochastic, Dresden Technical University, 01062, Dresden, Germany. Email: [email protected]

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 processes, which generalize the class of stochastic processes with stationary and independent increments or L´evy processes. A stochastic process (ξt )t≥0 in Rd is called Feller process if it generates a strongly continuous positivity preserving contraction semigroup (Tt )t≥0 on the space C∞ (Rd ) of continuous functions vanishing at infinity (i.e. Feller semigroup): Tt f (q) = Eq [f (ξt )] for any f ∈ C∞ (Rd ). Note that diffusion processes in Rd 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 Z Z −d exp{i(q − x)p}H(q, p)f (x) dx dp, Af (q) = H(q, D)f (q) = (2π) Rd

Rd

f ∈ Cc∞ (Rd ), with the symbol H : Rd × Rd → C, (q, p) 7→ H(q, p). In a similar way, each operator Tt can be represented as ¤a pseudo-differential £ operator λt (q, D) with symbol λt (q, p) = Eq ei(ξt −q)p . It is known that H(q, p) = limt→0 λt (q,p)−1 , see e.g. [9]. If (ξt )t≥0 is a L´evy process, we have t H(q, p) = H(p) and λt (q, p) = etH(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 straightforward 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 approximate the symbol λt (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 (ξt )t≥0 . The solution of the Cauchy problem for this equation with the initial data f (0, q) = f0 (q) can be obtained by the Feynman-Kac formula f (t, q) ≡ (Tt f0 )(q) = Eq [f0 (ξt )]. Here the expectation Eq [f0 (ξt )] is a functional (path) integral over the

2

set of paths of the process (ξt )t≥0 with respect to the measure generated by this process. If (ξt )t≥0 is a diffusion process, then Eq [f0 (ξt )] = R C([0,t],Rd ) f0 (ξt )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. Feynman, 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 respect 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 formulae (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 dimension of integrals tends to infinity.1 Obviously, the finite dimensional integrals in a Feynman formula, obtained for some problem, give approximations 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 (Tt )t≥0 on a Banach space can be represented as a strong limit: Tt = limn→∞ [F (t/n)]n where F (t) is an operator-valued function satisfying certain conditions. This equality is called a Feynman formula for the semigroup (Tt )t≥0 . We call this Feynman formula a Lagrangian Feynman formula, if the F (t), t > 0, are integral operators with elementary kernels; if the F (t) are ΨDOs, we speak of Hamiltonian Feynman formulae. In particular, we obtain Hamiltonian Feynman formulae for a semigroup Tt ≡ etH(q,D) generated by a ΨDO H(q, D) with symbol H(q, p) if £ t ¤n etH(q,D) = strong- lim e n H (q, D) , n→∞

t

t

where e n H (q, D) is the ΨDO with symbol e n H(q,p) . Note that, in general t e n H (q, D) is not a semigroup and that λt (q, p) 6= etH(q,p) . Our terminology is inspired by the fact that the Lagrangian Feynman formula gives approximations to a functional integral over a set of paths in the configuration space of a system (whose evolution is described by the semigroup (Tt )t≥0 ), while the Hamiltonian Feynman formula corresponds to a functional integral 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 |f (x)|. x∈R

We call ψ continuous negative definite function if ψ : Rd → C is cond ¡tinuous and for any choice ¢of k ∈ N and vectors ξ1 , ..., ξk ∈ R the matrix ψ(ξi ) + ψ(ξj ) − ψ(ξi − ξj ) i,j=1,...,k is positive Hermitian. The characteristic 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 Xt such that E[eiξXt ] = e−tψ(ξ) . Let (Tt )t≥0 be a strongly continuous contraction semigroup on C∞ (R) which is positivity preserving (i.e. 0 ≤ u implies 0 ≤ Tt u). Then (Tt )t≥0 is called Feller semigroup. If X, X1 , X2 are Banach spaces, then L(X1 , X2 ) denotes the space of continuous linear mappings from X1 to X2 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−1 (F (t)g − F (0)g), t&0

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 ≤ eat for some a ∈ [0, ∞) and all t ≥ 0. Let D be a linear subspace of D(F 0 (0)) such that the restriction of the operator F 0 (0) to this subspace is closable. Let (L, D(L)) be this closure. If (L, D(L)) is the generator of a strongly continuous semigroup (Tt )t≥0 , then for any t0 > 0 the sequence (F (t/n))n )n∈N converges to (Tt )t≥0 as n → ∞ in the strong operator topology, uniformly with respect to t ∈ [0, t0 ], i.e., Tt = limn→∞ (F (t/n))n locally uniformly in L(X). A family of operators (F (t))t≥0 is called Chernoff equivalent to the semigroup (Tt )t≥0 if this family satisfies the assertions of the Chernoff theorem with respect to this semigroup, i.e. by the Chernoff theorem locally uniformly in L(X) Tt = lim (F (t/n))n (1) n→∞

and the equality ( 1) is called Feynman formula for the semigroup (Tt )t≥0 .

5

3

Hamiltonian Feynman formula for Feller semigroups

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, ·) ∈ C2 (R) for each x ∈ R. Let for each p ∈ R the following estimates hold (the symbols ψ10 and ψ200 denote respectively the first and the second derivative of ψ w.r.t. its second variable): sup |ψ(x, p)| ≤ f0 (p),

(2)

sup |ψ20 (x, p)| ≤ f1 (p),

(3)

sup |ψ200 (x, p)| ≤ f2 (p),

(4)

x∈R

x∈R

x∈R

where functions f0 , f1 and f2 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 ∈ Cc∞ (R) we have Z Z −1 exp{ip(x − q)}ψ(x, p)ϕ(q)dqdp. H(x, D)ϕ(x) = 2π R R

Assumtion 3.1 (i) We assume that the function H(x, p) satisfies sufficient 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 Cc∞ (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 etH (x, D), i.e. for each ϕ ∈ Cc∞ (R) Z Z 1 F (t)ϕ(x) = exp{ip(x − q)} exp{−tψ(x, p)}ϕ(q)dqdp. 2π R R

6

Theorem 3.3 Under assumption (3.1) a family (F (t))t≥0 is Chernoff equivalent to a strongly continuous semigroup (Tt )t≥0 , generated by a closure of a ΨDO H(x, D) with a symbol £ H(x, ¤ p) = −ψ(x, p), and the Hamilt n tonian Feynman formula Tt = lim F ( n ) is valid in L(C∞ (R)), locally n→∞ uniformly with respect to t ≥ 0. To prove the Theorem it is sufficient to show that for each ϕ ∈ Cc∞ (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 0 ϕ ∈ C∞ c (R), x ∈ R the equality F (0)ϕ(x) = H(x, D)ϕ(x) holds. Then one needs only to apply the Chernoff theorem. Remark 3.4 For any ϕ ∈ Cc∞ (R) and any q0 ∈ R the Hamiltonian Feynman formula in the Theorem 3.3 has the following view: Z

1 (Tt ϕ)(q0 ) = lim n→∞ (2π)n

exp{i

n X

pk (qk−1 − qk )}×

k=1

R2n

n

tX ψ(qk−1 , pk )}ϕ(qn )dq1 dp1 ...dqn dpn . (5) × exp{− n k=1

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 H1 (q, p) = −a(q)p2 , where a(·) ∈ C ∞ (R) is a positive and bounded on R function. Then H1 (q, D) generates a Feller semigroup (Tt1 )t≥0 related to the process of diffusion with a variable diffusion coefficient. By the Hamiltonian Feynman formula (5) for any ϕ ∈ Cc∞ (R) and any q0 ∈ R we have: 1 (Tt1 ϕ)(q0 ) = lim n→∞ (2π)n

Z exp{i R2n

n X

pk (qk−1 − qk )}×

k=1 n

tX × exp{− a(qk−1 )p2k }ϕ(qn )dq1 dp1 ...dqn dpn . n k=1

p Example 3.7 Let us consider a symbol H2 (q, p) = p2 + m2 (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 (Tt2 )t≥0 : Z n X 1 2 (Tt ϕ)(q0 ) = lim exp{i pk (qk−1 − qk )}× n→∞ (2π)n k=1

R2n

n

tX × exp{− n

q

p2k + m2 (qk−1 ) − m(qk−1 )}ϕ(qn )dq1 dp1 ...dqn dpn ,

k=1

where ϕ ∈ Cc∞ (R) and q0 ∈ R. The operator H2 (q, D) can be considered as a Hamiltonian of a free relativistic (quasi)particle with a variable mass (cf. [5], [6]).

4

Lagrangian Feynman formula for the Feller semigroup 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 operator H0 (·, D) ≡ −a −∆ as a pseudo-differential operator with a symbol H0 (q, p) = −a(q)|p|, i.e. for each ϕ in Cc∞ (R) Z Z a(x) exp{ixp}|p| exp{−ipq}ϕ(q)dqdp. H0 (x, D)ϕ(x) = − 2π R R √ Since the operator H0 (·, D) is a composition of the operator − ∆, generating the Cauchy process, and the operator of multiplication on a bounded √ continuous function a(·), then H0 (·, D) has the same domain as − ∆ and the set Cc∞ (R) is a core for H0 (·, D). Assumtion 4.1 We assume that the coefficient a(·) satisfies sufficient conditions for the existence of a strongly continuous semigroup (Tt )t≥0 on the space C∞ (R) with the generator H0 (·, 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 H0 (·, 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) Z 1 a(x)t F (t)ϕ(x) = ϕ(x − y)dy. π (a(x)t)2 + y 2 R

8

Theorem 4.2 Let the Assumption 4.1 holds. Then the family (F (t))t≥0 is Chernoff equivalent to the semigroup (Tt )t≥0 , generated by the operator H0 (·, D) and for any ϕ ∈ C∞ (R) the Lagrangian Feynman formula holds: Tt ϕ(x) =

Z

= lim

p(t/n, x, x1 )p(t/n, x1 , x2 )...p(t/n, xn−1 , xn )ϕ(xn )dx1 ...dxn ,

n→∞ Rn

where p(t, x, y) =

a(x)t 1 π (a(x)t)2 +(x−y)2 .

To prove the Theorem it is sufficient to show that the family (Ft )t≥0 is strongly continuous, ||F (t)|| = 1 for any t ≥ 0, for any ϕ ∈ Cc∞ (R) and any x ∈ R the equality F 0 (0)ϕ(x) = H0 (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 0831).

References [1] Butko Ya., Grothaus M., Smolyanov O.G., Feynman Formula for a Class of Second-Order Parabolic Equations in a Bounded Domain. Doklady Math. 2008. V. 78. N. 1. P. 590-595. [2] Butko Ya. A., Feynman formulas and functional integrals for diffusion with drift in a domain on a manifold. Math. Notes. 2008. V. 83. N. 3-4. P. 301-316. [3] Chernoff P., Product formulas, nonlinear semigroups and addition of unbounded operators. Mem. Am. Math. Soc., 140 1974. [4] Feynman R. P., Space-time approach to nonrelativistic quantum mechanics. Rev. Mod. Phys. 20 (1948), 367–387. [5] Gadella M., Smolyanov O.G., Feynman Formulas for Particles with Position-Dependent Mass. Doklady Math. 2007. V. 77. N. 1. P. 120123.

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[6] Ichinose T., and Tamura H., Imaginary-time integral for a relativistic spinless particle in an electromagnetic field. Commun.Math.Phys., 105: 239–257, 1986. [7] Jacob N., Pseudo-differential operators and Markov processes. Vol. 1-3. Imperial College Press, 2001. [8] Sato K., Levy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press. 1999. [9] Schilling R.L. and Schnurr A., The symbol associated with the solution of a stochastic differential equation. Preprint 2009. [10] Smolyanov O.G., Weizs¨acker H.v. and Wittich O., Diffusion on compact Riemannian manifolds, and surface measures. Dokl. Math. 61 (2000), 230–234. [11] Smolyanov O.G., Weizsaecker H. v., Wittich O., Chernoff’s Theorem and Discrete Time Approximations of Brownian Motion on Manifolds. Potent. Anal. 2007. V. 26. N. 1. P. 1-29. [12] Smolyanov O.G., Tokarev A.G., Truman A., Hamiltonian Feynman path integrals via the Chernoff formula. J.Math. Phys. 2002. V.43. N. 10. P. 5161-5171. Yana A. Butko, Department of Fundamental Sciences, Bauman Moscow State Technical University, 105005, 2nd Baumanskaya str., 5, Moscow, Russia. Email: [email protected], Oleg G. Smolyanov, Department of Mechanics and Mathematics, Lomonosov Moscow State University, 119992, Vorob’evy gory, 1, Moscow, Russia. Email: [email protected], Ren´e L. Schilling Institute of Mathematical Stochastic, Dresden Technical University, 01062, Dresden, Germany. Email: [email protected]