Complex Obtuse Random Walks and their Continuous-Time Limits

Introduction and Motivations

Since this initial work of was only motivated by Probability Theory and Stochastic Process considerations, there was no real need for an extension of this notion to the complex case. The need for such an extension has appeared naturally through considerations in Quantum Statistical Mechanics. More precisely, the underlying motivation is to characterize the onset of classical noises emerging from quantum baths, in the so-called model of Repeated Quantum Interactions.

Repeated quantum interaction models are physical models, introduced and developed in , which consist in describing the Hamiltonian dynamics of a quantum system undergoing a sequence of interactions with an environment made of a chain of identical systems. These models were developed for they furnish toy models for quantum dissipative systems, they are at the same time Hamiltonian and Markovian, they spontaneously give rise to quantum stochastic differential equations in the continuous time limit. It has been proved in and that they constitute a good toy model for a quantum heat bath in some situations and that they can also give an account of the diffusive behavior of an electron in an electric field, when coupled to a heat bath. When adding to each step of the dynamics a measurement of the piece of the environment which has just interacted, we recover all the discrete-time quantum trajectories for quantum systems (, , ). Physically, this model corresponds exactly to physical experiments such as the ones performed by S. Haroche et al on the quantum trajectories of a photon in a cavity (, , ).

The discrete-time dynamics of these repeated interaction systems, as well as their continuous-time limits, give rise to time evolutions driven by quantum noises coming from the environment. These quantum noises emerging from the environment describe all the possible actions inside the environment (excitation, return to ground state, jumps in between two energy levels, …). It is a remarkable fact that these quantum noises can also be combined together in order to give rise to classical noises. In discrete-time they give rise to any random walk, in continuous-time they give rise to many well-known stochastic processes among which are all the Levy processes.

Surprisingly, the extension of obtuse random variables, obtuse random walks and their continuous-time limits, to the complex case is far from obvious. The algebraical properties of the associated 3-tensors give rise to the same kind of behaviors as in the real case, but, as we shall see in this article, many aspects (such as the diagonalization theorem) become now really non-trivial.

2 Examples

Let us have here a more detailed discussion on these physical motivations underlying our study. These motivations do not appear anymore in the rest of the article which is devoted entirely to the probabilistic properties of complex obtuse random walks and their continuous-time limits, but we have felt that it could be of interest for the reader to have a clearer picture of the physical motivations which have brought us to consider the complex case extension of obtuse random walks. This part can be skipped by the reader, it has no influence whatsoever on the rest of the article, these physical applications are developed in detail in .

Assume that the interaction between these two parts lasts for a small amount of time $h$, then the associated unitary evolution operator is $U=e^{-ihH_{\rm tot}}$ which can be decomposed as $U=\sum_{i,j=0}^{1}U^{i}_{j}\otimes a^{i}_{j}$ for some operators $U^{i}_{j}$ on $\mathcal{H}_{\mathcal{S}}$.

The action of the environment (the spin chain) by acting repeatedly of the system $\mathcal{H}_{\mathcal{S}}$, spin by spin, each time for a time duration $h$, gives rises to a time evolution driven by a sequence of unitary operators $(V_{n})$ which satisfies (cf for details)

This describes a rather general discrete time evolution for a quantum system and the operators $a^{i}_{j}(n)$ here play the role of discrete time quantum noises, they describe all the possible innovations brought by the environment.

In it is shown that if the total Hamiltonian $H_{\rm tot}$ is renormalized under the form

(this can be understood as follows: if the time duration of the interactions $h$ tends to 0, then then the interaction needs to be strengthen adequately if one wishes to obtain a non-trivial limit) then the time evolution $(V_{nh})$ converges, when $h$ tends to 0, to a continuous-time unitary evolution $(V_{t})$ satisfying an equation of the form

which is a quantum stochastic differential equation driven by quantum noises $da(t)$ and $da^{*}(t)$ on some appropriate Fock space. In other words, we obtain a perturbation of a Schrödinger equation by some additional quantum noise terms.

The point now is that in the special case where $L=L^{*}$ then the discrete-time evolution and its continuous-time limit are actually driven by classical noises, for some of the terms in the evolution equation factorize nicely and make appearing classical noises instead of quantum noises (the noises get grouped in order to furnish a family of commuting self-adjoint operators, that is, a classical stochastic process). Indeed, one can show (cf and ) that the discrete time evolution can be written under the form of classical random walk on the unitary group $\mathcal{U}(\mathcal{H}_{\mathcal{S}})$:

where $(X_{n})$ is a sequence of i.i.d. symmetric Bernoulli random variables. The continuous time limit, with the same renormalization as above, gives rise to a unitary evolution driven by a classical Brownian motion $(W_{t})$:

The equation above is the typical one for the perturbation of a Schrödinger equation by means of a Brownian additional term, if one wants the evolution to keep unitary at all times.

This example is a very simple one and belongs to those which were already well-known (cf ); they involve real obtuse random variables and real normal martingales.

which is self-adjoint under the condition $B=-(1/2)(A+(1+2i)A^{*})$.

In this case the quantum dynamics in discrete time happens to be driven by a classical noise too, but this does not appear obviously here! We will understand, with the tools developed in this article, that the resulting discrete time dynamics is of the form

with probabilities $p_{1}=1/3$, $p_{2}=1/4$ and $p_{3}=5/{12}$ respectively.

Putting a $1/\sqrt{h}$ normalization factor in front of $A$ and $B$ and taking the limit $h$ goes to 0, we will show in this article, that this gives rise to a continuous time dynamics of the form

where $(W^{1}\,,\,W^{2})$ is a 2-dimensional real Brownian motion.

The way these random walks and their characteristics are identified, the way the continuous-time limits and their characteristics are identified, are non-trivial and make use of all the tools we develop along this article: associated doubly-symmetric 3-tensor, diagonalisation of the 3-tensor, probabilistic characteristics of the associated random walk, passage to the limit on the tensor, passage to the limit on the discrete-time martingale, identification of the law of the limit martingale, etc.

3 Structure of the Article

In Section 3 we establish the important symmetries shared by the 3-tensors of obtuse random variables and we show one of our main results: these symmetries are the necessary and sufficient conditions for the 3-tensor to be diagonalizable in some orthonormal basis. We show how to recover the real case, which remarkably does not correspond to the real character of the 3-tensor but to a certain supplementary symmetry.

Section 4 is kind of preparatory to the continuous-time limit of complex obtuse random walks. In this section we show an important connection between complex obtuse random variables and real ones. This connection will be the key for understanding the continuous-time limits. In Section 4 we gather all the results concerning this connection with the real obtuse random variables and its consequences. We recall basic results on real normal martingales and deduce the corresponding ones for the complex normal martingales. In particular we establish what is the complex extension of a structure equation. We connect the behavior of the complex normal martingale to the diagonalization of its associated 3-tensor.

In Section 5 we finally prove our continuous-time convergence theorems. First of all, via the convergence of the tensors, exploiting the results of , we prove a convergence in law for the processes. Secondly, in the framework of Fock space approximation by spin chains developed in , we prove the convergence of the associated multiplication operators, with explicit formulas in terms of quantum noises.

We finally illustrate our results in Section 6 through 2 examples, showing up the different types of behavior.

Complex Obtuse Random Variables

As the $\widehat{v}_{j}$’s form a basis, this means that

This implies the two announced equalities. $\square$

that is, the announced equality. $\square$

2 Obtuse Random Variables

We shall also consider the deterministic random variable $X^{0}$ on $(\Omega,\mathcal{F},P)$, which is always equal to $1$. For $i=0,\ldots,N$ let $\widetilde{X}^{i}$ be the random variable defined by

for all $i=0,\ldots,N$ and all $j=1,\ldots,N+1$.

2) The $(N+1)\times(N+1)$-matrix $\left(\widetilde{X}^{i}(j)\right)_{i,j}$ is a unitary matrix.

3) The $(N+1)\times(N+1)$-matrix $\left(\sqrt{p_{i}}\,\widehat{v}_{i}^{j}\right)_{i,j}$ is a unitary matrix.

4) The family $\{v_{1},\dots,v_{N+1}\}$ is an obtuse system with

1) $\Rightarrow$ 2): Since the random variable $X$ is centered and normalized, each component $X^{i}$ has a zero mean and the scalar product between two components $X^{i}$, $X^{j}$ is given by the matrix $I$. Hence, for all $i$ in $\left\{1,\dots,N\right\}$, we get

Now, using Eqs. (4) and (5), we get, for all $i,j=1,\ldots,N$

2) $\Rightarrow$ 1): Conversely, if the matrix $\left(\widetilde{X}^{i}(j)\right)_{i,j}$ is unitary, the scalar products of column vectors give the mean and the covariance $I$ for the random variable $X$.

2) $\Leftrightarrow$ 3): The matrix $\left(\sqrt{p_{j}}\,\widehat{v}_{i}^{j}\right)_{i,j}$ is the transpose matrix of $\left(\widetilde{X}^{i}(j)\right)_{i,j}$. Therefore, if one of these two matrices is unitary, its transpose matrix is unitary too.

3) $\Leftrightarrow$ 4): The matrix $\left(\sqrt{p_{j}}\,\widehat{v}^{i}_{j}\right)_{i,j}$ is unitary if and only if

for all $i,j=1,\ldots,N+1$. On the other hand, the condition $\left\langle\sqrt{p_{i}}\,\widehat{v}_{i}\,,\,\sqrt{p_{i}}\,\widehat{v}_{i}\right\rangle=1$ is equivalent to $p_{i}\,(1+\|v_{i}\|^{2})=1$, whereas the condition $\left\langle\sqrt{p_{i}}\,\widehat{v}_{i}\,,\,\sqrt{p_{j}}\,\widehat{v}_{j}\right\rangle=0$ is equivalent to $\sqrt{p_{i}}\,\sqrt{p_{j}}\,(1+\left\langle v_{i}\,,\,v_{j}\right\rangle)=0$, that is, $\left\langle v_{i}\,,\,v_{j}\right\rangle=-1\,.$ This gives the result. $\square$

3 Generic Character of Obtuse Random Variables

In the converse direction, let $v_{1},\ldots,v_{N+1}$ be the possible values of $X$, associated to the probabilities $p_{1},\ldots,p_{N+1}$ respectively. Let $w_{1},\ldots,w_{N+1}$ be the ones associated to $Y$. In particular, the vectors

for all $i=1,\ldots,N+1$. This gives in particular

As the $\widehat{v_{i}}$’s are linearly independent then so are the $\sqrt{p_{1}}\widehat{v_{1}}-\sqrt{p_{i}}\widehat{v_{i}}$, for $i=2,\ldots,N+1$. Furthermore, we have

this means that the $v_{1}-v_{i}$’s, for $i=2,\ldots,N+1$, are linearly independent.

As a consequence the unique solution of the system (6) is $V^{0}_{j}=0$ for all $j=1,\ldots,N$. This implies $V^{0}_{0}=1$ obviously.

The same kind of reasoning applied to the relation $\widehat{v_{i}}=V^{*}\widehat{w_{i}}$ shows that the column coefficients $V^{j}_{0}$, $j=1,\ldots,N$ are also all vanishing. Finally the operator $V$ is of the form

for all $i=1,\ldots,d$, all $j=1,\ldots,n$. In particular, for each fixed $i=1,\ldots,d$, we have the following subsystem of $n-1$ equations with $n-1$ variables $A^{i}_{1},\ldots,A^{i}_{n-1}$:

are linearly independent. Thus so are the vectors

Hence the system (8) can be solved and furnishes the coefficients $A^{i}_{k}$, $k=1,\ldots,n-1$. We have to check that these coefficients are compatible with all the equations of (8). Actually, the only equation from (7) that we have forgotten in (8) is

But this equation comes easily from the $n-1$ first equations if we sum them after multiplication by $p_{j}$:

4 Associated 3-Tensors

Obtuse random variables are naturally associated to some 3-tensors with particular symmetries. This is what we shall prove here.

for all $i,j=0,\ldots,N$. This $3$-tensor $S$ is given by

We also have the relation, for all $i,j=0,\ldots,N$

for all $i,j=0,\ldots,N$. In particular we have

Finally, we have, by the orthonormality of the $X^{k}$’s

by (10). This gives the last identity. $\square$

This 3-tensor $S$ has quite some symmetries, let us detail them.

Equation (13) comes directly from Formula (10) which shows a clear symmetry in $(i,j)$.

But the left hand side is clearly symmetric in $(i,k)$ and (14) follows.

In order to prove (15), we write, using (11)

But the left hand side is clearly symmetric in $(i,k)$ and (15) is proved. $\square$

5 Representation of Multiplication Operators

These multiplication operators carry all the probabilistic informations on $X$, even through a unitary transform such as $U_{S}$, for we have, by the usual functional calculus for normal operators

The operator of multiplication by $\overline{X^{i}}$ is given by

Proof: We have, for any fixed $i\in\{0,\ldots,N\}$, for all $j=0,\ldots,N$

Hence the operator $U_{S}\,\mathcal{M}_{X^{i}}\,U_{S}^{*}$ has the same action on the orthonormal basis $\{e_{0},\ldots,e_{N}\}$ as the operator

The last identity is just an immediate translation of the relation (11). $\square$

6 Back to the Example

Let us illustrate the previous subsections with our example. To the obtuse system

These matrices are not symmetric (we shall see in Subsection 3.3 what the symmetry of the matrices $S^{j}$ corresponds to). We recognize the particular form of $S^{0}$, for it corresponds to $\mathcal{M}_{X^{0}}=I$.

Complex Doubly-Symmetric 3-Tensors

We are going to leave for a moment the obtuse random variables and concentrate on the symmetries we have obtained above. The relation (12) is really specific to obtuse random variables, we shall leave it for a moment. We concentrate on the relation (13), (14) and (15) which have important consequences for the 3-tensor.

Hence the $\lambda_{m}$’s are only determined up to a phase; only their modulus is determined by the representation (19).

for all $m$. In terms of the $v_{m}$’s, the decomposition (19) of $S$ becomes

This is the form of diagonalization we shall retain for 3-tensors. Be aware that in the above representation the vectors are orthogonal, but not normalized anymore. Also note that they represent the eigenvectors of $S$ associated only to the non-vanishing eigenvalues of $S$.

and the symmetry in $(i,k)$ is obvious. This gives (14).

and the symmetry in $(i,k)$ is obvious. This gives (15).

defines a complex doubly-symmetric 3-tensor if $\mathcal{V}$ is any family of (non-vanishing) orthogonal vectors.

Second step: now given a complex doubly-symmetric 3-tensor $S$ of the form (21), we shall prove that the set $\mathcal{V}$ coincides with the set

Clearly, if $y\in\mathcal{V}$ we have by (21)

This proves that $\mathcal{V}\subset\widehat{\mathcal{V}}$. Now, let $v\in\widehat{\mathcal{V}}$. On one side we have

In particular, applying $\langle y|\in\mathcal{S}^{*}$ to both sides, we get

and thus either $v$ is orthogonal to $y$ or $v=y$. This proves that $v$ is one of the elements $y$ of $\mathcal{V}$, for it were orthogonal to all the $y\in\mathcal{S}$ we would get $v\otimes v=S(v)=0$ and $v$ would be the null vector.

We have proved that $\mathcal{V}$ coincides with the set

are symmetric. But, as they are complex-valued matrices, this does not imply any property of diagonalization. Rather we have the following theorem ().

Let M be a complex symmetric matrix, there exist a unitary $U$ matrix and a diagonal matrix $D$ such that

Secondly, we shall need to simultaneously “factorize” the $S_{k}$’s as above. We shall make use of the following criteria (same reference).

This is the first part of Step three: proving that in our case the matrices $\overline{S_{i}}\,S_{j}$ commute. Using the 3 symmetry properties of $S$ we get

This proves that $\overline{S_{i}}\,S_{j}\,\overline{S_{k}}\,S_{l}=\overline{S_{k}}\,S_{l}\,\overline{S_{i}}\,S_{j}$. The family \big{\{}\overline{S_{i}}S_{j},i,j=1,\cdots,N\big{\}} is commuting. Thus, by Theorem 3.3, the matrices $S_{k}$ can be simultaneously Takagi-factorized. There exists then a unitary matrix $U=(u^{ij})_{i,j=0,\cdots,N}$ such that, for all $k$ in $\left\{0,\cdots,N\right\}$,

where the matrix $D_{k}$ is a diagonal matrix, $D_{k}=diag(\lambda_{k}^{1},\cdots,\lambda_{k}^{N})$. Thus, the coefficient $S^{ij}_{k}$ can be written as

Our aim now is to prove that $\lambda_{m}$ is proportional to $\overline{a_{m}}$. To this end, we shall use the symmetry properties of $S$. From the simultaneous reduction (23), we get

where ${}^{t}U$ is the transpose matrix of $U$. Thus, we have

In particular we have, for all $p\in\{0,\ldots,N\}$

But applying the symmetry (15) this is also equal to

We have obtained the orthonormal diagonalization of $S$. The proof is complete. $\square$

2 Back to Obtuse Random Variables

The theorem above is a general diagonalization theorem for 3-tensors. For the moment it does not take into account the relation (12). When we make it enter into the game, we see the obtuse systems appearing.

for all $i,k=0,\ldots,N$. Then the orthogonal system $\mathcal{V}$ such that

Proof: First assume that $\mathcal{V}=\{v_{1},\ldots,v_{K}\}$. By hypothesis, we have

for all $i,j,k=0,\ldots,N$. With hypothesis (12) we have in particular

This proves the first part of the theorem. The last part concerning obtuse systems is now obvious and was already noticed when we have introduced obtuse systems. $\square$

In particular we have proved the following theorem.

– The random variable $X$ is the only random variable satisfying

3 Recovering the Real Case

In have been introduced the notions of real obtuse random variables and their associated real doubly-symmetric 3-tensors. In the same way they obtained certain symmetries on the tensor which corresponded exactly to the condition for being diagonalizable in some real orthonormal basis. Note that in the situation for the diagonalization theorem was much easier, for the symmetries associated to the 3-tensor came down to simultaneous diagonalization of commuting symmetric real matrices.

The question we want to answer here is: How do we recover the real case from the complex case? By this we mean: On what condition a complex doubly-symmetric 3-tensor correspond to a real one, that is, corresponds to real-valued random variables? Surprisingly enough, the answer is not: When the coefficients $S^{ij}_{k}$ are all real! Let us see that with a counter-example.

Let us consider the one dimensional random variable $X$ which takes values $i$, $-i$ with probability $1/2$. As usual denote by $X^{0}$ the constant random variable equal to 1 and by $X^{1}$ the random variable $X$. We have the relations

which give us the following matrices for the associated 3-tensor $S$:

They are real-valued matrices, but they are associated to a complex (non real) random variable.

In fact, the major difference between a complex (non real) doubly-symmetric 3-tensor and a real doubly-symmetric 3-tensor is the commutation property of indices $i$ and $k$ in the coefficients $S^{ij}_{k}$. Let us make this more precise.

Proof: The commutation relation implies that

for all $i,k$. Then $\overline{X^{k}}\,X^{i}$ is almost surely real for all $i,k$. Considering the case $k=0$ implies that $X^{i}$ is almost surely real and the result follows. $\square$

In the counter-example above, one can check that $S^{01}_{1}=1$ and $S^{11}_{0}=-1$. The commutation condition is not satisfied.

4 From Complex to Real Obtuse Random Variables

Proof: This is essentially the same argument as in Theorem 2.4, at least for the second property. For the first property one has to write that, if $Y$ is real-valued and $Y=UX$ then

Now recall the following classical result.

We repeat the procedure until all the coordinates are exhausted. $\square$

Now, here is an independence property specific shared by the obtuse systems.

Every strict sub-family of an obtuse family is linearly free.

then, taking the scalar product with $v_{N}$ we would get

whereas, taking the scalar product with $v_{N+1}$ would give

This would imply ${\left\|v_{N}\right\|}^{2}=-1$, which is impossible. $\square$

Finally, using Proposition 3.8 we make an important step towards the main result.

Then there exist $\phi_{1},\ldots,\phi_{N}$, modulus 1 complex numbers, such that for every $i=1,\ldots N+1$ we have

Proof: First note that the $z^{i}_{i}$’s, $i=1,\ldots,N$, cannot vanish, for otherwise, the family $\{w_{1},\ldots,w_{N}\}$ would not be linearly free, contradicting Proposition 3.9.

Secondly, the scalar product conditions $\left\langle w_{1}\,,\,w_{j}\right\rangle=-1$, for $j=2,\ldots,N+1$, imply

In particular, all the $z^{1}_{i}$’s, $i=1,\ldots,N+1$, have the same argument.

With the conditions $\left\langle w_{2}\,,\,w_{j}\right\rangle=-1$, for $j=3,\ldots,N+1$, we get

Hence all the $z^{2}_{i}$’s are equal for $i=3,\ldots,n+1$ and all $z^{2}_{i}$’s have same argument ($i=2,\ldots,N+1$).

One easily obtains the result in the same way, line by line. $\square$

Altogether we have proved the following theorem.

5 Unitary Transforms of Obtuse Random Variables

As every complex obtuse random variable $X$ can be obtained as $UY$ for some unitary operator $U$ and some real obtuse random variable $Y$, we shall concentrate for a while on the unitary transformations of obtuse random variables and their consequences on the associated 3-tensors, on the multiplication operators, etc.

As a first step, let us see how is transformed the associated 3-tensor under a unitary map of the random variable.

If $S$ and $T$ are the 3-tensors of $X$ and $Y$ respectively, we then have

Conversely, the tensor $T$ can be deduced from the tensor $S$ by

The converse formula is obvious, replacing $U$ by $U^{*}$. $\square$

In the following if two 3-tensors $S$ and $T$ are connected by a formula of the form (26) we shall denote it by

Under the conditions and notations above, we have

An explicit formula for the operator $U_{S}\,U_{T^{*}}\,a^{n}_{p}\,U_{T}\,U_{S^{*}}$ is obtained easily by acting on the basis:

Injecting this in the previous identity, we get

That is, we get (27) and (28) immediately. $\square$

The point is that this unitary operator has not been yet obtained very constructively. The following theorem gives it a little more explicitly, from the associated 3-tensor.

By Takagi Theorem 3.2, this matrix $S_{0}$ can be decomposed as $U\,D\,U^{t}$ for some unitary $U$ and some diagonal matrix $D$. But as $S_{0}$ is unitary we have

and the matrix $D$ is unitary too. In particular its entries are complex numbers of modulus 1. Let $L$ be the diagonal matrix whose entries are the square root of the entries of $D$, they are also of modulus 1, so that $L\,\overline{L}=I$

We have proved the announced decomposition of $S_{0}$.

We now check the last assertion. Let $v_{ij}$ be the coefficients of $V$. Define the 3-tensor $R=V^{*}\circ S$, that is,

Injecting this relation in the expression of $R^{ij}_{k}$ above, we get

But the above expression is clearly symmetric in $(i,j)$, for $S^{p\alpha}_{n}$ is symmetric in $(p,\alpha)$. By Proposition 3.6 this means that the 3-tensor $R$ is real. The theorem is proved. $\square$

Complex Normal Martingales

In the next section of this article we wish to obtain two types of time-continuous results:

– a limit in distribution for the processes, for which we would like to rely on the results of where is proved that the convergence of the 3-tensors associated to the discrete time obtuse random walks implies the convergence in law of the processes;

– a limit theorem for the multiplication operators, for which we would like to rely on the approximation procedure developed in , where is constructed an approximation of the Fock space by means of spin chains and where is proved the convergence of the basic operators $a^{i}_{j}(n)$ to the increments of quantum noises.

When considering the complex case we had two choices: either develop a complex theory of normal martingales and structure equations, extend all the results of , of and of to the complex case and prove the limit theorems we wished to obtain; or find a way to connect the complex obtuse random walks to the real ones and rely on the results of the real case, in order to derive the corresponding one for the complex case. We have chosen the second scenario, for we have indeed the same connection between the complex obtuse random variables and the complex ones as we have obtained in the discrete time case. In this section we shall present, complex normal martingales and their structure equations, the connection between the complex and the real case, together with their consequences. Only in next section we shall apply these results in order to derive the continuous-time limit theorems.

The following theorem is proved in . It establishes the fundamental link between the 3-tensors $\Phi(s)$ associated to the martingale $X$ and the behavior of $X$.

If one denotes by $\mathcal{V}_{s}(\omega)$ the orthogonal family associated to the non-vanishing eigenvalues of $\Phi(s,\omega)$ and by $\Pi_{s}(\omega)$ the orthogonal projector onto $\mathcal{V}_{s}(\omega)^{\perp}$, that is, on the null-egeinvalue subspace of $\Phi(s,\omega)$, then the continuous part of $X$ is

the jumps of $X$ only happen at totally inaccessible times and they satisfy

The case we are concerned with is a simple case where the process $\Phi$ is actually constant. In that case, things can be made much more explicit, as is proved in again.

Conversely, any solution of (32) has the same law as $X$.

The martingale $X$ solution of (32) possesses the chaotic representation property.

– the angle bracket $\langle\overline{X^{i}}\,,\,X^{j}\rangle_{t}$ is equal to $\delta_{ij}\,t$,

– the martingale $X$ has the Predictable Representation Property.

To these conditions we add the following simplifying condition:

If $Y$ is the null process then $B^{k}_{s}=0$ almost surely, for a.a. $s$ and for all $k$. This means that

for all $t$ and thus $A$ vanishes too. $\square$

We now detail the symmetry properties of $S$, $T$ and $\Lambda$, together with some intertwining relations between $S$ and $\Lambda$.

Proof: The proof is a rather simple adaptation of the arguments used in Proposition 2.7 and Proposition 2.8. First of all, the symmetry $[X^{i}\,,\,X^{j}]_{t}=[X^{j}\,,\,X^{i}]_{t}$ gives

for all $t$. By the uniqueness Lemma 4.3 this gives the symmetry of the matrices $\Lambda_{s}$ and the first symmetry relation (13) for the 3-tensors $S(s)$.

Computing $[[X^{i}\,,\,X^{j}]\,,\,\overline{X^{k}}]_{t}$ we get

Again, by the uniqueness lemma, and the symmetry (13), we get the relation (36).

Now, in the same way as in the proof of Proposition 2.8, we compute $[[X^{i}\,,\,\overline{X^{j}}]\,,\,[X^{k}\,,\,X^{l}]]_{t}$ in two ways, using the symmetry in $(i,k)$ of that quadruple bracket:

By uniqueness again, the time integral part gives the relation

The relation (15) is obtained exactly in the same way, from the symmetry of $[[X^{i}\,,\,\overline{X^{j}}]\,,\,[\overline{X^{l}}\,,\,X^{k}]]_{t}$ in $(i,k)$. We have proved that the 3-tensors $S(s)$ are doubly-symmetric.

Computing $[X^{i}\,,\,[X^{j}\,,\,X^{k}]]_{t}$ in two different ways we get

on the other hand. Identifying the time integrals, we get the relation (37).

Finally, computing $[X^{i}\,,\,[X^{j}\,,\,\overline{X^{k}}]]_{t}$ in two different ways we get

on the other hand. Identifying the time integrals, we get the relation (38).

for some predictable processes $H^{ik}$. We write

The unicity lemma gives the relation $H^{ij}_{s}=\overline{\Lambda^{ij}_{s}}$, almost surely, for a.a. $s$.

This proves the announced unitarity. $\square$

Then the matrix $\Lambda$ admits a decomposition of the form

Proof: This is exactly the same proof as for Theorem 3.14 : the decomposition of $\Lambda$ comes from Takagi’s Theorem, the expression of $R^{ij}_{k}$ in terms of the coefficients of $V$ and of the $S^{ij}_{k}$’s is transformed with the help of the relation XX. One then see that $R$ satisfies the symmetry property which makes it real. $\square$

3 Complex Unitary Transforms of Real Normal Martingales

From the result above concerning real normal martingales, we shall deduce easily the corresponding behavior of complex normal martingales, as they are obtained by unitary transforms of real normal martingales.

With the notations above, the complex martingale $X$ satisfies the following two “structure equations”

The process $X$ possesses the chaotic representation property.

Proof: The martingale $Y$ satisfies the structure equation

But as $R$ is a real-valued 3-tensor, symmetric in $i,j,k$, the last expression gives

Decomposing each $A^{j}_{t}$ as $B^{j}_{t}+iC^{j}_{t}$ (real and imaginary parts), the last two relations ought to

The part (41) of the theorem is obvious, again by application of the map $U$.

for some deterministic functions $f_{i_{1},\ldots,i_{n}}$’s. But decomposing each $Y^{j}_{t}$ as $\sum_{m=1}^{N}\overline{u_{mj}}\,X^{m}_{t}$ shows clearly that $F$ can also be decomposed as

where the $g_{i_{1},\ldots,i_{n}}$’s are linear combinations of the $f_{i_{1},\ldots,i_{n}}$’s. This proves the chaotic representation property for $X$ and the theorem is completely proved. $\square$

Continuous-Time Limit of Complex Obtuse Random Walks

We are now ready to consider the convergence theorem for complexe obtuse random walks.

We are now given a time parameter $h>0$ which is meant to tend to 0 later on. This time parameter is the time step of the obtuse random walk we want to study, but note that $h$ may also appear in the internal parameters of the walk, that is, in the probabilities $p_{i}$ and the values $v_{i}$ of $X$.

for all $j=1,\ldots,n$. Putting $\varepsilon_{0}=1$ and $\varepsilon_{i}=1/2$ for all $i=1,\ldots,n$, we then have, for all $i,j=0,\ldots,N$

Proof: These are direct applications of the definitions and the symmetries verified by the $S^{ij}_{k}(h)$’s. For example:

This gives immediately that $M^{0j}_{k}=0$. And so on for all the other cases. $\square$

Proof: Let us check that $(M^{ij}_{k})_{i,j,k=1,\ldots,N}$ satifies the three conditions for being a doubly-symmetric 3-tensor. Recall that for these indices, we have

The first condition $M^{ij}_{k}=M^{kj}_{i}$ is obvious from the same property of $S^{ij}_{k}(h)$ and passing to the limit.

We wish now to prove that $\sum_{m=1}^{N}M^{im}_{j}\,{M^{kl}_{m}}$ is symmetric in $(i,k)$. The corresponding property for $S(h)$ gives

In particular, multiplying by $h$, we get

By hypothesis $\lim_{h\rightarrow 0}S^{kl}_{0}(h)$ and $\lim_{h\rightarrow 0}S^{il}_{0}(h)$ exist hence, passing to the limit, we get

which is the second symmetry asked to $M$ for being doubly-symmetric.

The third symmetry is obtained in a similar way. Indeed, we have

Now, passing to the limit as $h$ tends to 0, we get

This gives the last required symmetry. $\square$

2 Convergence in Distribution

We can now give our convergence in distribution theorem.

Applying the unitary operators $U_{h_{n_{k_{i}}}}$, which converge to $V$, we have the convergence in law of the process $Z^{h_{n_{k_{i}}}}$ to the process $Z=VY$. By Theorem 4.6 the process $Z$ is solution of the complex structure equations associated to the tensor $S$.

3 Convergence of the Multiplication Operators

Let us first recall very shortly the main elements of the construction and approximation developed in , which will now serve us in order to prove the convergence of the multiplication operators. This convergence of multiplication operators is not so usual in a probabilistic framework, but it is the one interesting in the framework of applications in Quantum Statistical Mechanics, for it shows the convergence of the quantum dynamics of repeated interactions towards a classical Langevin equation, when the unitary interaction is unitary (cf ).

When dealing with the associated random walk with time step $h$

the canonical space is naturally isomorphic to

via a unitary isomorphism denoted by $U_{A}$. This space is the natural space for the quantum noises $a^{i}_{j}(t)$, made of the time operator $a^{0}_{0}(t)=tI$, the creation noises $a^{0}_{i}(t)$, the annihilation noises $a^{i}_{0}(t)$ and the exchange processes $a^{i}_{j}(t)$, with $i,j=1,\ldots N$ (cf ).

The main constructions and results developed in are the following:

– each of the spaces $T\Phi(h)$ can be naturally seen as concrete subspace of $\Phi$;

– when $h$ tends to 0 the subspace $T\Phi(h)$ fills in the whole space $\Phi$, that is, concretely, the orthogonal projector $P_{h}$ onto $T\Phi(h)$ converges strongly to the identity $I$;

– the basic operators $a^{i}_{j}(nh)$, now concretely acting on $\Phi$, converge to the quantum noises, that is, more concretely the operator

converges strongly to $a^{i}_{j}(t)$ on a certain domain $\mathcal{D}$ (which we shall not make explicit here, please cf ), where

if one extends the coefficients $N^{ij}_{k}$ to the 0 index, by putting $N^{ij}_{0}=\delta_{ij}$.

Once this is recalled, the rest is now rather easy. We can prove the convergence theorem for the multiplication operators.

The operators of multiplication $\mathcal{M}_{Z^{h}_{t}}$, acting of $\Phi$, converge strongly on $\mathcal{D}$ to the operators

These operators are the operators of multiplication by $Z$ the complex martingale satisfying

Proof: The convergence toward the operator $\mathcal{Z}_{t}$ given by (44) is a simple application of the convergence theorems of , let us detail the different cases.

If $j,k\not=0$, we know that $\sqrt{h}\,S^{ij}_{k}$ converges to $M^{ij}_{k}$ and by we have that $\sum_{m=1}^{\left[t/h\right]}a^{j}_{k}(m)$ converges to $a^{j}_{k}(t)$.

If $j=0$ and $k\not=0$, we know that $S^{ij}_{0}$ converges to $M^{ij}_{0}$ and that $\sum_{m=1}^{\left[t/h\right]}\sqrt{h}\,a^{j}_{0}(m)$ converges to $a^{j}_{0}(t)$.

If $k=0$ and $j\not=0$, we know that $S^{i0}_{k}$ converges to $M^{i0}_{k}$ (actually their are all equal to $\delta_{ik}$) and that $\sum_{m=1}^{\left[t/h\right]}\sqrt{h}\,a^{0}_{k}(m)$ converges to $a^{0}_{k}(t)$.

The fact that $\mathcal{Z}_{t}$ is indeed the multiplication operator by the announced normal martingale comes as follows. The martingale $Z$ is the image $UA$, under a unitary operator $U$ of some real normal martingale $A$. The 3-tensor $M$ is the image $U\circ N$, under the unitary operator $U$, of some real tensor $N$. The real normal martingale $A$ associated to the real 3-tensor $N$ has its multiplication operator equal to

by Theorem 5.4. As $Z_{t}$ is equal to $UA_{t}$ its canonical space is the same as the one of $A$, only the canonical isomorphism is modified by a change of basis. The rest of the proof is then exactly similar to the one of Proposition 3.13. $\square$

Examples

We shall detail 2 examples in dimension 2, showing up typical different behaviors.

with probabilities $p_{1}=1/3$, $p_{2}=1/4$ and $p_{3}=5/{12}$ respectively. Then the 3-tensor $S$ associated to $X$ is given by

It is then rather easy to find a unitary matrix $V$ such that $V\,V^{t}=M_{0}$, we find

for example. Following our results on complex normal martingales, this means that the process $Z$ has the following distribution: given a 2-dimensional real Brownian motion $W=(W^{1}\,,\,W^{2})$ then

with probabilities $p_{1}=1/2$, $p_{2}=h/(1+2h)$ and $p_{3}=1/(2+4h)$ respectively. Then the 3-tensor $S$ associated to $X$ is given by the following, where we have only detailed the leading orders in $h$

The renormalized 3-tensor converges to the 3-tensor

In order to diagonalize the 3-tensor, we solve

This means that the continuous-time limit process $Z$ is a compensated Poisson process in the direction $v$.

This is all for the information which is given by the 3-tensor. If we want to know the direction where the process is Brownian, we need to look at the decomposition of $M^{0}$ as $V\,V^{t}$ for a unitary $V$. We easily find

The process $Z$ is finally described as follows, let $N$ and $W$ be a standard Poisson process and a Brownian motion, respectively, independant of each other. Then

References