$L^{p}$-solutions of QSDE driven by Fermion fields with nonlocal conditions under non-Lipschitz coefficients

The noncommutative $L^{p}$-space and associated harmonic analysis have been deeply studied in [7, 11, 26, 27, 28, 29] and references therein. In particular, martingale inequalities in the non-commutative case have been greatly developed since Pisier and Xu [23] proved the analogy of the classical Burkholder-Gundy inequality for non-commutative martingales, where they defined the noncommutative $H^{p}$ martingale space by introducing the row and column square functions. Subsequently, combined Khintchine inequalities of operator values obtained by Lust-Piquard [20], Junge, Randrianantoanina and Xu et al generalized almost all the martingale inequalities such as Doob maximal inequality and Rosenthal inequalities so on of classical martingale theory to the noncommutative case [13, 14, 16, 15, 17, 24, 25], which lays a foundation for the study of quantum stochastic analysis.

Inspired by Burkholder-Gundy inequality, it is natural to solve the QSDE (1.1) in non-commutative $L^{p}$-spaces. Problems with nonlocal conditions are more applicable to real life problems than problems with traditional local conditions. Analogously, we study the $L^{p}$-solution and the basic properties of solution to QSDE with nonlocal initial conditions for $p>2$. To state our results, we give some assumptions on the QSDE. Some basic notations of noncommutative filtration $\{\mathscr{C}_{t}\}_{t\geq 0}$ will be introduced in Section 2.

It is easy to check that if $X:\mathbb{R}^{+}\rightarrow L^{p}(\mathscr{C})$ and $F:L^{p}(\mathscr{C})\times\mathbb{R}^{+}\rightarrow L^{p}(\mathscr{C})$ are both adapted, so is the map $t\mapsto F(X_{t},t)$.

In the rest of this section, we consider the following equation with nonlocal conditions in the interval $[0,T]$ for some fixed $T>0$,

where $F(\cdot,\cdot),\ G(\cdot,\cdot),\ H(\cdot,\cdot):L^{p}(\mathscr{C})\times[0,T]\rightarrow L^{p}(\mathscr{C})$, $R(\cdot):L^{p}(\mathscr{C})\to L^{p}(\mathscr{C})$ constitutes the nonlocal condition, and $X_{t_{0}}\in L^{p}(\mathscr{C}_{t_{0}})$ for fixed $p>2$.

Throughout the paper, we shall make use of the following conditions:

This paper is organized as follows. In Section 2, we review some fundamental notations and preliminaries on Fermion fields, and introduce several useful inequalities which are the main techniques of subsequent proof. In section 3 and section 4, we obtain the existence, uniqueness and the stability of $L^{p}$-solution to the QSDE (1.1) and (1.5) with nonlocal conditions. In section 5, we get the self-adjointness and the Markov property of the solution to QSDE.

Let $\mathscr{H}$ be a separable complex Hilbert space. For any $z\in\mathscr{H}$, the creation operator $C(z):\Lambda_{n}(\mathscr{H})\rightarrow\Lambda_{n+1}(\mathscr{H})$ defined by $u\mapsto\sqrt{n+1}\ z\wedge$$u$, is a bounded operator on $\Lambda(\mathscr{H})$ with $\|C(z)\|=\|z\|$, where $\Lambda_{0}(\mathscr{H}):=\mathbb{C}$. Meanwhile, define the annihilation operator $A(z)=C^{*}(z)$. Then the antisymmetric Fock space $\Lambda(\mathscr{H}):=\oplus_{n=0}^{\infty}\Lambda_{n}(\mathscr{H})$ over $\mathscr{H}$ is a Hilbert space. Moreover, define the fermion field $\Psi(z)$ on $\Lambda(\mathscr{H})$ by $\Psi(z):=C(z)+A(Jz)$, where $J:\mathscr{H}\rightarrow\mathscr{H}$ is the conjugation operator (i.e., $J$ is antilinear, antiunitary and $J^{2}=1$). Denote by $\mathscr{C}$ the von Neumann algebra generated by the bounded operators $\{\Psi(z):z\in\mathscr{H}\}$. For the Fock vacauum $\mathds{1}\in\Lambda(\mathscr{H})$, define

on $\mathscr{C}$. Obviously, $m$ is a normal faithful state on $\mathscr{C}$, and $(\Lambda(\mathscr{H}),\mathscr{C},m)$ is a quantum probability space by [30].

Now, let $\mathscr{H}=L^{2}(\mathbb{R}^{+})$, and $\mathscr{C}_{t}$ be the von Neumann subalgebra of $\mathscr{C}$ generated by $\{\Psi(u):u\in L^{2}(\mathbb{R}_{+}),{\rm~{}ess~{}supp}~{}u\subseteq[0,t]\},$ then $\{\mathscr{C}_{t}\}_{t\geq 0}$ is an increasing family of von Neumann subalgebras of $\mathscr{C}$ which is the noncommutative analogue of filtration in stochastic analysis. Let

then $\{W_{t}:t\in\mathbb{R}^{+}\}$ is a Fermion Brownian motion adapted to the family $\{\mathscr{C}_{t}:t\in\mathbb{R}^{+}\}$. For any $1\leq p<\infty$, define the noncommutative $L^{p}$-norm on $\mathscr{C}$ by

where $|f|=(f^{*}f)^{1\over 2}$, then $L^{p}(\mathscr{C},m)$ is the completion of $(\mathscr{C},\|\cdot\|_{p})$, which is the noncommutative $L^{p}$-space, abbreviated as $L^{p}(\mathscr{C})$. For any interval $[0,T]\subset\mathbb{R^{+}}$, set

It is easy to check that $C_{\mathbb{A}}(0,T;L^{p}(\mathscr{C}_{T}))$ is a Banach space with the norm given by

For $p\geq 1$, let ${\cal S}^{p}_{\mathbb{A}}(\mathbb{R}^{+})$ be the linear space of all simple adapted $L^{p}$-processes, i.e.

Then ${\cal S}^{p}_{\mathbb{A}}([0,t])$ is subspace of ${\cal S}^{p}_{\mathbb{A}}(\mathbb{R}^{+})$ whose processes vanish in $(t,\infty)$. It is clear that $\int_{t_{0}}^{t}f(s)dW_{s}$ is a Clifford $L^{p}$-martingale for any $f\in{\cal S}^{p}_{\mathbb{A}}([0,t])$, i.e. $\mathbb{E}\left(\int_{t_{0}}^{t}f(\tau)dW_{\tau}\mid\mathscr{C}_{s}\right)=\int_{t_{0}}^{s}f(\tau)dW_{\tau}$ for any $t_{0}\leq s\leq t$. For all $f\in{\cal S}^{p}_{\mathbb{A}}([0,t])$, let

and denote by the non-commutative Hardy space $\mathcal{H}^{p}([0,t])$ the completion of ${\cal S}^{p}_{\mathbb{A}}([0,t])$ under the norm $\|\cdot\|_{\mathcal{H}^{p}([0,t])}$. For simplicity, we use $\mathcal{H}^{p}_{loc}(\mathbb{R}^{+})$ to represent the set of stochastic processes $f:\mathbb{R}^{+}\to L^{p}(\mathscr{C})$ and $\chi_{[0,t]}f\in\mathcal{H}^{p}([0,t])$. Moreover, we have the following the Burkholder-Gundy inequality (2.4) of Clifford $L^{p}$-martingale first established in [23].

The stochastic integral (2.3) is also called right stochastic integral. Analogously, we can define left stochastic integrals, and have the Burkholder-Gundy inequality with respect to left stochastic integrals.

By Lemma 2.1 and Lemma 2.2, the Itô-Clifford integral can be defined for any element of $\mathcal{H}^{p}([0,t])$, and Burkholder-Gundy inequality holds ture. For any $t\geq 0$, let $L^{q}_{\mathbb{A}}(0,t;L^{p}(\mathscr{C}_{t}))$ be the completion of ${\cal S}^{p}_{\mathbb{A}}([0,t])$ with the norm

Similarly, $L^{p}_{\mathbb{A}}(\mathscr{C}_{t};L^{q}(0,t))$ is the completion of ${\cal S}^{p}_{\mathbb{A}}([0,t])$ with the norm

As an application of Minkowski inequality, we have the following result.

In general, the Burkholder-Gundy inequality (2.4) cannot be directly used to study $L^{p}$-solution to QSDE. Instead, we need the following easy corollary.

Now, we give the parity of each element of $L^{p}(\mathscr{C})$. Let the parity operator $P$ be automorphism map on von Neumann algebra $\mathscr{C}$ generated by bounded linear operators on $\Lambda(\mathscr{H})$ as is in [3, 5, 23].

Furthermore, for any $1<p<\infty$,

where $L^{p}(\mathscr{C}_{e}),\ L^{p}(\mathscr{C}_{o})$ denote the even part and the odd part, respectively. More precisely, for any $h\in L^{p}(\mathscr{C})$, $h=\frac{h+Ph}{2}+\frac{h-Ph}{2}=h_{o}+h_{e}$, where $h_{e}$ and $h_{o}$ are even and odd, respectively. Since $P$ is isometric on $L^{p}(\mathscr{C})$,

Let $\mathscr{E}$ denote the algebra of even polynomials in the fields $\{\Psi(u):u\in\mathscr{H}\}$, and let $\mathscr{C}_{e}$ be the $W^{*}$-subalgebra of $\mathscr{C}$ generated by $\mathscr{E}$. If $h\in L^{p}(\mathscr{C})$ is even there is a sequence $\{h_{n}\}$ in $\mathscr{E}$ such that $h_{n}\rightarrow h$ in $L^{p}(\mathscr{C})$, and therefore $h_{n}^{*}\rightarrow h^{*}$ in $L^{p}(\mathscr{C})$. It follows that $h^{*}$ is also even. Similarly, if $g$ is odd in $L^{p}(\mathscr{C})$, there is a sequence $\{g_{n}\}$ of odd polynomials in the fields with $g_{n}\rightarrow g$ and thus $g_{n}^{*}\rightarrow g^{*}$ in $L^{p}(\mathscr{C})$, that is, $g^{*}$ is odd as well. It follows that if $h=h^{*}$ in $L^{p}(\mathscr{C})$ and $h=h_{e}+h_{o}$, then $h_{e}=h_{e}^{*}$ and $h_{o}=h_{o}^{*}$ in $L^{p}(\mathscr{C})$.