1 Introduction
We are intended to discuss the probabilistic interpretation of the backward Wigner equation [1, 2, 3], arising from the recently developed particlebased simulation of the Wigner quantum dynamics [4, 5, 6, 3]. The backward Wigner equation is a partial integrodifferential equation defined in phase space with an “initial” condition .
(1.1)  
(1.2) 
Here is the dual Wigner function, is the mass, represents the reduced Planck constant and the pseudodifferential operator (PDO) reads
(1.3) 
with (i.e., the central difference of the external potential ). Obviously, is antisymmetric in variable,
(1.4) 
It is well known that , a nonlocal operator with an antisymmetric symbol, actually characterizes a deformation of the classical Poisson bracket [7] and exactly reflects the nonlocal nature of quantum mechanics [8, 9, 10, 11].
The subsequent analysis will be based on two equivalent representations of the PDO. The first form is the kernel representation:
(1.5) 
with the realvalued kernel function (termed the Wigner kernel)
(1.6) 
Here
denotes the Fourier transform of the potential function
in variable, and(1.7) 
It is realized that the kernel is antisymmetric in variable
(1.8) 
due to the antisymmetry of (see Eq. (1.4)).
The second form is the oscillatory integral representation:
(1.9) 
which facilitates the derivation of its asymptotic expansion (see Theorem 2).
In order to extend to a bounded operator from to itself, say, there exists a uniform upper bound such that
(1.10) 
we make the following assumptions for a finite time interval .

and is localized in space for any , with the minimal compact support denoted by .

Suppose either of the following conditions holds:

;

,
and there exist a radial function , such that in and holds for sufficiently large and given constants and ;

Here is short for norm, say, .
The prototypes for the latter condition in (A2) arise from quantum molecular systems and fractional diffusion problems [12, 13]. When the potential is of the Coulomb type , it is easy to verify that and , so that the symbol functions may have singularities at and . Therefore, we need to focus on the weakly singular convolution [14], instead of solely treating it in the classical symbol class .
Now we turn to the probabilistic perspective. The starting point of the stochastic solution is to cast Eq. (1.1) into its equivalent integral formulation by adding a term on both sides of Eq. (1.1) [3],
(1.11) 
the derivation of which will be put in Section 2. The constant parameter
turns out to be the intensity of an exponential distribution as follows,
(1.12) 
The main problem is how to resolve the negative values of kernel . In constrast to nonlocal operators with nonnegative and symmetric kernels [12, 15, 13], the existing stochastic approach is based on the unique HahnJordan decomposition (HJD) [16]:
(1.13)  
(1.14)  
(1.15) 
so that become positive semidefinite kernels. Moreover, we assume that there exists a uniform normalizing bound for ,
(1.16) 
It follows that the probabilistic interpretation is to seek a branching random walk model (BRW) such that its first moment satisfies the renewaltype Wigner (W) equation (1.11) [17, 5, 3], dubbed WBRWHJD hereafter. Such model describes a mass distribution of a random cloud starting at
and frozen at random states and exhibiting both random motion and random growth. The random variable is a family history
, a denumerable random sequence corresponding to a unique family tree [18], and is the Borel extension of cylinder sets on . The particles in the family history move according to the following five rules.
(Markov property) The motion of each particle is described by a right continuous Markov process.

(Frozen state) The particle at is frozen at the state when its lifelength .

(Branching property) The particles at , carrying a weight , dies at age at state when , and produces at most five new offsprings at states , , endowed with updated weights , , respectively.

(Independence) The only interaction between the particles is that the birth time and state of offsprings coincide with the death time and state of their parent.
We are able to define a probability measure on the measurable space and thus the stochastic process based on a specific setting of the transition kernels and particle weights in the fourth rule (vide post). Roughly speaking, WBRWHJD can be categorized into the weightedparticle (wp) [3] and signedparticle (sp) [17, 5, 19] implementations, denoted by and associated with the probability laws and , respectively. It has been shown in [3] that (taking as an example),
(1.17) 
holds on some kind of probability space , where means the expectation of with respect to the probability measure
. However, to the best of our knowledge, the related variance estimation has not been established. To this end, our first contribution is to estimate the variance of WBRWHJD, as stated in Theorem
1.Theorem 1 (Variance of WBRWHJD)
Suppose (A1) and (A2) are satisfied and let . Then the variances of and satisfy
(1.18)  
(1.19) 
Two key observations are readily seen from Theorem 1. One is the exponential rate for spWBRWHJD is , that depends on the volume of the support and thus cannot be improved. This poses a huge challenge for high dimensional problems since usually depends on exponentially. By contrast, the rate for wpWBRWHJD can be reduced by increasing , and the optimal exponential rate is . Definitely, is usually far less than , implied by Eqs. (1.10) and (1.16). In this sense, the latter outperforms the former. The other is the large exponential rates and , introduced by HJD (1.13), lead to a rapid growth of variance. Such phenomenon is called “numerical sign problem” [20] as the HahnJordan decomposition of a signed measure totally ignores the nearcancellation of positive and negative weights.
Our second contribution is to formulate a new class of BRW solutions, dubbed WBRWSPA, to diminish the variance growth. The motivation comes from the stationary phase method, a useful technique in microlocal analysis [21], which makes full use of the essential contribution from the localized parts (see Theorem 2). As a consequence, the upper bounds in Eqs. (1.18) and (1.19) can be significantly reduced especially in the region where the module is sufficiently large (see Theorem 3).
Theorem 2 (Stationary phase approximation)
Suppose and the amplitude function . Then for a sufficiently large , we have a stationary phase approximation to PDO
(1.20)  
(1.21) 
where
in the sense that there exists a positive constant , which depends on and its first derivate but is independent on , such that
(1.22) 
Here the norm is and is a closed ball with radius centered at the origin, (short for represent two critical points on the
dimensional unit spherical surface with normal vectors pointing in (or opposite to) the direction of
, which can be parameterized bywith
(1.23) 
and is the central difference operator
(1.24) 
Intuitively speaking, the parameter serves as a filter to decompose PDO into a lowfrequency component and a highfrequency one, the leading terms of which are , and use the resulting nonlocal operator to directly formulate WBRWSPA, instead of as adopted in WBRWHJD. Specifically, we still use HJD to deal with and tackle by another two branches of particles, yielding two stochastic processes: the “wp” implementation and the “sp” implementation , associated with the probability measures and , respectively. In order to estimate the effect of lowfrequency parts, we further need the following assumptions.

and is localized in space for any with the minimal compact support denoted by ;

For the positive constant in Eq. (1.16) there exist positive constants and such that
(1.25)
The assumption (A4) indicates that the normalizing bound for can be diminished when is restricted in a smaller domain, which holds if is large enough. For instance, , it requires the displacement is sufficiently large. Accordingly, we are able to show that the first moment of WBRWSPA turns out to be an asymptotic approximation to the solution of Eq. (1.1). We also study its deviation from the dual Wigner function by estimating the second moment (also termed “variance” hereafter) and find that, in contrast to Eqs. (1.18) and (1.19), the exponential growth rate in the upper bound is suppressed, so that a moderate increase of variance can be achieved.
Theorem 3 (WbrwSpa)
Suppose (A2)(A4) are satisfied and let . Then for a sufficient large , there exist a weightedparticle branching random walk model and a signedparticle one on the probability spaces and , respectively, such that
(1.26) 
and their variances satisfy
(1.27)  
(1.28) 
The rest is organized as follows. Section 2 briefly reviews the basic of the Wigner equation. Section 3 derives the boundedness and the stationary phase approximation to PDO. WBRWHJD and WBRWSPA are analyzed in Sections 4 and 5, respectively. In Section 6, a typical numerical experiment is performed to verify our theoretical analysis. This paper is concluded in Section 7.
2 The Wigner equation
The Wigner equation, introduced by Wigner in his pioneering work [1], provides a fundamental phase space description of quantum mechanics, and quantum behavior is completely characterized by the nonlocal pseudodifferential operator defined in Eq. (1.3). Mathematically speaking, it is a partial integrodifferential equation defined in phase space
(2.1) 
with an initial value . The weak formulation of the Wigner equation is of great importance since any quantum observable can be expressed by its Weyl symbol averaged by the Wigner function [8], namely, with
(2.2) 
Thus it motivates to study the dual system [2] and derive the adjoint equation of Eq. (2.1) under a nondegenerate inner product:
(2.3) 
where is a fixed time instant and is a test function with a compact support in . Using the antisymmetry (1.8) of the Wigner kernel, we have
(2.4) 
and integration by parts directly leads to
where and are short for and , respectively. Therefore the adjoint correspondence, i.e., the backward Wigner equation (1.1), is immediately derived by setting
(2.5) 
Formally, Eq. (2.5) allows us to evaluate the quantum mechanical observable only by the “initial” data [3].
The backward Wigner equation (1.1) can be cast into a renewaltype equation by adding a term on both sides,
(2.6) 
with being a prescribed constant (see Eq. (1.16)), and the mild solution reads
by the variationofconstant formula [22]. Here is short for the semigroup generated by , and its action on a given function can be now readily performed. For instance, we have
(2.7) 
where gives the forwardintime trajectory of with a positive time increment . That is, the backward renewaltype equation Eq. (1.11) is thus verified.
3 boundedness and stationary phase approximation
Before proceeding to the probabilistic aspect, we first need to establish the boundedness of under the assumptions (A1) and (A2). For , Eq. (1.10) is readily verified by Young’s convolution inequality, whereas the boundedness for weakly singular kernels is obtained by the HardyLittlewoodSobolev theorem [23]. After that, we present the stationary phase approximation and detail its remainder estimate.
Suppose and the second condition of hold, then for due to the Hölder’s inequality:
(3.1) 
as has a finite measure. Next we introduce a smooth cutoff function :
(3.2) 
and let , . Here is introduced to remove the singularity at and is chosen sufficient large to ensure , as stated in assumption . Then it is readily verified that the truncated operator has the following estimate
(3.3) 
The first term is bounded from to itself as is locally integrable, say,
(3.4) 
and the bound is independent of . The second term is also bounded from to , with , owing to the HardyLittlewoodSobelev theorem,
(3.5) 
Now let in Eq. (3.3). By combining Eq. (3.1), we obtain that there exists a uniform such that
(3.6) 
A remarkable feature of the oscillatory integral operator is the decay property as the integrand becomes more and more oscillating. As stated by Hörmander’s theorem [21, 24], when is sufficiently smooth and compactly supported, it has a sharp estimate for a sufficiently large
(3.7) 
The physical meaning of Eq. (3.7) is also clear. When we consider the twobody interacting potential like , turns out to be the spatial displacement between two bodies, so that the estimate (3.7) characterizes the decay rate of quantum interaction as the distance increases. A similar result like Eq. (3.7) can also be found in our framework, and the decay property will be fully utilized by the stationary phase method as presented in Theorem 2, which is definitely ignored by HJD (1.13).
Proof (Proof of Theorem 2)
It starts by splitting the wavevector into its modulus and orientation parts with the modulus and the orientation , where denotes the dimensional spherical surface, and focusing on the highfrequency component
(3.8) 
where represents the orientation of , and denotes the induced Lebesgue measure on . After choosing the equatorial plane normal to , the unit sphere can be decomposed into an upper hemisphere and a lower one satisfying . Accordingly, the inner surface integral of the first kind over in Eq. (3.8) equals to the sum of those over and . Without loss of generality, it suffices to assume that , which be realized by a rotation otherwise. Let us start from the graph
(3.9) 
with
(3.10) 
and take the surface integral of the first kind over the upper hemisphere as an example. Now the phase function of the integrand becomes
(3.11) 
For such phase function, it can be easily verified that there is only one critical point satisfying , and the determinant of its Hessian matrix at turns out to be
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