Keywords

19.1 Introduction

Studies in Quantum mechanics have posed the problem of completely positive applications on C*-algebra of continuous linear operators on a Hilbert space. We consider completely positive applications because they describe the evolution of a quantum system, a high-physics energy system and we assume that this evolution is not affected by the existence of other systems that do not interact with the given one. Details concerning the way that the high-physics energy has come to pose this problem may be found in [2] and [3]. Starting from a semigroup of positive operators or from its infinitesimal operator, we can construct a homogenous Markov process. The construction of these processes is done through different methods of which we emphasize on solving stochastic integral equations [16]. Hence the theory of quantum probabilities has developed as a noncommutative theory of probabilities in [1] with motivations in high-physics energy [10]. The corresponding stochastic processes were constructed only in the case of infinitesimal operators and are expressed as finite sums. These are called Evans-Hudson diffusions [5]. This article builds these processes on the antisymmetric Fock space (called fermionic) in which the infinitesimal operator is an infinite sum. The case of the symmetric Fock space (called bosonic) was treated in [12]. This article shows how the obtained processes as solutions of certain stochastic integral equations are noncommutative Markov processes and appear as a particular case of certain Evans-Hudson diffusions [14] with an infinite number of components, a notion yet to be defined.

This paper shows how noncommutative Markov processes are obtained as solutions of certain stochastic equations, being particular cases of Evans-Hudson Diffusions with an infinity of components.

The case of Markov processes on symmetric Fock spaces for infinitesimal operator as an infinite sum was studied by Hudson and Parthasarathy [10]. This paper aims to build noncommutative Markov processes on antisymmetric Fock spaces where we do not have exponential commutative vectors and where the commutative property does not occur between operators describing disjoint time intervals. Unlike the symmetric Fock space, the defined operators are continuous and the integral is a particular case of Bochner integral [8].

The Brownian Motion and the Poisson Process were given as examples.

19.2 Fermion Stochastic Integrals of Simple Processes

First we construct a stochastic integral on Fermion Fock space [9, 13] by analogy with the same kind of integral on Boson Fock space [4], first of simple processes. We define the Fermion stochastic integral for square-integrable integrands. We present the infinitesimal operators like infinite sums, but we assume they are continuous. Because of the canonical anticommutation relation we have left, right and mixed stochastic integrals.

The noncommutative stochastic calculus was developed on Fermion Fock space.

Definition 2.1

Let H be a Hilbert space. We define the antisymmetric Fock space Γ a (H) over H as the linear hull of all \( {x}_1\wedge {x}_2\wedge \dots \wedge {x}_n,\ n\ge 0,\ {x}_i\in H \) (where for \( n=0 \), we have the unit element, namely 1) with the following inner product:

$$ \left\langle {x}_1\wedge {x}_2\wedge \dots \wedge {x}_n,{y}_1\wedge {y}_2\wedge \dots \wedge {y}_k\right\rangle = {\delta}_{n,k}\kern0.5em \det {\left(\left\langle {x}_i,{y}_i\right\rangle \right)}_{i,j=1,\dots, n} $$

for \( n=k=0 \) the determinant is considered to be 1.

About this space we mention the following:

  1. (i)

    \( {\Gamma}_a(H)=\underset{n\ge 0}{\oplus }{H}_{\wedge}^n \), where \( {H}_{\wedge}^n \) is the closed linear hull of all \( {x}_1\wedge {x}_2\wedge \dots \wedge {x}_n,\ {x}_i\in H \)

  2. (ii)

    \( {x}_{y(1)}\wedge \dots \wedge {x}_{y(n)}=\varepsilon \left(\gamma \right){x}_{y(1)}\wedge \dots \wedge {x}_{y(n)} \) where ε(γ) is 1 or −1 if γ is even or odd.

    If two x i with different indexes i are equal this product is null.

  3. (iii)

    \( {\Gamma}_a(H)=\left\{{\displaystyle \sum_{n\ge 0}{x}_n}:{x}_n\in {H}_{\wedge}^n,\kern0.5em \left\{n:{x}_n\ne 0\right\}\kern0.5em finite\right\} \) is an associative algebra, with unit element 1 and \( {x}_1\wedge \dots \wedge {x}_n \) is the product of \( {x}_1,\dots, {x}_n,\ {x}_i\in H={H}_{\wedge}^1 \), in established order.

  4. (iv)

    If \( H\subset K \), then \( {\Gamma}_a(H)\subset {\Gamma}_a(K) \).

Definition 2.2

Let H be a Hilbert space

  1. (a)

    By a filtration in H we mean a family \( {\left({H}_t\right)}_{t\in \left[0,\infty \right)} \) of closed subspaces of H such that

  2. (b)

    We say that it is a right continuous filtration if \( {H}_t={\displaystyle \underset{u>t}{\cap }{H}_u} \)

    $$ {H}_s\subset {H}_t,\kern1em \forall s<t. $$
  3. (c)

    We say that it is a left continuous filtration if H t is the closure of \( {\displaystyle \underset{u>s}{\cup }{H}_s} \)

  4. (d)

    We say that \( {\left({H}_t\right)}_{t>0} \) is continuous if the filtration is right and left continuous.

The idea of defining these processes may be found in [6].

Definition 2.3

An adapted process is a family of operators \( F=\left(F(t);t\ge 0\right) \) on h such that for each \( t\ge 0 \):

  1. (a)

    \( D\left(F(t)\right)={h}_0\underset{\bar{\mkern6mu}}{\otimes }{\varepsilon}_t\underset{\bar{\mkern6mu}}{\otimes }{h}^t \).

  2. (b)

    There is an operator \( {F}^{+}(t):{h}_0\underset{\bar{\mkern6mu}}{\otimes }{\varepsilon}_t\underset{\bar{\mkern6mu}}{\otimes }{h}^t\to {h}_0 \) such that

    \( \left\langle F(t)\zeta, \eta \right\rangle =\left\langle \zeta, F{(t)}^{+}\eta \right\rangle \) for \( \forall \zeta \eta \in {h}_0\underset{\bar{\mkern6mu}}{\otimes }{\varepsilon}_t\underset{\bar{\mkern6mu}}{\otimes }{h}^t \)

  3. (c)

    There are operators F 1(t) and \( {F}_{{}_1}^{+}(t) \) on \( {h}_0\otimes {\varepsilon}_t \) such that:

    $$ F(t)={F}_1(t)\otimes 1 $$
    $$ {F}^{+}(t)={F}_{{}_1}^{+}(t)\otimes 1 $$
  4. (d)

    For each t 0 and \( x\in \varepsilon \) we have:

    $$ \underset{\left\Vert u\right\Vert \le 1}{ \sup}\left\Vert \left(F\left({t}_0+h\right)-F(t)\right)\left(u\otimes x\right)\right\Vert \overset{h\to 0}{\to }0 $$

    hence \( \forall x\in \varepsilon \) and \( t\in \left[0,\infty \right) \) \( \underset{s\to t}{ \lim }{\left\Vert F(t)-F(s)\right\Vert}_x=0 \)

    where \( {\left\Vert T\right\Vert}_x=\underset{\left\Vert u\right\Vert \le 1}{ \sup}\left\Vert T\left(u\otimes x\right)\right\Vert \)

Definition 2.4

A simple process is an adapted process of the form:

\( F(t)={\displaystyle \sum_{n=0}^{\infty }{F}_n{\chi}_{\left[{t}_n,{t}_{n+1}\right)}(t);t\ge 0} \) for some sequence \( 0={t}_0<{t}_1<\dots <{t}_n\to \infty \)

We denote by A 0 and A, respectively, the sets of simple and adapted processes.

Definition 2.5

Let \( F,G,H\in {A}_0 \) and write

$$ F={\displaystyle \sum_{n=0}^{\infty }{F}_n{\chi}_{\left[{t}_n,{t}_{n+1}\right)}}\kern0.5em ,G={\displaystyle \sum_{n=0}^{\infty }{G}_n{\chi}_{\left[{t}_n,{t}_{n+1}\right)}}\kern0.5em ,H={\displaystyle \sum_{n=0}^{\infty }{H}_n{\chi}_{\left[{t}_n,{t}_{n+1}\right)}} $$
$$ 0={t}_0<{t}_1<\dots <{t}_n\to \infty $$

The family of operators \( M=\left(M(t),t\ge 0\right) \) with \( D\left(M(t)\right)={h}_0\underset{\bar{\mkern6mu}}{\otimes }{\varepsilon}_t\underset{\bar{\mkern6mu}}{\otimes }{h}^t \) defined by \( M(0)=0 \)

\( M(t)=M\left({t}_n\right)+\left({A}_L^{+}(t)-{A}_L^{+}\left({t}_n\right)\right){F}_n+{G}_n\left({A}_L(t)-{A}_L\left({t}_n\right)\right)+\left(t-{t}_n\right){H}_n \) for \( {t}_n<t<{t}_{n+1} \) is called stochastic integral of (F, G, H) and are denoted by:

$$ M(t)={\displaystyle \underset{0}{\overset{t}{\int }}d{A}_L^{+}F+Gd{A}_L+Hds} $$

19.3 Stochastic Integrals of Continuous Processes

Now we want to estimate the norm of \( M(t)\left(u\otimes x\right) \), in order to define the stochastic integrals [11].

We consider three possibilities, where the first is:

\( M(t)={\displaystyle \sum_{n=0}^b\left({A}_L^{+}\left({s}_{n+1}\right)-{A}_L^{+}\left({s}_n\right)\right){F}_n} \) for \( {t}_b<t<{t}_{b+1} \), \( {s}_i={t}_i \) for \( i=0,\cdots, b \) and \( {s}_{b+1}=t \).

We denote \( F(t)={\displaystyle \sum_{n=0}{\chi}_{\left[{t}_n,{t}_{n+1}\right]}(t){F}_n} \) and we write briefly

\( dM=\left(d{A}_L^{+}\right)F \) or \( M(t)={\displaystyle \underset{0}{\overset{t}{\int }}\left(d{A}_L^{+}\right)F} \).

We write as follows:

$$ {\left\Vert M(t)\left(u\otimes x\right)\right\Vert}^2\le {\left\Vert {\displaystyle \sum_p{L}_p*{L}_p}\right\Vert}^2{\displaystyle \underset{0}{\overset{t}{\int }}{\left\Vert F\left(u\otimes x\right)\right\Vert}^2da}+{\displaystyle \underset{0}{\overset{t}{\int }}{\left\Vert F\left(\theta u\otimes x\right)\right\Vert}^2da}+{\displaystyle \sum_k\underset{a\le t}{ \sup }{\left\Vert M(a)\left(u\otimes {x}_{ck}\right)\right\Vert}^2{\left\Vert {x}_k\right\Vert}^2} $$

Now we deduce that if \( {F}_c=1,2,\dots \) are “simple integrands” like before and if every \( u\in {h}_0 \), \( x\in \varepsilon \) and \( t>0 \), \( {\displaystyle \underset{0}{\overset{t}{\int }}{\left\Vert {F}_c\left(u\otimes x\right)-{F}_{c\hbox{'}}\left(u\otimes x\right)\right\Vert}^2da}\to 0 \) for

\( c,{c}^{\prime}\to \infty \), then for every \( t>0 \), \( u\in {h}_0 \) and \( x\in \varepsilon \),

\( \underset{a\le t}{ \sup }{\left\Vert {M}_c\left(u\otimes x\right)-{M}_{c\hbox{'}}\left(u\otimes x\right)\right\Vert}^2\to 0 \), where \( d{M}_c=\left(d{A}_L^{+}\right){F}_c \).

We consider \( x={x}_1\wedge \dots \wedge {x}_r \) and by induction on r, the term

\( {\displaystyle \sum_k\underset{a\le t}{ \sup }{\left\Vert M(a)\left(u\otimes {x}_{ck}\right)\right\Vert}^2{\left\Vert {x}_k\right\Vert}^2} \) vanishes for \( r=0 \).

This is the way we define \( {\displaystyle \underset{0}{\overset{t}{\int }}\left(d{A}_L^{+}\right)F} \) for those F for which it is a sequence F c of simple integrands with \( {\displaystyle \underset{0}{\overset{t}{\int }}{\left\Vert {F}_c\left(u\otimes x\right)-{F}_{c\hbox{'}}\left(u\otimes x\right)\right\Vert}^2da}\to 0 \) for \( {c}^{\prime}\to \infty \) for every \( t>0 \), \( u\in {h}_0 \) and \( x\in \varepsilon \).

We remark that if \( F(t)={F}_1(t)\otimes 1 \) with respect to \( h={h}_t\otimes {h}^t \) and if \( F(t)\left(u\otimes x\right) \) is continuous in t for every \( u\in {h}_0 \), \( x\in \varepsilon \), then there exists a sequence F c namely \( {F}_c(t)={\displaystyle \sum_{k\ge 0}{\chi}_{\left[\frac{k}{2^n},\frac{k+1}{2^n}\right]}F\left(\frac{k}{2^n}\right)} \).

We also mention that:

$$ {\left\Vert M(t)\right\Vert}_x^2\le {\left\Vert {\displaystyle \sum_p{L}_p*{L}_p}\right\Vert}^2{\displaystyle \underset{0}{\overset{t}{\int }}{\left\Vert F(a)\right\Vert}_x^2da}+{\displaystyle \underset{0}{\overset{t}{\int }}{\left\Vert F(a)\right\Vert}_x^2da}+{\displaystyle \sum_k\underset{a\le t}{ \sup }{\left\Vert M(a)\right\Vert}_{x_{ck}}^2{\left\Vert {x}_k\right\Vert}^2}. $$

Writing the formulas \( {\left\Vert M(t)\left(u\otimes x\right)\right\Vert}^2 \) for \( M(t)-M(s) \), we shall obtain:

$$ \begin{array}{l}{\left\Vert M(t)-M(s)\right\Vert}_x^2\le {\left\Vert {\displaystyle \sum_p{L}_p*{L}_p}\right\Vert}^2{\displaystyle \underset{s}{\overset{t}{\int }}{\left\Vert F(a)\right\Vert}_x^2da}+r{\displaystyle \underset{s}{\overset{t}{\int }}{\left\Vert F(a)\right\Vert}_x^2da}\\ {}+{\displaystyle \sum_k\underset{a\le t}{ \sup }{\left\Vert M(a)-M(s)\right\Vert}_{x_{ck}}^2{\left\Vert {x}_k\right\Vert}^2}\end{array} $$

and by induction we show that F is continuous hence \( \underset{s\to t}{ \lim }{\left\Vert F(s)-F(t)\right\Vert}_x=0 \) for every x, then M(t) follows continuous similarly.

Definition 3.1

The integrals can be defined separately and we have:

$$ M(t)={\displaystyle \underset{0}{\overset{t}{\int }}\left(d{A}_L^{+}F+Gd{A}_L+Hds\right)}={\displaystyle \underset{0}{\overset{t}{\int }}d{A}_L^{+}F}+{\displaystyle \underset{0}{\overset{t}{\int }}Gd{A}_L}+{\displaystyle \underset{0}{\overset{t}{\int }}Hds}. $$

19.4 Stochastic Equations

Theorem 4.1

Let be the operators X(0), B and D in L(h 0).

We show that the stochastic differential equation:

$$ X(t)=X(0)+{\displaystyle \underset{0}{\overset{t}{\int }}d{A}_L^{*}\left({B}_FX{D}_F\right)}+{\displaystyle \underset{0}{\overset{t}{\int }}\left({B}_GX{D}_G\right)d{A}_L}+{\displaystyle \underset{0}{\overset{t}{\int }}\left({B}_HX{D}_H\right)ds} $$

has a unique solution which is a continuous process.

Proof

We remark that the integrands are “allowable”, hence the stochastic integrals are well defined.

Unicity: if X and Y are two solutions, with \( X(0)=Y(0) \), then \( Z=X-Y \) will be a solution of the equation with \( Z(0)=0 \).

Since \( {\left\Vert BFD\right\Vert}_x\le \left\Vert B\right\Vert {\left\Vert F\right\Vert}_x\left\Vert D\right\Vert \) for \( B,D\in L\left({h}_0\right) \), we have for \( t\le T \):

$$ \begin{array}{l}{\left\Vert Z(t)\right\Vert}_x^2\le c{\displaystyle \underset{0}{\overset{t}{\int }}{\left\Vert Z(a)\right\Vert}_x^2da}+{\displaystyle \sum_k\underset{a\le t}{ \sup }{\displaystyle \underset{0}{\overset{a}{\int }}{\left\Vert {B}_FZ(a){D}_F\right\Vert}^2X{C}_k{\left\Vert {x}_k\right\Vert}^2da}}\\ {}+\left({\displaystyle \underset{0}{\overset{t}{\int }}{\displaystyle \sum_k{\left\Vert \left({B}_GZ(a){D}_G\right)\right\Vert}^2X{C}_kda}}\right){e}^{{\displaystyle \sum_k{\left\Vert {x}_k\right\Vert}^2}}\end{array} $$

We give the proof by induction on r, if \( x={x}_1\wedge {x}_2\wedge \dots \wedge {x}_r \).

Knowing that

\( Z(a)\left(u\otimes \left({x}_1\wedge {x}_2\wedge \dots \wedge {x}_{r-1}\right)\right)=0 \) for all u and x i , we deduce:

\( {\left\Vert Z(t)\right\Vert}_x^2\le c{\displaystyle \underset{0}{\overset{t}{\int }}{\left\Vert Z(a)\right\Vert}_x^2da}+0 \) (for \( k=0 \) this is obvious) and using Gronwall’s lemma, we obtain \( {\left\Vert Z(t)\right\Vert}_x^2\le 0{e}^{ct} \), hence \( Z(a)\left(u\otimes \left({x}_1\wedge {x}_2\wedge \dots \wedge {x}_{r-1}\right)\right)=0 \) for all u and x i .

Existence: We establish the existence iteratively.

We fix T > 0 and we consider \( X(0)(t)=X(0) \) for every \( t\le T \) and then inductively:

$$ {X}_{n+1}(t)=X(0)+{\displaystyle \underset{0}{\overset{t}{\int }}d{A}_L^{*}\left({B}_F{X}_n{D}_F\right)}+{\displaystyle \underset{0}{\overset{t}{\int }}\left({B}_G{X}_n{D}_G\right)d{A}_L}+{\displaystyle \underset{0}{\overset{t}{\int }}\left({B}_H{X}_n{D}_H\right)ds} $$

We have:

$$ \begin{array}{l}{\left\Vert {X}_{n+1}(t)-Xn(t)\right\Vert}_y=\\ {}\left\Vert {\displaystyle \underset{0}{\overset{t}{\int }}d{A}_L^{*}\left({B}_F\left({X}_n-{X}_{n-1}\right){D}_F\right)}+\right.{\displaystyle \underset{0}{\overset{t}{\int }}\left({B}_G\left({X}_n-{X}_{n-1}\right){D}_G\right)d{A}_L}+{\left.{\displaystyle \underset{0}{\overset{t}{\int }}\left({B}_H\left({X}_n-{X}_{n-1}\right){D}_H\right)ds}\right\Vert}_y\\ {}\le {d}^{\hbox{'}}{q_{n-1}}^{\left(p-1\right)}+{d}^{\hbox{'}}{q_{n-2}}^{\left(p-1\right)}cT+\dots +{d}^{\hbox{'}}{q_0}^{\left(p-1\right)}\left({(cT)}^{n-1}/\left(n-1\right)!\right)+{c}^{\hbox{'}}\left({c}^n{T}^n/n!\right)\end{array} $$

If we denote with q n the last expression which doesn’t depend on t, we have, if k and \( n-k \) converge to ∞ with n:

$$ \begin{array}{l}{q}_n={d}^{\prime }{\displaystyle \sum_{j=1}^{k-1}{q_{n-j}}^{\left(p-1\right)}{(cT)}^{j-1}/\left(j-1\right)!}+{d}^{\prime}\left({q_{n-k}}^{\left(p-1\right)}{(cT)}^{k-1}/\left(k-1\right)!+{q_{n-k-1}}^{\left(p-1\right)}{(cT)}^k/k!\right)\\ {}+{\displaystyle \sum_{j=k+2}^n{q_{n-j}}^{\left(p-1\right)}{(cT)}^{j-1}/\left(j-1\right)!+}{c}^{\prime}\left({c}^n{T}^n/n!\right)\end{array} $$

Now we use \( \left({\displaystyle \sum_k{a}_k}\right)/\left({\displaystyle \sum_k{b}_k}\right)\le \max \left({a}_k/{b}_k\right) \), and we have: \( \begin{array}{l}{ \max}_{k+2\le j\le n}\left((cT)/\left(j-1\right),cT/n\right)=\\ {} \max \left({ \max}_{j\ge n-k}\left({q}_{j+1}^{\left(p-1\right)}/{q}_j^{\left(p-1\right)}\right),\right.\left.\left({q}_{n-k}^{\left(p-1\right)}/{q}_{n-1-k}^{\left(p-1\right)}\right)+cT/k,{ \max}_{k+1\le j}\left(cT/j\right)\right)\end{array} \)

which converges to 0.

Hence \( {\displaystyle \sum \left(\left({X}_n-{X}_{n-1}\right)(t)\right)} \) is the solution of the equation.

Theorem 4.2

We consider the stochastic integral equation: \( U(t)=1+{\displaystyle \underset{0}{\overset{t}{\int }}\left(U\theta \left(d{A}_L^{+}\right)+U\theta \left(d{A}_L\right)+ UXds\right)} \),

where \( X=-\left({\displaystyle \sum_p{L}_p+{L}_p}\right)/2 \).

Then there exists a unique unitary process satisfying this equation.

Proof

We have \( {U}^{+}(t)=1+{\displaystyle \underset{0}{\overset{t}{\int }}\left(\left(d{A}_L^{+}\right)\theta {U}^{+}+\left(d{A}_L\right)\theta {U}^{+}+X{U}^{+}ds\right)} \) (since \( X={X}^{*} \)).

19.5 Noncommutative Markov Processes as Stochastic Equation Solutions

Definition 5.1

A noncommutative Markov process is a system which includes:

  1. (i)

    A Hilbert space h 0.

  2. (ii)

    A C*-algebra \( A\subset L\left({h}_0\right) \) with 1.

  3. (iii)

    A family of completely positive mappings: \( {T}_1:A\to A,\kern0.5em t\ge 0 \) with \( {T}_t1=1 \), \( {T}_0=1 \) and \( {T}_{t+s}={T}_t{T}_s \) (briefly a semigroup of completely positive mappings on A with \( {T}_t1=1 \)).

  4. (iv)

    Another Hilbert space h, in which h 0 is a closed subspace.

  5. (v)

    A family \( {\left({j}_t\right)}_{t\ge 0} \) of *-homomorphisms \( {j}_t:A\to L(h) \), such that:

    1. 1.

      \( {j}_0(x)=x\oplus 0 \) relatively to \( h={h}_0\oplus {h}_0^{\perp } \).

    2. 2.

      \( {j}_s(1){j}_{s+t}(x){j}_s(1)={j}_s\left({T}_tx\right) \).

Remark

  1. (a)

    j s (1) is a projector.

  2. (b)

    \( {j}_s(1)\le {j}_{s+t}(1) \) for \( t\ge 0 \), hence denoting \( {h}_t=\mathrm{I}\mathrm{m}{j}_t(1) \), notation which is not incompatible with h 0, we obtain a filtration (h t ).

  3. (c)

    There results that T t is completely positive:

    \( {j}_s\left({T}_tx\right)={j}_s(1){j}_{s+t}(x){j}_s(1) \) for \( s=0 \) we have \( {j}_0\left({T}_tx\right)={j}_0(1){j}_t(x){j}_0(1) \) and \( \left\langle {j}_0\left({T}_tx\right)u,v\right\rangle =\left\langle {j}_0(1){j}_t(x){j}_0(1)u,v\right\rangle \) and j 0 is a projector.

    We also have:

    $$ \left\langle \left({T}_tx\right)u,v\right\rangle =\left\langle {j}_t(x)\left(u\otimes 1\right),\left(v\otimes 1\right)\right\rangle . $$

    Let be \( {S}_i\in A \), \( {V}_i\in A \), then we have:

    $$ \begin{array}{l}\left\langle {\displaystyle {\sum}_{i,j}{S}_i*{T}_t\left({V}_i^{*}{V}_j\right){S}_ju,v}\right\rangle ={\displaystyle {\sum}_{i,j}\left\langle {T}_t\left({V}_i^{*}{V}_j\right){S}_ju,{S}_iv\right\rangle }=\\ {}{\displaystyle {\sum}_{i,j}\left\langle {j}_t\left({V}_i^{*}{V}_j\right)\left({S}_ju\otimes 1\right),\left({S}_iv\otimes 1\right)\right\rangle }={\displaystyle {\sum}_{i,j}\left\langle {j}_t*\left({V}_i\right){j}_t{(V)}_j\left({S}_ju\otimes 1\right),{S}_iv\otimes 1\right\rangle }=\\ {}{\displaystyle {\sum}_{i,j}\left\langle {j}_t\left({V}_j\right)\left({S}_ju\otimes 1\right),{j}_t\left({V}_i\right){S}_iv\otimes 1\right\rangle }={\left\Vert {\displaystyle {\sum}_j\left\langle {j}_t\left({V}_i\right){S}_iv\otimes 1\right\rangle}\right\Vert}^2\ge 0\end{array} $$
  4. (d)

    If \( S:A\to A \) is continuous and linear, then \( {T}_t={e}^{tS} \) defines a semigroup, but generally T t are not completely positive. If \( S1=0 \) then \( {T}_t1=1 \).

  5. (e)

    If \( U\in L(h) \) is unitary, then \( T\to UT{U}^{*} \) is a *-homomorphism:

    $$ L(h)\to L(h). $$

We shall use the following formulas:

  1. I.

    If \( M(t)=M(0)+{\displaystyle \underset{0}{\overset{t}{\int }}d{A}_L^{+}F+{(dA)}_LG+(ds)H} \) then

    $$ \begin{array}{l}\left\langle M(t)\left(u\otimes x\right),M(t)\left(v\otimes y\right)\right\rangle =\left\langle M(0)\left(u\otimes x\right),M(0)\left(v\otimes y\right)\right\rangle +\\ {}{\displaystyle \underset{0}{\overset{t}{\int }}\left({\displaystyle {\sum}_p\left\langle {L}_pF(a)\left(u\otimes x\right),{L}_pF(a)\left(v\otimes y\right)\right\rangle}\right.}\\ {}+{\displaystyle {\sum}_{p,j}{\left(-1\right)}^{j+r+w}\left({x}_{jp}(a)\left\langle M(a)\left(u\otimes {x}_{cj}\right),\right.\right.}\left.{L}_p IF(a)I\left(v\otimes y\right)\right\rangle \\ {}+\overline{y_{jp}}(a)\left\langle {L}_p IF(a)I\left(u\otimes x\right),M(a)\left(v\otimes {y}_{cj}\right)\right\rangle \\ {}+{\left(-1\right)}^{j-1}\left(\overline{y_{jp}}(a)\left\langle M(a)\left(u\otimes x\right),\right.\right.\left.{L}_pIG(a)I\left(v\otimes {y}_{cj}\right)\right\rangle +\\ {}\left.{x}_{jp}(a)\left\langle {L}_pIG(a)I\left(u\otimes {x}_{cj}\right),M(a)\left(v\otimes y\right)\right\rangle \right)+\left.\left\langle H(a)\left(u\otimes x\right),M(a)\left(v\otimes y\right)\right\rangle \right)da\end{array} $$
  2. II.

    For \( S\in L\left({h}_0\right) \), we have \( \left(S\otimes 1\right)={\displaystyle \underset{0}{\overset{t}{\int }}d{A}_L^{+}F+\left(d{A}_L\right)G+Hds}={\displaystyle \underset{0}{\overset{t}{\int }}\left(d{A}_{SL}^{+}\right)F+\left(d{A}_{L{S}^{*}}\right)G+ SHds} \) as we know from the definition of the stochastic integral.

  3. III.

    From \( {U}^{*}(t)=1+{\displaystyle \underset{0}{\overset{t}{\int }}\left(\left(d{A}_L^{+}\right)\theta {U}^{*}+\left(d{A}_L\right)\theta {U}^{*}+X{U}^{*}ds\right)} \) we deduce that

    $$ \left(S\otimes 1\right){U}^{*}(t)=\left(S\otimes 1\right)+{\displaystyle \underset{0}{\overset{t}{\int }}\left(\left(d{A}_L^{+}\right)\theta {U}^{*}+\left(d{A}_{L{S}^{*}}\right)\theta {U}^{*}+X{U}^{*}ds\right)} $$

    We consider

    $$ U(t)=1+{\displaystyle \underset{0}{\overset{t}{\int }}\left(U\theta \left(d{A}_L^{+}\right)+U\theta \left(d{A}_L\right)+ UXds\right)} $$

    we write it as follows:

    $$ U\left(t+s\right)=U(s)+{\displaystyle \underset{0}{\overset{s+t}{\int }}\left(U\theta \left(d{A}_L^{+}\right)+U\theta \left(d{A}_L\right)+ UXds\right)} $$

The integral can be considered as \( {\displaystyle \underset{0}{\overset{t}{\int }}} \) of the same integrant with h s instead of h 0 and h s instead of h 0. Using “III.”, the equation can be written: \( U{(s)}^{-1}U\left(t+s\right)=1+{\displaystyle \underset{0}{\overset{s+t}{\int }}\left(U{(s)}^{-1}U\theta \left(d{A}_L^{+}\right)+U{(s)}^{-1}U\theta \left(d{A}_L\right)\right.}\left.+U{(s)}^{-1} UXds\right) \) and then \( U{(s)}^{-1}U\left(.+s\right) \) appears as U(.).

Lemma 5.2

We consider the equation

\( U(t)=1+{\displaystyle \underset{0}{\overset{t}{\int }}\left(U\theta \left(d{A}_L^{+}\right)+U\theta \left(d{A}_L\right)+ UXds\right)} \) where \( {X}^{*}=X=-\left({\displaystyle \sum {L}_p+{L}_p}\right)/2 \).

If we define:

$$ A=\left\{S;\in L\left({h}_0\right),S\theta =\theta S\right\},\kern1em {T}_t(S)={e}^{ty}(S) $$

where \( Y(S)=\left({\displaystyle {\sum}_p{L_p}^{*}S{L}_p}\right)+XS+SX \) .

Then we have \( \left\langle {T}_t(S)u,v\right\rangle =\left\langle U(t)\left(S\otimes 1\right)U(t)\left(u\otimes 1\right),\left(v\otimes 1\right)\right\rangle \) .

Proposition 5.3

We consider the equation

$$ U(t)=1+{\displaystyle \underset{0}{\overset{t}{\int }}\left(U\theta \left(d{A}_L^{+}\right)+U\theta \left(d{A}_L\right)+ UXds\right)} $$

where \( X=-\left({\displaystyle \sum {L}_p+{L}_p}\right)/2 \).

Then, if we define

$$ A=\left\{S;\in L\left({h}_0\right),S\theta =\theta S\right\},\kern1em {T}_t(S)={e}^{ty}(S) $$

where

\( Y(S)=\left({\displaystyle {\sum}_p{L_p}^{*}S{L}_p}\right)+XS+SX \) and \( {j}_t(S)=\left({U}_t\left(S\otimes 1\right){U_t}^{*}\right){P}_t \), where P t is the projector on h t , the system h 0, L(h 0), T t and j t is a noncommutative Markov process.

Proof

Writing the equation

\( {U}^{+}(t)=1+{\displaystyle \underset{0}{\overset{t}{\int }}\left(\left(d{A}_L^{+}\right)\theta {U}^{+}+\left(d{A}_L\right)\theta {U}^{+}+X{U}^{+}ds\right)} \) and since U*(t) appears as \( \otimes 1 \) relatively to \( h={h}_t\otimes {h}^t \) and \( {P}_{s+t}=1\otimes \) our relation becomes

$$ \begin{array}{l}\left\langle \left(S\otimes 1\right)U{\left(s+t\right)}^{\ast}\left(u\otimes x\right),U{\left(s+t\right)}^{\ast}\left(v\otimes y\right)\right\rangle =\\ {}\left\langle \left({T}_t(S)\otimes 1\right)U{(s)}^{\ast}\left(u\otimes x\right),U{(s)}^{\ast}\left(v\otimes y\right)\right\rangle \end{array} $$

We have \( U{(s)}^{*}\left(u\otimes x\right),U{(s)}^{*}\left(v\otimes y\right)\in {h}_s \) and it suffices to show that \( \left\langle \left(S\otimes 1\right)U{\left(s+t\right)}^{\ast}\left(u\otimes x\right),U{\left(s+t\right)}^{\ast}\left(v\otimes y\right)\right\rangle =\left\langle \left({T}_t(S)\otimes 1\right)u,v\right\rangle \).

Hence we obtain the formula from Lemma 5.2, that is

$$ \left\langle {T}_t(S)u,v\right\rangle =\left\langle U(t)\left(S\otimes 1\right)U{(t)}^{\ast}\left(u\otimes 1\right),\left(v\otimes 1\right)\right\rangle . $$

Then \( T\to UT{U}^{*} \) is a *-homomorphism: \( L(h)\to L(h) \).

19.5.1 The Brownian Motion as Noncommutative Markov Process

We consider H a Hilbert space, a Brownian x t on a probability space (E, K, P), \( A={A}^{*}\in L(H) \) and \( U\left(t,\omega \right)={e}^{i{x}_t\left(\omega \right)A} \). Hence U(t, ω) is a unitary operator of L(H). Let be \( {T}_t:L(H)\to L(H) \) defined as \( {T}_t(x)={\displaystyle \int U(t)XU{(t)}^{*}}dP \).

We have \( U(t)={\displaystyle {\sum}_{n\ge 0}{i}^n{\left({x}_t\right)}^n{A}^n/n!} \), hence

$$ \begin{array}{l}U{(t)}^{*}={\displaystyle {\sum}_{n\ge 0}{\left(-1\right)}^n{\left({x}_t\right)}^n{A}^n/n!},U(t)X(t)U{(t)}^{*}=\\ {}={\displaystyle {\sum}_{n,k}{\left(-1\right)}^{n-k}{\left({x}_t\right)}^{n+k}{A}^nX{A}^k/n!k!}={\displaystyle {\sum}_u{\displaystyle {\sum}_{n+k=u}{\left({x}_t\right)}^{n+k}{i}^u{i}^{-2k}{A}^nX{A}^k/n!k!}}=\\ {}={\displaystyle {\sum}_u{\left(i{x}_t\right)}^u{\displaystyle {\sum}_{n+k=u}{\left(-1\right)}^k{A}^nX{A}^k/n!k!}}={\displaystyle {\sum}_u{\left(i{x}_t\right)}^u{D}^u(X)/u!}\end{array} $$

where \( D(X)= AX-XA \).

Indeed \( D=P-Q \).

\( P(X)= AX \), \( Q(X)=XA \), hence

\( {D}^u(X)={\left(P-Q\right)}^u(X)={\displaystyle {\sum}_{n+k=u}{C}_u^n{\left(-1\right)}^k{P}^n{Q}^kX} \), since P and Q commute.

Hence \( {T}_t(X)={\displaystyle {\sum}_u{i}^uE\left({\left({x}_t\right)}^u\right){D}^u(X)/u!} \).

From \( E\left({e}^{i\lambda xt}\right)={e}^{-t{\lambda}^2/2}E\left({e}^{{}^{i\lambda x}t}\right)={e}^{-t{\lambda}^2/2} \) we deduce that

$$ {\displaystyle {\sum}_n{i}^nE\left({\left({x}_t\right)}^n\right){\lambda}^n/n!}={\displaystyle {\sum}_n{\left(-{\lambda}^2t/2\right)}^n/n!} $$

and replacing \( \lambda =D \) we obtain \( {T}_t(X)={e}^{-t{D}^2/2}(X) \), that is \( {T}_t={e}^{-t{D}^2/2} \).

We have

\( -\left({D}^2/2\right)(X)=-\left({A}^2X-X{A}^2\right)/2+AXA \), hence it is of the considerate form with only one term \( L={L}^{*}=A \).

19.5.2 The Poisson Process as Noncommutative Markov Process

We consider H a Hilbert space, a sequence U n of unitary operators, a convergent sum with positive terms \( {\displaystyle {\sum}_n} \) \( {\lambda}_n=\lambda \), \( {p}_n={\lambda}_n/\lambda \), a probability space (E, K, P) and a particular composite Poisson process on it, that is \( {x}_t={y}_{zt} \), where (z t ) is a Poisson process of parameter λ and \( {\left({y}_n\right)}_{n\ge 1} \) is a sequence of independent variables, independent of (z t ), all having the repartition \( \varLambda =\varSigma \) p n ε n . We consider \( {Y}_0=1 \).

For every t we consider \( {U}_t\left(\omega \right)={U}_{y_{zt}}\dots {U}_0 \) and we define for \( X\in L(H) \),

$$ {T}_t(X)={\displaystyle \int U(t)XU{(t)}^{*}}dP. $$

We have

$$ \begin{array}{l}{T}_t(X)={\displaystyle {\sum}_k{\displaystyle \int {\chi}_{\left({z}_t=k\right)}U(t)XU{(t)}^{*}}dP}\\ {}={\displaystyle {\sum}_k\left({\left(\lambda t\right)}^k{e}^{-\lambda t}/k!\right){\displaystyle \int {\chi}_{\left({z}_t=k\right)}{U}_{y_k}\dots {U}_{y_0}X{U^{*}}_{y_0}}dP}=\\ {}{\displaystyle {\sum}_k\left({\left(\lambda t\right)}^k{e}^{-\lambda t}/k!\right){\displaystyle {\sum}_{n_1,\dots, {n}_k}{p}_{n_1}\dots {p}_{n_k}{U}_{n_k}\dots {U}_{n_1}X{U^{*}}_{n_1}\dots {U^{*}}_{n_k}}}\end{array} $$

We denote \( {L}_i(X)={U}_iX{U_i}^{*} \) and we have:

$$ \begin{array}{l}{T}_t(X)={\displaystyle {\sum}_k\left({(t)}^k{e}^{-\lambda t}/k!\right){\displaystyle {\sum}_{n_1,\dots, {n}_k}{\lambda}_{n_1}\dots {\lambda}_{n_k}{L}_{n_k}\dots {L}_{n_1}(X)}}\\ {}={{\displaystyle {\sum}_k\left({(t)}^k{e}^{-\lambda t}/k!\right)\left({\displaystyle {\sum}_n{\lambda}_n{L}_n}\right)}}^k(X)={e}^{t\left(-\lambda +{\displaystyle {\sum}_n{\lambda}_n{L}_n}\right)}(X)\end{array} $$

hence \( {T}_t={e}^{t\left(-\lambda +{\displaystyle {\sum}_n{\lambda}_n{L}_n}\right)} \).

We remark now that

$$ \left(-\lambda +{\displaystyle {\sum}_n{\lambda}_n{L}_n}\right)(X)={{\displaystyle {\sum}_n\left({\lambda_n}^{1/2}{U^{*}}_n\right)}}^{*}X\left({\lambda_n}^{1/2}{U^{*}}_n\right)+ZX+XZ, $$

where \( Z=-\lambda /2=-{\displaystyle {\sum}_n{\left(-{\lambda}_n^{1/2}{U^{*}}_n\right)}^{*}\left({\lambda}_n^{1/2}{U^{*}}_n\right)/2} \), hence T t is a particular case of the considerate semigroups.

19.6 Conclusions

The need to build the non-commutative Markov processes was given by the evolution of probabilities in quantum mechanics. This paper aims to build these processes on antisymmetric Fock space where we do not have exponential commutative vectors and where the commutative property does not occur between operators describing disjoint time intervals. For this reason the processes are obtained as solutions of stochastic integral equations. This mathematical model creates the possibility to construct physical processes as stochastic integral equations solutions, being at the same time a new method of proving that certain processes are noncommutative. The model may be used in diffusion processes.