# Probabilistic Representations for the Solution of Higher Order Differential Equations

Authors: S. Mazzucchi
Journal: International Journal of Partial Differential Equations
 (2013)

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## 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.


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## Body

## 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∞(ℝd) (the continuous functions vanishing at infinity):
(1)∂∂tu(t,x)=12Δu(t,x)-V(x)u(t,x),t∈ℝ+,x∈ℝd,u(0,x)=u0(x)
in terms of an integral with respect to the measure of the Wiener process, the mathematical model of the Brownian motion [2]:
(2)u(t,x)=∫Ct‍e-∫0t‍V(ω(s)+x)dsu0(ω(t)+x)dW(ω).If the Laplacian in (1) is replaced with a higher order differential operator, that is, if we consider a Cauchy problem of the form
(3)∂∂tu(t,x)=(-1)N+1ΔNu(t,x)-V(x)u(t,x),t∈ℝ+,x∈ℝd,u(0,x)=u0(x),
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 ΔN 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
(4)u(t,x)=∫ω(0)=x‍e-∫0t‍V(ω(s))dsu0(ω(t))dP(ω)
(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 P2N 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 P2N. 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∈C2N, 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 P2N 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 ofcomplex 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 {Xt}t≥0 by composing two independent Brownian motions {B(t)}t≥0 and {w(t)}t≥0 in the following way:
(5)Xt:={B(w(t))ifw(t)≥0iB(-w(t))ifw(t)<0
and proving that, for a suitable class of analytic initial datum u0, the solution of the Cauchy problem
(6)∂u∂t=18∂4u∂x4-∞<x<∞,0<t<∞,u(0,x)=u0(x)
is given by the expectation
(7)u(t,x)=𝔼[u0(x+Xt)].In fact this result can be generalized to partial differential equations of order2n, 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{Xt} 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/∂xn)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 equationΔku=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:
(8)∂u∂t(x,t)=-α8∂4∂x4u(x,t)+V(t,x)u(t,x)-∞<x<∞,0≤t<∞,u(x,0)=u0(x),
with α∈ℂ and for V,u0 satisfying suitable analyticity assumptions.The paper is organized as follows. Section2 presents the construction of a complex random variable ztα and the representation of the solution of (8) with V≡0 in terms of the expectation with respect to the probability measure associated with ztα. 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,
(9)∂u∂t(x,t)=-α8∂4u∂x4(x,t)-∞<x<∞,0≤t<∞.
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 spaceX and a strongly continuous group of operators {TA(t)}t∈ℝ on X with generator A, it is possible to construct the holomorphic semigroup et(A2/2) with generator A2/2 in terms of a Gaussian expectation of the group T(t):
(10)et(A2/2)f=𝔼N(0,t)[T(s)f]=(2πt)-1/2∫-∞∞‍e-s2/2tT(s)fds,f∈X.
More generally, given a polynomial P(A) in A with complex coefficients, whose leading term has the form c2mA2m, with (-1)m+1Re(c2m)>0, then P(A) generates a holomorphic semigroup on X. Its action on a vector f∈X belonging to the domain of A2m is given by
(11)etP(A)f=∫-∞∞‍g^t(s)TA(s)fds,f∈X,
with g^t(s)=(1/2π)∫-∞∞‍e-isξetP(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:
(12)e-(t/8)Δ2=∫-∞∞‍eis(Δ/2)e-s2/2t2πtds,
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 eis(Δ/2). By applying (12) to an initial datum u0∈S(ℝ) one can write the following chain of equalities:
(13)e-(t/8)Δ2u0(x)=∫-∞∞‍e-s2/2t2πteis(Δ/2)u0(x)ds=∫0∞‍e-s2/2t2πt(eis(Δ/2)u0(x)+e-is(Δ/2)u0(x))ds=∫0∞‍e-s2/2t2πt∫ℝ‍(ei(y2/2s)2πisu0(x+y)+e-i(y2/2s)-2πisu0(x+y))dyds.
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
(14)e-(t/8)Δ2u0(x)=∫0∞‍e-s2/2t2πt∫ℝ‍e-y2/2s2πs(u0(x+eiπ/4y)+u0(x+e-iπ/4y))dyds.
Analogously, if α is a real positive constant, a change of variable in the latter integral gives
(15)e-(αt/8)Δ2u0(x)=∫0∞‍e-s2/2t2πt∫ℝ‍e-y2/2s2πs(u0(x+α1/4eiπ/4y)+u0(x+α1/4e-iπ/4y))dyds.
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
(16)∂u∂t(x,t)=-α8∂4u∂x4(x,t)-∞<x<∞,0≤t<∞,u(x,0)=u0(x),
with α∈ℂ. The next theorems give a characterization of some classes of functions u0 such that the integral
(17)∫0∞‍e-s2/2t2πt∫ℝ‍e-y2/2s2πs(u0(x+α1/4eiπ/4y)+u0(x+α1/4e-iπ/4y))dyds
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.
Letu0 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 anyh∈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 thate-h|z|2|u0(z)| is a bounded functions on ℂ for any h∈ℝ+, one can easily verify that the integral
(18)u(t,x)=∫0∞‍e-s2/2t2πt∫ℝ‍e-y2/2s2πs(u0(x+α1/4eiπ/4y)+u0(x+α1/4e-iπ/4y))dyds
is absolutely convergent. By estimates
(19)e-h|z|2|∂u0(z)∂x|<C,e-h|z|2|∂u0(z)∂y|<C
and an integration by parts formula, one has that (∂4/∂x4)u(t,x) is given by
(20)∂4∂x4u(t,x)=-8α∫0∞‍e-s2/2t2πt(s22t4-12t)×∫ℝ‍e-y2/2s2πs(u0(x+α1/4eiπ/4y)+u0(x+α1/4e-iπ/4y))dyds
which is equal to -(8/α)(∂/∂t)u(t,x).Remark 2.
The functionsu0 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
(21)u0(x)=∫ℝ‍eikxdμ(k),
where μ is a complex bounded variation measure such that ∀x,y∈ℝ:  (i)
∫ℝ‍ekyd|μ|(k)<∞, ∫ℝ‍ekyd|μ|(k)<∞, |μ| being the total variation measure of μ,(ii)
for anyh∈ℝ+, the functions (x,y)↦e-h(x2+y2)|∫ℝ‍eikx-kydμ(k)| and (x,y)↦e-h(x2+y2)|∫ℝ‍eikx-kydμ(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:
(22)u(t,x)=∫0∞‍e-y2/2t2πt∫ℝ‍e-z2/2y2πy(u0(x+z)+u0(x+iz))dzdy
and can be obtained as a special case of (17).The following theorem describes a particular class of functionsu0 such that the integral (17) is absolutely convergent and provides a representation for the solution of the Cauchy problem (16).Theorem 4.
Letu0 be a real valued function of the form
(23)u0(x)=∫ℝ‍eikxdμ(k),x∈ℝ,
where μ is a complex bounded variation measure on ℝ satisfying the following bound:
(24)∫ℝ‍e(|α|t/8)y4d|μ|(y)<∞,|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 onu0, the integral (17) assumes the following form:
(25)∫0∞‍e-s2/2t2πt∫ℝ‍e-y2/2s2πs∫ℝ‍(eik(x+α1/4eiπ/4y)+eik(x+α1/4e-iπ/4y))dμ(k)
by Fubini theorem, which holds because of condition (24); the latter is equal to
(26)∫ℝ‍eikxeαty4/8dμ(y)
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 ztα:
(27)e-α(t/8)Δ2u0(x)=𝔼[u0(x+ztα)],
where ztα has the following distribution:
(28)P(ztα∈A)=∫0∞‍e-s2/2t2πt∫ℝ‍e-y2/2s2πs(χA(α1/4eiπ/4y)+χA(α1/4e-iπ/4y))dyds,A being a Borel subset of the complex plane and χA being its characteristic function. Clearly the measure is concentrated on two rays of the complex plane α1/4eiπ/4ℝ and α1/4e-iπ/4ℝ.One can easily verify that random variableztα has the following properties:  (i)
ztα~t1/4z1α,(ii)
𝔼[(ztα)k]=0, k=1,2,3,(iii)
𝔼[(ztα)4]=-3t,(iv)
𝔼[|ztα|2]=2|αt|1/2∫0∞‍(e-s2/2/2π)sds<+∞,(v)
𝔼[eiλztα]=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 {Xtα}t≥0 as
(29)Xtα:={eiπ/4α1/4B(w(t))ifw(t)≥0e-iπ/4α1/4B(-w(t))ifw(t)<0.
Then (17) can then be written in the following form:
(30)e-α(t/8)Δ2u0(x)=𝔼[u0(x+Xtα)].Remark 5.
Analogous results can be obtained also in the case whereΔ2 is replaced with higher powers of the Laplacian, namely, Δ4,Δ8,…,Δ2n. 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:
(31)e-(t/26)Δ6f(x)=∫-∞∞‍g^t(s)eis(Δ/2)u0(x)ds,
with
(32)g^t(s)=12π∫-∞∞‍e-isξe-tξ6dξ.
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{Xtα}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
(33)u(t,x)=𝔼[u0(x+Xtα)e-∫0t‍V(x+Xsα)ds].
Indeed, by applying formally the Trotter product formula, one gets
(34)e-t((αΔ2/8)+V)u0(x)=limn→∞(e-(t/n)(αΔ2/8)e-(t/n)V)nu0(x)=limn→∞𝔼[{∑j=1n‍}e-(t/n)∑k=1nV(x+∑j=1kzjα(t/n))×u0(x+∑j=1n‍zjα(tn))],
where zjα(t/n), j=1,…n, is a family  of  n  independent identically distributed random variables, distributed as zt/nα (see (28)). In the latter line one would be tempted to interpret the random variables zjα(t/n) as the independent increments of a complex-valued stochastic process {Ztα}t≥0, different from {Xtα}t≥0, that is, to interpret the limit (34) as the cylindrical approximations of an integral of the following form:
(35)𝔼[u0(x+Ztα)e-∫0t‍V(x+Zsα)ds]=limn→∞𝔼[{∑j=1n‍}e-(t/n)∑k=1nV(x+∑j=1kzjα(t/n))×u0(x+∑j=1n‍zjα(tn))].
In fact such a process cannot exist, as its construction would be possible  providing the weak convergence of the sequence of complex random variables ∑j=1n‍zjα(t/n) as n→∞. By using the scaling properties of the random variable ztα, this is equivalent to the weak convergence of the sequence n-1/4∑j=1n‍zjα(1). As zjα(1) are independent identically distributed complex random variables with finite covariance, then the sequence n-1/2∑j=1n‍zjα(1) has a Gaussian limit. Consequently, the sequence n-1/4∑j=1n‍zjα(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 zα(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.
Leta:ℝ→ℂ be a continuous function and let us denote  M:=maxs∈[0,t]|a(s)|. Let u0 be of the form u0(x)=∫ℝ‍eixydμ0(y), where μ0 is a complex bounded variation measure on ℝ satisfying the following condition:
(36)∫ℝ‍e(|α|t/8)(|y|+Mt)4d|μ0|(y)<∞.
Then the solution of the Cauchy problem:
(37)∂u∂t(x,t)=-α8∂4∂x4u(x,t)-ia(t)xu(t,x)-∞<x<∞,0≤t<∞,u(x,0)=u0(x),
is given by
(38)u(t,x)=𝔼[u0(x+Ztα)e-i∫0t‍a(t-s)(x+Zsα)ds]:=limn→∞𝔼[u0(x+∑j=1n‍zjα(tn))×e-it/n∑k=1na(t-(kt/n))(x+∑j=1kzjα(t/n)){(x+∑j=1n‍zjα(tn))}],Proof.
Under the stated assumptions, the finite dimensional integrals appearing in formula (38) assume the following form:
(39)e-(itx/n)∑k=1na(t-(kt/n))𝔼[×∫ℝ‍eixyeiy∑j=1nzjα(t/n)dμ0(y)e-i(t/n)∑k=1na(t-(kt/n))∑j=1kzjα(t/n)×∫ℝ‍eixyeiy∑j=1nzjα(t/n)dμ0(y)],
where zjα(t/n), j=1,…n, is a family of  n  independent identically distributed random variables, distributed as zt/nα. Now by applying the Fubini theorem, which holds because of condition (36), the latter is equal to
(40)e-(itx/n)∑k=1na(t-(kt/n))×∫ℝ‍eixy𝔼[e-i∑j=1nzjα(t/n)(-y+t/n∑k=jna(t-(kt/n)))]dμ0(y)=e-(itx/n)∑k=1na(t-(kt/n))×∫ℝ‍eixye-(α/8)(t/n)∑j=1n(-y+t/n∑k=jna(t-(kt/n)))4dμ0(y).
By dominated convergence theorem, the limit as n→∞ of the last line is equal to
(41)e-ix∫0t‍a(s)ds∫ℝ‍eixye-(α/8)∫0t‍(-y+∫st‍a(t-u)du)4dsdμ0(y),
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 equationu˙=-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.



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