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Spiking and collapsing in large noise limits of SDE's

We analyze strong noise limit of some stochastic differential equations. We focus on the particular case of Belavkin equations, arising from quantum measurements, where Bauer and Bernard pointed out an intriguing behavior. As the noise grows larger, the solutions exhibits locally a collapsing, that is to say converge to jump processes, very reminiscent of a metastability phenomenon. But surprisingly the limiting jump process is decorated by a spike process. We completely prove these statements for an archetypal one dimensional diffusion. The proof is robust and can easily be adapted to a large class of one dimensional diffusions.


Introduction
Belavkin equations, also called stochastic master equations, are particular stochastic differential equations (SDE's) arising from quantum measurement process. Briefly speaking, they describe the evolution of a quantum system subject to continuous indirect measurement [Dav76a,Car99,BP02,HR06,Car08,BG09,WM10,Jac14]. The general picture is a quantum system interacting with an environment, which is continuously measured. Due to entanglement between the quantum system and the environment, one gains information on the quantum system 1 .
The Belavkin equations we are interested in are matrix valued SDE's in the form The process (ρ t ; t ≥ 0) is called a "quantum trajectory" and is valued in the set of the so-called density matrices  [Lin76,GKS76]. While L encapsulates the evolution due to the Hamiltonian evolution of the system and the environment interaction, the term L is produced by the indirect measurement, so that it is weighted by the strength γ of the measurement process. The last term is defined in terms of a super-operator D : M n (C) → M n (C) (called the innovation term) and a one dimensional Brownian motion W . It is also due to the indirect measurement, and depends thus also on γ.
Motivations from quantum physics: From a physical point of view such equation are at the cornerstone of the understanding of quantum optics experiment: photon counting, heterodyne or homodyne detection [Car08,Car99]. Recent experiment, from Serge Haroche group in LKB, where manipulation of small systems have been implemented [HR06,Har13,GKG + 07] can be completely explained in terms of Belavkin equations [BBB13, BP14, ASD + 13, AMR12].
From a mathematical point of view such equations offer intriguing theoretical problem. The existence, uniqueness and properties of solutions are not straightforward and a large literature has considered this question. Typically such equations are non linear and usual techniques are useless to answer the question. In order to solve the problems, one can consider three different approaches. One is based on notion of quantum stochastic differential equations i.e quantum analogue of Langevin equation involving a theory of quantum noises and quantum filtering [Bel99,AJP06,BvHJ09,Bel12]. This approach is "operator algebras oriented". Another approach, more probabilistic, derives the equations from nonlinear transformations of linear stochastic differential equations. Well-posedness is established using Itô calculus with Girsanov techniques [BH95,BG09]. The last approach is based on the approximation of such equations via discrete time models.
In particular considering quantum repeated interactions models, that is a quantum system interacting with a sequence of auxiliary systems (called probes), one can develop models of discrete time quantum measurement. After each interaction between the quantum system and a probe, a measurement is performed at the level of the probe (this is typically the setup of experiments of Serge Haroche). Then introducing proper time scaling, one can obtain equations of the form (1.0.1) as a continuous limit of discrete time models [Pel08,Pel10a,Pel10b,BBB12,BBB13].
In the particular case where L = 0 and where the super-operators L-D come from a "non demolition" condition (see Section 2.2 for a precise definition), it can be shown that the solution (ρ t ; t ≥ 0) collapses in the long time. More precisely it means that (ρ t ; t ≥ 0) converges when t goes to infinity to a random pure state. The random pure state is valued in some particular set of rank one projectors associated to an orthonormal basis, called the pointer basis, and its law reproduces the distribution which describes the direct Von Neumann measurement of the initial state.
The problem: In this paper we will focus on a case where complete collapse is prevented by the non-measurement term L(ρ t ), which makes the time behavior of (ρ t ; t ≥ 0) more tricky and interesting. Motivated by the recent works [TBB15,BBT16,BB18], we are interested in the behavior of (ρ t ; t ≥ 0) in the strong measurement limit γ → ∞. Formal as well as rigorous studies on the convergence of the solution of the Belavkin equation to a Markov jump process have appeared during the last years [BB14, BBT15, BCF + 17]. We refer to this phenomenon as a local collapse. Even more interestingly it turns out that fluctuations around this typical local collapse do persist in the strong noise limit and take the form of "spikes" decorating the jump process [TBB15,BBT16,BB18,KL18]. For an illustration of the spikes, see Figure 1. These are indeed very thin as shown by smoothing.
So far, there is only a limited understanding of the convergence topology and the precise statistics of these spikes. Only Belavkin equations which can be reduced to one dimensional ones have been considered. Furthermore even in these cases, complete mathematical proofs are missing.
Our results: The aim of this work is to make progresses in this field by providing a rigorous analysis of the fluctuations of some "one-dimensional" Belavkin equations. We provide a general technique to study the strong noise limit γ → ∞ of one dimensional SDE's in the form is a Wiener process and the drift term b and the diffusion coefficient σ satisfy To the best of our knowledge, the previous studies have been restricted to SDE's living in [0, ∞) with σ(x) > 0 for all x ∈ (0, ∞), σ(0) = 0, b (0) > 0.
Our methods are simple, robust and can be applied to the class of SDE's described above, which appear also in other fields like chemistry, biology, population dynamics ... To present the proofs in a readable way, we will however where p ∈ (0, 1) is a parameter. But our proof can be applied also to other Belavkin equations like for example the one describing Rabi oscillations [BBB13].
Our main results are provided in Theorem 4.2. It shows first the convergence of the process (X γ t ; t ≥ 0) to a jump Markov process (X ∞ t ; t ≥ 0) as γ → ∞. A reader used to problems of weak convergence of stochastic processes will notice that the previous convergence cannot hold in the usual Skorohod topology since (X γ t ; t ≥ 0) has continuous paths while (X ∞ t ; t ≥ 0) has only càdlàg trajectories. The statement holds only upon smoothing (see Figure 1). Hence the precise statement is that for every compactly supported continuous function f of time and space Almost sure convergence is due to a particular coupling of X γ for different γ.
The previous convergence does not capture the spikes that are observed in numerical simulations. Therefore, in order to see them, we have to find the right topology. Our solution uses the Hausdorff metric on the graphs of functions. Indeed the second part of our theorems establishes the convergence of (X γ t ; t ≥ 0) to a spike process defined in terms of excursions.
Organization of the paper: In Section 2 we start with some generalities of the multidimensional Belavkin equation. A particular class of Belavkin equation, which arise in the thermal weak coupling regime, are presented in Section 3, where we focus on the n = 2 case. Last section contains the main result of the paper with its proof.
Notations: If M is a matrix, its entries are denoted by M i,j , its adjoint is M † and Tr(M ) is the trace. Matrices are denoted by capital letters or by Greek letters. The complex conjugate of z is denoted z.

A process is denoted by
The local time (see [RY99] for the definition) of a real valued process X at the point a is written (L a t (X) ; t ≥ 0).

Belavkin equation
2.1. Generalities. Let (ρ t ; t ≥ 0) be the density matrix of a n-dimensional quantum system. It is a process living in S + n , which is defined in (1.0.2). If such a quantum system is simultaneously subject to continuous measurements and to other interactions (ex : free Hamiltonian evolution, contact with some other system ...), then under appropriate hypothesis [Car99, BP02, HR06, Car08, BG09, WM10, Jac14], (ρ t ; t ≥ 0) is expected to follow a stochastic Lindblad equation, also called a Belavkin equation: Here N ∈ M n (C) is called a measurement operator and the application L is called a Lindbladian. It is a super operator i.e an application mapping a matrix M ∈ M n (C) to a linear operator L[M ] : M n (C) → M n (C). The first term involving the hermitian matrix H in (2.0.1) is due to the free Hamiltonian evolution, the terms n k,l=1 L[M k,l ] are due to the interaction with some environment (thermal bath for example) while the two last terms are result from the measurement process. In particular, the two last terms depend on some parameter γ > 0 which represents the intensity of the measurement process. The equation (2.0.1) is driven by a single Wiener process but in a more general setting, it makes sense to consider Belavkin equations driven by several Wiener processes.
2.2. Normal measurement matrices. The goal in this note is to consider the large γ limit when the two following conditions are satisfied: • the matrix N is a normal matrix (this can be taken as the definition of the "non-demolition hypothesis"), i.e. it is diagonalisable in an orthogonal basis |n i , called the pointer basis, • the matrices M kl or/and H are not diagonal in this orthonormal basis. In particular, observe that if the second condition is not satisfied while the first one is, then explicit calculations (similar to next section) show that and that the diagonal elements n i | ρ t |n i satisfy drift less SDE's and are then bounded martingales, and therefore convergent processes. One can then show that under appropriate assumptions, the limit is consistent with quantum collapse. This is the typical situation we want to avoid and explain the previous hypothesis.
Since N is diagonalisable in the orthonormal basis |n i with corresponding eigenvalues n i , we have and by decomposing all the matrices M ∈ M n (C) in this basis: the Belavkin equation takes the following form, component-wise: For convenience, we introduce the following notation for the diagonal terms q i t ≡ ρ i,i t and for the of-diagonal terms r i,j t ≡ ρ i,j t (for i = j). It is customary to refer to the q t 's as "populations" and to r t 's as"phases". In this new parametrization, we have that: Observe that a priori the populations and the phases are coupled.
3. Thermal weak coupling regime and diagonal Hamiltonian 3.1. The n-dimensional case. In this section we consider a particular case where populations become decoupled from phases in (2.0.2). This simplification occurs in the so-called physical context of weak coupling regime [Dav76b]. For a first read, the content of this subsection can be seen as algebraic simplifications aimed at making the problem more tractable. We consider a diagonal Hamiltonian in the pointer basis |n i and assume the operator M k,l are taken to be rank 1: Some of the complex constants Γ k,l may vanish. An explicit computation yields: which yields, in the parametrization of ρ by populations and phases: Then we can rewrite the evolution of the diagonal of the density matrix as: 3.2. Two state model with diagonal Hamiltonian. We consider a particular case where n = 2 and for notational convenience we use the parameterization: Observe that the positivity property is equivalent to i.e. (2p t , 2q t − 1) (considered as a point of R 3 ) is in the unit ball of R 3 . We make the choice with only two non-vanishing operators M k,l and: Here λ ± > 0 and w ∈ R are parameters. Note that we have then: After the simplifications detailed in Subsection 3.1, we get: Upon renaming variables, including γ, this gives the following one dimensional SDE On the one hand, there is time change via the Dambis-Dubins-Schwartz (DDS) theorem [RY99, Chapter V, Theorem 1.6]. The natural time scale for the process (q γ t , t ≥ 0) will be referred as real time. And the changed scale, which is natural for a DDS Brownian motion β, will be referred to as effective time. This follows the denomination of Bauer, Bernard and Tilloy; and it is helpful in explaining proofs where several time scales interact.
On the other hand, let (σ t ; t ≥ 0) be the jump process defined as the inverse of mixed local times accumulated by β at levels 0 and 1: This will be the time change of interest from real time to effective time.
We start with a simple: Proposition 4.1. Let β be a standard one-dimensional Brownian motion started at x 0 ∈ [0, 1]. The process is a càdlàg {0, 1}-valued Markov process with P (Q 0 = 1) = x 0 and jump rates W where: Proof. Firstly, given that σ is càdlàg with jumps corresponding to excursions of β away from 0 and 1, Q t = β σt is indeed càdlàg {0, 1}-valued. Moreover, since (σ t ; t ≥ 0) is an increasing collection of stopping times, where T a is the hitting time of a by β. Since h is a harmonic function for β with boundary conditions h(1) = 1, h(0) = 0, we deduce that h(x 0 ) = x 0 .
Finally, let us prove that Q is Markov with jump rates as described. Suppose we run the Brownian motion β started from 0 and killed upon hitting 1. Let T 1 be first time that β reaches 1 -this is effective time. And let τ be first time that β σ. reaches 1 -this is real time. Given the definition of σ, σ τ = T 1 if and only if and hence the jumping rate from 0 to 1 is W 0,1 = λp.
The proof that W 1,0 = λ(1 − p) is follows exactly the same lines.
Note that this Markov process can be written as X Nt when (X k ) k≥0 are independent Bernoulli variables, and (N t ) t≥0 is an independent Poisson process of parameter λ. X 0 has parameter x 0 , while X k have parameter p for k ≥ 1.

Our result in this section is:
Theorem 4.2. Consider the process (q γ t , t ≥ 0), unique strong solution of the SDE (3.0.4) starting from q 0 . There exists a Brownian motion β (reflected in [0, 1]) such that the following two statements hold almost surely.
On the one hand, for every continuous and compactly supported function f on R + × R: On the other hand, we have for all H > 0 the Hausdorff convergence of graphs: Notice that the RHS is not the graph of a function, but rather the graph of a multi-valued function [0, H] → P([0, 1]).

Remark 4.3 (Explanations)
. The first part of the theorem can be loosely reformulated by saying that the convergence: holds upon smoothing, which amounts to deleting the spikes. Nevertheless, one needs an appropriate notion of convergence in order to capture the spikes, which are infinitely thin in the limit. Thus, we resort to the Hausdorff metric on the collection of closed sets of [0, H] × [0, 1] [Mun00, Ex 7, p.280]: where B is the unit ball. The second part of theorem says that d H (A, B) → 0 where: The fact that B is a closed set comes as a by-product of the proof.
As a corollary, one can give a description of the spike process via excursions. This description isolates the underlying Markov chain on {0, 1} and gives a very efficient simulation scheme.
(2) Simulation of the first spike: If q 0 = 0 the spike at t = 0 is {0} and if q 0 = 1, the spike at t = 1 is {1}. If q 0 = x ∈ (0, 1), the spike at t = 0 is an interval of the form [y, 1] with probability x, and [0, y] with probability 1 − x. The probability density of y in the range of the first case is (1 − x)(1 − y) −2 1 0<y<x , and the probability density of y in the range of the second case is xy −2 1 x<y<1 .
(3) Simulation of the other spikes: Sample spikes (t, M t ) following a Poisson point process with intensity Leb ⊗ dm m 2 1 0<m<1 . Then when the current state is 0, rescale time by a factor λp and every spike (t, M t ) yields an upward segment [0, M t ]. When the current state is 1, rescale time by a factor λ(1−p) and the spike is a downward segment [1 − M t , 1].
Proof. The second point concerning (2) comes as a consequence of standard martingale and stopping time arguments. Now for the third point describing the spikes. Thanks to the Markov property, we only need to give an excursion point of view of the process for q 0 = 0 and up to hitting 1, and for q 0 = 1 and up to hitting 0. We shall focus on the first case and leave the other one to the reader. Let τ = L 0 (β) −1 be the inverse of local time. On the segment [0, T 1 ], we have: Now recall that (see [RY99,Chapter XII]) the Brownian path β can be broken into a Poisson process of excursions away from zero (e t , t > 0). As such (t, e t ) has intensity (Leb ⊗ n), where n is the Itô measure on excursions and the time scale is that of τ i.e there is an excursion for every t such that τ t − τ t − > 0. By changing to the time scale of σ via Eq. (4.4.1), (t, e t ) has intensity (2λp Leb ⊗ n) and there is an excursion every t such that σ t − σ t − > 0.
Moreover the Itô measure restricted to positive excursions gives intensity Scale function: Consider the scale function h := h γ (depending on all parameters γ, λ and p) which is harmonic with respect to the process (q t ; t ≥ 0). It solves: Since h γ is unique up to affine transformation, we can take: It is easy to check that h γ is a strictly increasing diffeomorphism from (0, 1) to R. Moreover, as γ → ∞, h γ tends to the identity on ( , 1 − ) for any ∈ (0, 1/2) fixed. We deduce that uniformly in y ∈ R: The expression h −1 ∞ should just be understood as a convenient notation. It is not by any means the inverse of a real valued function.
Time change: As announced, we invoke the Dambis-Dubins-Schwarz Theorem in order to write h γ (q t ) = h γ (q 0 ) + β Tt = q 0 + β Tt where β is a Brownian motion and Taking the inverse, we get For the sake of simplicity h γ (q 0 ) = 0. Since where ϕ γ : R → R + is the function In the end: where T t can be defined by 4.3. The original process and real time. Let us start by a Lemma: Lemma 4.5. We have the weak convergence: Proof. Let f : R → R be a continuous compactly supported function. Using the change of variables x = h γ (q), we have: γu 2 (1−u) 2 du is increasing on (0, p] and decreasing on [p, 1). Its range is (0, 1]. As such: where L a (β) is the local time accumulated by β at the point a. As customary, we are considering a version of local time so that the map a → L a (β) is continuous and compactly supported. Therefore, the previous Lemma 4.5 immediately yields the almost sure convergence: L 0 (β) + 1 2λ(1 − p) L 1 (β) . This convergence holds uniformly in ∈ [0, L]. Notice that by doing so, we avoid invoking "the approximate Skorohod reflection theorem" from [BBT16].
We are now ready to conclude the proof by analyzing q γ t = β Tt .

4.4.
Concluding the proof of Theorem 4.2. We start by writing: For the first statement, taking f compactly supported and performing a change of variable: Because of Eq. (4.5.1), we have the weak convergence dT Notice that the above weak convergence argument does blur the fine properties of the spike process. Now for the second statement, we shall use the notation from the explanatory Remark 4.3. As such let A be the graph of q γ . For γ > 0, we have via time change:  In all cases, we recognize the same description as in the theorem, which discriminates only on the value of Q t .