Probabilistic Representations for the Solution of Higher Order Differential Equations

Abstract

A probabilistic representation for the solution of the partial differential equation (𝜕/𝜕𝑡)𝑢(𝑡,𝑥)=−𝛼Δ²𝑢(𝑡,𝑥),𝛼∈ℂ, is constructed in terms of the expectation with respect to the measure associated to a complex-valued stochastic process.

1. Introduction

The connection between the solution of parabolic equations associated to second-order elliptic operators and the theory of stochastic processes is a largely studied topic [1]. The main instance is the Feynman-Kac formula, providing a representation of the solution of the heat equation with potential 𝑉∈𝐶_∞(ℝ^𝑑) (the continuous functions vanishing at infinity): 𝜕𝑢()=1𝜕𝑡𝑡,𝑥2Δ𝑢(𝑡,𝑥)−𝑉(𝑥)𝑢(𝑡,𝑥),𝑡∈ℝ⁺,𝑥∈ℝ^𝑑,𝑢(0,𝑥)=𝑢₀(𝑥) (1) in terms of an integral with respect to the measure of the Wiener process, the mathematical model of the Brownian motion [2]: 𝑢(𝑡,𝑥)=_𝐶𝑡𝑒^{−∫𝑡0𝑉(𝜔(𝑠)+𝑥)𝑑𝑠}𝑢₀(𝜔(𝑡)+𝑥)𝑑𝑊(𝜔). (2)

If the Laplacian in (1) is replaced with a higher order differential operator, that is, if we consider a Cauchy problem of the form 𝜕𝑢()=()𝜕𝑡𝑡,𝑥−1^𝑁+1Δ^𝑁𝑢(𝑡,𝑥)−𝑉(𝑥)𝑢(𝑡,𝑥),𝑡∈ℝ⁺,𝑥∈ℝ^𝑑,𝑢(0,𝑥)=𝑢₀(𝑥), (3) with 𝑁∈ℕ, 𝑁≥2, then a formula analogous to (2), giving the solution of (3) in terms of the expectation with respect to the measure associated to a Markov process, is lacking. In fact, such a formula cannot be proved for semigroups whose generator does not satisfy the maximum principle, as in the case of Δ^𝑁 with 𝑁>1. In fact in the case where 𝑁>1 the Cauchy problem (3) is not well posed on the space of continuous bounded functions [3]. In other words it is not possible to find a stochastic process which plays for the parabolic equation (3) the same role that the Wiener process plays for the heat equation.

We would like to point out that the problem of the probabilistic representation of the solution of the Cauchy problem (3) presents some similarities with the problem of the mathematical definition of Feynman path integrals (see [4–7] for a discussion of this topic). Indeed in both cases it is not possible to implement an integration theory of Lebesgue type in terms of a bounded variation measure on a space of continuous paths [8]. An analogous of the Feynman-Kac formula for the parabolic equation (3), namely, an equation of the form 𝑢(𝑡,𝑥)=_𝜔(0)=𝑥𝑒^{−∫𝑡0𝑉(𝜔(𝑠))𝑑𝑠}𝑢₀(𝜔(𝑡))𝑑𝑃(𝜔) (4) (where 𝑃 should be some “measure” on a space of “paths” 𝜔∶[0,𝑡]→ℝ), can be obtained only under some restrictions on 𝑢₀ and 𝑉 and by giving up a traditional integration theory in the Lebesgue sense with respect to a bounded variation measure on a space of (real) continuous paths.

In the mathematical literature two main approaches have been proposed. The first one [9, 10] realizes formula (4) in terms of the expectation with respect to a signed measure 𝑃_2𝑁 on a space of paths on the interval [0,𝑡].

It is worthwhile to mention that an analogous of the arc-sine law [10, 11], of the central limit theorem [12], and of Ito formula and Ito stochastic calculus [10, 13] have been proved for the (finite additive) signed measure 𝑃_2𝑁. Moreover, a Feynman-Kac formula has been proved [9–11], for the representation of the solution of the Cauchy problem (3) in the case where 𝑉 is a bounded piecewise continuous function and for an initial datum 𝑢₀∈𝐶^2𝑁, by realizing formula (4) as limit of finite dimensional cylindrical approximations [14]. We also mention the work by Levin and Lyons [15] on rough paths, conjecturing that the signed measure 𝑃_2𝑁 could exist on the quotient space of equivalence classes of paths corresponding to different parametrization of the same path.

A different approach is based on the construction of a stochastic process on a space of complex paths. In this case the integration is performed with respect to a well-defined positive probability measure on a complex space. One of the first results was given by Funaki [16], who constructed a complex stochastic process {𝑋_𝑡}_𝑡≥0 by composing two independent Brownian motions {𝐵(𝑡)}_𝑡≥0 and {𝑤(𝑡)}_𝑡≥0 in the following way: 𝑋_𝑡𝐵(𝑤(𝑡(𝑡)((𝑡(𝑡)∶=))if𝑤≥0𝑖𝐵−𝑤))if𝑤<0 (5) and proving that, for a suitable class of analytic initial datum 𝑢₀, the solution of the Cauchy problem 𝜕𝑢=1𝜕𝑡8𝜕⁴𝑢𝜕𝑥⁴𝑢()−∞<𝑥<∞,0<𝑡<∞,0,𝑥=𝑢₀(𝑥) (6) is given by the expectation 𝑢(𝑡,𝑥)=𝔼𝑢₀𝑥+𝑋_𝑡. (7)

In fact this result can be generalized to partial differential equations of order 2^𝑛, by multiple iterations of suitable processes [16–18]. These results are also related to Bochner subordination [19].

There are also similarities between the Funaki’s process {𝑋_𝑡} and the “iterated Brownian motion” [20], but the latter is not connected to the probabilistic representation of the solution of a partial differential equation with regular coefficients. In fact the processes constructed by iterating copies of independent BMs (or other process) are associated to higher order PDE of particular form, where the initial datum plays a particular role and enters also in the differential equation [21].

Complex-valued processes, connected to PDE of the form (3), have been also proposed by other authors by means of different techniques. In [22–24] Madrecki et al. consider the fourth degree heat-type equation (6) and construct a probabilistic representation for its solution in terms of a stable probabilistic Borel measure 𝑚 on the space Ω=𝐶([0,𝑡],ℂ^∞) of continuous mappings on [0,𝑡] with values in the set ℂ^∞ of complex-valued sequences, endowed with the product topology. In this setting a Feynman-Kac type formula is proved, for the fourth order heat equation with linear potential.

Another probabilistic approach is presented by Sainty in [25], where a representation for the solution of (𝜕/𝜕𝑡)𝑢(𝑡,𝑥)=(𝜕^𝑛/𝜕𝑥^𝑛)𝑢(𝑡,𝑥) is given in terms of the expectation with respect to a particular complex-valued process 𝑋_[𝑛](𝑡),𝑡≥0, called “Brownian motion of order 𝑛”. It is worthwhile also to mention a completely different approach proposed by Léandre [26, 27], which has some analogies with the mathematical realization of Feynman path integrals by means of white noise calculus [28]. Indeed Léandre has recently constructed a “probabilistic representation” of the solution of the Cauchy problem (3) in terms of an infinite dimensional distribution on the Connes space [27, 29, 30]. Another interesting approach related to the theory of pseudoprocesses introduced by Daletsky and Fomin [31] has been recently proposed by Smorodina and Faddeev [32].

We eventually mention an interesting probabilistic approach to the equation Δ^𝑘𝑢=0 described in [33].

The present paper presents the construction of an alternative complex-valued stochastic process generalizing Funaki's result [16] and a corresponding probabilistic representation for the solution of the Cauchy problem: 𝜕𝑢()𝛼𝜕𝑡𝑥,𝑡=−8𝜕⁴𝜕𝑥⁴𝑢()()𝑢()𝑢()𝑥,𝑡+𝑉𝑡,𝑥𝑡,𝑥−∞<𝑥<∞,0≤𝑡<∞,𝑥,0=𝑢₀(𝑥), (8) with 𝛼∈ℂ and for 𝑉,𝑢₀ satisfying suitable analyticity assumptions.

The paper is organized as follows. Section 2 presents the construction of a complex random variable 𝑧^𝛼_𝑡 and the representation of the solution of (8) with 𝑉≡0 in terms of the expectation with respect to the probability measure associated with 𝑧^𝛼_𝑡. Section 3 presents the proof of a Feynman-Kac type formula for the solution of (8) in the cases where 𝑉 is linear in the space variables and presents an explicit time dependence.

2. A Complex-Valued Random Variable Associated to the 4-Order Heat-Type Equation

In the present section we construct a probabilistic representation for the solution of (8) in the case where 𝑉=0, namely, 𝜕𝑢()𝛼𝜕𝑡𝑥,𝑡=−8𝜕⁴𝑢𝜕𝑥⁴(𝑥,𝑡)−∞<𝑥<∞,0≤𝑡<∞. (9) An equation of this form in the case where 𝛼∈ℝ, as mentioned in the introduction, has been studied by several authors by means of different techniques [9, 10, 16, 25]. In this section we show that the results in [9, 10, 16] can be seen as particular cases of a general theory presented in [34–37], connecting the solution of parabolic problems with the solution of related hyperbolic problems.

Given a Banach space 𝑋 and a strongly continuous group of operators {𝑇_𝐴(𝑡)}_𝑡∈ℝ on 𝑋 with generator 𝐴, it is possible to construct the holomorphic semigroup 𝑒^{𝑡(𝐴2/2)} with generator 𝐴²/2 in terms of a Gaussian expectation of the group 𝑇(𝑡): 𝑒^{𝑡(𝐴2/2)}𝑓=𝔼_𝑁(0,𝑡)𝑇(𝑠)𝑓=(2𝜋𝑡)^−1/2∞_−∞𝑒^{−𝑠2/2𝑡}𝑇(𝑠)𝑓𝑑𝑠,𝑓∈𝑋. (10) More generally, given a polynomial 𝑃(𝐴) in 𝐴 with complex coefficients, whose leading term has the form 𝑐_2𝑚𝐴^2𝑚, with (−1)^𝑚+1Re(𝑐_2𝑚)>0, then 𝑃(𝐴) generates a holomorphic semigroup on 𝑋. Its action on a vector 𝑓∈𝑋 belonging to the domain of 𝐴^2𝑚 is given by 𝑒^{𝑡𝑃(𝐴)}𝑓=^∞_−∞𝑔_𝑡(𝑠)𝑇_𝐴(𝑠)𝑓𝑑𝑠,𝑓∈𝑋, (11) with 𝑔_𝑡(𝑠)=(1/2𝜋)∫^∞_−∞𝑒^{−𝑖𝑠𝜉}𝑒^{𝑡𝑃(𝑖𝜉)}𝑑𝜉 (see [36]). Equation (11) can be regarded as a particular example of the “method of transmutation” described in [38], which allows to relate different classes of problems by means of a suitable transformation technique.

Let us consider (11) in the case where 𝑋 is the Hilbert space 𝐿²(ℝ), 𝐴=𝑖(Δ/2), and 𝑃(𝑥)=𝑥²/2. One gets the following representation: 𝑒^{−(𝑡/8)Δ2}=^∞_−∞𝑒^{𝑖𝑠(Δ/2)}𝑒^{−𝑠2/2𝑡}√2𝜋𝑡𝑑𝑠, (12) giving the semigroup 𝑒^{−(𝑡/8)Δ2} in terms of a Gaussian expectation, with respect to the time variable 𝑠∈ℝ, of the Schrödinger group 𝑒^{𝑖𝑠(Δ/2)}. By applying (12) to an initial datum 𝑢₀∈𝑆(ℝ) one can write the following chain of equalities: 𝑒^{−(𝑡/8)Δ2}𝑢₀(𝑥)=^∞_−∞𝑒^{−𝑠2/2𝑡}√𝑒2𝜋𝑡^{𝑖𝑠(Δ/2)}𝑢₀(𝑥)=𝑑𝑠_∞0𝑒^{−𝑠2/2𝑡}√𝑒2𝜋𝑡^{𝑖𝑠(Δ/2)}𝑢₀(𝑥)+𝑒^{−𝑖𝑠(Δ/2)}𝑢₀(𝑥)=𝑑𝑠_∞0𝑒^{−𝑠2/2𝑡}√2𝜋𝑡^ℝ𝑒^{𝑖(𝑦2/2𝑠)}√𝑢2𝜋𝑖𝑠⁰+𝑒𝑥+𝑦_{−𝑖(𝑦2/2𝑠)}√𝑢−2𝜋𝑖𝑠⁰𝑥+𝑦𝑑𝑦𝑑𝑠. (13) Under the assumptions that 𝑢₀ can be extended to an entire function on the complex plane ℂ, denoted again with 𝑢₀, such that for any ℎ∈ℝ⁺ one has that 𝑒^{−ℎ|𝑧|2}|𝑢₀(𝑧)| is a bounded function on ℂ, a rotation of the integration path in the complex 𝑦-plane gives 𝑒^{−(𝑡/8)Δ2}𝑢₀(𝑥)=^∞₀𝑒^{−𝑠2/2𝑡}√2𝜋𝑡^ℝ𝑒^{−𝑦2/2𝑠}√𝑢2𝜋𝑠⁰𝑥+𝑒^𝑖𝜋/4𝑦+𝑢₀𝑥+𝑒^{−𝑖𝜋/4}𝑦𝑑𝑦𝑑𝑠. (14) Analogously, if 𝛼 is a real positive constant, a change of variable in the latter integral gives 𝑒^{−(𝛼𝑡/8)Δ2}𝑢₀(𝑥)=^∞₀𝑒^{−𝑠2/2𝑡}√2𝜋𝑡^ℝ𝑒^{−𝑦2/2𝑠}√𝑢2𝜋𝑠⁰𝑥+𝛼^1/4𝑒^𝑖𝜋/4𝑦+𝑢₀𝑥+𝛼^1/4𝑒^{−𝑖𝜋/4}𝑦𝑑𝑦𝑑𝑠. (15) Now the integral on the right hand side of (15) makes sense for a class of functions 𝑢₀ larger than those considered so far, as well as for complex values of the parameter 𝛼. Let us consider the Cauchy problem 𝜕𝑢()𝛼𝜕𝑡𝑥,𝑡=−8𝜕⁴𝑢𝜕𝑥⁴()𝑢()𝑥,𝑡−∞<𝑥<∞,0≤𝑡<∞,𝑥,0=𝑢₀(𝑥), (16) with 𝛼∈ℂ. The next theorems give a characterization of some classes of functions 𝑢₀ such that the integral ^∞₀𝑒^{−𝑠2/2𝑡}√2𝜋𝑡^ℝ𝑒^{−𝑦2/2𝑠}√𝑢2𝜋𝑠⁰𝑥+𝛼^1/4𝑒^𝑖𝜋/4𝑦+𝑢₀𝑥+𝛼^1/4𝑒^{−𝑖𝜋/4}𝑦𝑑𝑦𝑑𝑠 (17) is absolutely convergent and provides a representation for the solution of the Cauchy problem (16). The following theorem is a generalization of Funaki's result [16] to the case where 𝛼 is an arbitrary complex parameter.

Theorem 1. Let 𝑢₀ be a real valued function satisfying the following properties: (1)𝑢₀ can be extended to an entire function on the complex plane ℂ, denoted again with 𝑢₀;(2)for any ℎ∈ℝ⁺, one has that 𝑒^{−ℎ|𝑧|2}|𝑢₀(𝑧)|, 𝑒^{−ℎ|𝑧|2}|𝜕𝑢₀(𝑧)/𝜕𝑥|, and 𝑒^{−ℎ|𝑧|2}|𝜕𝑢₀(𝑧)/𝜕𝑦|, with 𝑧=𝑥+𝑖𝑦, are bounded functions on ℂ.
Then the solution of (16) is given by (17).

Proof. By the assumption that 𝑒^{−ℎ|𝑧|2}|𝑢₀(𝑧)| is a bounded functions on ℂ for any ℎ∈ℝ⁺, one can easily verify that the integral 𝑢(𝑡,𝑥)=^∞₀𝑒^{−𝑠2/2𝑡}√2𝜋𝑡^ℝ𝑒^{−𝑦2/2𝑠}√𝑢2𝜋𝑠⁰𝑥+𝛼^1/4𝑒^𝑖𝜋/4𝑦+𝑢₀𝑥+𝛼^1/4𝑒^{−𝑖𝜋/4}𝑦𝑑𝑦𝑑𝑠 (18) is absolutely convergent. By estimates 𝑒^{−ℎ|𝑧|2}𝜕𝑢₀(𝑧)𝑒𝜕𝑥<𝐶,_{−ℎ|𝑧|2}𝜕𝑢₀(𝑧)𝜕𝑦<𝐶 (19) and an integration by parts formula, one has that (𝜕⁴/𝜕𝑥⁴)𝑢(𝑡,𝑥) is given by 𝜕⁴𝜕𝑥⁴𝑢()8𝑡,𝑥=−𝛼^∞₀𝑒^{−𝑠2/2𝑡}√𝑠2𝜋𝑡2𝑡²⁴−1×2𝑡_ℝ𝑒^{−𝑦2/2𝑠}√𝑢2𝜋𝑠⁰𝑥+𝛼^1/4𝑒^𝑖𝜋/4𝑦+𝑢₀𝑥+𝛼^1/4𝑒^{−𝑖𝜋/4}𝑦𝑑𝑦𝑑𝑠 (20) which is equal to −(8/𝛼)(𝜕/𝜕𝑡)𝑢(𝑡,𝑥).

Remark 2. The functions 𝑢₀ satisfying the assumptions (1) and (2) of Theorem 1 include, for instance, polynomials of arbitrary degree, as well as the functions 𝑢₀ that are Fourier transform of measures, that is, of the form 𝑢₀(𝑥)=_ℝ𝑒^𝑖𝑘𝑥𝑑𝜇(𝑘), (21) where 𝜇 is a complex bounded variation measure such that ∀𝑥,𝑦∈ℝ: (i)∫_ℝ𝑒^𝑘𝑦𝑑|𝜇|(𝑘)<∞, ∫_ℝ𝑒^𝑘𝑦𝑑|𝜇|(𝑘)<∞, |𝜇| being the total variation measure of 𝜇, (ii)for any ℎ∈ℝ⁺, the functions (𝑥,𝑦)↦𝑒^{−ℎ(𝑥2+𝑦2)}|∫_ℝ𝑒^{𝑖𝑘𝑥−𝑘𝑦}𝑑𝜇(𝑘)| and (𝑥,𝑦)↦𝑒^{−ℎ(𝑥2+𝑦2)}|∫_ℝ𝑒^{𝑖𝑘𝑥−𝑘𝑦}𝑑𝜇(𝑘)| are bounded on ℝ².

Remark 3. The Funaki formula (7) for the solution of (16) in the case where 𝛼=−1 can be written in the following form: 𝑢(𝑡,𝑥)=^∞₀𝑒^{−𝑦2/2𝑡}√2𝜋𝑡^ℝ𝑒^{−𝑧2/2𝑦}𝑢2𝜋𝑦⁰(𝑥+𝑧)+𝑢₀(𝑥+𝑖𝑧)𝑑𝑧𝑑𝑦 (22) and can be obtained as a special case of (17).

The following theorem describes a particular class of functions 𝑢₀ such that the integral (17) is absolutely convergent and provides a representation for the solution of the Cauchy problem (16).

Theorem 4. Let 𝑢₀ be a real valued function of the form 𝑢₀(𝑥)=_ℝ𝑒^𝑖𝑘𝑥𝑑𝜇(𝑘),𝑥∈ℝ, (23) where 𝜇 is a complex bounded variation measure on ℝ satisfying the following bound: _ℝ𝑒^{(|𝛼|𝑡/8)𝑦4}𝑑𝜇𝑦<∞, (24)|𝜇| being the total variation measure of 𝜇.
Then the solution of the Cauchy problem (16) is given by (17).

Proof. Under the stated assumption on 𝑢₀, the integral (17) assumes the following form: ^∞₀𝑒^{−𝑠2/2𝑡}√2𝜋𝑡^ℝ𝑒^{−𝑦2/2𝑠}√2𝜋𝑠^ℝ𝑒^{𝑖𝑘(𝑥+𝛼1/4𝑒𝑖𝜋/4𝑦)}+𝑒^{𝑖𝑘(𝑥+𝛼1/4𝑒−𝑖𝜋/4𝑦)}𝑑𝜇(𝑘) (25) by Fubini theorem, which holds because of condition (24); the latter is equal to _ℝ𝑒^𝑖𝑘𝑥𝑒^{𝛼𝑡𝑦4/8}𝑑𝜇𝑦 (26) which is the solution of the Cauchy problem (16).

Equation (17) can be written in terms of the expectation with respect to the measure associated to a complex random variable 𝑧^𝛼_𝑡: 𝑒^{−𝛼(𝑡/8)Δ2}𝑢₀(𝑥)=𝔼𝑢₀𝑥+𝑧^𝛼_𝑡, (27) where 𝑧^𝛼_𝑡 has the following distribution: 𝑃𝑧^𝛼_𝑡∈𝐴=^∞₀𝑒^{−𝑠2/2𝑡}√2𝜋𝑡^ℝ𝑒^{−𝑦2/2𝑠}√𝜒2𝜋𝑠^𝐴𝛼^1/4𝑒^𝑖𝜋/4𝑦+𝜒_𝐴𝛼^1/4𝑒^{−𝑖𝜋/4}𝑦𝑑𝑦𝑑𝑠, (28)𝐴 being a Borel subset of the complex plane and 𝜒_𝐴 being its characteristic function. Clearly the measure is concentrated on two rays of the complex plane 𝛼^1/4𝑒^𝑖𝜋/4ℝ and 𝛼^1/4𝑒^{−𝑖𝜋/4}ℝ.

One can easily verify that random variable 𝑧^𝛼_𝑡 has the following properties: (i)𝑧^𝛼_𝑡∼𝑡^1/4𝑧^𝛼₁, (ii)𝔼[(𝑧^𝛼_𝑡)^𝑘]=0, 𝑘=1,2,3, (iii)𝔼[(𝑧^𝛼_𝑡)⁴]=−3𝑡, (iv)𝔼[|𝑧^𝛼_𝑡|²]=2|𝛼𝑡|^1/2∫^∞₀(𝑒^−𝑠2/2/√2𝜋)𝑠𝑑𝑠<+∞, (v)𝔼[𝑒^{𝑖𝜆𝑧𝛼𝑡}]=𝑒^{−(𝑡/8)𝛼𝜆4}.

Moreover formula (27) can be written in Funaki's notation (see (6) and (7)). Indeed let us consider two independent Brownian motions {𝐵(𝑡)}_𝑡≥0 and {𝑤(𝑡)}_𝑡≥0 and define the process {𝑋^𝛼_𝑡}_𝑡≥0 as 𝑋^𝛼_𝑡𝑒∶=^𝑖𝜋/4𝛼^1/4𝐵(𝑤(𝑡(𝑡)𝑒))if𝑤≥0_{−𝑖𝜋/4}𝛼^1/4𝐵(−𝑤(𝑡))if𝑤(𝑡)<0. (29) Then (17) can then be written in the following form: 𝑒^{−𝛼(𝑡/8)Δ2}𝑢₀(𝑥)=𝔼𝑢₀𝑥+𝑋^𝛼_𝑡. (30)

Remark 5. Analogous results can be obtained also in the case where Δ² is replaced with higher powers of the Laplacian, namely, Δ⁴,Δ⁸,…,Δ^2𝑛. It is sufficient to iterate 𝑛-times formula (10). One obtains a formula with multiple Gaussian integrations, similar to the one proposed, for instance, in [17]. As in the Funaki approach, the probability measure of the complex random variable can also be obtained by composing three independent Brownian motions in a suitable way. In fact any even power of the Laplacian can be handled by means of the general formula (11), but a probabilistic interpretation in terms of the composition of several independent Brownian motions is not always possible. For instance, in the case where one considers Δ⁶, (11) gives the following result: 𝑒^{−(𝑡/26)Δ6}𝑓(𝑥)=^∞_−∞𝑔_𝑡(𝑠)𝑒^{𝑖𝑠(Δ/2)}𝑢₀(𝑥)𝑑𝑠, (31) with 𝑔_𝑡(𝑠)=12𝜋^∞_−∞𝑒^{−𝑖𝑠𝜉}𝑒^−𝑡𝜉6𝑑𝜉. (32) We will not further develop these formulae here, but we will only focus on the case of Δ².

3. Feynman-Kac Type Formulae

The process {𝑋^𝛼_𝑡}_𝑡≥0 defined in (29) and appearing in (30) provides a probabilistic representation for the solution of (16) (under suitable analyticity assumptions on the initial datum 𝑢₀). On the other hand it has not independent increments, so it does not naturally give rise to generalizations of formula (30) to the case where (16) contains also a potential 𝑉, that is, to a Feynman-Kac formula of the form 𝑢(𝑡,𝑥)=𝔼𝑢₀𝑥+𝑋^𝛼_𝑡𝑒^{−∫𝑡0𝑉(𝑥+𝑋𝛼𝑠)𝑑𝑠}. (33) Indeed, by applying formally the Trotter product formula, one gets 𝑒^{−𝑡((𝛼Δ2/8)+𝑉)}𝑢₀(𝑥)=lim_𝑛→∞𝑒^{−(𝑡/𝑛)(𝛼Δ2/8)}𝑒^{−(𝑡/𝑛)𝑉𝑛}𝑢₀(𝑥)=lim_𝑛→∞𝔼𝑒^{−(𝑡/𝑛)∑𝑛𝑘=1𝑉(𝑥+∑𝑘𝑗=1𝑧𝛼𝑗(𝑡/𝑛))}×𝑢₀𝑥+^𝑛_𝑗=1𝑧^𝛼_𝑗𝑡𝑛, (34) where 𝑧^𝛼_𝑗(𝑡/𝑛), 𝑗=1,…𝑛, is a family of  𝑛  independent identically distributed random variables, distributed as 𝑧^𝛼_𝑡/𝑛 (see (28)). In the latter line one would be tempted to interpret the random variables 𝑧^𝛼_𝑗(𝑡/𝑛) as the independent increments of a complex-valued stochastic process {𝑍^𝛼_𝑡}_𝑡≥0, different from {𝑋^𝛼_𝑡}_𝑡≥0, that is, to interpret the limit (34) as the cylindrical approximations of an integral of the following form: 𝔼𝑢₀𝑥+𝑍^𝛼_𝑡𝑒^{−∫𝑡0𝑉(𝑥+𝑍𝛼𝑠)𝑑𝑠}=lim_𝑛→∞𝔼𝑒^{−(𝑡/𝑛)∑𝑛𝑘=1𝑉(𝑥+∑𝑘𝑗=1𝑧𝛼𝑗(𝑡/𝑛))}×𝑢₀𝑥+^𝑛_𝑗=1𝑧^𝛼_𝑗𝑡𝑛. (35) In fact such a process cannot exist, as its construction would be possible providing the weak convergence of the sequence of complex random variables ∑^𝑛_𝑗=1𝑧^𝛼_𝑗(𝑡/𝑛) as 𝑛→∞. By using the scaling properties of the random variable 𝑧^𝛼_𝑡, this is equivalent to the weak convergence of the sequence 𝑛^−1/4∑^𝑛_𝑗=1𝑧^𝛼_𝑗(1). As 𝑧^𝛼_𝑗(1) are independent identically distributed complex random variables with finite covariance, then the sequence 𝑛^−1/2∑^𝑛_𝑗=1𝑧^𝛼_𝑗(1) has a Gaussian limit. Consequently, the sequence 𝑛^−1/4∑^𝑛_𝑗=1𝑧^𝛼_𝑗(1) cannot converge weakly, as erroneously stated in [25], and formula (35) cannot be interpreted as an integral with respect to the measure associated to a process but just as the limit (34). Indeed the present section is devoted to the proof, for a suitable class of continuous functions 𝑉 and initial datum 𝑢₀, of a Feynman-Kac formula representing the solution of the Cauchy problem (8) as the limit of a sequence of finite dimensional approximations.

The implementation of formula (35) presents some technical problems, which do not appear in the proof of the classical Feynman-Kac formula (for the heat equation with potential). The first one is the definition of the integrals involved. In fact, since the random variables 𝑧^𝛼(𝑡/𝑛) are complex valued, the real valued function 𝑉 as well as the initial datum 𝑢₀ has to admit an analytic extension to an entire function on the complex plane. We cannot require that it is bounded on ℂ; otherwise we could consider only the trivial case. Consequently we will integrate unbounded function, and in principle the convergence of the integrals has to be checked. In fact, for a large class of potentials, the integrals are not absolutely convergent and have to be defined in a suitable way.

The second problem concerns the proof that the integral (35) represents the solution of the Cauchy problem (8). Even if the second line of (35) recalls Trotter’s product, this formula cannot be directly applied since it does not hold for general 𝛼∈ℂ and 𝑉 continuous real valued function.

The problem of the proof of a Feynman-Kac type formula for equations of the form (8) has been analyzed in [22], where the case of a linear potential 𝑉 is handled, and in [39], but a detailed proof for a sufficiently large class of potentials 𝑉 is still lacking. We generalized these results to the case where 𝑉 is linear in the space variable and presents an explicit time dependence.

Theorem 6. Let 𝑎∶ℝ→ℂ be a continuous function and let us denote  𝑀∶=max_{𝑠∈[0,𝑡]}|𝑎(𝑠)|. Let 𝑢₀ be of the form 𝑢₀(𝑥)=∫_ℝ𝑒^𝑖𝑥𝑦𝑑𝜇₀(𝑦), where 𝜇₀ is a complex bounded variation measure on ℝ satisfying the following condition: _ℝ𝑒^{(|𝛼|𝑡/8)(|𝑦|+𝑀𝑡)4}𝑑𝜇₀𝑦<∞. (36) Then the solution of the Cauchy problem: 𝜕𝑢()𝛼𝜕𝑡𝑥,𝑡=−8𝜕⁴𝜕𝑥⁴𝑢()(𝑡)()𝑢()𝑥,𝑡−𝑖𝑎𝑥𝑢𝑡,𝑥−∞<𝑥<∞,0≤𝑡<∞,𝑥,0=𝑢₀(𝑥), (37) is given by 𝑢(𝑡,𝑥)=𝔼𝑢₀𝑥+𝑍^𝛼_𝑡𝑒^{−𝑖∫𝑡0𝑎(𝑡−𝑠)(𝑥+𝑍𝛼𝑠)𝑑𝑠}∶=lim_𝑛→∞𝔼𝑢₀𝑥+^𝑛_𝑗=1𝑧^𝛼_𝑗𝑡𝑛×𝑒^{−𝑖𝑡/𝑛∑𝑛𝑘=1𝑎(𝑡−(𝑘𝑡/𝑛))(𝑥+∑𝑘𝑗=1𝑧𝛼𝑗(𝑡/𝑛))}, (38)

Proof. Under the stated assumptions, the finite dimensional integrals appearing in formula (38) assume the following form: 𝑒^{−(𝑖𝑡𝑥/𝑛)∑𝑛𝑘=1𝑎(𝑡−(𝑘𝑡/𝑛))}𝔼𝑒^{−𝑖(𝑡/𝑛)∑𝑛𝑘=1𝑎(𝑡−(𝑘𝑡/𝑛))∑𝑘𝑗=1𝑧𝛼𝑗(𝑡/𝑛)}×_ℝ𝑒^𝑖𝑥𝑦𝑒^{𝑖𝑦∑𝑛𝑗=1𝑧𝛼𝑗(𝑡/𝑛)}𝑑𝜇₀𝑦, (39) where 𝑧^𝛼_𝑗(𝑡/𝑛), 𝑗=1,…𝑛, is a family of  𝑛  independent identically distributed random variables, distributed as 𝑧^𝛼_𝑡/𝑛. Now by applying the Fubini theorem, which holds because of condition (36), the latter is equal to 𝑒^{−(𝑖𝑡𝑥/𝑛)∑𝑛𝑘=1𝑎(𝑡−(𝑘𝑡/𝑛))}×_ℝ𝑒^𝑖𝑥𝑦𝔼𝑒^{−𝑖∑𝑛𝑗=1𝑧𝛼𝑗(𝑡/𝑛)(−𝑦+𝑡/𝑛∑𝑛𝑘=𝑗𝑎(𝑡−(𝑘𝑡/𝑛)))}𝑑𝜇₀𝑦=𝑒^{−(𝑖𝑡𝑥/𝑛)∑𝑛𝑘=1𝑎(𝑡−(𝑘𝑡/𝑛))}×_ℝ𝑒^𝑖𝑥𝑦𝑒^{−(𝛼/8)(𝑡/𝑛)∑𝑛𝑗=1(−𝑦+𝑡/𝑛∑𝑛𝑘=𝑗𝑎(𝑡−(𝑘𝑡/𝑛)))4}𝑑𝜇₀𝑦. (40) By dominated convergence theorem, the limit as 𝑛→∞ of the last line is equal to 𝑒^{−𝑖𝑥∫𝑡0𝑎(𝑠)𝑑𝑠}_ℝ𝑒^𝑖𝑥𝑦𝑒^{−(𝛼/8)∫𝑡0(−𝑦+∫𝑡𝑠𝑎(𝑡−𝑢)𝑑𝑢)4𝑑𝑠}𝑑𝜇₀𝑦, (41) which is the solution of Cauchy problem (37).

4. Conclusions

In this paper we have proposed the construction of a particular probabilistic representation for the solution of the equation 𝑢̇=−𝛼Δ²𝑢+𝑉 in terms of a Feynman-Kac type formula. The class of potentials 𝑉 which can be handled by requiring that the probabilistic integrals are defined in Lebesgue sense, that is, as absolutely convergent integrals, is rather restricted because of the complex nature of the process. A generalization of these results to more general potentials requires the implementation of an integration technique, in infinite dimensions, of a different type, by relaxing the absolute convergence of the integrals, as in the cases handled, for instance, in [6] concerning the functional integral representation for the solution of Schrödinger equations. This problem will be handled in a forthcoming paper.

Acknowledgments

Interesting discussions with Sergio Albeverio, Giuseppe Da Prato, Paolo Dai Pra, Franco Flandoli, Enrico Priola, and Luciano Tubaro are gratefully acknowledged, as well as the financial support of Fondazione Bruno Kessler, Trento, Italy.