Stochastic differential equations for models of non-relativistic matter interacting with quantized radiation fields

Abstract

We discuss Hilbert space-valued stochastic differential equations associated with the heat semi-groups of the standard model of non-relativistic quantum electrodynamics and of corresponding fiber Hamiltonians for translation invariant systems. In particular, we prove the existence of a stochastic flow satisfying the strong Markov property and the Feller property. To this end we employ an explicit solution ansatz. In the matrix-valued case, i.e., if the electron spin is taken into account, it is given by a series of operator-valued time-ordered integrals, whose integrands are factorized into annihilation, preservation, creation, and scalar parts. The Feynman–Kac formula implied by these results is new in the matrix-valued case. Furthermore, we discuss stochastic differential equations and Feynman–Kac representations for an operator-valued integral kernel of the semi-group. As a byproduct we obtain analogous results for Nelson’s model.

Appendices

Appendix 1: Examples

1.1 Non-relativistic quantum electrodynamics

Example 12.1

In all items below we choose $\mathcal {M}=\mathbb {R}^3\times \{1,2\}$, equipped with the product of the Lebesgue and counting measures, i.e., $\mathfrak {h}=L^2(\mathbb {R}^3\times \{1,2\})$.

• (1) In the standard model of NRQED for one electron interacting with the electromagnetic radiation field with sharp ultra-violet cut-off one chooses $\nu =3$, $\omega ({\varvec{k}},j)=|{\varvec{k}}|$, for $({\varvec{k}},j)\in \mathbb {R}^3\times \{1,2\}$, ${\varvec{m}}={\varvec{0}}$, and ${\varvec{G}}$ is given by

  $$\begin{aligned} {\varvec{G}}_{{\varvec{x}}}^\Lambda ({\varvec{k}},j) :=(\alpha /2)^{{1}/{2}}(2\pi )^{-{3}/{2}}|{\varvec{k}}|^{-{1}/{2}}\,\chi _\Lambda ({\varvec{k}})\,e^{-i{\varvec{k}}\cdot {\varvec{x}}}{\varvec{\varepsilon }}({\varvec{k}},j), \quad \text {a.e. }({\varvec{k}},j), \end{aligned}$$

  where $\alpha >0$ and $\chi _\Lambda$ is the characteristic function of a ball of radius $\Lambda >0$ about the origin in $\mathbb {R}^3$. The vectors $|{\varvec{k}}|^{-1}{\varvec{k}}$, ${\varvec{\varepsilon }}({\varvec{k}},1)$, and ${\varvec{\varepsilon }}({\varvec{k}},2)$ form a.e. an oriented orthonormal basis of $\mathbb {R}^3$, so that the Coulomb gauge condition ${\mathrm {div}}_{{\varvec{x}}}{\varvec{G}}_{{\varvec{x}}}^\Lambda =0$ is satisfied in $\mathfrak {h}$. If the electron spin is neglected, then one chooses $L=1$ and ${\varvec{F}}={\varvec{0}}$. To include the electron spin one takes $L=2$, $S=3$, $\sigma _1$, $\sigma _2$, and $\sigma _3$ are the $2\times 2$-Pauli-spin matrices, and for ${\varvec{F}}$ one chooses

  $$\begin{aligned} {\varvec{F}}_{{\varvec{x}}}^\Lambda ({\varvec{k}},j):=-\tfrac{i}{2}{\varvec{k}}\times {\varvec{G}}_{{\varvec{x}}}^\Lambda ({\varvec{k}},j),\quad {\varvec{x}}\in \mathbb {R}^3,\;\text {a.e.}\;({\varvec{k}},j). \end{aligned}$$

  Applying a suitable unitary transformation to the total Hamiltonian, if necessary, one may always assume that the polarization vectors are given by

  $$\begin{aligned} {\varvec{\varepsilon }}({\varvec{k}},1)=|{\varvec{e}}\times {\varvec{k}}|^{-1}{\varvec{e}}\times {\varvec{k}},\quad {\varvec{\varepsilon }}({\varvec{k}},2)=|{\varvec{k}}|^{-1}{\varvec{k}}\times {\varvec{\varepsilon }}({\varvec{k}},1),\quad \text {a.e.}\;{\varvec{k}}, \end{aligned}$$

  where ${\varvec{e}}$ is some unit vector in $\mathbb {R}^3$. Then a suitable conjugation is given by $(Cf)({\varvec{k}},j):=(-1)^{j}\overline{f(-{\varvec{k}},j)}$, for a.e. $({\varvec{k}},j)$ and $f\in \mathfrak {h}$.

• (2) To cover the standard model of NRQED for $N\in \mathbb {N}$ electrons we choose $\nu =3N$, write $\underline{{\varvec{x}}}=({\varvec{x}}_1,\ldots ,{\varvec{x}}_N)\in (\mathbb {R}^3)^N$ instead of ${\varvec{x}}$, and set

  $$\begin{aligned} {\varvec{G}}_{\underline{{\varvec{x}}}}^{\Lambda ,N}({\varvec{k}},j):= ({\varvec{G}}_{{\varvec{x}}_1}^\Lambda ({\varvec{k}},j),\ldots ,{\varvec{G}}_{{\varvec{x}}_N}^\Lambda ({\varvec{k}},j))\in (\mathbb {C}^3)^N. \end{aligned}$$

  If spin is neglected, then we again set $L=1$ and ${\varvec{F}}={\varvec{0}}$. To include spin, we choose $L=2^N$, so that ${\mathbb {C}^L}=(\mathbb {C}^2)^{\otimes _N}$, $S=3N$, and

  $$\begin{aligned} \sigma _{3\ell +j}:=\mathbbm {1}_{\mathbb {C}^2}^{\otimes _\ell }\otimes \sigma _j\otimes \mathbbm {1}_{\mathbb {C}^2}^{\otimes _{N-\ell -1}}, \quad \ell =0,\ldots ,N-1,\;j=1,2,3, \end{aligned}$$

  with the Pauli matrices $\sigma _1$, $\sigma _2$, and $\sigma _3$, as well as

  $$\begin{aligned} {\varvec{F}}_{\underline{{\varvec{x}}}}^{\Lambda ,N}({\varvec{k}},j):= ({\varvec{F}}_{{\varvec{x}}_1}^\Lambda ({\varvec{k}},j),\ldots ,{\varvec{F}}_{{\varvec{x}}_N}^\Lambda ({\varvec{k}},j))\in (\mathbb {C}^3)^N. \end{aligned}$$

• (3) In the standard model of NRQED for N electrons in the electrostatic potential of $K\in \mathbb {N}$ nuclei with atomic numbers $\mathscr {Z}=(Z_1,\ldots ,Z_K)\in (0,\infty )^K$ located at the sites $\mathscr {R}=({\varvec{R}}_1,\ldots ,{\varvec{R}}_K)\in (\mathbb {R}^3)^K$, the potential V is given by the Coulomb interaction potential,

  $$\begin{aligned} V^N_{\mathscr {R},\mathscr {Z}}({\varvec{x}}_1,\ldots ,{\varvec{x}}_N)&:=-\sum _{i=1}^N\sum _{\varkappa =1}^K\frac{\alpha \,Z_\varkappa }{|{\varvec{x}}_i-{\varvec{R}}_\varkappa |} +\sum _{1\leqslant i<j\leqslant N}\frac{\alpha }{|{\varvec{x}}_i-{\varvec{x}}_j|}. \end{aligned}$$

  It is infinitesimally Laplace-bounded [24]. The corresponding total Hamiltonain acts in $\mathscr {H}=L^2((\mathbb {R}^{3})^N,(\mathbb {C}^{2})^{\otimes _N}\otimes \mathscr {F})$ and attains the form

  $$\begin{aligned} H_{\mathscr {R},\mathscr {Z}}^{\Lambda ,N}&:=\sum _{\ell =1}^N\left\{ \tfrac{1}{2}(-i\nabla _{{\varvec{x}}_\ell }-\varphi ({\varvec{G}}_{{\varvec{x}}_\ell }^\Lambda ))^2 -{\varvec{\sigma }}^{(\ell )}\cdot \varphi ({\varvec{F}}_{{\varvec{x}}_\ell }^\Lambda )\right\} +\mathrm {d}\Gamma (\omega )+ {V}^N_{\mathscr {R},\mathscr {Z}}, \end{aligned}$$

  where ${\varvec{\sigma }}^{(\ell )}:=(\sigma _{3\ell -2},\sigma _{3\ell -1},\sigma _{3\ell })$. Here we abuse notation: all terms in the previous formula have to be considered as operators in $\mathscr {H}$ in the canonical way; see, e.g., [11, 27] for careful discussions. According to the Pauli principle the physical Hamiltonian is actually given by the restriction of $H_{\mathscr {R},\mathscr {Z}}^{\Lambda ,N}$ to the reducing subspace of functions which are anti-symmetric under simultaneous permutations of the N position-spin degrees of freedom. By the permutation symmetry of the Hamiltonian the Feynman–Kac formula for the restricted, physical Hamiltonian is, however, the same as for the non-restricted one.

• (4) Fiber decompositions in the translation-invariant case. Consider again the situation in Part (1) of this example. Let $H^0$ be the corresponding total Hamiltonian for one electron interacting with the quantized photon field and with a vanishing electrostatic potential. Then it turns out that $H^0$ is unitarily equivalent to a direct integral, $\int _{\mathbb {R}^3}^\oplus \widehat{H}({\varvec{\xi }})\,\mathrm {d}{\varvec{\xi }}$, of fiber Hamiltonians attached to the total momenta ${\varvec{\xi }}\in \mathbb {R}^3$ of the system,

  $$\begin{aligned} \widehat{H}({\varvec{\xi }})&=\tfrac{1}{2}({\varvec{\xi }}-\mathrm {d}\Gamma ({\varvec{m}})-\varphi ({\varvec{G}}^\Lambda _{{\varvec{0}}}))^2 -{\varvec{\sigma }}\cdot \varphi ({\varvec{F}}^\Lambda _{{\varvec{0}}})+\mathrm {d}\Gamma (\omega ). \end{aligned}$$

  (12.1)

  In (12.1) we have ${\varvec{m}}({\varvec{k}},j)={\varvec{k}}$. The transformation is achieved by applying first $\int _{\mathbb {R}^3}^\oplus e^{i{\varvec{x}}\cdot \mathrm {d}\Gamma ({\varvec{m}})}\mathrm {d}{\varvec{x}}$ and then a ($\mathbb {C}^2\otimes \mathscr {F}$-valued) Fourier transform acting on the ${\varvec{x}}$-variables; recall that $e^{i{\varvec{x}}\cdot \mathrm {d}\Gamma ({\varvec{m}})}\varphi (e^{-i{\varvec{m}}\cdot {\varvec{x}}}f)e^{-i{\varvec{x}}\cdot \mathrm {d}\Gamma ({\varvec{m}})}=\varphi (f)$.

1.2 The Nelson model

Example 12.2

Let $L=S=1$, $\sigma _1=-1$, and ${\varvec{G}}={\varvec{0}}$. Then ${\varvec{F}}_{{\varvec{x}}}$ has only one component which we denote by $F_{{\varvec{x}}}$. With the usual abuse of notation, the total Hamiltonian then attains the general form of the Nelson Hamiltonian,

$$\begin{aligned} H^V_\mathrm {N}&:=-\tfrac{1}{2}\Delta +\varphi (F_{{\varvec{x}}})+\mathrm {d}\Gamma (\omega )+{V}. \end{aligned}$$

The easiest way to treat Nelson’s model is to adapt the proof of Theorem 4.7 by replacing $-i\varphi (q)$ by $\varphi (F)$ in the computations. To illustrate the involved formulas of Definition 5.1 we shall, however, demonstrate how they simplify in the above situation: Of course, ${\varvec{G}}={\varvec{0}}$ entails $K_t=U_t^{\pm }=U_{s,t}^-=0$. Recalling also that $w_{s,t}=\iota _t^*\iota _s$, if $s\leqslant t$, we see that the quantity defined in (5.8) satisfies

$$\begin{aligned} \mathscr {Q}_t^{(n)}(g,h;t_{[n]})&=(-1)^n\mathop {\mathop {\sum }_{{\mathcal {A}\cup \mathcal {B}\cup \mathcal {C}=[n]}}}\limits _{\#\mathcal {C}\in 2\mathbb {N}_0} \sum _{{\mathcal {C}=\cup \{c_p,c_p'\}}} \left( \prod _{p=1}^{\#\mathcal {C}/2}\frac{1}{2}\langle \iota _{t_{c_p'}}F_{{\varvec{X}}_{t_{c_p'}}}| \iota _{t_{c_p}}F_{{\varvec{X}}_{t_{c_p}}} \rangle \right) \nonumber \\&\quad \times \left( \prod _{a\in \mathcal {A}}-i\langle \iota _tg|\iota _{t_a}{F}_{{\varvec{X}}_{t_a}} \rangle \right) \prod _{b\in \mathcal {B}}i\langle \iota _{t_b}{F}_{{\varvec{X}}_{t_b}}|\iota _0h \rangle . \end{aligned}$$

(12.2)

Here we dropped the condition $c_p<c_p'$ in the partitions of $\mathcal {C}$, i.e. we sum over all possibilities to partition $\mathcal {C}$ into ordered pairs; thus the new factors $\frac{1}{2}$ appearing in (12.2). In doing so we exploited that the scalar products in the first line of (12.2) are real. Written in this way the sum on the right hand side of (12.2) becomes permutation symmetric as a function of $t_1,\ldots ,t_n$. Instead of integrating it over the simplex $t{\triangle }_n$, we may just as well integrate it over the cube $[0,t]^n$ and multiply the result by 1 / n!. Therefore,

$$\begin{aligned}&\int _{t{\triangle }_n}\mathscr {Q}_t^{(n)}(g,h;t_{[n]})\mathrm {d}t_{[n]}\\&\quad =\frac{(-1)^n}{n!}\mathop {\mathop {\sum }_{{\mathcal {A}\cup \mathcal {B}\cup \mathcal {C}=[n]}}}\limits _{\#\mathcal {C}\in 2\mathbb {N}_0} \frac{(\#\mathcal {C})!}{(\#\mathcal {C}/2)!}\left( \frac{\Vert K^{\mathrm {N}}_t\Vert ^2}{2}\right) ^{\frac{\#\mathcal {C}}{2}} \langle ig|U^{\mathrm {N},+}_t \rangle ^{\#\mathcal {A}}\langle U^{\mathrm {N},-}_t|ih \rangle ^{\#\mathcal {B}}, \end{aligned}$$

where the analogs of the basic processes for Nelson’s model are given by

$$\begin{aligned} K^{\mathrm {N}}_t&:=\int _0^t\iota _sF_{{\varvec{X}}_s}\,\mathrm {d}s,\quad U^{\mathrm {N},+}_t:=\iota _t^*K^{\mathrm {N}}_t,\quad U^{\mathrm {N},-}_t:=\iota _0^*K^{\mathrm {N}}_t, \end{aligned}$$

i.e. only by Bochner–Lebesgue integrals. A little combinatorics reveals that

$$\begin{aligned}&\sum _{n=0}^\infty \int _{t{\triangle }_n}\mathscr {Q}_t^{(n)}(g,h;t_{[n]})\,\mathrm {d}t_{[n]}\\&\quad =\sum _{n=0}^\infty \frac{1}{n!}\sum _{P=0}^{\lfloor {n}/{2}\rfloor }\frac{(-1)^{n-2P}n!}{(n-2P)!P!} \left( \frac{\Vert K^{\mathrm {N}}_t\Vert ^2}{2}\right) ^P (\langle ig|U^{\mathrm {N},+}_t \rangle +\langle U^{\mathrm {N},-}_t|ih \rangle )^{n-2P}\\&\quad =\sum _{P=0}^\infty \frac{1}{P!}\left( \frac{\Vert K^{\mathrm {N}}_t\Vert ^2}{2}\right) ^P \sum _{\ell =0}^\infty \frac{(-1)^\ell }{\ell !} (\langle ig|U^{\mathrm {N},+}_t \rangle +\langle U^{\mathrm {N},-}_t|ih \rangle )^{\ell }. \end{aligned}$$

Combining this formula with (4.2) and (5.9) we arrive at

$$\begin{aligned} \lim _{M\rightarrow \infty }\langle \zeta (g)|\mathbb {W}_{{\varvec{\xi }},t}^{V,(0,M)}\,\zeta (h) \rangle&=e^{-u^{\mathrm {N},V}_{-{\varvec{\xi }},t}+\langle g|w_{0,t}h \rangle +i\langle g|U^{\mathrm {N},+}_t \rangle -i\langle U^{\mathrm {N},-}_t|h \rangle }\\&=\langle \zeta (g)|W^{\mathrm {N},V}_{{\varvec{\xi }},t}\zeta (h) \rangle , \end{aligned}$$

where (observe the flipped sign of the first term in comparison to (3.6))

$$\begin{aligned} u_{{\varvec{\xi }},\bullet }^{\mathrm {N},V}:=-\frac{1}{2}\,\Vert K_{\bullet }^{\mathrm {N}}\Vert ^2+\int _0^\bullet V({\varvec{X}}_s)\,\mathrm {d}s-i{\varvec{\xi }}\cdot ({\varvec{X}}_\bullet -{\varvec{X}}_0), \end{aligned}$$

and where (the exponentials converge strongly on the normed space $\mathscr {C}[\mathfrak {d}_C]$)

$$\begin{aligned} W^{\mathrm {N},V}_{{\varvec{\xi }},\bullet }\psi&:=e^{-u^{\mathrm {N},V}_{-{\varvec{\xi }},\bullet }}\exp \{-a^\dagger (U^{\mathrm {N},+}_\bullet )\}\Gamma (w_{0,\bullet }) \exp \{-a(U^{\mathrm {N},-}_\bullet )\}\psi ,\quad \psi \in \mathscr {C}[\mathfrak {d}_C]. \end{aligned}$$

Applying (2.15) we see that

$$\begin{aligned} W^{\mathrm {N},V}_{{\varvec{\xi }},t}=\Gamma (\iota _t^*)e^{-\varphi (K^{\mathrm {N}}_t)} \Gamma (\iota _0)e^{i{\varvec{\xi }}\cdot ({\varvec{X}}_t-{\varvec{X}}_0)-\int _0^tV({\varvec{X}}_s)\mathrm {d}s} \quad \text {on }\mathscr {C}[\mathfrak {d}_C], \end{aligned}$$

which is the formula appearing, e.g., in [27].

Appendix 2: Self-adjointness of fiber Hamiltonians

The following short proof of Proposition 2.6 combines three observations: a first one by Könenberg (see [25]) who noticed that, by putting an artificial, large constant in front of $\mathrm {d}\Gamma (\omega )$ (instead of assuming weak coupling), one obtains a manifestly self-adjoint and surprisingly useful comparison operator. The second one is borrowed from [11] where a double commutator analog to the one in (13.2) appears. The third ingredient is the following result [41, Thm. 2.b)]:

Theorem 13.1

If A is a self-adjoint operator in some Hibert space $\mathscr {K}$, B is symmetric in $\mathscr {K}$ and A-bounded, and $A+tB$ is closed, for all $t\in [0,1]$, then $A+B$ is self-adjoint.

The following proof can mutatis mutandis also be used for the total Hamiltonian.

Proof of Proposition 2.6

Step 1 Starting from (2.29) and the representation of the scalar Hamiltonian in the second and third lines of (2.30) the bounds (2.32) and (2.33) follow, for sufficiently large $a\geqslant 1$, from (2.18), (2.19), and brief and elementary estimations using

$$\begin{aligned} \Vert (1+\mathrm {d}\Gamma (\omega ))^{{1}/{2}}({\varvec{\xi }}-\mathrm {d}\Gamma ({\varvec{m}}))\psi \Vert ^2 \leqslant \Vert ({\varvec{\xi }}-\mathrm {d}\Gamma ({\varvec{m}}))^2\psi \Vert \,\Vert (1+\mathrm {d}\Gamma (\omega ))\psi \Vert . \end{aligned}$$

By virtue of the Kato-Rellich theorem the bound (2.32) shows that $T:=\widehat{H}^0({\varvec{\xi }},{\varvec{x}})+(a-1)\,\mathrm {d}\Gamma (\omega )$ is closed (resp. self-adjoint if $q_{{\varvec{x}}}=0$) on $\widehat{\mathcal {D}}$ and that every core of ${M}_a({\varvec{\xi }})$ is a core of T; in fact, $T-{M}_a({\varvec{\xi }})=\widehat{H}^0({\varvec{\xi }},{\varvec{x}})-{M}_{1}({\varvec{\xi }})$. We further have

$$\begin{aligned} a\,\Vert \mathrm {d}\Gamma (\omega )\,\psi \Vert \leqslant \Vert {M}_a({\varvec{\xi }})\,\psi \Vert \leqslant 2\,\Vert T\psi \Vert +\mathfrak {c}\,\Vert \psi \Vert , \end{aligned}$$

(13.1)

and, hence, $\Vert \widehat{H}^0({\varvec{\xi }},{\varvec{x}})\,\psi \Vert \leqslant 3\Vert T\,\psi \Vert +\mathfrak {c}\,\Vert \psi \Vert$, for all $\psi \in \widehat{\mathcal {D}}$. Since $\mathbb {C}^L\otimes \mathscr {C}[\mathfrak {d}_C]$ is a core of T, this implies $\widehat{H}^0({\varvec{\xi }},{\varvec{x}})\subset \overline{\widehat{H}^0({\varvec{\xi }},{\varvec{x}})\upharpoonright _{\mathbb {C}^L\otimes \mathscr {C}[\mathfrak {d}_C]}}$.

Abbreviate ${\varvec{v}}:={\varvec{\xi }}-\mathrm {d}\Gamma ({\varvec{m}})-\varphi ({\varvec{G}}_{{\varvec{x}}})$ and assume that $\omega$ is bounded for the moment. Then (2.9), (2.10), and (2.20) yield

$$\begin{aligned} 2\mathrm {Re}\big \langle \mathrm {d}\Gamma (\omega )\,\psi \big |{\varvec{v}}^2\,\psi \big \rangle&=2\langle {\varvec{v}}\,\psi |\mathrm {d}\Gamma (\omega )\,{\varvec{v}}\,\psi \rangle +\big \langle \psi \big |\big [{\varvec{v}},[{\varvec{v}},\mathrm {d}\Gamma (\omega )]\big ]\,\psi \big \rangle \nonumber \\&\geqslant -2\,\Vert \omega ^{{1}/{2}}{\varvec{G}}_{{\varvec{x}}}\Vert ^2\,\Vert \psi \Vert ^2+\langle \psi |\varphi (\omega \,{\varvec{m}}\cdot {\varvec{G}}_{{\varvec{x}}})\,\psi \rangle \nonumber \\&\geqslant -(2\,\Vert \omega ^{{1}/{2}}{\varvec{G}}_{{\varvec{x}}}\Vert ^2+\Vert \omega ^{{1}/{2}}{\varvec{m}}\cdot {\varvec{G}}_{{\varvec{x}}}\Vert ^2)\,\Vert \psi \Vert ^2 -\langle \psi |\mathrm {d}\Gamma (\omega )\,\psi \rangle ,\nonumber \\ \end{aligned}$$

(13.2)

for all $\psi \in \mathbb {C}^L\otimes \mathscr {C}[\mathfrak {d}_C]$. Returning to our general assumptions on $\omega$ we apply (13.2) with $\omega \wedge n$ instead of $\omega$, for every $n\in \mathbb {N}$, and pass to the limit $n\rightarrow \infty$ on the left hand side and in the last line. (Notice that Hypothesis 2.3 does not imply that $\omega \,{\varvec{m}}\cdot {\varvec{G}}_{{\varvec{x}}}\in \mathfrak {h}$.) In combination with (2.18) the so-obtained estimate entails

$$\begin{aligned} \Vert T\psi \Vert ^2&\leqslant 2\left( \left\| \tfrac{1}{2}{\varvec{v}}^2\,\psi \right\| ^2 +2a\,\mathrm {Re}\left. \left\langle {\tfrac{1}{2}{\varvec{v}}^2\,\psi }\right| {\mathrm {d}\Gamma (\omega )\,\psi }\right\rangle +a^2\,\Vert \mathrm {d}\Gamma (\omega )\,\psi \Vert ^2\right) \\&\quad +\mathfrak {c}\,\Vert (\mathrm {d}\Gamma (\omega )+1)^{{1}/{2}}\,\psi \Vert ^2\\&\leqslant 2a^2\,\left\| \left( \tfrac{1}{2}{\varvec{v}}^2+\mathrm {d}\Gamma (\omega )\right) \,\psi \right\| ^2 +\mathfrak {c}'\langle \psi |(\mathrm {d}\Gamma (\omega )+1)\,\psi \rangle \\&\leqslant 4a^2\,\Vert \widehat{H}^0({\varvec{\xi }},{\varvec{x}})\,\psi \Vert ^2+\mathfrak {c}''\langle \psi |(\mathrm {d}\Gamma (\omega )+1)\,\psi \rangle , \end{aligned}$$

for all $\psi \in \mathbb {C}^L\otimes \mathscr {C}[\mathfrak {d}_C]$. Since, by (13.1), $\langle \psi |\mathrm {d}\Gamma (\omega )\,\psi \rangle \leqslant \varepsilon \,\Vert T\,\psi \Vert ^2+\mathfrak {c}(\varepsilon )\,\Vert \psi \Vert ^2$, we obtain $\Vert T\psi \Vert \leqslant \mathfrak {c}(a\Vert \widehat{H}^0({\varvec{\xi }},{\varvec{x}})\psi \Vert +\Vert \psi \Vert )$, for all $\psi \in \mathbb {C}^L\otimes \mathscr {C}[\mathfrak {d}_C]$. Together with the above remarks this implies that $\widehat{H}^0({\varvec{\xi }},{\varvec{x}})=\overline{\widehat{H}^0({\varvec{\xi }},{\varvec{x}})\upharpoonright _{\mathbb {C}^L\otimes \mathscr {C}[\mathfrak {d}_C]}}$ and that the graph norms of T and $\widehat{H}^0({\varvec{\xi }},{\varvec{x}})$ are equivalent.

Step 2 Assume that $q_{{\varvec{x}}}=0$ in the rest of this proof. To conclude that $\widehat{H}^0({\varvec{\xi }},{\varvec{x}})$ is selfadjoint in this case we apply Theorem 13.1 with $A=T$ and $B=(1-a)\mathrm {d}\Gamma (\omega )$. In fact, we then have

$$\begin{aligned} A+tB=\tfrac{1}{2}({\varvec{\xi }}-\mathrm {d}\Gamma ({\varvec{m}})-\varphi ({\varvec{G}}_{{\varvec{x}}}))^2 -{\varvec{\sigma }}\cdot \varphi ({\varvec{F}}_{{\varvec{x}}})+\mathrm {d}\Gamma (\omega _t) \end{aligned}$$

on $\widehat{\mathcal {D}}$, where $\omega _t:=(1-t)a\omega +t\omega$, $t\in [0,1]$. In particular, $A+tB$ is closed by Step 1, since the tuple $(\omega _t,{\varvec{m}},{\varvec{G}},{\varvec{F}})$ satisfies Hypothesis 2.3, for every $t\in [0,1]$. $\square$

Appendix 3: Commutator estimates

In the next lemma we prove a number of commutator estimates which have been used in Sect. 7.

Lemma 14.1

Define $\theta _\varepsilon$ by (7.17), $\varUpsilon _\varepsilon$ by (7.27), and let $\theta :=\theta _0=1+\mathrm {d}\Gamma (\omega )$. Then the following bounds hold true, for all $E\geqslant 1$, $\varepsilon \in (0,1/E]$, $\alpha \in [{1}/{2},1]$, and $f\in \mathfrak {k}$,

$$\begin{aligned} \Vert \theta _\varepsilon ^{-{1}/{2}}\,\mathrm {ad}_{\varphi (f)}\theta _\varepsilon \Vert =\Vert (\mathrm {ad}_{\varphi (f)}\theta _\varepsilon )\,\theta _\varepsilon ^{-{1}/{2}}\Vert&\leqslant \mathfrak {c}\,\Vert \omega ^{{1}/{2}}(1+\omega )^{{1}/{2}}f\Vert _{\mathfrak {h}}, \end{aligned}$$

(14.1)

$$\begin{aligned} \Vert \mathrm {ad}_{\varphi (f)}^2\theta _\varepsilon \Vert&\leqslant \mathfrak {c}\,\Vert \omega ^{{1}/{2}}f\Vert _{\mathfrak {h}}^2, \end{aligned}$$

(14.2)

$$\begin{aligned} \Vert \theta _\varepsilon ^{-1}\,(\mathrm {ad}_{\varphi (f)}\,\theta _\varepsilon ^2)\,\theta _\varepsilon ^{-1}\Vert&\leqslant \mathfrak {c}\,\Vert \omega ^{{1}/{2}}(1+\omega )^{{1}/{2}}f\Vert _{\mathfrak {h}}^2, \end{aligned}$$

(14.3)

$$\begin{aligned} \Vert (\mathrm {ad}_{\varphi (f)}\varUpsilon _\varepsilon )\varUpsilon _\varepsilon ^{-\alpha }\theta ^{-{1}/{2}}\Vert&\leqslant \mathfrak {c}\,E^{{{1}/{2}}-\alpha }\,\Vert f\Vert _{\mathfrak {k}}, \end{aligned}$$

(14.4)

$$\begin{aligned} \Vert \theta ^{-{1}/{2}}(\mathrm {ad}_{\varphi (f)}\varUpsilon _\varepsilon )\varUpsilon _\varepsilon ^{-\alpha }\Vert&\leqslant \mathfrak {c}\,E^{{{1}/{2}}-\alpha }\,\Vert f\Vert _{\mathfrak {k}}, \end{aligned}$$

(14.5)

$$\begin{aligned} \Vert \theta ^{-{1}/{4}}(\mathrm {ad}_{\varphi (f)}\varUpsilon _\varepsilon )\varUpsilon _\varepsilon ^{-\alpha }\theta ^{-{1}/{4}}\Vert&\leqslant \mathfrak {c}\,E^{{{1}/{2}}-\alpha }\,\Vert f\Vert _{\mathfrak {k}}, \end{aligned}$$

(14.6)

$$\begin{aligned} \Vert \theta ^{-{1}/{2}}\varUpsilon _\varepsilon ^{-\alpha }\mathrm {ad}_{\varphi (f)}\varUpsilon _\varepsilon \Vert&\leqslant \mathfrak {c}\,E^{{{1}/{2}}-\alpha }\,\Vert f\Vert _{\mathfrak {k}}, \end{aligned}$$

(14.7)

$$\begin{aligned} \Vert \theta ^{-{1}/{2}}\mathrm {Re}[\varUpsilon _\varepsilon ^{-1}(\mathrm {ad}_{\varphi (f)}^2\varUpsilon _\varepsilon )]\theta ^{-{1}/{2}}\Vert&\leqslant \mathfrak {c}\,E^{{-{1}/{2}}}\,\Vert f\Vert _{\mathfrak {k}}^2, \end{aligned}$$

(14.8)

$$\begin{aligned} \Vert \theta ^{-{1}/{2}}\varUpsilon _\varepsilon ^{-1}(\mathrm {ad}_{\varphi (f)}^2\varUpsilon _\varepsilon ^2)\varUpsilon _\varepsilon ^{-1}\theta ^{-{1}/{2}}\Vert&\leqslant \mathfrak {c}\,E^{{-{1}/{2}}}\,\Vert f\Vert _{\mathfrak {k}}^2. \end{aligned}$$

(14.9)

Proof

We remark that all algebraic identities between various combinations of operators below are valid on the dense domain $\mathscr {C}[\mathfrak {d}_C]$. All norms have to be read as norms of operators which are densely defined and bounded on $\mathscr {C}[\mathfrak {d}_C]$.

First, we observe that, if $\varTheta$ denotes one of the weights $\theta _\varepsilon$ or $\varUpsilon _\varepsilon$, then

$$\begin{aligned} \varTheta ^{-1}\,(\mathrm {ad}_{\varphi (f)}\,\varTheta ^2)\,\varTheta ^{-1}&=2\{\varTheta ^{-1}\,(\mathrm {ad}_{\varphi (f)}\varTheta )\}\{(\mathrm {ad}_{\varphi (f)}\varTheta )\,\varTheta ^{-1}\}\\&\quad +\varTheta ^{-1}\,\mathrm {ad}_{\varphi (f)}^2\varTheta +(\mathrm {ad}_{\varphi (f)}^2\varTheta )\,\varTheta ^{-1}, \end{aligned}$$

so that (14.3) follows from (14.1) and (14.2) and (14.9) from (14.4), (14.7), and (14.8). Writing

$$\begin{aligned} \varTheta _\varepsilon =(1+\mathrm {d}\Gamma (\omega ))(1+\varepsilon \mathrm {d}\Gamma (\omega ))^{-1}= \varepsilon ^{-1}\big (\mathbbm {1}-(1-\varepsilon E)(1+\varepsilon \mathrm {d}\Gamma (\omega ))^{-1}\big ) \end{aligned}$$

and applying (2.8), (2.11), and $\mathrm {ad}_S(TT')=T\mathrm {ad}_ST'+(\mathrm {ad}_ST)T'$ as well as $\mathrm {ad}_ST^{-1}=-T^{-1}(\mathrm {ad}_ST)\,T^{-1}$, repeatedly we obtain

$$\begin{aligned} \mathrm {ad}_{\varphi (f)}\varTheta _\varepsilon&=(\varepsilon E-1)(1+\varepsilon \mathrm {d}\Gamma (\omega ))^{-1}\,i\varphi (i\omega f) (1+\varepsilon \mathrm {d}\Gamma (\omega ))^{-1},\\ \mathrm {ad}_{\varphi (f)}^2\varTheta _\varepsilon&=2\varepsilon \,(1-\varepsilon E)(1+\varepsilon \mathrm {d}\Gamma (\omega ))^{-1}\,(\varphi (i\omega f) (1+\varepsilon \mathrm {d}\Gamma (\omega ))^{-1})^2\nonumber \end{aligned}$$

(14.10)

$$\begin{aligned}&\quad +(1-\varepsilon E)\,\Vert \omega ^{{1}/{2}}f\Vert ^2\,(1+\varepsilon \mathrm {d}\Gamma (\omega ))^{-2}. \end{aligned}$$

(14.11)

As a consequence of (2.16) we have $\varepsilon ^{{1}/{2}}\Vert a(\omega f)(1+\varepsilon \mathrm {d}\Gamma (\omega ))^{-{1}/{2}}\Vert \leqslant \Vert \omega ^{{1}/{2}}f\Vert$, which together with (2.8) and (14.11) implies (14.2). From (2.18) and (14.10) we readily infer that (14.1) is satisfied.

Likewise, by writing

$$\begin{aligned} \varUpsilon _\varepsilon =(E+\mathrm {d}\Gamma (m_j)^2)(1+\varepsilon \mathrm {d}\Gamma (m_j)^2)^{-1} =\varepsilon ^{-1}(\mathbbm {1}-(1-\varepsilon E)\,R_\varepsilon ) \end{aligned}$$

with $R_\varepsilon :=(1+\varepsilon \mathrm {d}\Gamma (m_j)^2)^{-1}$, we deduce that

$$\begin{aligned} \mathrm {ad}_{\varphi (f)}\varUpsilon _\varepsilon&=(1-\varepsilon E)\,R_\varepsilon \,\{\mathrm {ad}_{\varphi (f)}(\mathrm {d}\Gamma (m_j)^2)\}\,R_\varepsilon , \end{aligned}$$

where, on account of (2.11),

$$\begin{aligned} \mathrm {ad}_{\varphi (f)}(\mathrm {d}\Gamma (m_j)^2)&=2i\mathrm {d}\Gamma (m_j)\,\varphi (im_jf)+\varphi (m_j^2f)\\&=2i\varphi (im_jf)\,\mathrm {d}\Gamma (m_j)-\varphi (m_j^2f). \end{aligned}$$

Consequently, for $\alpha \in [{1}/{2},1]$, $\gamma \in [0,1]$, and $\beta :=1-\gamma$,

$$\begin{aligned} \big \Vert&\theta ^{-{\beta }/{2}}(\mathrm {ad}_{\varphi (f)}\varUpsilon _\varepsilon )\varUpsilon _\varepsilon ^{-\alpha } \theta ^{-{\gamma }/{2}}\big \Vert \\&\leqslant |1-\varepsilon E|\,\Vert \theta ^{-{\beta }/{2}}\varphi (m_j^2f)\,\theta ^{-{\gamma }/{2}}\Vert \, \Vert (E+\mathrm {d}\Gamma (m_j)^2)^{-\alpha }\Vert \\&\quad +2|1-\varepsilon E|\,\big \Vert \mathrm {d}\Gamma (m_j)(E+\mathrm {d}\Gamma (m_j)^2)^{-\alpha }\big \Vert \, \Vert \theta ^{-{\beta }/{2}}\varphi (im_jf)\,\theta ^{-{\gamma }/{2}}\Vert , \end{aligned}$$

which proves (14.4), (14.5), and (14.6). Here we use that (2.18) implies the bounds

$$\begin{aligned} \Vert \theta ^{-{1}/{4}}\varphi (g)\,\theta ^{-{1}/{4}}\Vert ,\Vert \theta ^{-{1}/{4}}\varphi (ig)\,\theta ^{-{1}/{4}}\Vert \leqslant 2^{{1}/{2}}\Vert (1+\omega ^{-1})^{{1}/{2}}g\Vert . \end{aligned}$$

(14.12)

Using the above identities for a single commutator we further find

$$\begin{aligned} \mathrm {ad}_{\varphi (f)}^2\varUpsilon _\varepsilon&=2(1-\varepsilon E)R_\varepsilon \,2i\varphi (im_jf)\,\frac{\varepsilon \mathrm {d}\Gamma (m_j)^2}{1+\varepsilon \mathrm {d}\Gamma (m_j)^2} \,2i\varphi (im_jf)\,R_\varepsilon \\&\quad +8(1-\varepsilon E)\,\mathrm {Re}[ R_\varepsilon \,2i\varphi (im_jf)\,\frac{\varepsilon \mathrm {d}\Gamma (m_j)}{1+\varepsilon \mathrm {d}\Gamma (m_j)^2}\,\varphi (m_j^2f)\, R_\varepsilon ]\\&\quad -2\varepsilon (1-\varepsilon E)R_\varepsilon \,\varphi (m_j^2f)R_\varepsilon \,\varphi (m_j^2f)\, R_\varepsilon \\&\quad -2(1-\varepsilon E)\,(R_\varepsilon \,\varphi (im_jf)^2\,R_\varepsilon +\Vert \,|m_j|^{{1}/{2}}f\Vert ^2\,R_\varepsilon \,\mathrm {d}\Gamma (m_j)\,R_\varepsilon ). \end{aligned}$$

Now, we multiply the previous identity both from the left and from the right with $\theta ^{-{1}/{2}}=(1+\mathrm {d}\Gamma (\omega ))^{-{1}/{2}}$. By (2.18) the latter operators control all unbounded fields. Multiplying the previous identity in addition with $\varUpsilon _{\varepsilon }^{{-{1}/{2}}}$ from the left or from the right we can also control the operator $\mathrm {d}\Gamma (m_j)$ in the last line, where no power of $\varepsilon$ can be employed to control it by means of the resolvents $R_\varepsilon$. From these remarks we readily infer (14.8). $\square$

Appendix 4: Admissibility of Brownian bridges

In this “Appendix” we verify that the semi-martingale realizations of Brownian bridges satisfy the technical condition (2.37) of Hypothesis 2.7. After that we also present a detailed proof of Lemma 10.5 on time-reversals of Brownian bridges.

In all what follows, ${\varvec{y}}\in \mathbb {R}^\nu$ and ${\varvec{q}}:\varOmega \rightarrow \mathbb {R}^\nu$ is $\mathfrak {F}_0$-measurable such that $\mathbb {E}[|{\varvec{q}}|^n]<\infty$, for all $n\in \mathbb {N}$. Recall that the (up to indistinguishability unique) solution of ${\varvec{b}}_\bullet ={\varvec{q}}+{\varvec{B}}_\bullet +\int _0^\bullet \frac{{\varvec{y}}-{\varvec{b}}_s}{\mathcal {T}-s}\,\mathrm {d}s$ is explicitly given by

$$\begin{aligned}&{\varvec{b}}_t^{\mathcal {T};{\varvec{q}},{\varvec{y}}}:={\left\{ \begin{array}{ll} \; \frac{t}{\mathcal {T}}{\varvec{y}}+\frac{\mathcal {T}-t}{\mathcal {T}}{\varvec{q}}+{\varvec{B}}_t -(\mathcal {T}-t)\int ^t_0\frac{{\varvec{B}}_s}{(\mathcal {T}-s)^2}\mathrm {d}s, &{}\quad \text {if }0\leqslant t<\mathcal {T},\\ \;{\varvec{y}},&{}\quad \text {if }t=\mathcal {T}.\end{array}\right. } \end{aligned}$$

(15.1)

Lemma 15.1

The drift vector field in the SDE ${\varvec{b}}_\bullet ^{\mathcal {T};{\varvec{q}},{\varvec{y}}}={\varvec{B}}_\bullet ^{{\varvec{q}}}+\int _0^\bullet {\varvec{Y}}_s\mathrm {d}s$ satisfied by the process in (15.1) can $\mathbb {P}$-a.s. be written as

$$\begin{aligned} {\varvec{Y}}_t&:=\frac{{\varvec{y}}-{\varvec{b}}_t^{\mathcal {T};{\varvec{q}},{\varvec{y}}}}{\mathcal {T}-t}= \frac{{\varvec{y}}-{\varvec{q}}}{\mathcal {T}}\,-\int _0^t\frac{1}{\mathcal {T}-s}\,\mathrm {d}{\varvec{B}}_s,\quad 0\leqslant t<\mathcal {T}. \end{aligned}$$

(15.2)

Proof

Plugging the formula (15.1) for ${\varvec{b}}_t^{\mathcal {T};{\varvec{q}},{\varvec{y}}}$ into the expression in the middle in (15.2) and taking the following consequence of Itō’s formula into account,

$$\begin{aligned} -\frac{{\varvec{B}}_t}{\mathcal {T}-t}+\int _0^t\frac{{\varvec{B}}_s}{(\mathcal {T}-s)^2}\,\mathrm {d}s&=-\int _0^t\frac{1}{\mathcal {T}-s}\,\mathrm {d}{\varvec{B}}_s,\quad 0\leqslant t<\mathcal {T},\;\;\mathbb {P}\text {-a.s.,} \end{aligned}$$

we arrive at the formula on the right hand side of (15.2). $\square$

Lemma 15.2

For all $\mathcal {T}>0$, $p\in \mathbb {N}$, and $t\in [0,\mathcal {T})$,

$$\begin{aligned} \mathbb {E}[|{\varvec{Y}}_t|^{2p}]&= \sum _{\ell =0}^p\frac{(2p-2+\nu )!!}{(2(p-\ell )-2+\nu )!!} {p\atopwithdelims ()\ell }\frac{\mathbb {E}[|{\varvec{q}}-{\varvec{y}}|^{2(p-\ell )}]}{\mathcal {T}^{2(p-\ell )}} \,\frac{t^\ell }{\mathcal {T}^\ell (\mathcal {T}-t)^\ell }, \end{aligned}$$

(15.3)

where $(2j)!!:=2^j j!$, $(2j+1)!!=(2j+1)!/(2j)!!$, and in particular

$$\begin{aligned} \int _0^\mathcal {T}(\mathcal {T}-s)^{\varkappa }\mathbb {E}[|{\varvec{Y}}_s|^{2\varkappa }]\,\mathrm {d}s \leqslant \mathfrak {c}(\nu ,\varkappa ,\mathcal {T})\,\mathbb {E}[(1+|{\varvec{q}}-{\varvec{y}}|)^{2\varkappa }], \quad \varkappa >0. \end{aligned}$$

(15.4)

Proof

By (15.2) and Itō’s formula (ignore the last integral, if $p=1$)

$$\begin{aligned} |{\varvec{Y}}_t|^{2p}&=\frac{|{\varvec{q}}-{\varvec{y}}|^{2p}}{\mathcal {T}^{2p}}-2p\int _0^t|{\varvec{Y}}_s|^{2(p-1)}\frac{{\varvec{Y}}_s}{\mathcal {T}-s}\,\mathrm {d}{\varvec{B}}_s\\&\;\;\, +p\,\nu \int _0^t|{\varvec{Y}}_s|^{2(p-1)}\,\frac{\mathrm {d}s}{(\mathcal {T}-s)^2} +\frac{p(p-1)}{2}\int _0^t|{\varvec{Y}}_s|^{2(p-2)}\,\frac{4|{\varvec{Y}}_s|^2}{(\mathcal {T}-s)^2}\,\mathrm {d}s, \end{aligned}$$

for all $t\in [0,\mathcal {T})$, $\mathbb {P}$-a.s., and therefore,

$$\begin{aligned} \mathbb {E}[|{\varvec{Y}}_t|^{2p}]&=\frac{\mathbb {E}[|{\varvec{q}}-{\varvec{y}}|^{2p}]}{\mathcal {T}^{2p}} +(2p-2+\nu )p\int _0^t\mathbb {E}[|{\varvec{Y}}_s|^{2(p-1)}]\,\frac{\mathrm {d}s}{(\mathcal {T}-s)^2}. \end{aligned}$$

Iterating this we find

$$\begin{aligned} \mathbb {E}[|{\varvec{Y}}_t|^{2p}]&=\sum _{\ell =0}^p\frac{(2p-2+\nu )!!\,p!}{(2(p-\ell )-2+\nu )!!} \,\frac{\mathbb {E}[|{\varvec{q}}-{\varvec{y}}|^{2(p-\ell )}]}{(p-\ell )!\,\mathcal {T}^{2(p-\ell )}} \int _{t{\triangle }_\ell }\prod _{j=1}^\ell \frac{\mathrm {d}t_j}{(\mathcal {T}-t_j)^2}, \end{aligned}$$

which is (15.3) since the integral over $t{\triangle }_n$ equals $(\int _0^t\mathrm {d}s/(\mathcal {T}-s)^2)^\ell /\ell !$. $\square$

As announced above, we shall now work out the details on the time-reversal of a Brownian bridge:

Proof of Lemma 10.5

In this proof the letter $\mathcal {T}$ plays the role of the letter t in the statement of Lemma 10.5, i.e., we consider the bridge ${\varvec{b}}^{\mathcal {T};{\varvec{x}},{\varvec{y}}}$ reversed at $\mathcal {T}$.

Let $\mathfrak {N}\subset \mathfrak {F}$ be the set of $\mathbb {P}$-zero sets. Recall that, for every $t\in [0,\mathcal {T}]$, we defined $\mathfrak {H}_t$ to be the smallest $\sigma$-algebra containing $\mathfrak {N}$ and the $\sigma$-algebra $\sigma ({\varvec{b}}^{\mathcal {T};{\varvec{x}},{\varvec{y}}}_{\mathcal {T}-t};\,{\varvec{B}}_{s}-{\varvec{B}}_{\mathcal {T}}:\,\mathcal {T}-t\leqslant s\leqslant \mathcal {T}) =\sigma ({\varvec{b}}^{\mathcal {T};{\varvec{x}},{\varvec{y}}}_{\mathcal {T}-t};\,{\varvec{B}}_{s}-{\varvec{B}}_{r}:\,\mathcal {T}-t\leqslant r\leqslant s\leqslant \mathcal {T})$.

Step 1 We claim that $(\mathfrak {H}_t)_{t\in [0,\mathcal {T}]}$ is a filtration and that ${\varvec{b}}^{\mathcal {T};{\varvec{x}},{\varvec{y}}}_{\mathcal {T}-s}$ is $\mathfrak {H}_t$-measurable, for all $0\leqslant s\leqslant t\leqslant \mathcal {T}$.

Of course, the second assertion implies the first. Since ${\varvec{b}}^{\mathcal {T};{\varvec{x}},{\varvec{y}}}_{\mathcal {T}}={\varvec{y}}$, $\mathbb {P}$-a.s., and $\sigma (\mathfrak {N})=\mathfrak {H}_0\subset \mathfrak {H}_t$, $t\in (0,\mathcal {T}]$, we see that ${\varvec{b}}^{\mathcal {T};{\varvec{x}},{\varvec{y}}}_{\mathcal {T}}$ is $\mathfrak {H}_t$-measurable, for all $t\in [0,\mathcal {T}]$. Let $0<s<t\leqslant \mathcal {T}$. Then, up to indistinguishability, $({\varvec{b}}_{\mathcal {T}-t+s}^{\mathcal {T};{\varvec{x}},{\varvec{y}}})_{s\in [0,t]}$ is the unique semi-martingale with respect to $\mathbb {B}_{\mathcal {T}-t}$ on [0, t] which $\mathbb {P}$-a.s. solves

$$\begin{aligned} {\varvec{X}}_\bullet&={\varvec{b}}_{\mathcal {T}-t}^{\mathcal {T};{\varvec{x}},{\varvec{y}}}+{\varvec{B}}_{\mathcal {T}-t+\bullet }-{\varvec{B}}_{\mathcal {T}-t}+ \int _0^\bullet \frac{{\varvec{y}}-{\varvec{X}}_r}{t-r}\mathrm {d}r\;\;\text {on } [0,t),\quad {\varvec{X}}_t={\varvec{y}}. \end{aligned}$$

The standard solution theory for SDE thus implies that, for every $\varepsilon \in (0,s)$, the random variable ${\varvec{b}}_{\mathcal {T}-s}^{\mathcal {T};{\varvec{x}},{\varvec{y}}}$ is measurable with respect to the smallest $\sigma$-algebra containing $\mathfrak {N}$ and $\sigma ({\varvec{b}}^{\mathcal {T};{\varvec{x}},{\varvec{y}}}_{\mathcal {T}-t};\,{\varvec{B}}_{r}-{\varvec{B}}_{\mathcal {T}-t}:\mathcal {T}-t\leqslant r\leqslant \mathcal {T}-s+\varepsilon )$. In particular, ${\varvec{b}}_{\mathcal {T}-s}^{\mathcal {T};{\varvec{x}},{\varvec{y}}}$ is $\mathfrak {H}_t$-measurable.

Step 2 Next, we claim that (10.13) defines a continuous martingale with respect to $(\mathfrak {H}_t)_{t\in [0,\mathcal {T})}$ starting at zero.

Obviously, all paths of $\hat{{\varvec{B}}}$ are continuous on $[0,\mathcal {T})$ and, by Step 1 and (10.13), $\hat{{\varvec{B}}}$ is adapted to $(\mathfrak {H}_t)_{t\in [0,\mathcal {T})}$. Using $\mathbb {E}[{\varvec{b}}_t^{\mathcal {T};{\varvec{x}},{\varvec{y}}}]=\frac{t}{\mathcal {T}}{\varvec{y}}+\frac{\mathcal {T}-t}{\mathcal {T}}{\varvec{x}}$, $t\in [0,\mathcal {T}]$, which is obvious from (15.1), we read off from (10.13) that $\hat{{\varvec{B}}}_t$ is integrable and $\mathbb {E}[\hat{{\varvec{B}}}_t]={\varvec{0}}$, for all $t\in [0,\mathcal {T}]$. Of course, $\hat{{\varvec{B}}}_0={\varvec{0}}$, $\mathbb {P}$-a.s., and

$$\begin{aligned} \mathbb {E}^{\mathfrak {H}_0}[\hat{{\varvec{B}}}_t]=\mathbb {E}^{\sigma (\mathfrak {N})}[\hat{{\varvec{B}}}_t] =\mathbb {E}[\hat{{\varvec{B}}}_t]={\varvec{0}},\quad t\in (0,\mathcal {T}). \end{aligned}$$

Let $0<s<t<\mathcal {T}$. Taking the SDE solved by ${\varvec{b}}^{\mathcal {T};{\varvec{x}},{\varvec{y}}}$ into account we see that

$$\begin{aligned} \hat{{\varvec{B}}}_t-\hat{{\varvec{B}}}_s&={\varvec{B}}_{\mathcal {T}-t}-{\varvec{B}}_{\mathcal {T}-s}-\int _{\mathcal {T}-t}^{\mathcal {T}-s} \left( \frac{{\varvec{x}}-{\varvec{b}}^{\mathcal {T};{\varvec{x}},{\varvec{y}}}_u}{u}+\frac{{\varvec{y}}-{\varvec{b}}^{\mathcal {T};{\varvec{x}},{\varvec{y}}}_u}{\mathcal {T}-u}\right) \mathrm {d}u \nonumber \\&={\varvec{B}}_{\mathcal {T}-t}-{\varvec{B}}_{\mathcal {T}-s}-\int _{\mathcal {T}-t}^{\mathcal {T}-s}\nabla \ln d_u^{\mathcal {T};{\varvec{x}},{\varvec{y}}}({\varvec{b}}^{\mathcal {T};{\varvec{x}},{\varvec{y}}}_u) \mathrm {d}u,\quad \mathbb {P}\text {-a.s.,} \end{aligned}$$

(15.5)

where

$$\begin{aligned} d_u^{\mathcal {T};{\varvec{x}},{\varvec{y}}}:=p_u({\varvec{x}},\cdot )p_{\mathcal {T}-u}({\varvec{y}},\cdot )/p_{\mathcal {T}}({\varvec{x}},{\varvec{y}}),\quad u\in (0,\mathcal {T}). \end{aligned}$$

(15.6)

We may now employ the arguments of [33, §4] to show that $\mathbb {E}^{\sigma ({\varvec{b}}_{\mathcal {T}-s}^{\mathcal {T};{\varvec{x}},{\varvec{y}}})}[\hat{{\varvec{B}}}_t-\hat{{\varvec{B}}}_s]={\varvec{0}}$, $\mathbb {P}$-a.s.; see Lemma 15.3 below. For all $\mathcal {T}-s\leqslant r\leqslant u\leqslant \mathcal {T}$, we further know that $\sigma ({\varvec{B}}_u-{\varvec{B}}_r)$ and $\sigma ({\varvec{b}}_{\mathcal {T}-s}^{\mathcal {T};{\varvec{x}},{\varvec{y}}};\hat{{\varvec{B}}}_t-\hat{{\varvec{B}}}_s)$ are independent since ${\varvec{b}}_{\mathcal {T}-s}^{\mathcal {T};{\varvec{x}},{\varvec{y}}}$ and $\hat{{\varvec{B}}}_t-\hat{{\varvec{B}}}_s$ are $\mathfrak {F}_{\mathcal {T}-s}$-measurable while ${\varvec{B}}_u-{\varvec{B}}_r$ is $\mathfrak {F}_{\mathcal {T}-s}$-independent. For trivial reasons, $\sigma (\mathfrak {N})$ and $\sigma ({\varvec{b}}_{\mathcal {T}-s}^{\mathcal {T};{\varvec{x}},{\varvec{y}}};\hat{{\varvec{B}}}_t-\hat{{\varvec{B}}}_s)$ are independent as well. These remarks entail $\mathbb {E}^{\mathfrak {H}_s}[\hat{{\varvec{B}}}_t-\hat{{\varvec{B}}}_s]=\mathbb {E}^{\sigma ({\varvec{b}}_{\mathcal {T}-s}^{\mathcal {T};{\varvec{x}},{\varvec{y}}})} [\hat{{\varvec{B}}}_t-\hat{{\varvec{B}}}_s]={\varvec{0}}$, $\mathbb {P}$-a.s. Since $\hat{{\varvec{B}}}_s$ is $\mathfrak {H}_s$-measurable, we arrive at

$$\begin{aligned} \mathbb {E}^{\mathfrak {H}_{s}}[\hat{{\varvec{B}}}_t]=\hat{{\varvec{B}}}_s,\quad \mathbb {P}\text {-a.s.} \end{aligned}$$

Hence, $\hat{{\varvec{B}}}$ is a continuous martingale with respect to $(\mathfrak {H}_t)_{t\in [0,\mathcal {T})}$ starting $\mathbb {P}$-a.s. at zero.

Step 3 Invoking a martingale convergence theorem, we see that $(\hat{{\varvec{B}}}_t)_{t\in [0,\mathcal {T})}$ has a unique extension to a continuous $(\mathfrak {H}_t)_{t\in [0,\mathcal {T}]}$-martingale (starting at zero), again denoted by $\hat{{\varvec{B}}}$. Furhermore, a glance at (10.13) reveals that the quadratic variation process of $\hat{{\varvec{B}}}$ is $(t\mathbbm {1})_{t\in [0,\mathcal {T}]}$. In view of Lévy’s characterization we now see that $\hat{{\varvec{B}}}$ is a Brownian motion with respect to $(\mathfrak {H}_t)_{t\in [0,\mathcal {T}]}$ and, hence, also with respect to its standard extension $(\bar{\mathfrak {F}}_t)_{t\in [0,\mathcal {T}]}$; see [10, p. 219].

Step 4 Substituting $u(s):=\mathcal {T}-s$ pathwise in (10.13) we obtain (10.14). By Step 1, $({\varvec{b}}^{\mathcal {T};{\varvec{x}},{\varvec{y}}}_{\mathcal {T}-t})_{t\in [0,\mathcal {T}]}$ is adapted to $(\bar{\mathfrak {F}}_t)_{t\in [0,\mathcal {T}]}$ and we conclude. $\square$

Lemma 15.3

Let $0<s<t<\mathcal {T}$ and $f:\mathbb {R}^\nu \rightarrow \mathbb {R}$ be bounded and Borel measurable. With $d^{\mathcal {T};{\varvec{x}},{\varvec{y}}}$ defined by (15.6), we then have

$$\begin{aligned} \mathbb {E}[f({\varvec{b}}_t^{\mathcal {T};{\varvec{x}},{\varvec{y}}})]&=\int _{\mathbb {R}^\nu }d_t^{\mathcal {T};{\varvec{x}},{\varvec{y}}}({\varvec{z}})f({\varvec{z}})\mathrm {d}{\varvec{z}},\\ \mathbb {E}[f({\varvec{b}}_{\mathcal {T}-s}^{\mathcal {T};{\varvec{x}},{\varvec{y}}})({\varvec{B}}_{\mathcal {T}-s}-{\varvec{B}}_{\mathcal {T}-t})]&=\mathbb {E}\left[ f({\varvec{b}}_{\mathcal {T}-s}^{\mathcal {T};{\varvec{x}},{\varvec{y}}})\int _{\mathcal {T}-t}^{\mathcal {T}-s} \nabla \ln d_u^{\mathcal {T};{\varvec{x}},{\varvec{y}}}({\varvec{b}}^{\mathcal {T};{\varvec{x}},{\varvec{y}}}_u)\mathrm {d}u\right] . \end{aligned}$$

Proof

We shall show the second asserted identity with $0<\mathcal {T}-t<\mathcal {T}-s<\mathcal {T}$ replaced by $0<s<t<\mathcal {T}$. By an approximation argument, we may actually assume f to be continuous with compact support, which we do from now on.

For fixed $\mathcal {T}>0$ and ${\varvec{y}}\in \mathbb {R}^\nu$, we set

$$\begin{aligned} \varrho _{s,t}({\varvec{z}},{\varvec{a}})&:=p_{t-s}({\varvec{z}},{\varvec{a}})p_{\mathcal {T}-t}({\varvec{a}},{\varvec{y}})\big /p_{\mathcal {T}-s}({\varvec{z}},{\varvec{y}}), \quad {\varvec{a}},{\varvec{z}}\in \mathbb {R}^\nu ,\,0\leqslant s<t<\mathcal {T}. \end{aligned}$$

Then

$$\begin{aligned} \left( \partial _s+\tfrac{1}{2}\Delta _{{\varvec{z}}}+\frac{{\varvec{y}}-{\varvec{z}}}{\mathcal {T}-s}\cdot \nabla _{{\varvec{z}}}\right) \varrho _{s,t}({\varvec{z}},{\varvec{a}})=0, \end{aligned}$$

for all ${\varvec{a}},{\varvec{z}}\in \mathbb {R}^\nu$ and $0\leqslant s<t<\mathcal {T}$. We set

$$\begin{aligned} (\pi _{s,t}f)({\varvec{z}}):=\int _{\mathbb {R}^\nu }\varrho _{s,t}({\varvec{z}},{\varvec{a}})f({\varvec{a}})\mathrm {d}{\varvec{a}},\quad {\varvec{z}}\in \mathbb {R}^\nu ,\,0\leqslant s<t<\mathcal {T}. \end{aligned}$$

Since f is bounded, it is then also clear that $(s,{\varvec{z}})\mapsto (\pi _{s,t}f)({\varvec{z}})$ belongs to $C^2([0,t)\times \mathbb {R}^\nu )$ with

$$\begin{aligned} \left( \partial _s+\tfrac{1}{2}\Delta _{{\varvec{z}}}+\frac{{\varvec{y}}-{\varvec{z}}}{\mathcal {T}-s}\cdot \nabla _{{\varvec{z}}}\right) (\pi _{s,t}f)({\varvec{z}})&=0,\quad {\varvec{z}}\in \mathbb {R}^\nu ,\,0\leqslant s<t<\mathcal {T}. \end{aligned}$$

Hence, Itō’s formula (applied with respect to the time-shifted basis $\mathbb {B}_s$) $\mathbb {P}$-a.s. entails, for all $0\leqslant s\leqslant r<t$,

$$\begin{aligned} (\pi _{r,t}f)({\varvec{b}}_r^{\mathcal {T};{\varvec{x}},{\varvec{y}}})-(\pi _{s,t}f)({\varvec{b}}_s^{\mathcal {T};{\varvec{x}},{\varvec{y}}})&=\int _s^r\nabla (\pi _{u,t}f)({\varvec{b}}_u^{\mathcal {T};{\varvec{x}},{\varvec{y}}})\mathrm {d}{\varvec{B}}_u. \end{aligned}$$

(15.7)

Since $f\in C_0(\mathbb {R}^\nu )$, we further know that $(s,{\varvec{z}})\mapsto \pi _{s,t}f$ has a unique bounded and continuous extension onto $[0,t]\times \mathbb {R}^\nu$ with $\pi _{t,t}f({\varvec{z}})=f({\varvec{z}})$, ${\varvec{z}}\in \mathbb {R}^\nu$. The function $(s,{\varvec{z}})\mapsto \nabla (\pi _{s,t}f)({\varvec{z}})$ is bounded on every set $[0,r]\times \mathbb {R}^\nu$ with $r\in [0,t)$. Let $F:\varOmega \rightarrow \mathbb {R}$ be bounded and $\mathfrak {F}_s$-measurable. Then the dominated convergence theorem and (15.7) yield

$$\begin{aligned} \mathbb {E}[F\big (f({\varvec{b}}_t^{\mathcal {T};{\varvec{x}},{\varvec{y}}})-(\pi _{s,t}f)({\varvec{b}}_s^{\mathcal {T};{\varvec{x}},{\varvec{y}}})\big )] =\lim _{r\uparrow t}\mathbb {E}\left[ \int _s^rF\,\nabla (\pi _{u,t}f)({\varvec{b}}_u^{\mathcal {T};{\varvec{x}},{\varvec{y}}})\mathrm {d}{\varvec{B}}_u\right] =0. \end{aligned}$$

This proves the following relation,

$$\begin{aligned} \mathbb {E}^{\mathfrak {F}_s}[f({\varvec{b}}_t^{\mathcal {T};{\varvec{x}},{\varvec{y}}})]=(\pi _{s,t}f)({\varvec{b}}_s^{\mathcal {T};{\varvec{x}},{\varvec{y}}}),\;\mathbb {P}\text {-a.s.,} \quad 0\leqslant s<t. \end{aligned}$$

(15.8)

In particular, $\mathbb {E}[f({\varvec{b}}_t^{\mathcal {T};{\varvec{x}},{\varvec{y}}})]=\mathbb {E}[\mathbb {E}^{\mathfrak {F}_0}[f({\varvec{b}}_t^{\mathcal {T};{\varvec{x}},{\varvec{y}}})]] =(\pi _{0,t}f)({\varvec{x}})$, which is the first asserted identity. Applying (15.7) first and Itō’s formula with respect to $\mathbb {B}_s$ afterwards, we $\mathbb {P}$-a.s. obtain

$$\begin{aligned} (\pi _{r,t}f)({\varvec{b}}_r^{\mathcal {T};{\varvec{x}},{\varvec{y}}})({B}_{\ell ,r}-{B}_{\ell ,s})&=\int _s^r\partial _\ell (\pi _{u,t}f)({\varvec{b}}_u^{\mathcal {T};{\varvec{x}},{\varvec{y}}})\mathrm {d}u \nonumber \\&\quad +\int _s^r({B}_{\ell ,u}-{B}_{\ell ,s})\nabla (\pi _{u,t}f)({\varvec{b}}_u^{\mathcal {T};{\varvec{x}},{\varvec{y}}})\mathrm {d}{\varvec{B}}_u\nonumber \\&\quad +\int _s^r(\pi _{u,t}f)({\varvec{b}}_u^{\mathcal {T};{\varvec{x}},{\varvec{y}}})\mathrm {d}B_{\ell ,u}, \end{aligned}$$

(15.9)

for all $r\in [s,t)$.

The dominated convergence theorem and (15.9) now imply

$$\begin{aligned} \mathbb {E}[f({\varvec{b}}_{t}^{\mathcal {T};{\varvec{x}},{\varvec{y}}})({\varvec{B}}_{t}-{\varvec{B}}_{s})]&=\lim _{r\uparrow t} \mathbb {E}[(\pi _{r,t}f)({\varvec{b}}_r^{\mathcal {T};{\varvec{x}},{\varvec{y}}})({\varvec{B}}_{r}-{\varvec{B}}_{s})] \nonumber \\&=\lim _{r\uparrow t}\int _s^r\mathbb {E}[\nabla (\pi _{u,t}f)({\varvec{b}}_u^{\mathcal {T};{\varvec{x}},{\varvec{y}}})]\mathrm {d}u. \end{aligned}$$

(15.10)

Here we further infer from the first asserted identity (extended to bounded measurable f) and from (15.8) that

$$\begin{aligned} \int _s^r\mathbb {E}\big [\nabla (p_{u,t}f)({\varvec{b}}_u^{\mathcal {T};{\varvec{x}},{\varvec{y}}})\big ]\mathrm {d}u&=\int _s^r\int _{\mathbb {R}^\nu }\varrho _{0,u}({\varvec{x}},{\varvec{z}})\nabla (p_{u,t}f)({\varvec{z}})\mathrm {d}{\varvec{z}}\mathrm {d}u \nonumber \\&=-\int _s^r\int _{\mathbb {R}^\nu }(\nabla _{{\varvec{z}}}\varrho _{0,u})({\varvec{x}},{\varvec{z}})(p_{u,t}f)({\varvec{z}})\mathrm {d}{\varvec{z}}\mathrm {d}u \nonumber \\&\xrightarrow {\;\,r\uparrow t\;\,}-\int _s^t\int _{\mathbb {R}^\nu }(\varrho _{0,u})({\varvec{x}},{\varvec{z}})(p_{u,t}f)({\varvec{z}}) (\nabla \ln d_u^{\mathcal {T};{\varvec{x}},{\varvec{y}}})({\varvec{z}})\mathrm {d}{\varvec{z}}\mathrm {d}u \nonumber \\&=-\int _s^t\mathbb {E}[(p_{u,t}f)({\varvec{b}}_u^{\mathcal {T};{\varvec{x}},{\varvec{y}}}) (\nabla \ln d_u^{\mathcal {T};{\varvec{x}},{\varvec{y}}})({\varvec{b}}_u^{\mathcal {T};{\varvec{x}},{\varvec{y}}})]\mathrm {d}u \nonumber \\&=-\int _s^t\mathbb {E}[\mathbb {E}^{\mathfrak {F}_u}[f({\varvec{b}}_t^{\mathcal {T};{\varvec{x}},{\varvec{y}}})] (\nabla \ln d_u^{\mathcal {T};{\varvec{x}},{\varvec{y}}})({\varvec{b}}_u^{\mathcal {T};{\varvec{x}},{\varvec{y}}})]\mathrm {d}u \nonumber \\&=-\int _s^t\mathbb {E}[f({\varvec{b}}_t^{\mathcal {T};{\varvec{x}},{\varvec{y}}}) (\nabla \ln d_u^{\mathcal {T};{\varvec{x}},{\varvec{y}}})({\varvec{b}}_u^{\mathcal {T};{\varvec{x}},{\varvec{y}}})]\mathrm {d}u \nonumber \\&=-\mathbb {E}\left[ f({\varvec{b}}_t^{\mathcal {T};{\varvec{x}},{\varvec{y}}})\int _s^t (\nabla \ln d_u^{\mathcal {T};{\varvec{x}},{\varvec{y}}})({\varvec{b}}_u^{\mathcal {T};{\varvec{x}},{\varvec{y}}})\mathrm {d}u\right] . \end{aligned}$$

(15.11)

Combinig (15.10) and (15.11) we arrive at the second asserted identity. $\square$

Appendix 5: On time-ordered integration of a stochastic integral

After the application of the stochastic calculus in Sect. 6 we obtain the relation (6.18) on the complement of a $\mathbb {P}$-zero set which depends inter alia on the parameters $t_{[n]}=(t_1,\dots ,t_n)$. Hence, it is clear a priori that, $\mathbb {P}$-a.s., (6.18) is available for all rational $t_{[n]}\in I{\triangle }_n\cap \mathbb {Q}^n$ at the same time, where $I{\triangle }_n:=\{0\leqslant t_1\leqslant \cdots \leqslant t_n\in I\}\subset \mathbb {R}^{n}$. To obtain (6.18) for all $t_{[n]}\in I{\triangle }$ on the complement of one fixed $\mathbb {P}$-zero set, we shall exploit the continuity in $t_{[n]}$ of the various terms in (6.18). To this end we have to show in particular that the stochastic integrals in (6.18) posses modifications which define a process that is jointly continuous in $(t,t_{[n]})$. This is essentially what is done in the proof of the first of the two following lemmas. In the second one we justify the use of the stochastic Fubini theorem in the proof of Lemma 6.1 at the end of Sect. 6. In this appendix the results of Sects. 3 and 4 may be used without producing logical inconsistences, and the vectors $g,h\in \mathfrak {d}_C$ are fixed.

Lemma 16.1

On the complement of some $(t,t_{[n]})$-independent $\mathbb {P}$-zero set, the stochastic integral formula (6.18) holds true, for all $t\in [0,\sup I)$, $n\in \mathbb {N}$, and $t_{[n]}\in t{\triangle }_n$.

Proof

Step 1 Employing (2.9), (2.12), (2.6), (4.2), and (5.6) we first observe that the integrand $\langle \zeta (g)|i{\varvec{v}}({\varvec{\xi }},{\varvec{X}}_\tau )\,\mathscr {Q}_{\tau }^{(n)}(h;t_{[n]})W_{{\varvec{\xi }},\tau }^{0}\zeta (h) \rangle$ of the stochastic integral in (6.18) is a linear combination of terms of the form

$$\begin{aligned} {\varvec{L}}_\tau [t_{[n]}]&:=\mathscr {I}_{\alpha _{\mathcal {C}}}(t_{\mathcal {C}})\,\mathscr {L}^{\alpha _{\mathcal {A}}}_{\tau }(t_{\mathcal {A}};g)\,\mathscr {R}_{\alpha _{\mathcal {B}}}(t_{\mathcal {B}};h) \nonumber \\&\quad \cdot \langle i{\varvec{m}}g+{\varvec{G}}_{{\varvec{X}}_\tau }|w_{t_d,\tau }F_{\alpha _d,{\varvec{X}}_{t_d}} \rangle ^\varkappa \langle \zeta (g)|W_{{\varvec{\xi }},\tau }^{0}\zeta (h) \rangle , \end{aligned}$$

(16.1)

with disjoint (and possibly empty) subsets $\mathcal {A},\mathcal {B},\mathcal {C},\{d\}\subset [n]$ and $\varkappa \in \{0,1\}$. As a consequence, if we define

$$\begin{aligned} \mathscr {I}_t[t_{[n]}]&:=\int _{0}^t1_{\tau > t_n}{\varvec{L}}_\tau [t_{[n]}]\,\mathrm {d}{\varvec{B}}_\tau , \quad t\in I,\quad t_{[n]}\in I{\triangle }_n, \end{aligned}$$

then it suffices to verify the following:

Claim There exists a $\mathfrak {B}(I{\triangle }_n)\otimes \mathfrak {F}$-measurable map $\mathscr {J}^\sharp :(t_{[n]},t,{\varvec{\gamma }})\mapsto \mathscr {I}_t^\sharp [t_{[n]}]({\varvec{\gamma }})\in \mathbb {R}$ such that, for each fixed $(t_{[n]},t)\in I{\triangle }_n\times [0,\sup I)$, we $\mathbb {P}$-a.s. have $\mathscr {I}_t^\sharp [t_{[n]}]=\mathscr {I}_t[t_{[n]}]$, and such that, for all ${\varvec{\gamma }}\in \varOmega$, the map $I{\triangle }_n\times [0,\sup I)\ni (t_{[n]},t)\mapsto \mathscr {I}_t^\sharp [t_{[n]}]({\varvec{\gamma }})$ is continuous.

Step 2 To begin with we argue that we may additionally assume that ${\varvec{X}}={\varvec{X}}^{{\varvec{q}}}$, for some bounded $\mathfrak {F}_0$-measurable ${\varvec{q}}:\varOmega \rightarrow \mathbb {R}^\nu$, so that (2.37) is available. For, if ${\varvec{X}}_0={\varvec{q}}$ is unbounded, then we can set ${\varvec{q}}_m:=1 {\{}|{\varvec{q}}|\leqslant m\}{\varvec{q}}$, $m\in \mathbb {N}$, and verify the claim in Step 1 for each ${\varvec{X}}^{{\varvec{q}}_m}$. After that we invoke the pathwise uniqueness property ${\varvec{X}}_\bullet ^{{\varvec{q}}}={\varvec{X}}_\bullet ^{{\varvec{q}}_m}$, $\mathbb {P}$-a.s. on $\{|{\varvec{q}}|\leqslant m\}$, which entails $u_{{\varvec{\xi }},\bullet }^0[{\varvec{X}}^{{\varvec{q}}}]=u_{{\varvec{\xi }},\bullet }^0[{\varvec{X}}^{{\varvec{q}}_m}]$, $U_\bullet ^+[{\varvec{X}}^{{\varvec{q}}}]=U_\bullet ^+[{\varvec{X}}^{{\varvec{q}}_m}]$, and $U_{s,\bullet }^-[{\varvec{X}}^{{\varvec{q}}}]=U_{s,\bullet }^-[{\varvec{X}}^{{\varvec{q}}_m}]$, $\mathbb {P}$-a.s. on $\{|{\varvec{q}}|\leqslant m\}$, for each $s\in I$. (Here we use the notation explained in the second paragraph of Sect. 8.)

So let ${\varvec{q}}$ be bounded. Then the claim in Step 1 follows from the Kolmogoroff-Neveu lemma (see, e.g., [10, Satz 2.11$'$] or [30, Exercise E.4 of Chap. 8]) as soon as we can find (n-dependent) $p,\varepsilon >0$ and some function $c:I\rightarrow (0,\infty )$ such that

$$\begin{aligned} \mathbb {E}\left[ \sup _{\tau \leqslant \sigma }\Vert \mathscr {I}_\tau [t_{[n]}]-\mathscr {I}_\tau [s_{[n]}]\Vert ^p\right] \leqslant c(\sigma )\,|t_{[n]}-s_{[n]}|^{n+\varepsilon },\quad \sigma \in [0,\sup I), \end{aligned}$$

for all $t_{[n]},s_{[n]}\in I{\triangle }_n$ with $|t_{[n]}-s_{[n]}|\leqslant 1$. To this end we shall prove that

$$\begin{aligned} \mathbb {E}\left[ \left( \int _0^\sigma \big |1_{\tau >t_n}{\varvec{L}}_\tau [t_{[n]}]- 1_{\tau >s_n}{\varvec{L}}_\tau {[s_{[n]}]} \big |^2\mathrm {d}\tau \right) ^{{p}/{2}}\right] \leqslant c_{n,p}(\sigma )\,|t_{[n]}-s_{[n]}|^{\frac{p-2}{2}}, \end{aligned}$$

(16.2)

for all $\sigma ,t_{[n]},s_{[n]}$ as above and for all $p\geqslant 2$.

Step 3 First, we derive suitable bounds on the scalar products whose products define ${\varvec{L}}_\tau [t_{[n]}]$; recall (5.1) and (5.2). In fact, by Hypothesis 2.3 the terms

$$\begin{aligned} {\varvec{a}}^{(\ell )}_{s,t}:=\big \langle i{\varvec{m}}\,g+{\varvec{G}}_{{\varvec{X}}_t}\big |w_{s,t}{F}_{\ell ,{\varvec{X}}_{s}} \big \rangle \quad \text {and}\quad {\varvec{a}}^{(S+\ell )}_{s,t}:=\langle g|w_{s,t}{F}_{\ell ,{\varvec{X}}_{s}} \rangle ,\quad \ell =1,\ldots ,S, \end{aligned}$$

are bounded on $\varOmega$, uniformly in $0\leqslant s\leqslant t\in I$. Moreover, it is straightforward to infer the following bounds from Hypothesis 2.3,

$$\begin{aligned} |{\varvec{a}}^{(\ell )}_{s,t}-{\varvec{a}}^{(\ell )}_{\tilde{s},t}|&\leqslant \mathfrak {c}(|s-\tilde{s}|+|{\varvec{X}}_s-{\varvec{X}}_{\tilde{s}}|),\quad s,\tilde{s}\leqslant t\in I, \;\ell =1,\ldots ,2S, \end{aligned}$$

on $\varOmega$ with a t-independent constant $\mathfrak {c}>0$, where (2.35) $\mathbb {P}$-a.s. implies

$$\begin{aligned} |{\varvec{X}}_s-{\varvec{X}}_{\tilde{s}}|&\leqslant |{\varvec{B}}_s-{\varvec{B}}_{\tilde{s}}|+\int _{\tilde{s}}^s|{\varvec{\beta }}(\tau ,{\varvec{X}}_\tau )|\,\mathrm {d}\tau , \quad 0\leqslant \tilde{s}\leqslant s<\sup I, \end{aligned}$$

(16.3)

Taking (2.37) into account we deduce that, for all $p\geqslant 2$ and $\sigma \in [0,\sup I)$,

$$\begin{aligned}&\mathbb {E}\left[ \int _0^\sigma |{\varvec{a}}^{(\ell )}_{s,\tau }-{\varvec{a}}^{(\ell )}_{\tilde{s},\tau }|^p\mathrm {d}\tau \right] \nonumber \\&\quad \leqslant \mathfrak {c}(p)\left( |s-\tilde{s}|^p+\mathbb {E}[|{\varvec{B}}_s-{\varvec{B}}_{\tilde{s}}|^p] +|s-\tilde{s}|^{p-1}\int _0^\sigma \mathbb {E}[|{\varvec{\beta }}(\tau ,{\varvec{X}}_\tau )|^p]\,\mathrm {d}\tau \right) \nonumber \\&\quad \leqslant \mathfrak {c}'(p,\sigma )\,|s-\tilde{s}|^{{p}/{2}},\quad s,\tilde{s}\in [0,\sigma ],\; |s-\tilde{s}|\leqslant 1. \end{aligned}$$

(16.4)

Furthermore, in view of (3.4) and (3.5) the scalar products

$$\begin{aligned} {\varvec{a}}_{s,t}^{(\ell )}:=\langle U_{s,t}^-|{F}_{\ell ,{\varvec{X}}_s} \rangle ,\quad \ell =2S+1,\ldots ,3S, \end{aligned}$$

satisfy, for all $p\geqslant 2$, $\sigma \in [0,\sup I)$, and $s\in [0,\sigma ]$,

$$\begin{aligned}&\mathbb {E}\left[ \sup _{t\leqslant \sigma }|{\varvec{a}}_{s,t}^{(\ell )}|^p\right] \nonumber \\&\quad \leqslant \mathfrak {c}^p\,\mathbb {E}\left[ \sup _{t\leqslant \sigma } \left\| \int _0^t1_{r>s}\iota _r{\varvec{G}}_{{\varvec{X}}_r}\mathrm {d}{\varvec{B}}_r +\int _0^t1_{r>s}\iota _r({\varvec{G}}_{{\varvec{X}}_r}\cdot {\varvec{\beta }}(r,{\varvec{X}}_r)+\breve{q}_{{\varvec{X}}_r})\mathrm {d}r\right\| ^p\right] \nonumber \\&\quad \leqslant \mathfrak {c}'(p)\,\sigma ^{\frac{p-2}{2}} \mathbb {E}\left[ \int _0^\sigma \Vert {\varvec{G}}_{{\varvec{X}}_r}\Vert ^p\mathrm {d}r\right] +\mathfrak {c}'(p)\,\sigma ^{p-1}\int _0^\sigma \mathbb {E}[1+|{\varvec{\beta }}(r,{\varvec{X}}_r)|^p]\,\mathrm {d}r \nonumber \\&\quad \leqslant \mathfrak {c}''(p,\sigma ),\quad \ell =2S+1,\ldots ,3S. \end{aligned}$$

(16.5)

For all $p\geqslant 2$ and $\tilde{s}\leqslant s\leqslant \sigma <\sup I$ with $|s-\tilde{s}|\leqslant 1$, we likewise have

$$\begin{aligned}&\mathbb {E}\left[ \sup _{0\leqslant \tau \leqslant \sigma }\Vert U_{s,\tau }^--U_{\tilde{s},\tau }^-\Vert ^p\right] \nonumber \\&\quad =\mathbb {E}\left[ \sup _{0\leqslant \tau \leqslant \sigma } \left\| \int _0^\tau 1_{\tilde{s}< r\leqslant s}\iota _r{\varvec{G}}_{{\varvec{X}}_r}\mathrm {d}{\varvec{B}}_r +\int _0^\tau 1_{\tilde{s}< r\leqslant s}\iota _r({\varvec{G}}_{{\varvec{X}}_r}{\varvec{\beta }}(r,{\varvec{X}}_r) +\breve{q}_{{\varvec{X}}_r})\mathrm {d}r\right\| ^p\right] \nonumber \\&\quad \leqslant \mathfrak {c}(p)\mathbb {E}\left[ \left( \int _0^\sigma 1_{\tilde{s}<r\leqslant s}\Vert {\varvec{G}}_{{\varvec{X}}_r}\Vert ^2\mathrm {d}r\right) ^{{p}/{2}}\right] +\mathfrak {c}(p)|s-\tilde{s}|^{p-1}\int _{0}^\sigma \mathbb {E}[1+|{\varvec{\beta }}(r,{\varvec{X}}_r)|^p]\mathrm {d}r \nonumber \\&\quad \leqslant \mathfrak {c}'(p,\sigma )|s-\tilde{s}|^{{p}/{2}}. \end{aligned}$$

(16.6)

Together with the global Lipschitz continuity of ${\varvec{x}}\mapsto {\varvec{F}}_{{\varvec{x}}}$, (16.3), and an estimate analog to (16.4), the bound (16.6) implies

$$\begin{aligned} \mathbb {E}\left[ \sup _{0\leqslant \tau \leqslant \sigma }|{\varvec{a}}_{s,\tau }^{(\ell )}-{\varvec{a}}_{\tilde{s},\tau }^{(\ell )}|^p\right] \leqslant \mathfrak {c}(p,\sigma )\,|s-\tilde{s}|^{{p}/{2}}, \quad \ell =2S+1,\ldots ,3S, \end{aligned}$$

(16.7)

under the above conditions on s, $\tilde{s}$, $\sigma$, and p.

Let us finally consider the $\tau$-independent terms in (16.1). It is clear that

$$\begin{aligned} {a}_{r,s}^{(j,\ell )}:=\langle F_{j,{\varvec{X}}_s}|w_{r,s}F_{\ell ,{\varvec{X}}_r} \rangle \quad \text {and}\quad {\varvec{a}}^{(3S+1)}_{s,\tau }:={\varvec{a}}^{(3S+1)}_s:=\langle {\varvec{F}}_{{\varvec{X}}_s}|w_{0,s}h \rangle , \end{aligned}$$

are bounded on $\varOmega$ uniformly in $r,s\in I$ (and $\tau$, of course). Thanks to the above discussion it is also clear that ${\varvec{a}}^{(3S+1)}_{s,\tau }$ satisfies a bound analog to (16.4) and that

$$\begin{aligned} \mathbb {E}\big [|a_{r,s}^{(j,\ell )}-a_{\tilde{r},\tilde{s}}^{(j,\ell )}|^p\big ]&\leqslant \mathfrak {c}(p,\sigma )\big (|r-\tilde{r}|+|s-\tilde{s}|\big )^{{p}/{2}}, \end{aligned}$$

(16.8)

for all $r,\tilde{r},s,\tilde{s}\in [0,\sigma ]$ with $|r-\tilde{r}|\leqslant 1$ and $|s-\tilde{s}|\leqslant 1$. Finally, setting ${\varvec{a}}^{(3S+2)}_{s,\tau }:={\varvec{a}}^{(3S+2)}_s:=\langle {\varvec{F}}_{{\varvec{X}}_s}|U^+_s \rangle$, we get $\mathbb {E}[|{\varvec{a}}^{(3S+2)}_s|^p]\leqslant \mathfrak {c}(p,\sigma )$ and a bound analog to (16.7).

Step 4 Next, we derive the bound (16.2) assuming that $s_n\leqslant t_n\leqslant \sigma <\sup I$ with $|t_n-s_n|\leqslant 1$ without loss of generality: Notice that $L_\tau [t_{[n]}]$ is the product of $m\leqslant n$ scalar products which are either uniformly bounded or can be estimated as in (16.5), whence

$$\begin{aligned}&\mathbb {E}\left[ \left( \int _{s_n}^{t_n}(|{\varvec{L}}_\tau [t_{[n]}]| +|{\varvec{L}}_\tau [s_{[n]}]|)^2\mathrm {d}\tau \right) ^{{p}/{2}}\right] \\&\quad \leqslant \mathfrak {c}(n,p)\,|t_n-s_n|^{\frac{p-2}{2}} \mathop {\mathop {\sup }_{{s\leqslant \sigma }}}\limits _{j=1,\ldots ,3S+2}\int _{s_n}^{t_n}\mathbb {E}[ 1+|{\varvec{a}}^{(j)}_{s,\tau }|^{np}]\mathrm {d}\tau \leqslant \mathfrak {c}'(n,p,\sigma )\,|t_n-s_n|^{\frac{p-2}{2}}. \end{aligned}$$

Furthermore, representing the difference ${\varvec{L}}_\tau [t_{[n]}]-{\varvec{L}}_\tau [s_{[n]}]$ as a telescopic sum and using the bound (4.3), we readily deduce that

$$\begin{aligned}&\mathbb {E}\left[ \left( \int _{t_n}^{\sigma }\big |{\varvec{L}}_\tau [t_{[n]}] -{\varvec{L}}_\tau [s_{[n]}]\big |^2\mathrm {d}\tau \right) ^{{p}/{2}}\right] \leqslant \sigma ^{\frac{p-2}{2}}\mathbb {E}\left[ \int _0^\sigma \big |{\varvec{L}}_\tau [t_{[n]}]-{\varvec{L}}_\tau [s_{[n]}]\big |^p\mathrm {d}\tau \right] \\&\quad \leqslant \mathfrak {c}\max _{{1\leqslant j,k\leqslant 3S+2}}\mathbb {E}\left[ \int _0^\sigma \left( \sum _{m=1}^n |{\varvec{a}}^{(j)}_{t_m,\tau }-{\varvec{a}}^{(j)}_{s_m,\tau }|\mathop {\mathop {\prod }_{{\ell =1}}}\limits _{\ell \not =m}^n \big (1+|{\varvec{a}}_{s_\ell ,\tau }^{(k)}|+|{\varvec{a}}_{t_\ell ,\tau }^{(k)}|\big )\right) ^p \mathrm {d}\tau \right] \\&\quad \quad +\mathfrak {c}\mathop {\mathop {\max }_{{1\leqslant k\leqslant 3S+2}}}\limits _{1\leqslant j,\ell \leqslant S}\mathbb {E}\left[ \int _0^\sigma \left( \mathop {\mathop {\sum }_{{a,b=1}}}\limits _{a<b}^n|a_{t_a,t_b}^{(j,\ell )}-a_{s_a,s_b}^{(j,\ell )}| \mathop {\mathop {\prod }_{{c=1}}}\limits _{c\not =a,b}^n \big (1+|{\varvec{a}}_{s_c,\tau }^{(k)}|+|{\varvec{a}}_{t_c,\tau }^{(k)}|\big )\right) ^p\mathrm {d}\tau \right] \\&\quad \leqslant \mathfrak {c}'\mathop {\mathop {\max }_{{1\leqslant j,k\leqslant 3S+2}}}\limits _{m=1,\ldots ,n} \mathbb {E}\left[ \int _0^\sigma |{\varvec{a}}^{(j)}_{t_m,\tau }-{\varvec{a}}^{(j)}_{s_m,\tau }|^{pn}\mathrm {d}\tau \right] ^{{1}/{n}} \sup _{s\in [0,\sigma ]} \mathbb {E}\left[ \int _0^\sigma \big (1+|{\varvec{a}}_{s,\tau }^{(k)}|^{pn}\big )\mathrm {d}\tau \right] ^{\frac{n-1}{n}}\\&\quad \quad +\mathfrak {c}'\mathop {\mathop {\mathop {\max }_{{{1\leqslant k\leqslant 3S+2}}}}\limits _{1\leqslant a<b\leqslant n}}\limits _{1\leqslant j,\ell \leqslant S}\mathbb {E}[ |a^{(j,\ell )}_{t_a,t_b}-a^{(j,\ell )}_{s_a,s_b}|^{np}]^{{1}/{n}} \sup _{s\in [0,\sigma ]}\mathbb {E}\left[ \int _0^\sigma \big (1+|{\varvec{a}}_{s,\tau }^{(k)}|^{np}\big )\mathrm {d}\tau \right] ^{\frac{n-2}{n}}\\&\quad \leqslant \mathfrak {c}''\,|t_{[n]}-s_{[n]}|^{{p}/{2}}. \end{aligned}$$

Here the constants $\mathfrak {c},\mathfrak {c}',\mathfrak {c}''>0$ depend on g, h, n, p, and $\sigma$. Altogether this proves (16.2), where $|t_{[n]}-s_{[n]}|\leqslant 1$.

Conclusion A priori we know that (6.18) is valid, for all $t\in [t_n,\sup I)$ and all rational $t_{[n]}\in I{\triangle }_n\cap \mathbb {Q}^n$, ouside some $(t,t_{[n]})$-independent $\mathbb {P}$-zero set. The above steps show, however, that the stochastic integral appearing in (6.18) has a suitable modification which is jointly continuous in $(t,t_{[n]})$. Using Hypothesis 2.7(2), (5.7), and Remark 5.2(2) it is straightforward to see that all remaining terms on both sides of (6.18) have continuous modifications as well. Hence, we can extend (6.18) by continuity to all $t_{[n]}\in I{\triangle }_n$ and $t_n\leqslant t<\sup I$ such that it holds outside of one fixed $\mathbb {P}$-zero set. $\square$

Lemma 16.2

The following relation holds $\mathbb {P}$-a.s., for all $t\in [0,\sup I)$,

$$\begin{aligned}&\int _{I^n}\int _{0}^t1_{\tau {\triangle }_n}(t_{[n]}) \big \langle \zeta (g)\big |i{\varvec{v}}({\varvec{\xi }},{\varvec{X}}_\tau )\, \mathscr {Q}_\tau ^{(n)}(h;t_{[n]})\,W_{{\varvec{\xi }},\tau }^{0}\zeta (h) \big \rangle \,\mathrm {d}{\varvec{B}}_\tau \,\mathrm {d}t_{[n]}\\&\quad =\int _{0}^t\int _{I^n}1_{\tau {\triangle }_n}(t_{[n]}) \big \langle \zeta (g)\big |i{\varvec{v}}({\varvec{\xi }},{\varvec{X}}_\tau )\, \mathscr {Q}_\tau ^{(n)}(h;t_{[n]})\,W_{{\varvec{\xi }},\tau }^{0}\zeta (h) \big \rangle \,\mathrm {d}t_{[n]}\,\mathrm {d}{\varvec{B}}_\tau . \end{aligned}$$

Proof

Since both sides of the asserted identity are continuous in t (according to Lemma 16.1), it suffices to prove it ($\mathbb {P}$-a.s.) for some fixed t. So, let $t\in [0,\sup I)$ in what follows. By the remark in the very beginning of the proof of Lemma 16.1, we then have to show that, $\mathbb {P}$-a.s.,

$$\begin{aligned} \int _{I^n}\int _{0}^t&1_{\tau {\triangle }_n}(t_{[n]}){\varvec{L}}_\tau [t_{[n]}]\,\mathrm {d}{\varvec{B}}_\tau \,\mathrm {d}t_{[n]} =\int _{0}^t\int _{I^n}1_{\tau {\triangle }_n}(t_{[n]}){\varvec{L}}_\tau [t_{[n]}]\,\mathrm {d}t_{[n]}\,\mathrm {d}{\varvec{B}}_\tau , \end{aligned}$$

(16.1)

where ${\varvec{L}}_\tau [t_{[n]}]$ is given by (16.1). Invoking the pathwise uniqueness properties discussed in Step 2 of the proof of Lemma 16.1 and the pathwise uniqueness property of stochastic integrals with respect to Brownian motion, we may again argue that it suffices to prove (16.1) under the additional assumption that the initial condition ${\varvec{q}}$ in the SDE solved by ${\varvec{X}}={\varvec{X}}^{{\varvec{q}}}$ be bounded. In order to justify the application of the stochastic Fubini theorem it then suffices (see, e.g., [6, Rem. 4.35]) to check that

$$\begin{aligned} \int _{t{\triangle }_n}\left( \mathbb {E}\left[ \int _{t_n}^t\big |{\varvec{L}}_\tau [t_{[n]}]\big |^2\mathrm {d}\tau \right] \right) ^{{1}/{2}}\mathrm {d}t_{[n]}<\infty . \end{aligned}$$

(16.2)

Since ${\varvec{q}}$ is assumed to be bounded we know, however, from the arguments in the proof of Lemma 16.1 that, for all $t\in [0,\sup I)$,

$$\begin{aligned} \mathbb {E}\left[ \int _{t_n}^t\big |{\varvec{L}}_\tau [t_{[n]}]\big |^2\mathrm {d}\tau \right]&\leqslant \mathfrak {c}(n)\mathop {\mathop {\sup }_{{s\leqslant t}}}\limits _{j=1,\ldots ,3S+2}\int _{t_n}^t \mathbb {E}[1+|{\varvec{a}}_{s,\tau }^{(j)}|^{np}]<\infty , \end{aligned}$$

where we use the notation introduced in Step 3 of the proof of Lemma 16.1. Clearly, this implies (16.2) and we conclude. $\square$

Appendix 6: Measurability of the operator-valued map $\mathbb {W}_{{\varvec{\xi }},t}^{0}$

Recall that a measurable map from a measurable space into a Banach space equipped with its Borel $\sigma$-algebra can be (a.e.) approximated by measurable simple functions, if and only if its range is (a.e.) separable. In particular, it is not possible to define its Bochner–Lebesgue integral, if its range is not (a.e.) separable. Since $\mathscr {B}(\hat{\mathscr {H}})$ is a non-separable Banach space, we shall therefore prove the following two propositions in this appendix:

Proposition 17.1

Let ${\varvec{\xi }}\in \mathbb {R}^\nu$ and assume, in addition to our standing hypotheses, that $|{\varvec{m}}|\leqslant c\omega$, for some $c>0$. Then, after a suitable modification, the operator-valued map $\mathbb {W}_{{\varvec{\xi }}}^{V}:I\times \varOmega \rightarrow \mathscr {B}(\hat{\mathscr {H}})$ has a separable image and defines an adapted $\mathscr {B}(\hat{\mathscr {H}})$-valued process whose paths are continuous on $I{\setminus }\{0\}$. In particular, it is predictable.

Proposition 17.2

Let ${\varvec{\xi }}\in \mathbb {R}^\nu$ and let $\mathscr {T}$ be a locally compact metric space. Assume that V is continuous and that $|{\varvec{m}}|\leqslant c\omega$, for some $c>0$. Assume further that the driving process depends parametrically on $x\in \mathscr {T}$, which we indicate by writing ${\varvec{X}}^x$, such that $I\times \mathscr {T}\ni (t,x)\mapsto {\varvec{X}}_t^x({\varvec{\gamma }})$ is continuous, for all ${\varvec{\gamma }}\in \varOmega$. Finally, assume that the basic processes can and have been chosen such that

$$\begin{aligned} I^2\times \mathscr {T}\ni (\tau ,t,x)\longmapsto (u_{-{\varvec{\xi }},t}^V[{\varvec{X}}^x],U^+_t[{\varvec{X}}^x],U_{\tau ,t}^-[{\varvec{X}}^x])({\varvec{\gamma }})\in \mathbb {C}\oplus \mathfrak {k}^2 \end{aligned}$$

is continuous, for all ${\varvec{\gamma }}\in \varOmega$. Then we can modify each process $\mathbb {W}_{{\varvec{\xi }}}^{V}[{\varvec{X}}^x]$, $x\in \mathscr {T}$, such that $(t,x,{\varvec{\gamma }})\mapsto \mathbb {W}_{{\varvec{\xi }}}^{V}[{\varvec{X}}^x]({\varvec{\gamma }})$ is measurable from $I\times \mathscr {T}\times \varOmega$ to $\mathscr {B}(\hat{\mathscr {H}})$ with a separable image, $\mathbb {W}_{{\varvec{\xi }},t}^{V}[{\varvec{X}}^x]:\varOmega \rightarrow \mathscr {B}(\hat{\mathscr {H}})$ is $\mathfrak {F}_t$-$\mathfrak {B}(\mathscr {B}(\hat{\mathscr {H}}))$-measurable, for all $(t,x)\in I\times \mathscr {T}$, and $(I{\setminus }\{0\})\times \mathscr {T}\ni (t,x)\mapsto \mathbb {W}_{{\varvec{\xi }},t}^{V}[{\varvec{X}}^x]({\varvec{\gamma }})$ is operator norm continuous, for all ${\varvec{\gamma }}\in \varOmega$.

Remark 17.3

• (1) Note that, in the trivial case where ${\varvec{m}}$, ${\varvec{G}}$, and ${\varvec{F}}$ are all equal to zero, we have $\mathbb {W}_{{\varvec{0}},t}^{0}=e^{-t\mathrm {d}\Gamma (\omega )}$, which is not continuous at $t=0$ with respect to the operator norm.

• (2) Employing the bounds derived in Lemma 17.4 below, we can actually verify, without using Theorem 5.3, that the series of time-ordered integrals (5.13) converges with respect to the operator norm pointwise on $\varOmega$. The bounds on the norm of $\mathbb {W}_{{\varvec{\xi }},t}^{V}$ thus obtained are, however, not $\mathbb {P}$-integrable in general and way too rough in order to discuss the SDE (5.15).

To prove the above propositions we shall employ the bound

$$\begin{aligned} \Vert a^\dagger (f_1)\dots a^\dagger (f_m)\psi \Vert&\leqslant 2^{\frac{m}{2}}(m!)^{{1}/{2}}\left( \prod _{j=1}^m\Vert f_j\Vert _{\omega }\right) \left( \sum _{\ell =0}^m\frac{1}{\ell !}\big \Vert \mathrm {d}\Gamma (\omega )^{\frac{\ell }{2}}\psi \big \Vert ^2\right) ^{\frac{1}{2}} \end{aligned}$$

(17.1)

with $\Vert f\Vert _\omega ^2:=\Vert f\Vert ^2+\Vert \omega ^{-{1}/{2}}f\Vert ^2$ and where $f,f_1,\ldots ,f_m$ and $\psi$ are such that the norms on the right hand sides are well-defined. We leave the proof of (17.1) as an exercise to the reader.

For general information on analytic maps from one Hilbert space into another, like the one appearing in the next lemma, we refer again to [14, § III.3.3].

Lemma 17.4

Let $t>0$ and $m\in \mathbb {N}_0$. Then the map $F_{m,t}:\mathfrak {k}^{m+1}\rightarrow \mathscr {B}(\hat{\mathscr {H}})$,

$$\begin{aligned} F_{m,t}(f_1,\ldots ,f_m,g)&:=\sum _{n=0}^\infty \frac{1}{n!} a^\dagger (f_m)\ldots a^\dagger (f_1)a^\dagger (g)^ne^{-t\mathrm {d}\Gamma (\omega )}, \end{aligned}$$

(17.2)

is well-defined, analytic on $\mathfrak {k}^{m+1}$, and satisfies

$$\begin{aligned} \Vert F_{m,t}(f_1,\ldots ,f_m,g)\Vert&\leqslant (m!)^{{1}/{2}}\left( \prod _{j=1}^m2T^{{1}/{2}}\Vert f_j\Vert _{\omega }\right) s\big ((2T)^{{1}/{2}}\Vert g\Vert _\omega \big ), \end{aligned}$$

(17.3)

where $T:=1\vee (1/2t)$ and $s(z):=\sum _{n=0}^\infty (n!)^{-{1}/{2}}z^n$, $z\in \mathbb {C}$. If $\ell \in \mathbb {N}$ and $f_{1},\ldots ,f_{m+\ell },g\in \mathfrak {k}$, then $\mathrm {Ran}(F_{m,t}(f_1,\ldots ,f_m,g))\subset \mathcal {D}(a^\dagger (f_{m+\ell })\dots a^\dagger (f_{m+1}))$ and

$$\begin{aligned} a^\dagger (f_{m+\ell })\dots a^\dagger (f_{m+1})F_{m,t}(f_1,\ldots ,f_m,g) =F_{m+\ell ,t}(f_1,\ldots ,f_{m+\ell },g). \end{aligned}$$

(17.4)

In particular, we may write

$$\begin{aligned} F_{m,t}(f_1,\ldots ,f_m,g)=a^\dagger (f_m)\ldots a^\dagger (f_1)\exp \{a^\dagger (g)\}e^{-t\mathrm {d}\Gamma (\omega )} \end{aligned}$$

with $\exp \{a^\dagger (g)\}e^{-t\mathrm {d}\Gamma (\omega )}:=F_{0,t}(g)$. For every $s>0$, we finally have

$$\begin{aligned} F_{m,t+s}(f_1,\ldots ,f_m,g)=F_{m,t}(f_1,\ldots ,f_m,g)e^{-s\mathrm {d}\Gamma (\omega )}. \end{aligned}$$

(17.5)

Proof

Let $t>0$. It follows immediately from (17.1) that, for all $\ell \in \mathbb {N}_0$, the multi-linear map $\mathfrak {k}^{\ell }\ni (h_1,\ldots ,h_\ell )\mapsto a^\dagger (h_1)\ldots a^\dagger (h_\ell )e^{-t\mathrm {d}\Gamma (\omega )}\in \mathscr {B}(\mathscr {F})$ is bounded and, in particular, analytic. Therefore, to show analyticity of $F_{m,t}$, it suffices to show that the series in (17.2) converges uniformly on every bounded subset of $\mathfrak {k}^{m+1}$. Applying (17.1) we obtain, for all $\phi \in \bigcap _{\ell \in \mathbb {N}}\mathcal {D}(\mathrm {d}\Gamma (\omega )^{\ell })$,

$$\begin{aligned}&\frac{1}{n!}\big \Vert a^\dagger (f_1)\ldots a^\dagger (f_m)a^\dagger (g)^n\phi \big \Vert \\&\quad \leqslant (2T)^{\frac{m}{2}}\left( \prod _{j=1}^m\Vert f_j\Vert _{\omega }\right) \frac{((m+n)!)^{\frac{1}{2}}}{n!} (2T)^{\frac{n}{2}}\Vert g\Vert _\omega ^n\left( \sum _{\ell =0}^{m+n}\frac{T^{-\ell }}{\ell !} \big \langle \phi \big |\mathrm {d}\Gamma (\omega )^\ell \phi \big \rangle \right) ^{\frac{1}{2}}\\&\quad \leqslant (2T^{{1}/{2}})^m(m!)^{{1}/{2}}\left( \prod _{j=1}^m\Vert f_j\Vert _{\omega }\right) \frac{(2T^{{1}/{2}}\Vert g\Vert _\omega )^n}{(n!)^{{1}/{2}}}\left( \sum _{\ell =0}^{\infty }\frac{T^{-\ell }}{\ell !} \big \langle \phi \big |\mathrm {d}\Gamma (\omega )^\ell \phi \big \rangle \right) ^{\frac{1}{2}}. \end{aligned}$$

Here we used the bound $\frac{(m+n)!}{m!n!}<2^{m+n}$ in the second step. Since

$$\begin{aligned} \sum _{\ell =0}^{\infty }\frac{T^{-\ell }}{\ell !} \big \langle e^{-t\mathrm {d}\Gamma (\omega )}\psi \big |\mathrm {d}\Gamma (\omega )^\ell e^{-t\mathrm {d}\Gamma (\omega )}\psi \big \rangle =\big \Vert e^{-(t-1/2T)\mathrm {d}\Gamma (\omega )}\psi \big \Vert ^2\leqslant \Vert \psi \Vert ^2, \end{aligned}$$

for all $\psi \in \mathscr {F}$, this implies

$$\begin{aligned}&\frac{1}{n!}\big \Vert a^\dagger (f_1)\ldots a^\dagger (f_m)a^\dagger (g)^n e^{-t\mathrm {d}\Gamma (\omega )}\big \Vert \\&\quad \leqslant (2T^{\frac{1}{2}})^m(m!)^{\frac{1}{2}}\left( \prod _{j=1}^m\Vert f_j\Vert _{\omega }\right) \frac{(2T^{\frac{1}{2}}\Vert g\Vert _\omega )^n}{(n!)^{\frac{1}{2}}}. \end{aligned}$$

Therefore, the series in (17.2) converges absolutely in operator norm, uniformly on every bounded subset of $\mathfrak {k}^{m+1}$, and we also obtain (17.3). The relation (17.4) follows inductively from the fact that $a^\dagger (f)$ is closed, for every $f\in \mathfrak {h}$, and (17.5) is obvious from the fact that right multiplication with $e^{-s\mathrm {d}\Gamma (\omega )}$ is continuous on $\mathscr {B}(\mathscr {F})$. $\square$

Corollary 17.5

Let $r,s,\tau >0$ and $m\in \mathbb {N}_0$. Then, for all $f_1,\ldots ,f_m,g\in \mathfrak {k}$, the operator $G_{m,s}(f_1,\ldots ,f_m,g)$ defined on the dense domain $\mathscr {C}[\mathfrak {d}_C]$ by

$$\begin{aligned} G_{m,s}(f_1,\ldots ,f_m,g)\psi :=e^{-s\mathrm {d}\Gamma (\omega )}\exp \{a(g)\}a(f_1)\ldots a(f_m)\psi , \quad \psi \in \mathscr {C}[\mathfrak {d}_C], \end{aligned}$$

is bounded and its unique extension to an element of $\mathscr {B}(\mathscr {F})$ is given by

$$\begin{aligned} \overline{G_{m,s}(f_1,\ldots ,f_m,g)}=F_{m,s}(f_1,\ldots ,f_m,g)^*. \end{aligned}$$

If $n\in \mathbb {N}_0$ and $|{\varvec{m}}|\leqslant c\omega$, for some $c>0$, then the map $D_{r,s,\tau }^{(m,n)}:\mathbb {C}\times [0,\infty )\times \mathbb {R}^\nu \times \mathfrak {k}^{m+n+2}\rightarrow \mathscr {B}(\mathscr {F})$ defined by

$$\begin{aligned}&D_{r,s,\tau }^{(m,n)}(a,t,{\varvec{x}},f_1,\ldots ,f_m,\tilde{f}_1,\ldots ,\tilde{f}_n,g,\tilde{g})\nonumber \\&\quad := aF_{m,r}(f_1,\ldots ,f_m,g)\Gamma (e^{-(\tau +t)\omega +i{\varvec{m}}\cdot {\varvec{x}}}) F_{n,s}(\tilde{f}_1,\ldots ,\tilde{f}_n,\tilde{g})^* \end{aligned}$$

(17.6)

is uniformly continuous on every bounded subset of $\mathbb {C}\times [0,\infty )\times \mathbb {R}^\nu \times \mathfrak {k}^{m+n+2}$ and has a separable image. Moreover, $D_{r,s,\tau }^{(m,n)}=D_{\tilde{r},\tilde{s},\tilde{\tau }}^{(m,n)}$, for all $\tilde{r},\tilde{s},\tilde{\tau }>0$ satisfying $\tilde{r}+\tilde{s}+\tilde{\tau }=r+s+\tau$.

Proof

The first assertion follows from Lemma 17.4, and the continuity of the map (17.6) follows from Lemma 17.4 and the bound

$$\begin{aligned}&\Vert \Gamma (e^{-(\tau +t)\omega +i{\varvec{m}}\cdot {\varvec{x}}}) -\Gamma (e^{-(\tau +u)\omega +i{\varvec{m}}\cdot {\varvec{y}}})\Vert \\&\quad \leqslant \Vert (\mathbbm {1}-e^{(t-u)\mathrm {d}\Gamma (\omega )+i({\varvec{x}}-{\varvec{y}})\cdot \mathrm {d}\Gamma ({\varvec{m}})}) e^{-(t+\tau )\mathrm {d}\Gamma (\omega )}\Vert \;\\&\quad \leqslant (u-t)\Vert \mathrm {d}\Gamma (\omega )e^{-\tau \mathrm {d}\Gamma (\omega )}\Vert +|{\varvec{x}}-{\varvec{y}}|\Vert \mathrm {d}\Gamma (|{\varvec{m}}|)e^{-\tau \mathrm {d}\Gamma (\omega )}\Vert \\&\quad \leqslant (u-t+c|{\varvec{x}}-{\varvec{y}}|)\Vert \mathrm {d}\Gamma (\omega )e^{-\tau \mathrm {d}\Gamma (\omega )}\Vert , \end{aligned}$$

for all ${\varvec{x}},{\varvec{y}}\in \mathbb {R}^\nu$ and $u>t>0$. The map (17.6) has a separable image because it is continuous and its domain $\mathbb {C}\times [0,\infty )\times \mathbb {R}^\nu \times \mathfrak {k}^{m+n+2}$ is separable. The relation $D_{r,s,\tau }^{(m,n)}=D_{\tilde{r},\tilde{s},\tilde{\tau }}^{(m,n)}$ is a consequence of (17.5). $\square$

Corollary 17.6

Let $\mathscr {T}$ be a locally compact metric space, let $\mathscr {K}$ be a separable Hilbert space, and let $\mathsf {T}_{\mathscr {K}}$ be the set of measurable maps $X:I\times \mathscr {T}\times \varOmega \rightarrow \mathscr {K}$, $(t,x,{\varvec{\gamma }})\mapsto X_t^x({\varvec{\gamma }})$, such that $X^x$ is an adapted process, for every $x\in \mathscr {T}$, and $I\times \mathscr {T}\ni (t,x)\mapsto X_t^x({\varvec{\gamma }})$ is continuous, for all ${\varvec{\gamma }}\in \varOmega$.

Let $r,s,\tau >0$, $\ell ,m,n\in \mathbb {N}_0$, $\tilde{{\varvec{X}}}\in \mathsf {T}_{\mathbb {R}^\nu }$, $Z_1,\ldots ,Z_m,\widetilde{Z}_1,\ldots ,\widetilde{Z}_n,Y,\widetilde{Y}\in \mathsf {T}_{\mathfrak {k}}$, and $h:I^\ell \times \mathscr {T}\times \varOmega \rightarrow \mathbb {C}$, $(t_{[\ell ]},x,{\varvec{\gamma }})\mapsto h_{t_{[\ell ]}}^x({\varvec{\gamma }})$ be measurable such that its restriction to $[0,t]^\ell \times \mathscr {T}\times \varOmega$ is $\mathfrak {B}([0,t]^\ell )\otimes \mathfrak {B}(\mathscr {T})\otimes \mathfrak {F}_t$-measurable, for every $t\in I$, and such that $I^\ell \times \mathscr {T}\ni (t_{[\ell ]},x)\mapsto h_{t_{[\ell ]}}^x({\varvec{\gamma }})$ is continuous, for all ${\varvec{\gamma }}\in \varOmega$. For all $(t_{[\ell +m+n]},\rho _{[3]},t,x)\in \mathscr {G}:=I^{\ell +m+n+3}\times [0,\infty )\times \mathscr {T}$, define a function $\varOmega \rightarrow \mathscr {B}(\mathscr {F})$ by

$$\begin{aligned}&B_{t,\rho _{[3]}}^x(t_{[\ell +m+n]},\cdot )\nonumber \\&\quad :=D_{r,s,\tau }^{(m,n)}\big (h_{t_{[\ell ]}}^x,t,\tilde{{\varvec{X}}}_{\rho _1}^x,Z_{1,t_{\ell +1}}^x, \ldots ,Z_{m,t_{\ell +m}}^x,\widetilde{Z}_{1,t_{\ell +m+1}}^x,\ldots ,\widetilde{Z}_{n,t_{\ell +m+n}}^x,Y_{\rho _2}^x, \widetilde{Y}_{\rho _3}^x\big ).\nonumber \\ \end{aligned}$$

(17.7)

Then $B:\mathscr {G}\times \varOmega \rightarrow \mathscr {B}(\mathscr {F})$ is measurable, it has a separable image, its restriction to $[0,t]^{\ell +m+n+3}\times [0,\infty )\times \mathscr {T}\times \varOmega \rightarrow \mathscr {B}(\mathscr {F})$ is $\mathfrak {B}([0,t]^{\ell +m+n+3}\times [0,\infty )\times \mathscr {T})\otimes \mathfrak {F}_t-\mathfrak {B}(\mathscr {B}(\mathscr {F}))$-measurable, and the map $(t_{[\ell +m+n]},\rho _{[3]},t,x)\mapsto B_{t,\rho _{[3]}}^x(t_{[\ell +m+n]},{\varvec{\gamma }})$ is continuous on $\mathscr {G}$, for all ${\varvec{\gamma }}$. Furthermore, the $\mathscr {B}(\mathscr {F})$-valued Bochner–Lebesgue integrals in

$$\begin{aligned} J(\tilde{t},\rho _{[3]},t,x,{\varvec{\gamma }}):=\int _{\tilde{t}\triangle _{m+n+\ell }} B_{t,\rho _{[3]}}^x(t_{[\ell +m+n]},{\varvec{\gamma }})\mathrm {d}t_{[\ell +m+n]},\quad \tilde{t}\in I, \end{aligned}$$

(17.8)

are well-defined, the map $J:\mathscr {G}':=I^4\times [0,\infty )\times \mathscr {T}\times \varOmega \rightarrow \mathscr {B}(\mathscr {F})$ is measurable with a separable image and its restriction to $[0,t]^4\times [0,\infty )\times \mathscr {T}\times \varOmega \rightarrow \mathscr {B}(\mathscr {F})$ is $\mathfrak {B}([0,t]^4\times [0,\infty )\times \mathscr {T})\otimes \mathfrak {F}_t$-$\mathfrak {B}(\mathscr {B}(\hat{\mathscr {H}}))$-measurable, for every $t\in I$. Finally, for all ${\varvec{\gamma }}\in \varOmega$, the map $(\tilde{t},\rho _{[3]},t,x)\mapsto J(\tilde{t},\rho _{[3]},t,x,{\varvec{\gamma }})$ is continuous on $\mathscr {G}'$.

Proof

The measurability properties of B are clear by definition and Corollary 17.5, since B is the composition of two maps which are measurable in the appropriate sense. (Here we use that $\otimes _{i=1}^n\mathfrak {B}(\mathfrak {k})=\mathfrak {B}(\mathfrak {k}^n)$ which follows from the separability of $\mathfrak {k}$.) Since the image of B is contained in the image of (17.6), it is separable. Corollary 17.5 also shows that, at each fixed ${\varvec{\gamma }}$, B can be written as a composition of two continuous maps. In particular, the integral in (17.8) is a (well-defined) Bochner–Lebesgue integral of a continuous function over a compact simplex. The measurability properties of J thus follow from a standard result in integration theory and the image of J is contained in any closed separable subspace of $\mathscr {B}(\mathscr {F})$ containing the image of B. The continuity of J follows from the dominated convergence theorem and the local compactness of $\mathscr {G}$.   $\square$

Remark 17.7

Let $t>0$ and pick arbitrary $r,s,\tau >0$ with $r+s+\tau <t$. Then the following statements hold true on all of $\varOmega$:

• (1) In view of (2.15) and (4.2) we have the following factorization,

  $$\begin{aligned} W_{{\varvec{\xi }},t}^{V}\psi&=e^{-u_{-{\varvec{\xi }},t}^V}\exp \{ia^\dagger (U_t^+)\}\,\Gamma (w_{0,t})\exp \{ia(U_t^-)\}\psi ,\quad \psi \in \mathscr {C}[\mathfrak {h}]. \end{aligned}$$

  Thus, $W_{{\varvec{\xi }},t}^{V}=D_{r,s,\tau }^{(0,0)}(e^{-u_{-{\varvec{\xi }},t}^V},t-\tau ,{\varvec{X}}_t-{\varvec{X}}_0,U_t^+,U_t^-)$ with $D_{r,s,\tau }^{(0,0)}$ as in (17.6).

• (2) Let $n\in \mathbb {N}$. Then $\mathbb {W}_{{\varvec{\xi }},t}^{V,(n)}$ can be written as a linear combination (with coefficients in $\mathscr {B}(\mathbb {C}^L)$) of $\mathscr {B}(\mathscr {F})$-valued Bochner–Lebesgue integrals,

  $$\begin{aligned} \mathbb {W}_{{\varvec{\xi }},t}^{V,(n)}=\sum _{\alpha \in [S]^n} \sigma _{\alpha _n}\dots \sigma _{\alpha _1}\mathop {\mathop {\sum }_{{\mathcal {A}\cup \mathcal {A}'\cup \mathcal {B}\cup \mathcal {B}'\cup \mathcal {C}=[n]}}}\limits _{\#\mathcal {C}\in 2\mathbb {N}_0}\int _{t{\triangle }_n}D_{r,s,\tau }^{(\#\mathcal {A},\#\mathcal {B})}(\aleph (t,t_{[n]}))\mathrm {d}t_{[n]}, \end{aligned}$$

  (17.9)

  where the argument of the integrand is given by

  $$\begin{aligned}&\aleph (t,t_{[n]})\\&\quad :=(h_{t,t_{\mathcal {A}'\cup \mathcal {B}'\cup \mathcal {C}}},t-\tau , {\varvec{X}}_t-{\varvec{X}}_0,\{w_{t_a,t}{F}_{\alpha _a,{\varvec{X}}_{t_a}}\}_{a\in \mathcal {A}}, \{\overline{w}_{0,t_b}{F}_{\alpha _b,{\varvec{X}}_{t_b}}\}_{b\in \mathcal {B}},iU_t^+,iU_t^-),\\&h_{t,t_{\mathcal {A}'\cup \mathcal {B}'\cup \mathcal {C}}}\\&\quad :=\mathscr {I}_{\alpha _{\mathcal {C}}}(t_{\mathcal {C}})e^{-u_{-{\varvec{\xi }},t}^V} \left( {\prod _{a'\in \mathcal {A}'}}\{i\langle U_{t_{a'},t}^{-}|{F}_{\alpha _{a'},{\varvec{X}}_{t_{a'}}} \rangle \}\right) {\prod _{b'\in \mathcal {B}'}}\{i\langle {F}_{\alpha _{b'},{\varvec{X}}_{t_{b'}}}|U_{t_b}^{+} \rangle \}. \end{aligned}$$

• (3) We may compute the adjoint of $\mathbb {W}_{{\varvec{\xi }},t}^{V,(n)}$ by replacing the integrand in (17.9) by its adjoint. Hence, in combination with (17.6) we obtain a fairly detailed formula for $\mathbb {W}_{{\varvec{\xi }},t}^{V*}=\sum _{n=0}^\infty \mathbb {W}_{{\varvec{\xi }},t}^{V,(n)*}$ in terms of the basic processes.

Proof of Proposition 17.2

Since $\mathbb {W}_{{\varvec{\xi }},0}^{V,(n)}=\delta _{0,n}\mathbbm {1}$ on $\varOmega$, the $\mathfrak {F}_0$-measurability of $\mathbb {W}_{{\varvec{\xi }},0}^{V}$ is trivial. Thus, for every $n\in \mathbb {N}_0$, the statement of the proposition with $\mathbb {W}_{{\varvec{\xi }}}^{V}$ replaced by $\mathbb {W}_{{\varvec{\xi }}}^{V,(n)}$ follows immediately from Corollary 17.6 in combination with the formulas of Remark 17.7. Combining this result with the bound (7.10), we conclude that, $\mathbb {P}$-a.s., the convergence $\mathbb {W}_{{\varvec{\xi }},t}^{V}[{\varvec{X}}^x]=\lim _{N\rightarrow \infty }\mathbb {W}_{{\varvec{\xi }},t}^{V,(0,N)}[{\varvec{X}}^x]$ in $\mathscr {B}(\hat{\mathscr {H}})$ is locally uniform in $(t,x)\in I\times \mathscr {T}$. Since each measure space $(\varOmega ,\mathfrak {F}_t,\mathbb {P})$ with $t\in I$ is complete, this proves the proposition. $\square$

Proof of Proposition 17.1

Proposition 17.1 is proved in the same way as Proposition 17.2. $\square$

Appendix 7: General notation and list of symbols

$s\wedge t:=\min \{s,t\}$ and $s\vee t:=\max \{s,t\}$, for $s,t\in \mathbb {R}$.

$1_A$ is the characteristic function of a set A.

Vectors and vector spaces

$\mathcal {D}(\cdot )$ denotes the domain of linear operators, and $\mathcal {Q}(\cdot )$ the quadratic form domain of suitable linear operators. $\mathscr {B}(\mathscr {K}_1,\mathscr {K}_2)$ is the space of bounded linear operators between two normed linear spaces $\mathscr {K}_1$, $\mathscr {K}_2$; $\mathscr {B}(\mathscr {K}_1):=\mathscr {B}(\mathscr {K}_1,\mathscr {K}_1)$.

$x^{\otimes _n}$ denotes the n-fold tensor product of a vector x with itself.

$\mathfrak {h}=L^2(\mathcal {M},\mathfrak {A},\mu )$; $\mathfrak {k}$, $\mathfrak {d}$; $\mathfrak {h}_{+1}$, $\mathfrak {k}_{+1}$	(2.1); Hypothesis 2.3; Sect. 3
$\mathscr {F}=\Gamma _{\mathrm {s}}(\mathfrak {h})$; $\hat{\mathscr {H}}=\mathbb {C}^L\otimes \mathscr {F}$; $\mathscr {H}$	(2.2); (2.21); (10.26)
$\zeta (h)$; $\mathscr {E}[\mathfrak {v}]$, $\mathscr {C}[\mathfrak {v}]$	(2.3); (2.4)
$\widehat{\mathcal {D}}$; $\mathscr {D}_0$	(1.5); (11.4)
$\mathfrak {h}_C$; $\mathfrak {k}_C$, $\mathfrak {d}_C$; $\mathscr {F}_C$	Hypothesis 2.3; (2.26); (2.27)

Quantities determining the model, operators

$\mathscr {W}(f,U)$, $\mathscr {W}(f)$, $\Gamma (U)$	Section 2.1
$\varphi (f)$, $\mathrm {d}\Gamma (T)$, $a^\dagger (f)$, a(f)	Section 2.1
$\omega$, ${\varvec{m}}$, ${\varvec{G}}$, ${\varvec{F}}$, C, ${\varvec{\sigma }}$, $\nu$, L, S	Hypothesis 2.3 and preceding paragraphs
q, $\breve{q}$	(2.25)
$\widehat{H}^V({\varvec{\xi }},{\varvec{x}})$, $\widehat{H}_{\mathrm {sc}}^V({\varvec{\xi }},{\varvec{x}})$, $\widehat{H}({\varvec{\xi }})$, ${\varvec{v}}({\varvec{\xi }},{\varvec{x}})$	Definition 2.5
M; $M_a({\varvec{\xi }})$	(1.5); (2.31)
V; $H^V$	Hypothesis 2.4; (1.6) and Sect. 11
$p_t$; $\widehat{T}_t({\varvec{\xi }})$, $T_t^V$, $T_t^V({\varvec{x}},{\varvec{y}})$	(1.12); Definition 10.7

Measure theoretic and probabilistic objects, processes

$\mathfrak {B}(\mathscr {T})$ denotes the Borel $\sigma$-algebra of a topological space $\mathscr {T}$.

$\lambda ^\nu$ is the $\nu$-dimensional Lebesgue–Borel measure and $\lambda :=\lambda ^1$.

I, $\mathcal {T}$, $\mathbb {B}=(\varOmega ,\mathfrak {F},(\mathfrak {F}_t)_{t\in I},\mathbb {P})$, $\mathfrak {F}_{s,t}$, $\mathbb {E}$, $\mathbb {E}^{\mathfrak {H}}$	Beginning of Sect. 2.3
$I^s$, $\mathbb {B}_s$	(2.34)
${\varvec{B}}$, ${\varvec{X}}$, ${\varvec{X}}^{{\varvec{q}}}$, ${}^s\!{\varvec{X}}^{{\varvec{q}}}$, ${\varvec{\beta }}$, ${\varvec{\varXi }}$	Hypothesis 2.7
${\varvec{Y}}$; ${\varvec{B}}^{{\varvec{x}}}:={\varvec{x}}+{\varvec{B}}$	(7.15); (10.10)
${\varvec{b}}^{\mathcal {T};{\varvec{x}},{\varvec{y}}}; {\hat{{\varvec{b}}}}^{\mathcal {T};{\varvec{y}},{\varvec{x}}}$	Lemma 10.5 and “Appendix 4”; (10.15)
$\mathsf {S}_I(\mathscr {K})$	Beginning of Sect. 2.3
$\llbracket \cdot ,\, \cdot \cdot \rrbracket$	Remark 2.15
$j_t$; $\iota _t$; $w_{\tau ,t}$, $\overline{w}_{\tau ,t}$	(3.1); (3.3); (3.10)
$u_{{\varvec{\xi }}}^V$, $U^\pm$, $(U_{\tau ,t}^-)_{t\in I}$, $K_{\tau ,t}$, $K_t$	Definition 3.1
$W_{{\varvec{\xi }}}^{V}$; $\mathbb {W}_{{\varvec{\xi }}}^{V}$; $\mathbb {W}_{{\varvec{\xi }}}^{V,(n)}$, $\mathbb {W}_{{\varvec{\xi }}}^{V,(N,M)}$	(4.1); Theorem 5.3; Definition 5.1
$t{\triangle }_n$, $\mathscr {L}^{\alpha _{\mathcal {A}}}_{}(t_{\mathcal {A}})$, $\mathscr {R}_{\alpha _{\mathcal {B}}}(t_{\mathcal {B}})$, $\mathscr {I}_{\alpha _{\mathcal {C}}}(t_{\mathcal {C}})$	Definition 5.1
$\mathscr {L}^{\alpha _{\mathcal {A}}}_{}(t_{\mathcal {A}};g)$, $\mathscr {R}_{\alpha _{\mathcal {B}}}(t_{\mathcal {B}};h)$, $\mathscr {Q}_\tau ^{(n)}$	Remark 5.2
$\Lambda _{s,t}({\varvec{x}},\psi )$; $\Lambda _{s,t}[{\varvec{q}},\eta ]$, $P_{s,t}$	Theorem 9.2; Proposition 9.3
$\bar{{\varvec{X}}}$, $\bar{\mathfrak {F}}_\tau$, $\bar{\mathbb {B}}$	(10.1)

The meaning and use of an additional argument $[{\varvec{X}}]$, $[{\varvec{B}}^{{\varvec{x}}}]$, etc., of a process, e.g., $U^\pm [{\varvec{X}}]$ or $\mathbb {W}_{{\varvec{\xi }},t}^{V}[{\varvec{b}}^{t:{\varvec{y}},{\varvec{x}}}]$, is explained in the beginning of Sect. 8.