A note on stochastic Schrödinger equations with fractional multiplicative noise

A note on stochastic Schrödinger equations with fractional multiplicative noise

Olivier Pinaud 111pinaud@math.colostate.edu Department of Mathematics, Colorado State University
Fort Collins, CO 80523
Abstract

This work is devoted to non-linear stochastic Schrödinger equations with multiplicative fractional noise, where the stochastic integral is defined following the Riemann-Stieljes approach of Zähle. Under the assumptions that the initial condition is in the Sobolev space for a dimension less than three and an integer greater or equal to zero, that the noise is a fractional Brownian motion with Hurst index and spatial regularity , as well as appropriate hypotheses on the non-linearity, we obtain the local existence of a unique pathwise solution in , for any . Contrary to the parabolic case, standard fixed point techniques based on the mild formulation of the SPDE cannot be directly used because of the weak smoothing in time properties of the Schrödinger semigroup. We follow here a different route and our proof relies on a change of phase that removes the noise and leads to a Schrödinger equation with a magnetic potential that is not differentiable in time.

1 Introduction

This work is concerned with the existence theory for the stochastic Schrödinger equation with fractional multiplicative random noise of the form

(1)

where is an infinite dimensional fractional Brownian motion in time and smooth in the space variables. The sense of the stochastic integral will be precised later on and the term is non-linear. We limit ourselves to for physical considerations, but the theory should hold for arbitrary with adjustments of some hypotheses. Our interest for such a problem is motivated by the study of the propagation of paraxial waves in random media that are both strongly oscillating and slowly decorrelating in the variable associated to the distance of propagation. Such media are encountered for instance in turbulent atmosphere or in the earth’s crust [9, 25]. More precisely, it is well-known that the wave equation reduces in the paraxial approximation [27] to the Schrödinger equation on the enveloppe function , which reads in three dimensions

where is the direction of propagation of the collimated beam, is the transverse plane, , and is a random potential accounting for the fluctuations of the refraction index. If is stationary and its correlation function has the property that

where is a smooth function and , then the process presents long-range correlations in the variable since is not integrable. Rescaling as and invoking the non-central limit theorem, see e.g. [28], one may expect formally when that

where is a Gaussian process with correlation function

(2)

with . Proving this fact is an open problem, while the short-range case (when is integrable) was addressed in [1, 12], and the limiting wave function is shown to be a solution to the Itô-Schrödinger equation. Our starting point here is (1), where we added a non-linear term to account for possible non-linearities arising for instance in non-linear optics.

Let us be more precise now about the nature of the stochastic integral in (1). Since (1) is obtained after formal asymptotic limit of a norm preserving Schrödinger equation, one may legitimately expect the limiting equation to also preserve the norm. The appropriate stochastic integral should therefore be of Stratonovich type, which in the context of fractional Brownian motions are encountered in the literature as pathwise integrals of various types, e.g. symmetric, forward, or backward, [2, 32]. Since in our case of interest the Hurst index is greater than , all integrals are equivalent and can be seen as Riemann-Stieljes integrals of appropriate functions, see [2, 32] and section 2 for more details. Such an integral is well-defined for instance if both integrands are of Hölder regularity with respective indices and such that [32].

In the context of SPDEs, the infinite dimensional character of the Gaussian process is usually addressed within two frameworks, whether for standard or fractional Brownian motions: the (fractional) Brownian type, or the cylindrical type, see [4]. The first class is more restrictive and requires the correlation operator in the space variables to be a positive trace class operator (or even more for fractional Brownian motions, see [17]); in the second class, it is only supposed that is a positive self-adjoint operator on some Hilbert space with appropriate Hilbert-Schmidt embeddings. As was done in [17] for parabolic equations with multiplicative fractional noise, we will assume our noise is of fractional type, which yields direct pathwise (almost sure) estimates on in some functional spaces. The cylindrical case is more difficult and our approach does not seem to generalize to it. The fractional case actually excludes stationary in correlation functions of the form (2) since they lead to a cylindrical type noise, which is a drawback of our assumptions. This latter case, even in the more favorable situation of parabolic equations, seems to still be open. Standard Brownian motions are more amenable to cylindrical noises since the Itô isometry holds. In the case of fractional type integrals, the “Itô isometry” involves the Malliavin derivative of the process, which is difficult to handle in the context of SPDEs with multiplicative noise. Hence, an existence theory for the Schrödinger equation in some average sense seems more involved to achieve, and we thus focus on a pathwise theory which requires the Brownian assumption in our setting.

Stochastic ODEs with fractional Brownian motion were investigated in great generality in [21]. Stochastic PDEs with fractional multiplicative noise are somewhat difficult to study and to the best of our knowledge, the most advanced results in the field are that of Maslowski and al [17], Duncan et al [10] or Grecksch et al [14]. The reference [10] involves finite dimensional fractional noises, which is a limitation. Several other works deal with additive noise, which is a much more tractable situation as stochastic integrals are seen as Wiener integrals [11, 29, 13] and cylindrical type noises are allowed. References [17, 10, 14] consider variations of parabolic equations of the form

(3)

where is the generator of an analytic semigroup , and the equation can be complemented with non-linear terms and a time-dependency in in [10]. The noise in these references is a fractional Brownian with possibly additional assumptions. The difficulty is naturally to make sense of the term and to show that is Hölder in time. In that respect, the analyticity hypothesis is crucial: indeed, the standard technique to analyze (3) is to use mild solutions of the form

and for the integral to exist, one needs the term to be roughly of Hölder regularity in time with index greater than . This means that both and the semigroup need such a regularity. While the term can be treated in the fixed point procedure, the semigroup has to be sufficiently smooth in time, which holds for analytic semigroups, but not in the case of unitary groups generated by in the Schrödinger equation. In the latter situation, one can “trade” some regularity in time for with some spatial regularity on , but this procedure does not seem to be exploitable in a fixed point procedure. Another possibility could be to take advantage of the regularizing properties of the Schrödinger semigroup that provide a gain of almost half a derivative in space, and therefore to almost a quarter of a derivative in time [3]. It looked to us rather delicate to follow such an approach since the smoothing effects hold for particular topologies involving spatial weights which looked fairly intricate to handle in our problem, even by using the classical exchange regularity/decay for the Schrödinger equation. The strictly linear case (i.e. when in (1)) can likely be treated by somewhat brute force with iterated Wiener integrals and the Hu-Meyer formula, but this approach does not carry on to the non-linear setting.

We propose in this work a different route than the mild formulation and a quite simple remedy based on two direct observations: (i) the usual change of variables formula holds for the pathwise stochastic integral and (ii) using it along with a change of phase removes the noise and leads to a Schrödinger equation with magnetic vector potential . Forgetting for the moment the non-linear term , introducing the filtered wavefunction, and supposing without lack of generality that , this yields the system

(4)

which is a standard Schrödinger equation with a time-dependent Hamiltonian. There is a vast literature on the subject, see [24, 5, 15, 16, 19, 18, 31, 30, 20] for a non-exhaustive list. One of the most classical assumptions on for the existence of an evolution operator generated by the Hamiltonian is that is a function in time with values in . This is of course not verified for the fractional Brownian motion. The price to pay for that is to require additional spatial regularity, and one possibility (likely not optimal) is to suppose that has values in . Assuming such a strong regularity is naturally a drawback in this approach.

Regarding the treatment of the non-linearity, we suppose that it is invariant by a change of phase, that is , which is verified by power non-linearities of the form or by where is the Poisson potential. Contrary to the case of non-linear Itô-Schrödinger equations where various tools such as Strichartz estimates or Morawetz estimates have been successfully used to investigate focusing/defocusing phenomena in random and deterministic settings [3, 8, 7, 6], there are very few available techniques to study (4) with a potential vector not smooth in time and augmented with the term . There are Strichartz estimates in the context of magnetic Schrödinger equations, but some require to be in time [31], and some others avoid such an hypothesis but assume instead that is small in some sense [26], which has no reason to hold here. As a result, we are lead to make rather crude assumptions on in order to obtain a local existence result. Moreover, the analysis of non-linear Schrödinger equations generally relies in a crucial manner on energy methods. In our problem of interest, we are only able to obtain energy conservation for smooth solutions, which turns out to be of no use when trying to obtain a global-in-time result and limits us to local results, unless the non-linearity is globally Lipschitz in the appropriate topology. This is due to the fractional noise that does not allow us to obtain estimates for via the energy relation, as we explain further in remark 2.

The main result of the paper is therefore a local existence result of pathwise solutions to (1) with a smooth fractional noise and appropriate assumptions on the non-linearity . The article is structured as follows: in section 2, we recall basic results on fractional stochastic integration, and present our main result in section 3; section 4 is devoted to the magnetic Schrödinger equation (4), while section 5 concerns the proof of our main theorem.

2 Preliminaries

Notation. We denote by and , , the standard Sobolev spaces with the convention that . For a Banach space , , and , denote the space or mesurable functions equipped with the norm

The space denotes the classical Hölder space of functions with values in . When or , we will simply use the notations , and . Notice that for any , . For two Banach spaces and , denotes the space of bounded operators from to , with the convention . The inner product is denoted by where is the complex conjugate of .

Fractional Brownian motion. For some positive time , we denote by a standard fractional Brownian motion (fBm) over a probability space with Hurst index . We will denote by the space of square integrable random variables for the measure and will often omit the dependence of on for simplicity. The process is a centered Gaussian process with covariance

Since , admits a Hölder continuous version with index strictly less than . In order to definite the infinite dimensional noise , consider a sequence of independent fBm . Let be a positive trace class operator on and denote by its spectral elements. For , non-negative integer, and , we assume that

(5)

The process is then formally defined by

The sum is normally convergent in , almost surely for . Indeed, in the same fashion as [17], let

so that by monotone convergence

According to [21] Lemma 7.4, for every and , there exists a positive random variable where is finite for and independent of since the are identically distributed, such that almost surely. Hence, thanks to (5) and picking , we have ,

(6)

and defines almost surely an element of . As a contrast, a cylindrical fractional Brownian motion is defined for a positive self-adjoint , which does not provide us with almost sure bounds on in . Suppose indeed that is a convolution operator of the form for some smooth real-valued kernel and that is a real-valued basis of . Then, the resulting correlation function is stationary (this follows from the convolution and is motivated by (2)) and

so that belongs to for . As explained in the introduction, we are not able to handle such a noise since integration in the probability space is required beforehand in order to get some estimates. This is not an issue in the context of standard Brownian motions or additive fractional noise, but leads to technical difficulties here.

Fractional stochastic integration. We follow the approach of [17, 21] based on the work of Zähle [32] and introduce the so-called Weyl derivatives defined by, for any and :

whenever these quantities are finite. Above, stands for the Euler function. Following [32], the generalized Stieljes integral of a function against a function with , and is defined by

(7)

with . The definition does not depend on and

The integral can be extended to different classes of functions since, see [21],

(8)

where

so that the integral is well-defined if and . Besides, the fractional integral satisfies the following change of variables formula, see [32]: let , and with , then

(9)

where , denotes the partial derivative of with respect to the coordinate.

For some Banach space and an operator-valued random function almost surely for some , the stochastic integral of with respect to is then formally defined by

(10)

The integral defines almost surely an element of for all since by Jensen’s inequality for the second line

and as shown in [17],

(11)

Hence (10) is well-defined and the convergence of the sum has to be understood as the almost sure convergence in .

We will use the following two results: the first Lemma is a generalization of the change of variables formula (9) to the infinite dimensional setting, and the second a version a the Fubini theorem adapted to the stochastic integral. Their proofs are given in the appendix. Below, .

Lemma 2.1

Let be a continuously differentiable function. Let be the differential of with respect to the first argument and be its partial derivative with respect to the second. For every and for any , let . Assume that with , and that there exists a constant such that, for all with :

(12)

Then, we have the change of variables formula, , almost surely:

Lemma 2.2

Let with . Then we have:

3 Main result

We present in this section the main result of the paper. We precise first in which sense (1) is understood. We say that , for all , non-negative integer, is a solution to (1) if it verifies for all test function , for all and almost surely

(13)

where the term involving the stochastic integral is understood as

The latter is well-defined since the mapping belongs to thanks to standard Sobolev embeddings for . We assume the following hypotheses on the non-linear term :

H: We have for all real function , and for any in with , , there exist and positive constants and such that

The main result of this paper is the following:

Theorem 1

Assume that H is satisfied. Suppose moreover that (5) is verified for , non-negative integer. Then, for every , there exists a maximal existence time and a unique function , , verifying (3) for all almost surely. Moreover, admits the following representation formula:

(14)

where is the evolution operator generated by the operator

If in addition , then for all the charge conservation holds:

If is globally Lipschitz in , then the solution exists for all time .

When , a classical example of a non-linearity satisfying H for is , where is the Poisson potential defined by

Indeed, is locally Lipschitz in : let ; thanks to the Hardy-Littlewood-Sobolev inequality [24], Chapter IX.4, as well as standard Sobolev embeddings, we have

and direct computations yield

Hence,

Another example is given by power non-linearities of the form for some and . A bound is needed on for H to be verified. When , we set then and obtain, for all :

while it can be easily shown that H is verified for and for all .

Remark 2

In order to both lower the spatial regularity assumptions on , , and to obtain global-in-time results, it is natural to consider the energy conservation identity (derived formally by multiplying (26) by and integrating, and can be justified for classical solutions when using the regularity of of Theorem 8 and Lemma 2.1) that reads for for simplicity:

Unfortunately, it is not clear to us how this identity can be used in order to obtain estimates on for that would depend only on and , . Indeed, following the lines of the stochastic ODE case of [21] in order to treat the stochastic integral and use the Gronwall Lemma, what can be deduced from the above relation is an estimate of the form

for some positive integrable function and where the constant depends on and . This does not yield the desired bound since we cannot control the term by . Hence, as opposed to the standard Brownian case, energy methods do not provide us here with an global-in-time estimate.

Remark 3

When , then is a classical solution to (1) in the sense that it satisfies for all , a.s., a.e.:

A proof of this result is given in the appendix.

The rest of the paper is devoted to the proof of Theorem 1. The starting point is to define , to use the invariance of with respect to a change of phase and to formally apply Lemma 2.1 to arrive at

(15)

Remark that can formally be recast as

In section 4, we construct the evolution operator generated by and obtain the existence of a unique solution to the latter magnetic Schrödinger equation. In section 5, we use the regularity properties of the function together with Lemma 2.1 to prove that is the unique solution to (1). The existence follows from showing that is a solution to (3). The uniqueness stems from a reverse argument: owing a solution to with the corresponding regularity, we show that is a solution to (15). This requires some regularization since the function for smooth cannot be used as a test function in (3), as well as the interpretation of a classical integral involving a full derivative as a fractional integral.

4 Existence theory of the magnetic Schrödinger equation

The first part of this section consists in constructing the evolution operator . We follow the classical methods of Kato [16] and [22]. The second part is devoted to the existence theory for the linear magnetic Schrödinger equation, which is then used for the non-linear case.

4.1 Construction of the evolution operator

We follow here the construction of [22], Chapter 5. Let and be Banach spaces with norms and , where is densely and continuously embedded in . For , let be the infinitesimal generator of a semigroup on . Consider the following hypotheses:

  • is such that there are constants and , where for , denoting the resolvent set of , and

  • There is a family of isomorphisms of onto such that for every , is continously differentiable in on and

    where , , is a strongly continuous family of bounded operators on .

  • For , , is a bounded operator from into and is continuous in the norm.

We then have the following result, see [16], or [22], Chapter 5, Theorems 2.2 and 4.6:

Theorem 4

Assume that (H-1)-(H-2)-(H-3) are verified. Then, there exists a unique evolution operator , defined on the triangle such that

  • is strongly continuous on to , with ,

  • ,

  • , and is strongly continuous on to ,

  • , , which exist in the strong sense in , and are strongly continuous to .

In the next result, we show that for suitable functions , the operator generates an evolution operator .

Proposition 5

Let and , , and let . Then, the operator generates an evolution operator satisfying Theorem 4 and is an isometry on .

Proof. We verify hypotheses (H-1)-(H-2)-(H-3) for . Let with

(16)

First, for fixed in , the Kato-Rellich theorem [24] yields that is self-adjoint on . Indeed, using the regularity , it is straightforward to verify that is symmetric and -bounded with relative bound strictly less than one. We also obtain that , . Stone’s theorem [23] then implies that for fixed, is the generator of a unitary group on . Moreover, is positive, so that the spectrum of lies in . We therefore conclude that the family satisfies hypothesis (H-1).

Regarding (H-2), let , where is the identity operator and

The operator is a positive definite self-adjoint operator on , and an isomorphism from to . It is also obviously continuously differentiable since it does not depend on