SMSM References

# Ultra-robust topologically protected edge states in quasi-1D systems

Valerii Kachin
Institute of Informatics, Faculty of Mathematics, Informatics and Mechanics,
University of Warsaw
Banacha 2, 02–097 Warsaw, Poland  Juan Camilo López Carreño
Institute of Informatics, Faculty of Mathematics, Informatics and Mechanics,
University of Warsaw
Banacha 2, 02–097 Warsaw, Poland
Institute of Theoretical Physics, Faculty of Physics, University of Warsaw
Pasteura 5, 02–093 Warsaw, Poland  Magdalena Stobińska
Institute of Informatics, Faculty of Mathematics, Informatics and Mechanics,
University of Warsaw
Banacha 2, 02–097 Warsaw, Poland<EMAIL_ADDRESS>
###### Abstract

Topological materials yield robust edge states with potential applications in
electronics or quantum technologies. Yet, their simplest Su–Schrieffer–Heeger
model covers only coupling disorders, leaving other types unexplored. Here, by
studying a quasi-one-dimensional zigzag model with negative couplings, we show
non-chiral edge states simultaneously resilient to dissipation, hopping and
on-site energy disorders. To this end, we derive regularized values of the
topological invariant via a novel approach. Our work hints on constructing
topological phases even in the absence of usual symmetries.

_Introduction._ —Topologically protected one-dimensional edge states are
quantum states of matter whose wave function is concentrated at the edges of a
1D lattice, and vanishes in its bulk. Lattice symmetries shield them from
noise, crystal defects, or chemical composition changes [1, 2]. Their energy
levels lie inside the energy gap between the valence and conduction bands.
Beyond fundamental studies [3, 4, 5, 6, 7], they open prospects for fault-
tolerant quantum computation [8, 9, 10, 11, 12], robust information processing
[13, 14, 15, 16, 17, 18] and building numerous devices [19, 20, 21, 22, 23].

The best known edge states are protected by the chiral symmetry, described by
the Su–Schrieffer–Heeger (SSH) chain model [24, 25, 26]. Here, an electron can
either hop between the two sites within the unit cell, or to an adjoining one,
with strengths $v$ and $w$, respectively. If $w>v$, topologically protected
states appear.

Various models offering edge states have been proposed, including SSH
generalizations [27, 28, 29, 30, 31], quasicrystal structures [32, 33, 34],
the Aubry–André–Harper (AAH) model [35, 36], and Anderson topological
insulators [37, 38]. They exhibit robustness against disorders in couplings
that result from variations in distances between quantum particles [39, 40].
However, while the previous research addressed spectral disorders and on-site
imperfections, particularly in non-dissipative systems [41, 42], the effects
of site-dependent losses and phase disorder have received scant attention.
These imperfections can significantly degrade edge state localization, induce
energy gap closure, and prompt a shift from topological to conventional matter
[1, 2]. Therefore, developing edge states resilient to these defects is
crucial.

(a)
(b)

Figure 1: (a) Quasi-1D zigzag structure with interacting chains $A$ and $B$
connected by strength $v>0$ (dashed line). The coupling strengths between the
sites within chain $A$ is $f>0$ (black line), while within $B$ is $t<0$ (red
line). (b) Our system can be represented as a chain with next-neighbor and
long-range interactions. An elementary two-site unit cell is shown as a red
frame.

This letter presents the emergence of remarkably resilient topologically
protected edge states in a quasi-1D zigzag structure where some lattice
couplings are negative (with phase $\pi$), even _in the absence_ of chiral
symmetry. These states maintain their nature despite simultaneous disorders in
lattice couplings, variations in on-site phases and potentials, and
dissipation. Negative couplings play a crucial role in their concurrent
stabilizing, enhancing localization, and controlling energy gaps. We introduce
a novel approach to restore bulk-boundary correspondence, and to enable
computing a regularized topological invariant. It establishes a continuous
mapping between the original model and the “parental model”, with restored
chiral symmetry. To this end, we construct in the Fourier space a special
variant of the unit cell that we call “weighted”.

Negative couplings have gained attention for achieving non-trivial topological
phases and creating energy gaps [43, 44, 45, 46, 47, 48, 49]. An example is
the quadrupole topological insulator (QTI) exhibiting corner states limited by
resistance to interaction strength disorders [43, 50]. Negative couplings
naturally emerge in quantum particle interactions in alternative media: atoms
in optical potentials [51, 52], arrays of microwave resonators [53, 50],
optical waveguide lattices [54, 55, 56, 57, 44, 58, 59, 49], acoustic cavities
[46, 60, 47], synthetic dimensions [61, 62], or resonant electric circuits
[63, 64].

Our results pave the way for realistic implementation of topological quantum
computing, information processing, and quantum transport on numerous
platforms, including photonic, acoustic, mechanical metamaterials, and
resonant electric circuits.

_Edge states.—_ Let us consider the zigzag structure shown in Fig. 1a where
one of the interaction strength is negative. It is composed of two interacting
topologically trivial chains $A$ and $B$, and it is described by a finite
tight-binding Hamiltonian

$H\\!=\\!\\!\\!\sum_{j=1}^{N-1}\bigl{(}va_{j}^{\dagger}b_{j+1}+fa^{\dagger}_{j}a_{j+1}+tb^{\dagger}_{j}b_{j+1}\bigr{)}+\\!\\!\sum_{j=1}^{N}va^{\dagger}_{j}b_{j}+h.c.,$
(1)

where $N$ is the number of sites on each chain, $v\geq 0$ is the hopping
amplitude between them, $f\geq 0$ ($t\leq 0$) is the hopping amplitude within
the chain $A$ ($B$); and $a_{j}$ ($a_{j}^{\dagger}$) and $b_{j}$
($b_{j}^{\dagger}$) denote annihilation (creation) operators at the $j$-th
site of the chain $A$ ($B$). Note that we did not made any assumption of the
statistics that the operators $a$ and $b$ follow.

Our system can be viewed as a chain with uniform strength of next-neighbor
(nn) interactions, and two opposite in phase strengths of next-next-neighbor
(nnn) interactions, Fig. 1b. The presence of long-range repulsive and
attractive interactions renders this model distinct from the SSH and AAH. All
its non-trivial topological properties are due to the nnn-interactions; a
model without them becomes a topologically trivial gapless chain. The analysis
of energy bands of $H/f$ in terms of the inverse participation ratio
$\mathfrak{I}$ (IPR) [65] (see Supplemental Material, SM, Section S1) for a
chain of $90$ sites shown in Fig. 2a reveals the presence of two bands if
$v^{2}>tf(t+1)^{2}(f+1)^{2}$ and $f\neq t$, and edge states within the gap if
$v^{2}/4>tf$ and $v\neq 0$. The distance between the edge state and the
closest band is larger, and the above inequalities always hold true
simultaneously if $t/f<0$. Figs. 2b–c show the structure of edge states for
$t=-f$, when the system acquires chiral symmetry.

Figure 2: (a) Energy band structure of the zigzag system with $N=90$ sites
computed for $v/f=1$, as a function of the interaction strengths’ ratio $t/f$.
The edge (red) and bulk (blue) states here and further are highlighted using
the inverse participation ratio (IPR) $\mathfrak{I}$. (b) Mode amplitude
$\Psi_{k}$ on site $k$ of the edge and bulk states’ wave function evaluated
for $t/f=-1$. (c) Profile of an edge state eigenmode shown for the first $20$
sites.

To describe the bulk properties of the two-band spectrum shown in Fig. 2, it
is sufficient to consider a unit cell that consists of two sites, $A$ and $B$,
and the corresponding bulk Hamiltonian in the continuum space

$H_{2}(k_{x})=\begin{pmatrix}2f\cos{k_{x}}&v\bigl{(}1+e^{-ik_{x}}\bigr{)}\\\
v\bigl{(}1+e^{ik_{x}}\bigr{)}&2t\cos{k_{x}}\end{pmatrix}.$ (2)

Then, the bulk spectrum equals $\varepsilon=(f+t)\cos
k_{x}\break\pm\sqrt{(f-t)^{2}\cos^{2}k_{x}+2v^{2}(1+\cos k_{x})}$. If $t=-f$,
it exhibits the chiral symmetry,
$\Gamma_{2}H_{2}(k_{x})\Gamma_{2}^{*}=-H_{2}(k_{x})$, where
$\Gamma_{2}=\sigma_{y}T_{2}$, and the winding number is $W=0$ [66]. In a
general case, however, the $2\times 2$ bulk model in Eq. (2) possesses neither
chiral nor inversion symmetry. Thus, both the Zak phase and the winding number
are not quantized, and the use of bulk-edge correspondence is not justified.

_Topological invariant.—_ We propose to construct a “parental” model that
features the chiral symmetry, connected to our original zigzag model by a
continuous mapping, Fig. 3a. Here, this “parental” model is a ladder system,
Fig. 3b, with initial setting $w=0$, coinciding with Fig. 1a for $w=v$. Since
transition occurs without closing the energy gap, the two models can be
continuously connected by a mapping
$f(H_{4}):=(\Gamma_{4}H_{4}\Gamma_{4}^{*}-H_{4})/2$, where
$\Gamma_{4}=\sigma_{z}\otimes I_{2}$ is the chiral symmetry operator for the
“parental” model.

Let $H_{4}^{\text{par}}$ and $H_{4}^{\text{zig}}$ are the bulk Hamiltonians of
the “parental” and the zigzag models, respectively, both defined on a four-
site unit cell. Then,
$f(H_{4}^{\text{par}})=f(H_{4}^{\text{zig}})=H_{4}^{\text{par}}$. This allows
us to draw conclusions about topological edge states in the zigzag structure
solely based on the analysis of the ladder system. The construction of the
winding number for the latter requires a regularization step: the quasi-1D
nature necessitates a correct geometrical interpretation of couplings $v$
(perpendicular to the selected momentum direction) between two chains.

Figure 3: “Parental” model for the quasi-1D zigzag structure. (a) We
generalize the zigzag model by allowing the chains $A$ and $B$ to interact
with two alternating strengths, $v$ and $w$. For $f=t=0$ it restores the SSH
model. (b) If $w=0$, the model amounts to a ladder with four-site non-trivial
unit cell. (c) and (d) are two geometrical interpretations of the quasi-1D
ladder structure. Unit cells at (b), (c) and (d) correspond to Hamiltonians
$H_{4}\left(\raisebox{-0.25pt}{\includegraphics[width=11.9501pt]{Kachin_symbol3.pdf}}\right)$,
$H_{4}\left(\raisebox{-0.25pt}{\includegraphics[width=11.9501pt]{Kachin_symbol2.pdf}}\right)$
and
$H_{4}\left(\raisebox{-0.25pt}{\includegraphics[width=11.9501pt]{Kachin_symbol1.pdf}}\right)$,
respectively.

Let us now denote the unit cells of the ladder systems from Fig. 3b–d as , ,
and , respectively. While Hamiltonian
$H_{4}\left(\raisebox{-0.25pt}{\includegraphics[width=11.9501pt]{Kachin_symbol3.pdf}}\right)$
requires definition, the winding number integral calculated for Hamiltonians
$H_{4}\left(\raisebox{-0.25pt}{\includegraphics[width=11.9501pt]{Kachin_symbol2.pdf}}\right)$
and
$H_{4}\left(\raisebox{-0.25pt}{\includegraphics[width=11.9501pt]{Kachin_symbol1.pdf}}\right)$
yields $W=1$ in the presence of in-gap edge states, and does not converge in
their absence. Thus, we construct a “weighted” unit cell, and the
corresponding Hamiltonian
$H_{4}\left(\raisebox{-0.25pt}{\includegraphics[width=11.9501pt]{Kachin_symbol3.pdf}}\right)$

$H_{4}\left(\raisebox{-0.25pt}{\includegraphics[width=11.9501pt]{Kachin_symbol3.pdf}}\right)=\frac{H_{4}\left(\raisebox{-0.25pt}{\includegraphics[width=11.9501pt]{Kachin_symbol2.pdf}}\right)+\xi
H_{4}\left(\raisebox{-0.25pt}{\includegraphics[width=11.9501pt]{Kachin_symbol1.pdf}}\right)}{\xi+1},$
(3)

which properties in the basis of $\Gamma_{4}$ are determined by the upper-
right block

$h_{2}(k_{x},\xi)=\begin{pmatrix}f\bigl{(}1+e^{-ik_{x}}\bigr{)}&v\frac{e^{-ik_{x}}+\xi
e^{ik_{x}}}{\xi+1}\\\ v&t\bigl{(}1+e^{-ik_{x}}\bigr{)}\end{pmatrix}.$ (4)

It also governs the winding number which takes the form

$\displaystyle W={}$
$\displaystyle\frac{1}{i2\pi}\int_{BZ}dk_{x}\textrm{Tr}\left(h_{2}(k_{x},\xi)^{-1}\frac{d}{dk_{x}}h_{2}(k_{x},\xi)\right)={}$
$\displaystyle{}={}$
$\displaystyle\int_{0}^{2\pi}\frac{dk_{x}}{2\pi}\frac{\frac{2ft}{v^{2}}(\xi+1)\left(1+e^{-ik_{x}}\right)+\left(\xi
e^{2ik_{x}}-1\right)}{\frac{2ft}{v^{2}}(\xi+1)(1+\cos{k_{x}})-\left(\xi
e^{2ik_{x}}+1\right)}.$ (5)

For $\xi\ll 1$, the integral in $W$ does not converge. However, larger values
of $\xi$ lead to the identification of non-trivial region, namely $W=1$ for
$t\in(-\infty;v^{2}/(4f))$ and $W=0$ for $t\in[v^{2}/(4f);\infty)$.

Figure 4: Energy eigenmodes computed for a chiral ($t/f=-1$) zigzag system
with $N=90$ sites and $v/f=1$, averaged over 30 random instances of a disorder
in (a) nn interactions, (b) nnn interactions, and (c) on-site energy, as a
function of the disorder strength $\sigma/f$. The findings stem from analyzing
the finite Hamiltonian $H_{\sigma}=H+\sigma D$, featuring the tight-binding
Hamiltonian $H$ and the disorder operator $D$.

_Robustness to disorders.—_ We explored the robustness of edge states in the
zigzag system against disorders in the strengths of the nn and nnn
interactions, and against the spectral disorder, by computing the energy
spectrum of the Hamiltonian $H_{\sigma}=H+\sigma D$, where $H$ is defined by
Eq. (1), $\sigma$ is a constant that quantifies the strength of the disorder,
and $D$ is a matrix that defines the type of disorder. The matrix elements
$D_{l,m}=\Delta d\delta_{l\pm 1,m}$ and $D_{l,m}=\Delta d\delta_{l\pm 2,m}$
represent the disorder in the nn and nnn coupling constants, while
$D_{l,m}=\Delta d\delta_{l,m}$ model the spectral disorder. We sampled $\Delta
d$ uniformly in range $[-0.5;0.5]$.

Fig. 4 depicts the spectrum of $H_{\sigma}$ computed for the chiral ($t/f=-1$)
zigzag chain that was averaged over 30 random instances of a disorder in (a)
next-neighbor, (b) long-range interactions, and (c) on-site energy spectrum,
for $N=90$ sites and $v/f=1$. For each disorder type, as its strength
increases, the edge states become less localized and the the bulk bands
broaden. This effect becomes visible when disorders are as strong as the
system interaction, $\sigma/f=1$.

The edge states also persist in a zigzag system without the chiral symmetry.
Although they reveal similar level of robustness, the positive mode edge state
starts to delocalize at smaller disorders’ strength, $\sigma/f=0.5$ for
$t/f=-0.7$ and $\sigma/f=0.3$ for $t/f=-0.3$. Edge states persist in chains
with odd $N$, exhibiting degenerate energy levels, and higher disorders induce
a minor spectral spreading of these states (see SM, S2 and S3).

Figure 5: Eigenmodes of a finite zigzag structure, averaged over 30 random
realizations of loss, as a function of loss strength $\gamma/f$ for $v/f=1$.
Panels (a), (b), and (c) are computed for $t/f=-0.7$, with the numbers of
sites $N=40$, $N=60$, and $N=80$, respectively. In (d), the choice of
$t/f=-0.3$ facilitates resistant edge states for longer chains ($N=90$).
Optimizing system parameters allows well-localized edge states even with
losses’ strength as high as the system interaction. The findings stem from
analyzing the finite Hamiltonian $H_{\gamma}=H+i\gamma D^{\prime}$, featuring
the tight-binding Hamiltonian $H$ and the loss operator $D^{\prime}$.

_Resistance to losses.—_ We model dissipation in the zigzag structure by a
finite non-hermitian Hamiltonian $H_{\gamma}=H+i\gamma D^{\prime}$, where
$\gamma$ represents losses and $D^{\prime}$ is a diagonal matrix with elements
$D^{\prime}_{l,m}=\Delta d\delta_{l,m}$. We sample $\Delta d$ uniformly from
$[-0.5;0.5]$.

Figure 6: Our main result: robust topologically protected edge states in a
zigzag system affected simultaneously by losses and disorders in next-neighbor
and long-range interactions, and on-site energies; all quantified with
strength $\sigma$. Remarkably, the edge states remain well localized for the
disorders and losses comparable with the system interaction strength,
$\sigma/f\leq 0.3$. The plot is computed for a chain of $N=60$ sites, $v/f=1$,
(a) $t/f=-0.7$, (b) $t/f=-0.3$.

Fig. 5 shows eigenmodes of $H_{\gamma}$ computed for several zigzag chains,
and averaged over 30 random instances of losses, as a function of the loss
strength $\gamma/f$. We observe that the stability of edge states depends on
$N$ along with ratios $v/f$ and $t/f$. While larger $t/f$ increases the
resistance of edge states for bigger $N$, it simultaneously reduces the width
of the energy gap between the edge states and bulk bands (see SM, S4).
Moreover, the losses do not break the degeneracy of the edge states for a
system with odd $N$. (a), (b), and (c) highlight the system’s vulnerability to
losses as the number of sites increases for $t/f=-0.7$. (d) demonstrates the
robustness of edge states for a chain with $t/f=-0.3$ and $N=90$. As before,
$v/f=1$.

_Our main result_ is demonstrated in Fig. 6. It shows ultra-robust
topologically protected edge states obtained for the zigzag structure affected
simultaneously by dissipation and all three types of disorders considered. It
was obtained by an averaged simulation of the cumulative Hamiltonian
$H_{\sigma}+H_{\gamma}$, where for the simplicity of computations, all the
imperfections had the same strength $\sigma$. The computations were performed
for $N=60$ sites with $v/f=1$, for $t/f=-0.7$ – panel (a), and $t/f=-0.3$ –
panel (b). Remarkably, the edge states remain well localized for the disorders
and losses comparable with the system interaction strength, $\sigma/f\leq
0.5$. Here, the negative coupling is vital for securing a distinct, resilient
edge state against losses and various disorders in zigzag models of different
sizes. While a certain level of resilience can be attained by relying solely
on positive couplings, this comes at the cost of a smaller gap and potentially
weaker localization of the edge state, allowing for simultaneous excitations
in both edge and bulk states (see SM, S5). Tab. 1 compares the robustness of
our edge states to the SSH, AAH, and QTI models.

_Implementation.—_ There are several experimental platforms that can test our
results. In photonic lattices [55, 56, 59, 49] , each waveguide can correspond
to a site in either the A or B chains. An effective negative coupling can be
induced by inserting an auxiliary defect waveguide with a precisely tuned
refractive index between the two original waveguides, or by orbital-induced
synthetic flux in multiorbital ($s$ and $d$ orbitals) waveguide lattices. The
coupling noise can be controlled by adjusting waveguide spacing, with on-site
noise modulated by varying waveguide widths, and system loss achieved by
altering waveguide etch depth. Alternatively, one can use acoustic
metacrystals [46], fabricating the proposed chain through 3D printing, with
negative couplings formed by a special choice of unit cell connections, or
superconducting circuits [67], where adjacent sites in the lattice are
connected through SQUIDs and the magnetic flux can assist in manipulating the
coupling sign.

| SSH | AAH | QTI | Our model
---|---|---|---|---
System properties
Couplings | positive | positive | positive | positive
|  |  | & negative | & negative
Topological | 1-2 | 1-4 | 1-4 | 1-2
states | at $\varepsilon=0$ | at $\varepsilon\neq 0$ | at $\varepsilon=0$ | at $\varepsilon\neq 0$
Self-energy | – | yes | – | yes
Robustness to disorder type
In couplings | ✓ | ✓ | ✓ | ✓
On-site energy | – | ✓ | – | ✓
Losses | – | – | – | ✓
(dissipation) |  |  |  |
Table 1: Robustness of topologically protected edge states obtained in the
SSH, AAH, and quadrupole topological insulator (QTI) models, with these
observed in the discussed zigzag structure with negative couplings.
$\varepsilon$ denotes the energy level of an edge state.

_Discussion and conclusions.—_ We demonstrated an extreme robustness of
topologically protected edge states emerging from the interacting quasi-1D
zigzag structure, where some of the couplings are negative. These states
remain well localized in the simultaneous presence of dissipation and
disorders in short- and long-range interactions, as well as in on-site
energies, whose strengths are comparable with interactions in the system. We
also identified several physical platforms that are capable of testing our
results.

We proved non-trivial topological nature of the edge states using the bulk-
boundary correspondence, despite the lack of chiral symmetry in the system. To
this end, we proposed a new method that relates the original zigzag system via
a continuous mapping to a ladder model featuring chiral symmetry, which is
investigated in a larger, specially constructed unit cell.

The model’s simplicity and generality can make it a cornerstone for future
practical quantum topology-based applications. Realization of negative
couplings plays a pivotal role in broadening the range of system parameters,
enabling it to support edge states, develop new symmetries, and control the
energy band gap size.

_Acknowledgments.—_ V.K. and M.S. were supported by the European Union’s
Horizon 2020 research and innovation programme under the Marie Skłodowska-
Curie project “AppQInfo” No. 956071. M.S. was supported by the National
Science Centre “Sonata Bis” project No. 2019/34/E/ST2/00273, and the QuantERA
II Programme that received funding from the European Union’s Horizon 2020
research and innovation programme under Grant Agreement No. 101017733, project
“PhoMemtor” No. 2021/03/Y/ST2/00177. V.K. thanks Alexander Khanikaev for
initial discussions.

## References

  * [1] M. Z. Hasan and C. L. Kane.  Colloquium: Topological insulators.  Rev. Mod. Phys., 82:3045–3067, 11 2010.
  * [2] Xiao-Liang Qi and Shou-Cheng Zhang.  Topological insulators and superconductors.  Rev. Mod. Phys., 83:1057–1110, 10 2011.
  * [3] Klaus von Klitzing, Tapash Chakraborty, Philip Kim, Vidya Madhavan, Xi Dai, James McIver, Yoshinori Tokura, Lucile Savary, Daria Smirnova, Ana Maria Rey, Claudia Felser, Johannes Gooth, and Xiaoliang Qi.  40 years of the quantum hall effect.  Nat Rev Phys, 2(8):397–401, 08 2020.
  * [4] Cui-Zu Chang, Chao-Xing Liu, and Allan H. MacDonald.  Colloquium: Quantum anomalous hall effect.  Rev. Mod. Phys., 95:011002, 01 2023.
  * [5] Steven R. Elliott and Marcel Franz.  Colloquium: Majorana fermions in nuclear, particle, and solid-state physics.  Rev. Mod. Phys., 87:137–163, 02 2015.
  * [6] M. Zahid Hasan, Guoqing Chang, Ilya Belopolski, Guang Bian, Su-Yang Xu, and Jia-Xin Yin.  Weyl, dirac and high-fold chiral fermions in topological quantum matter.  Nat Rev Mater, 6(9):784–803, 09 2021.
  * [7] Dennis M. Nenno, Christina A. C. Garcia, Johannes Gooth, Claudia Felser, and Prineha Narang.  Axion physics in condensed-matter systems.  Nat Rev Phys, 2(12):682–696, 12 2020.
  * [8] A.Yu. Kitaev.  Fault-tolerant quantum computation by anyons.  Annals of Physics, 303(1):2–30, 2003.
  * [9] Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, and Sankar Das Sarma.  Non-abelian anyons and topological quantum computation.  Rev. Mod. Phys., 80:1083–1159, 09 2008.
  * [10] Jiannis K. Pachos.  Introduction to Topological Quantum Computation.  Cambridge University Press, 2012.
  * [11] Péter Boross, János K. Asbóth, Gábor Széchenyi, László Oroszlány, and András Pályi.  Poor man’s topological quantum gate based on the su-schrieffer-heeger model.  Phys. Rev. B, 100:045414, 07 2019.
  * [12] Ville Lahtinen and Jiannis K. Pachos.  A Short Introduction to Topological Quantum Computation.  SciPost Phys., 3:021, 2017.
  * [13] Sunil Mittal, Venkata Vikram Orre, and Mohammad Hafezi.  Topologically robust transport of entangled photons in a 2d photonic system.  Opt. Express, 24(14):15631–15641, 07 2016.
  * [14] Mikael C. Rechtsman, Yaakov Lumer, Yonatan Plotnik, Armando Perez-Leija, Alexander Szameit, and Mordechai Segev.  Topological protection of photonic path entanglement.  Optica, 3(9):925–930, 09 2016.
  * [15] Samuel Boutin, Pedro L. S. Lopes, Anqi Mu, Udson C. Mendes, and Ion Garate.  Topological Josephson bifurcation amplifier: Semiclassical theory.  J. Appl. Phys., 129(21):214302, 06 2021.
  * [16] Jason Alicea, Yuval Oreg, Gil Refael, Felix von Oppen, and Matthew P. A. Fisher.  Non-abelian statistics and topological quantum information processing in 1d wire networks.  Nat. Phys., 7(5):412–417, 05 2011.
  * [17] Andrea Blanco-Redondo, Bryn Bell, Dikla Oren, Benjamin J. Eggleton, and Mordechai Segev.  Topological protection of biphoton states.  Science, 362(6414):568–571, 2018.
  * [18] Fulvio Flamini, Nicolo Spagnolo, and Fabio Sciarrino.  Photonic quantum information processing: a review.  Rep. Prog. Phys., 82(1):016001, 2018.
  * [19] Jean-Luc Tambasco, Giacomo Corrielli, Robert J. Chapman, Andrea Crespi, Oded Zilberberg, Roberto Osellame, and Alberto Peruzzo.  Quantum interference of topological states of light.  Sci. Adv., 4(9):eaat3187, 2018.
  * [20] Zengji Yue, Xiaolin Wang, and Min Gu.  Topological insulator materials for advanced optoelectronic devices.  Advanced Topological Insulators, pages 45–70, 2019.
  * [21] Haoran Xue, Yihao Yang, and Baile Zhang.  Topological valley photonics: Physics and device applications.  Adv. Photonics Res., 2(8):2100013, 2021.
  * [22] Hannah Price, Yidong Chong, Alexander Khanikaev, Henning Schomerus, Lukas J Maczewsky, Mark Kremer, Matthias Heinrich, Alexander Szameit, Oded Zilberberg, Yihao Yang, Baile Zhang, Andrea Alù, Ronny Thomale, Iacopo Carusotto, Philippe St-Jean, Alberto Amo, Avik Dutt, Luqi Yuan, Shanhui Fan, Xuefan Yin, Chao Peng, Tomoki Ozawa, and Andrea Blanco-Redondo.  A. roadmap on topological photonics.  Journal Of Physics: Photonics, 4, Jun 2022.
  * [23] Viktor M. Puchnin, Olga V. Matvievskaya, Alexey P. Slobozhanyuk, Alena V. Shchelokova, and Nikita A. Olekhno.  Application of topological edge states in magnetic resonance imaging.  Phys. Rev. Appl., 20:024076, 08 2023.
  * [24] W. P. Su, J. R. Schrieffer, and A. J. Heeger.  Solitons in polyacetylene.  Phys. Rev. Lett., 42:1698–1701, 06 1979.
  * [25] Hajime Takayama, Y. R. Lin-Liu, and Kazumi Maki.  Continuum model for solitons in polyacetylene.  Phys. Rev. B, 21:2388–2393, 03 1980.
  * [26] Shinsei Ryu and Yasuhiro Hatsugai.  Topological origin of zero-energy edge states in particle-hole symmetric systems.  Phys. Rev. Lett., 89:077002, 07 2002.
  * [27] Linhu Li, Zhihao Xu, and Shu Chen.  Topological phases of generalized su-schrieffer-heeger models.  Phys. Rev. B, 89:085111, 02 2014.
  * [28] Maria Maffei, Alexandre Dauphin, Filippo Cardano, Maciej Lewenstein, and Pietro Massignan.  Topological characterization of chiral models through their long time dynamics.  New J. Phys., 20(1):013023, 01 2018.
  * [29] Jung-Wan Ryu, Sungjong Woo, Nojoon Myoung, and Hee Chul Park.  Topological edge states in bowtie ladders with different cutting edges.  Physica E: Low-dimensional Systems and Nanostructures, 137:114941, 2022.
  * [30] Chen-Shen Lee, Iao-Fai Io, and Hsien chung Kao.  Winding number and zak phase in multi-band ssh models.  Chinese Journal of Physics, 78:96–110, 2022.
  * [31] Beatriz Pérez-González, Miguel Bello, Álvaro Gómez-León, and Gloria Platero.  Interplay between long-range hopping and disorder in topological systems.  Phys. Rev. B, 99:035146, 01 2019.
  * [32] Mor Verbin, Oded Zilberberg, Yaacov E. Kraus, Yoav Lahini, and Yaron Silberberg.  Observation of topological phase transitions in photonic quasicrystals.  Phys. Rev. Lett., 110:076403, 02 2013.
  * [33] Yaacov E Kraus and Oded Zilberberg.  Quasiperiodicity and topology transcend dimensions.  Nat. Phys., 12(7):624–626, 2016.
  * [34] Oded Zilberberg.  Topology in quasicrystals.  Opt. Mater. Express, 11(4):1143–1157, 04 2021.
  * [35] Serge Aubry and Gilles André.  Analyticity breaking and anderson localization in incommensurate lattices.  Ann. Israel Phys. Soc, 3(133):18, 1980.
  * [36] Yaacov E. Kraus, Yoav Lahini, Zohar Ringel, Mor Verbin, and Oded Zilberberg.  Topological states and adiabatic pumping in quasicrystals.  Phys. Rev. Lett., 109:106402, 09 2012.
  * [37] Jian Li, Rui-Lin Chu, J. K. Jain, and Shun-Qing Shen.  Topological anderson insulator.  Phys. Rev. Lett., 102:136806, 04 2009.
  * [38] Simon Stützer, Yonatan Plotnik, Yaakov Lumer, Paraj Titum, Netanel H Lindner, Mordechai Segev, Mikael C Rechtsman, and Alexander Szameit.  Photonic topological anderson insulators.  Nature, 560(7719):461–465, 2018.
  * [39] Dmitry V. Zhirihin and Yuri S. Kivshar.  Topological photonics on a small scale.  Small Sci., 1(12):2100065, 2021.
  * [40] Mordechai Segev and Miguel A. Bandres.  Topological photonics: Where do we go from here?  Nanophotonics, 10(1):425–434, 2021.
  * [41] Rajesh Chaunsali, Haitao Xu, Jinkyu Yang, Panayotis G. Kevrekidis, and Georgios Theocharis.  Stability of topological edge states under strong nonlinear effects.  Phys. Rev. B, 103:024106, 01 2021.
  * [42] Xiaotian Shi, Ioannis Kiorpelidis, Rajesh Chaunsali, Vassos Achilleos, Georgios Theocharis, and Jinkyu Yang.  Disorder-induced topological phase transition in a one-dimensional mechanical system.  Phys. Rev. Res., 3:033012, 07 2021.
  * [43] Wladimir A. Benalcazar, B. Andrei Bernevig, and Taylor L. Hughes.  Quantized electric multipole insulators.  Science, 357(6346):61–66, 2017.
  * [44] Nianzu Fu, Ziwei Fu, Huaiyuan Zhang, Qing Liao, Dong Zhao, and Shaolin Ke.  Topological bound modes in optical waveguide arrays with alternating positive and negative couplings.  Opt Quant Electron, 52(2):61, 01 2020.
  * [45] Mark Kremer, Ioannis Petrides, Eric Meyer, Matthias Heinrich, Oded Zilberberg, and Alexander Szameit.  A square-root topological insulator with non-quantized indices realized with photonic aharonov-bohm cages.  Nat Commun, 11(1):907, 02 2020.
  * [46] Xiang Ni, Mengyao Li, Matthew Weiner, Andrea Alù, and Alexander B. Khanikaev.  Demonstration of a quantized acoustic octupole topological insulator.  Nat Commun, 11(1):2108, 04 2020.
  * [47] Danwei Liao, Zhiwang Zhang, Ying Cheng, and Xiaojun Liu.  Engineering negative coupling and corner modes in a three-dimensional acoustic topological network.  Phys. Rev. B, 105:184108, 05 2022.
  * [48] Valerii I. Kachin and Maxim A. Gorlach.  Three-dimensional higher-order topological insulator protected by cubic symmetry.  Phys. Rev. Appl., 16:024032, 08 2021.
  * [49] Chuang Jiang, Yiling Song, Xiaohong Li, Peixiang Lu, and Shaolin Ke.  Photonic möbius topological insulator from projective symmetry in multiorbital waveguides.  Opt. Lett., 48(9):2337–2340, 05 2023.
  * [50] Christopher W. Peterson, Wladimir A. Benalcazar, Taylor L. Hughes, and Gaurav Bahl.  A quantized microwave quadrupole insulator with topologically protected corner states.  Nature, 555(7696):346–350, 03 2018.
  * [51] Jean Dalibard, Fabrice Gerbier, Gediminas Juzeliūnas, and Patrik Öhberg.  Colloquium: Artificial gauge potentials for neutral atoms.  Rev. Mod. Phys., 83:1523–1543, 11 2011.
  * [52] Xiaopeng Li, Erhai Zhao, and W. Vincent Liu.  Topological states in a ladder-like optical lattice containing ultracold atoms in higher orbital bands.  Nat Commun, 4(1):1523, 02 2013.
  * [53] Matthieu Bellec, Ulrich Kuhl, Gilles Montambaux, and Fabrice Mortessagne.  Topological transition of dirac points in a microwave experiment.  Phys. Rev. Lett., 110:033902, 01 2013.
  * [54] Sung Hyun Nam, Antoinette J. Taylor, and Anatoly Efimov.  Diabolical point and conical-like diffraction in periodic plasmonic nanostructures.  Opt. Express, 18(10):10120–10126, 05 2010.
  * [55] J. M. Zeuner, M. C. Rechtsman, R. Keil, F. Dreisow, A. Tünnermann, S. Nolte, and A. Szameit.  Negative coupling between defects in waveguide arrays.  Opt. Lett., 37(4):533–535, 02 2012.
  * [56] Robert Keil, Charles Poli, Matthias Heinrich, Jake Arkinstall, Gregor Weihs, Henning Schomerus, and Alexander Szameit.  Universal sign control of coupling in tight-binding lattices.  Phys. Rev. Lett., 116:213901, 05 2016.
  * [57] Simon R. Pocock, Xiaofei Xiao, Paloma A. Huidobro, and Vincenzo Giannini.  Topological plasmonic chain with retardation and radiative effects.  ACS Photonics, 5(6):2271–2279, 06 2018.
  * [58] G. Cáceres-Aravena, L. E. F. Foa Torres, and R. A. Vicencio.  Topological and flat-band states induced by hybridized linear interactions in one-dimensional photonic lattices.  Phys. Rev. A, 102:023505, 08 2020.
  * [59] Julian Schulz, Jiho Noh, Wladimir A. Benalcazar, Gaurav Bahl, and Georg von Freymann.  Photonic quadrupole topological insulator using orbital-induced synthetic flux.  Nat Commun, 13(1):6597, 11 2022.
  * [60] Yajuan Qi, Chunyin Qiu, Meng Xiao, Hailong He, Manzhu Ke, and Zhengyou Liu.  Acoustic realization of quadrupole topological insulators.  Phys. Rev. Lett., 124:206601, 05 2020.
  * [61] Avik Dutt, Momchil Minkov, Ian A. D. Williamson, and Shanhui Fan.  Higher-order topological insulators in synthetic dimensions.  Light Sci Appl, 9(1):131, 07 2020.
  * [62] Xiang Ni and Andrea Alù.  Higher-order topolectrical semimetal realized via synthetic gauge fields.  APL Photonics, 6(5):050802, 05 2021.
  * [63] Jiacheng Bao, Deyuan Zou, Weixuan Zhang, Wenjing He, Houjun Sun, and Xiangdong Zhang.  Topoelectrical circuit octupole insulator with topologically protected corner states.  Phys. Rev. B, 100(20):201406(R), 11 2019.
  * [64] Shuo Liu, Shaojie Ma, Qian Zhang, Lei Zhang, Cheng Yang, Oubo You, Wenlong Gao, Yuanjiang Xiang, Tie Jun Cui, and Shuang Zhang.  Octupole corner state in a three-dimensional topological circuit.  Light Sci Appl, 9:145, 2020.
  * [65] D.J. Thouless.  Electrons in disordered systems and the theory of localization.  Physics Reports, 13(3):93–142, 1974.
  * [66] Shinsei Ryu, Andreas P Schnyder, Akira Furusaki, and Andreas WW Ludwig.  Topological insulators and superconductors: tenfold way and dimensional hierarchy.  New J. Phys., 12(6):065010, 2010.
  * [67] Xiu-Hao Deng, Chen-Yen Lai, and Chih-Chun Chien.  Superconducting circuit simulator of bose-hubbard model with a flat band.  Phys. Rev. B, 93:054116, 02 2016.

Supplemental Material: Ultra-robust topologically protected edge states in
quasi-1D systems

##  SM1 Inverse participation ratio

To quantify mode localization properties, we calculate an inverse
participation ratio $\mathfrak{I}$ (IPR), defined in the following way
THOULESS197493_1

$\mathfrak{I}=\left(\sum\limits_{m}\,|\psi_{m}|^{4}\right)\,\left(\sum\limits_{m}\,|\psi_{m}|^{2}\right)^{-2}.$
(6)

Here, $\psi_{m}$ represents a mode amplitude on site $m$ in the 1D system. The
mode corresponds to e.g., a field strength, voltage, pressure, or other
relevant physical quantities characteristic to a physical implementation under
consideration. The summation is performed over all $N$ sites within the chain.

##  SM2 Robustness for chains with even number of sites in the non-chiral
cases

In the figures presented in the main text, we illustrate the robustness
properties of a chain consisting of an even number of sites in the chiral
case. Interestingly, nonchiral cases exhibit nearly the same level of
robustness. However, the band gap and spectrum no longer maintain symmetry,
and the influence of the bulk states on the edge states becomes noticeable at
a disorder strength smaller than that in the chiral case, Fig. SM_1.

Figure SM_1: Energy eigenmodes computed for the zigzag system with $N=90$
sites, while setting $v/f=1$ and (a) $t/f=-0.7$ and (b) $t/f=-0.3$. Each case
was averaged over 30 random instances of a disorder $\sigma$, for (left to
right) the next-neighbour (nn) interactions, long-range (nnn) interactions,
and the on-site energy.

##  SM3 Robustness for chains with odd number of sites in the chiral case
$t/f=-1$

In the figures presented in the main text, we illustrate the robustness
properties of a chain consisting of an even number of sites in the chiral
case. Interestingly, the edge states can be spotted also in chains with an odd
number of sites, despite its degenerate energy levels. Higher disorders in
this system result in a gradual spectral spreading of edge states that remains
rather insignificant, e.g., less than $1\%$ of the band gap for
$\sigma/f<0.2$. It increases by $4\%$ for $\sigma/f=0.5$, and reaches $10\%$
for $\sigma/f=1$, Fig. SM_2.

Figure SM_2: Energy eigenmodes computed for the zigzag system with $N=91$
sites, while setting $v/f=1$ and $t/f=-1$. The three panels have been averaged
over 30 random instances of a disorder $\sigma$ for the (a) next-neighbour
(nn) interactions, (b) long-range (nnn) interactions, and (c) the on-site
energy.

##  SM4 Resistance to losses as optimization task

To investigate the dependence of edge state resistance to losses in the range
$\gamma\in[0.0,0.1]$, we conducted calculations for $2.3\times 10^{5}$
parameter sets ${v/f,t/f,N}$, where $v/f\in[0.3,3.0]$, $t/f\in[-3.0,0.5]$, and
$N\in[10,100]$. Each set underwent an initial assessment for the presence of
edge states using the IPR. The results are depicted in Fig. SM_3, where the
sets lacking edge states are marked in green.

Subsequently, the sets with edge states were analyzed for their ability to
maintain well-localized edge states at the same energy while increasing
$\gamma$ from $0$ to $0.1$. The ones exhibiting this property are labeled
yellow, while those without it are marked with violet color. For clarity, all
sets were grouped by their $N$ value. The black solid lines in Fig. SM_3
represent the parameter choices for the figures from the main text.

From these results, it is evident that the resistance of edge states to losses
is directly correlated with $N$. The green areas in the panels of Fig. SM_3
align with the parabolic prediction from the topological invariant. Another
observation is that most sets exhibit resistance within the ranges
$t/f\in[-1.0,0.5]$ and $v/f\in[1.0,3.0]$. However, such strong near
interactions lead to a reduction in the gap between the edge state and the
closest energy band.

Figure SM_3: Resistance to losses for $v/f\in\left[0.3;3.0\right]$ and
$t/f\in\left[-3.0;0.5\right]$. Green color represents the parameter region
where the system does not support edge states, while yellow and violet regions
correspond to areas with edge states, distinguishing between those that
exhibit and those that do not exhibit the resistance, respectively. For
simplicity, the loss resistance is defined as the property of an edge state to
remain selected for the same energy within the range
$\gamma\in\left[0.0;0.1\right]$. Each panel represents results based on
approximately $5\times 10^{4}$ random points within the specified range of
parameters $v/f$, $t/f$ and the system size (a) $N\in[20;40]$, (b)
$N\in[40;60]$, (c) $N\in[60;80]$, and (d) $N\in[80;100]$. The horizontal solid
line represents the chiral case $t/f=-1$, while the vertical one denotes the
case of $v/f=1$.

##  SM5 Simultaneous presence of all types of disorder and losses in systems
with positive and negative couplings

We would like to highlight that the studied system supports edge states when
$t$ is both positive and negative, and these states remain robust in the
presence of a certain level of disorders and losses. However, it is essential
to consider the distinct nature of these edge states. Systems with different
choices of $v/f$ and $t/f$ exhibit variations in localization and energy gap
between the edge state and the closest bulk state. This variability is of
significance for diverse implementations and applications.

In Fig. SM_4a, the system displays the largest energy gap,
$\Delta\tilde{\varepsilon}=(\varepsilon_{\text{edge
state}}-\varepsilon_{\text{bulk state}})/\max(\lvert v/f\rvert,\lvert
t/f\rvert,1)$, with normalization aimed at ensuring the largest couplings in
the Hamiltonian are less than or equal to 1. Notably, a substantial portion of
the brightest region resides in the diagram’s negative $t/f$ quadrant.

Panels (b)–(d) illustrate the potential strength $\sigma$ when the system
continues to support edge states at the same ($\pm 15\%$) energy level and
with identical localization. Disorder and losses are introduced to the system
Hamiltonian as $H_{\sigma}=H+\sigma(D+iD^{\prime})$ with operators $D$ and
$D^{\prime}$, respectively. Here, the reduction in the parameter region, where
the system maintains robust edge states as its size increases, is primarily
associated with a decrease in resistance to losses, as predicted in Section
SM4.

Figs. SM_4e–g represent the elementwise multiplication of panel (a) with
panels (b)–(d), respectively. These figures distinctly indicate that the best
edge states (considering localization, distance to the closest band, and
robustness) are observed in systems with negative couplings. They also reveal
a possibility of having edge states in zigzag chains exclusively with positive
couplings, albeit with certain limitations that require attention. This paves
the way to further research and simplifies the potential manufacturing process
of the samples.

Figure SM_4: Optimization of the band gap size and robustness against
simultaneous presence of all types of disorder and losses, with equal strength
$\sigma$, computed for $v/f\in\left[0.3;4.1\right]$ and
$t/f\in\left[-1;1\right]$. (a) Gap size $\varepsilon_{\text{edge
state}}-\varepsilon_{\text{bulk state}}$ computed for the relative distance
between the zigzag’s edge state and the closest band, normalized to
$\max(\lvert t/f\rvert,\lvert v/f\rvert,1)$. Panels (b)–(d) depict the
maximally possible strength $\sigma$ of disorder and losses, while maintaining
the edge states of the zigzag model at the same ($\pm 15\%$) energy level and
with an equivalent level of localization. The sizes of the systems in (b),
(c), and (d) are $N=50$, $N=70$, and $N=90$, respectively. In (e)–(g), the
results of element-wise multiplying the data from plot (a) by values from
(b)–(d) are presented. Remarkably, the optimal choice for the most stable and
remote edge states from the band edge can be achieved in models with negative
couplings. However, for specific purposes, where the distance between the edge
state and the band is less critical, it is possible to operate solely with
positive couplings to attain a good level of resistance.

unsrt our1


# A mild Girsanov formula

Giuseppe Da Prato
Scuola Normale Superiore, Pisa, Italy  Enrico Priola
University of Pavia, Pavia, Italy  Luciano Tubaro
University of Trento, Trento, Italy

(6 August 2023)

###### Abstract

We consider a well posed SPDE$\colon dZ=(AZ+b(Z))dt+dW(t),\,Z_{0}=x,$ on a
separable Hilbert space $H$, where $A\colon H\to H$ is self-adjoint, negative
and such that $A^{-1+\beta}$ is of trace class for some $\beta>0$, $b\colon
H\to H$ is Lipschitz continuous and $W$ is a cylindrical Wiener process on
$H$. We denote by $W_{A}(t)=\int_{0}^{t}e^{(t-s)A}\,dW(s),\,t\in[0,T],$ the
stochastic convolution. We prove, with the help of a formula for nonlinear
transformations of Gaussian integrals due to R. Ramer [15], the following
identity

$(\mathbb{P}\circ Z_{x}^{-1})(\Phi)=\int_{X}\Phi(h+e^{\cdot
A}x)\,\exp\left\\{-\tfrac{1}{2}|\gamma_{x}(h)|^{2}_{\mathscr{H}_{\mathbb{Q}_{T}}}+I(\gamma_{x})(h)\right\\}\mathscr{N}_{\mathbb{Q}_{T}}(dh),$

where $\mathscr{N}_{\mathbb{Q}_{T}}$ is the law of $W_{A}$ in $C([0,T],H)$,
$\mathscr{H}_{\mathbb{Q}_{T}}$ its Cameron–Martin space,

$[\gamma_{x}(k)](t)=\int_{0}^{t}e^{(t-s)A}b(k(s)+e^{sA}x)ds,\quad
t\in[0,T],\;k\in C([0,T],H)$

and $I(\gamma_{x})$ is the Itô integral of $\gamma_{x}$. Some applications are
discussed; in particular, when $b$ is dissipative we provide an explicit
formula for the law of the stationary process and the invariant measure $\nu$
of the Markov semigroup $(P_{t})$. Some concluding remarks are devoted to a
similar problem with colored noise.

2000 Mathematics Subject Classification AMS: 60H15, 60H07, 60H30.

Key words: Mild stochastic equation, Girsanov formula, Malliavin Calculus,
Gaussian divergence.

###### Contents

  1. 1 Introduction and setting of the problem
  2. 2 The Cameron–Martin space
    1. 2.1 Brownian Motion
  3. 3 Proof of the mild Girsanov formula
    1. 3.1 The Ramer identity
    2. 3.2 Some estimates
  4. 4 Invariant measure
    1. 4.1 Preliminaries on Gaussian measures on locally convex spaces
    2. 4.2 Existence of invariant measure
    3. 4.3 Law of $W_{A,-\infty}$ on $E_{-\infty}$
    4. 4.4 Representation formula for the invariant measure
  5. 5 Colored noise
  6. A Maximal regularity for linear evolution equations
  7. B A disintegration theorem

##  1 Introduction and setting of the problem

Let $H$ be a real separable Hilbert space, (norm $|\cdot|_{H}$, scalar product
$\langle\cdot,\cdot\rangle_{H}$). We are concerned with the following
stochastic differential equation on $H$,

$\left\\{\begin{array}[]{l}dZ_{x}(t)=(AZ_{x}(t)+b(Z_{x}(t))dt+dW(t),\quad
t\geq 0,\\\ \\\ Z_{x}(0)=x\in H,\end{array}\right.$ (1.1)

under the following assumptions.

######  Hypothesis 1.

(i) $A\colon D(A)\subset H\to H$ is a self–adjoint operator and there is
$\omega>0$ such that $\|e^{tA}\|_{\mathscr{L}(H)}\leq e^{-\omega t},\,t\geq
0.$ Moreover, $(-A)^{-1+\beta}$ is of trace class for some $\beta>0$.

(ii) $b\colon H\to H$ is Lipschitz continuous.

(iii) $W$ is an $H$–valued cylindrical Wiener process on a filtered
probability space $(\Omega,\mathscr{F},(\mathscr{F}_{t})_{t\geq
0},\mathbb{P})$.

We shall denote by $W_{A}$ the stochastic convolution

$W_{A}(t)=\int_{0}^{t}e^{(t-s)A}\,dW(s),\quad t\geq 0.$ (1.2)

Thanks to Hypothesis 1(i), $W_{A}$ is a continuous process, see, e.g., [4,
Theorem 2.6].

We fix $T>0$ and set $E_{[0,T]}=C([0,T];H)$, the space of all continuous
mapping $[0,T]\to H$ endowed with the sup norm; we shall denote by
$\mathscr{B}(E_{[0,T]})$ the $\sigma$–algebra of all Borel subsets of
$E_{[0,T]}$. Moreover, $B_{b}(E_{[0,T]})$ will represent the space of all
mappings $E_{[0,T]}\to\mathbb{R}$ which are Borel and bounded.

The law of $W_{A}$ in $E_{[0,T]}$ is Gaussian $\mathscr{N}_{\mathbb{Q}_{T}}$,
that is, it has mean $0$ and covariance $\mathbb{Q}_{T}$; it will be described
in Section 2 below.

The goal of this paper is to determine the law of $Z_{x}$ on $E_{[0,T]}$ for
all $x\in H$ represented as a special integral with respect to
$\mathscr{N}_{\mathbb{Q}_{T}}$, that we shall call a mild Girsanov formula,
and to deduce some consequences from this identity.

We shall proceed as follows. Setting $Z_{x}(t)-e^{tA}x=:K_{x}(t),\,t\in[0,T],$
equation (1.1) becomes

$K_{x}(t)=\int_{0}^{t}e^{(t-s)A}\,b(K_{x}(s)+e^{sA}x)ds+W_{A}(t),\quad
t\in[0,T].$ (1.3)

Then we associate with the stochastic equation (1.3) the following
deterministic one,

$k_{x}(t)=\int_{0}^{t}e^{(t-s)A}b(k_{x}(s)+e^{sA}x)ds+h(t),\quad h\in
E_{[0,T]},\,t\in[0,T],$ (1.4)

that, by a standard fixed point arguments, has a unique solution in
$E_{[0,T]}$ that we denote by $k_{x}(t),\,t\in[0,T]$. Next, for every $x\in H$
we consider the mapping

$F_{x}\colon E_{[0,T]}\to E_{[0,T]},\quad h\to F_{x}(h)=k_{x}.$ (1.5)

We shall denote by $G_{x}:E_{[0,T]}\to E_{[0,T]}$ the inverse mapping of
$F_{x}$, which reads as follows

$[G_{x}(h)](t)=h(t)-\int_{0}^{t}e^{(t-s)A}\,b(h(s)+e^{sA}x)\,ds,\quad
h\color[rgb]{0,0,0}\in E_{[0,T]}.$ (1.6)

Since $b$ is Lipschitz continuous, both $F_{x}$ and $G_{x}$ are continuous;
therefore they are homeomorphisms from $E_{[0,T]}$ onto $E_{[0,T]}$.

A key role will be plaid in what follows by the mapping
$\gamma_{x}(h)=h-G_{x}(h),\,h\in E_{[0,T]},$ (see e.g. identity (1.9)),

$[\gamma_{x}(k)](t)=\int_{0}^{t}e^{(t-s)A}b(k(s)+e^{sA}x)\,ds,\quad k\in
E_{[0,T]},\,t\in[0,T].$ (1.7)

Now the solution $K_{x}(\cdot)$ of (1.3) can be written as
$K_{x}(\cdot)=F_{x}(W_{A}(\cdot)).$ So, the law of $Z_{x}$ on $E_{[0,T]}$ is
given by

$(\mathbb{P}\circ
Z_{x}^{-1})(\Phi)=\mathbb{E}[\Phi(Z_{x}(\cdot))]=\mathbb{E}[\Phi(K_{x}(\cdot)+e^{\cdot
A}x)]=\mathbb{E}[\Phi(F_{x}(W_{A}(\cdot))+e^{\cdot A}x)],\quad\forall\,\Phi\in
B_{b}(\mathbb{E}_{[0,T]}).$

Therefore, by a change of variables, it results

$(\mathbb{P}\circ
Z_{x}^{-1})(\Phi)=\int_{E_{[0,T]}}\Phi(F_{x}(h(\cdot))+e^{\cdot
A}x)\,\mathscr{N}_{\mathbb{Q}_{T}}(dh),\quad\forall\,\Phi\in
B_{b}(E_{[0,T]}).$ (1.8)

This formula is usual when dealing with stochastic equations with additive
noise, see e.g. [9]. But we look for a formula involving $G_{x}$ rather than
$F_{x}$ because $G_{x}$ is explicit while $F_{x}$ is not.

So, we set $F_{x}(h)=k,$ and obtain

$(\mathbb{P}\circ Z_{x}^{-1})(\Phi)=\int_{E_{[0,T]}}\Phi(k+e^{\cdot
A}x)\,(\mathscr{N}_{\mathbb{Q}_{T}}\circ G_{x})(dk),\quad\forall\,\Phi\in
B_{b}(E_{[0,T]}).$ (1.9)

But now we need to show that

$N_{\mathbb{Q}_{T}}\circ G_{x}\ll N_{\mathscr{\mathbb{Q}}_{T}}.$ (1.10)

In fact, using an identity due to R. Ramer, [15](see Section 3.1 below) we
show that

$\frac{d(\mathscr{N}_{\mathbb{Q}_{T}}\circ
G_{x})}{d\mathscr{N}_{\mathbb{Q}_{T}}}\,(h)=\exp\left\\{-\tfrac{1}{2}|\gamma_{x}(h)|^{2}_{\mathscr{H}_{\mathbb{Q}_{T}}}+[I(\gamma_{x})](h)\right\\},\quad\mathscr{N}_{\mathbb{Q}_{T}}-a.s.$
(1.11)

where $\mathscr{H}_{\mathbb{Q}_{T}}$ denotes the Cameron–Martin space of
$\mathscr{N}_{\mathbb{Q}_{T}}$ and $I(\gamma_{x})$ the Itô integral of
$\gamma_{x}$, which results to be adapted.

So, we end up with, the identity

$(\mathbb{P}\circ Z_{x}^{-1})(\Phi)=\int_{E_{[0,T]}}\Phi(h+e^{\cdot
A}x)\,\exp\left\\{-\tfrac{1}{2}|\gamma_{x}(h)|^{2}_{\mathscr{H}_{\mathbb{Q}_{T}}}+(\gamma_{x})(h)\right\\}\mathscr{N}_{\mathbb{Q}_{T}}(dh),$
(1.12)

which it will be proved in Theorem 3.2 below.

As a consequence of (1.12) we obtain an expression for the transition
semigroup $P_{t},\;t\geq 0,$ corresponding to the process $Z_{x}$, see
Corollary 3.3,

$P_{t}\varphi(x)=\mathbb{E}[\varphi(Z_{x}(t))]=\int_{E_{[0,T]}}\varphi(k(t)+e^{tA}x)\,\exp\left\\{-\tfrac{1}{2}|\gamma_{x}(k)|^{2}_{\mathscr{H}_{\mathbb{Q}_{T}}}+I(\gamma_{x})(k)\right\\}\,\mathscr{N}_{\mathbb{Q}_{T}}(dk),\quad
t\in[0,T].$ (1.13)

######  Remark 1.1.

When $A=0$ identity (1.12) follows from the classical Girsanov formula, see
e.g. [2].

We end this section with some comments about identities (1.12) and (1.13).
First we note that the classical Girsanov formula describes the law of $Z_{x}$
in terms of the Wiener measure on $C([0,T];H)$, and so, does not provide any
information about the asymptotic behaviour of $Z_{x}$. Instead the mild
Girsanov formula allows to guess the explicit form of the invariant measure
$\nu$ of $P_{t},\,t\geq 0$ in case this measure exists.

To give an idea: consider the problem

$Z_{x,-n}(t)=e^{(t+n)A}x+\int_{-n}^{t}e^{(t-r)A}\,b(Z_{x,-n}(r))dr+W_{A,-n}(t),\quad
t\leq 0,\color[rgb]{0,0,0}\,n\in\mathbb{N},$ (1.14)

where $W_{A,-n}(t)$ denotes the modified stochastic convolution

$W_{A,-n}(t)=\int_{-n}^{t}e^{(t-r)A}\,dW(r),$

where $W$ is extended to $\mathbb{R}$ in the usual way. Letting $n\to\infty$,
we are able to obtain an explicit formula for the invariant measure, see
Theorem 4.3.

In Section 4 we will examine the case when $b$ is Lipschitz continuous and
dissipative. In this case it is known that there is a unique invariant mesure
$\nu$, see [8]. We shall prove explicit representation formulas both for $\nu$
and for its density with respect to the Ornstein–Uhlenbeck process that we
obtain setting $b=0$. A final result of this theorem yields a representation
formula for the law of the stationary process corresponding to (1.1).

Let us explain briefly the content of the paper.

The application of the Ramer formula requires some preliminaries as: the
Cameron–Martin space of $\mathscr{N}_{\mathbb{Q}_{T}}$, determined in Section
2 as well the Malliavin derivative and the Gaussian divergence operator. The
mild Girsanov formula, Theorem 3.2, is proven in Section 3.

Moreover, Section 4 is devoted to an explicit formula for the law of the
stationary process of $P_{t}$ on $(-\infty,0]$ (and consequently of the
invariant measure) when $b$ is dissipative. Here a difficulty arises because
the stationary process lives on the space of continuous functions on
$(-\infty,0]$, not necessarily bounded, which is a separable Fréchet space.
Then, as far as the Malliavin calculus is concerned on locally convex spaces,
we refer to the monograph [2]. In Section 5 we consider stochastic equations
with an additive colored noise. Finally, Appendix A is devoted to recall some
maximal regularity results for abstract evolution equations, which are needed
in what follows.

##  2 The Cameron–Martin space

Since

$\mathbb{E}\int_{0}^{T}|W_{A}(t)|^{2}_{H}\,dt=\frac{1}{2}\int_{0}^{T}\mbox{\rm
Tr}\,[(-A)^{-1}\,(1-e^{2tA})]dt<\infty,\color[rgb]{0,0,0}$

we can extend the measure $\mathscr{N}_{\mathbb{Q}_{T}}$ from $E_{[0,T]}$ to
$\mathbb{X}:=L^{2}(0,T;H)$ which we shall denote by
$\mathscr{N}_{\overline{\mathbb{Q}}_{T}}$. It is well known that
$\mathscr{N}_{\overline{\mathbb{Q}}_{T}}$ is Gaussian, with mean $0$ and
covariance $\overline{\mathbb{Q}}_{T}$ given by

$(\overline{\mathbb{Q}}_{T}\,h)(t)=\int_{0}^{T}K(t,s)h(s)\,ds,\quad
t\in[0,T],\;h\in\mathbb{X},$ (2.1)

where

$K(t,s)=\int_{0}^{\min\\{t,s\\}}\,e^{(t+s-2r)A}dr\quad t,s\in[0,T],$ (2.2)

see e.g. [8, Theorem 5.4]. The measure
$\mathscr{N}_{\overline{\mathbb{Q}}_{T}}$ is concentrated on $E_{[0,T]}$ and
its Cameron–Martin space
$\mathscr{H}_{\overline{\mathbb{Q}}_{T}}=\overline{\mathbb{Q}}_{T}^{1/2}(\mathbb{X})$
coincides with that of $\mathscr{N}_{\mathbb{Q}_{T}}$, see e.g. [8,
Proposition 2.10].

The following lemma is well known; we give a sketch of its proof, however, for
the reader convenience.

######  Lemma 2.1.

Setting $\mathbb{A}_{T}=-\overline{\mathbb{Q}}_{T}^{-1}$, it results

$\left\\{\begin{array}[]{l}\mathbb{A}_{T}f=f^{\prime\prime}(t)-A^{2}\,f(t),\quad\forall\,f\in
D(\mathbb{A}_{T})\\\ \\\ D(\mathbb{A}_{T})=\\{f\in L^{2}(0,T;D(A^{2}))\cap
W^{2,2}(0,T;H):\,\;f(0)=0,\;f^{\prime}(T)=Af(T)\\}.\end{array}\right.$ (2.3)

###### Proof.

Given $h\in\mathbb{X}$ set $\overline{\mathbb{Q}_{T}}\,h=f$ and write

$\int_{0}^{t}K(t,s)\,h(s)\,ds+\int_{t}^{T}K(t,s)\,h(s)ds=f(t),\quad
t\in[0,T].$

So,

$\int_{0}^{t}\left(\int_{0}^{s}e^{(t+s-2r)A}\,dr\right)\,h(s)\,ds+\int_{t}^{T}\left(\int_{0}^{t}e^{(t+s-2r)A}\,dr\right)\,h(s)ds=f(t).$
(2.4)

Note that $f(0)=0.$ Introducing
$Q_{t}=\int_{0}^{t}e^{2sA}ds=(-2A)^{-1}(I-e^{2tA}),$ $t\geq 0,$ and
differentiating (2.4) with respect to $t$ yields

$\begin{array}[]{l}\displaystyle
Q_{t}\,h(t)+A\int_{0}^{t}\left(\int_{0}^{s}e^{(t+s-2r)A}\,dr\right)\,h(s)\,ds\\\
\\\ \displaystyle-
Q_{t}\,h(t)+\int_{t}^{T}e^{(s-t)A}\,h(s)ds+A\int_{t}^{T}\left(\int_{0}^{t}e^{(t+s-2r)A}\,dr\right)\,h(s)\,ds=f^{\prime}(t).\end{array}$

It follows that

$Af(t)+\int_{t}^{T}e^{(s-t)A}\,\,h(s)ds=f^{\prime}(t),$ (2.5)

which implies $f^{\prime}(T)=Af(T).$ Now, differentiating (2.5) with respect
to $t$, yields

$Af^{\prime}(t)-h(t)-A\int_{t}^{T}e^{(s-t)A}\,h(s)ds=f^{\prime\prime}(t).$
(2.6)

Taking into account (2.5), we deduce

$f^{\prime\prime}(t)-A^{2}f(t)=-h(t).$ (2.7)

Since $h=-\mathbb{A}_{T}f$, we have

$\mathbb{A}_{T}f=f^{\prime\prime}(t)-A^{2}f(t),$

as required. ∎

Now we can determine the Cameron–Martin space of
$\mathscr{N}_{\overline{\mathbb{Q}}_{T}}$ which, as remarked earlier,
coincides with that of $\mathscr{N}_{\mathbb{Q}_{T}}$.

######  Proposition 2.2.

It results, see Appendix A,

$\mathscr{H}_{\mathbb{Q}_{T}\color[rgb]{0,0,0}}=\mathscr{H}_{\overline{\mathbb{Q}}_{T}}=\\{u\in
L^{2}(0,T;D(A))\cap W^{1,2}(0,T;H):\,u(0)=0\\}:=\Gamma_{A}(H).$

Moreover, if $u\in\mathscr{H}_{\overline{\mathbb{Q}}_{T}}$, we have

$|u|^{2}_{\mathscr{H}_{\overline{\mathbb{Q}}_{T}}}:=|\overline{Q_{T}}^{-1/2}u|^{2}_{\mathbb{X}}=|(-A)^{1/2}\,u(T)|_{H}^{2}+\int_{0}^{T}(|u^{\prime}(t)|_{H}^{2}+|Au(t)|_{H}^{2})\,dt.$
(2.8)

######  Remark 2.3.

Note that by interpolation it results, see e.g. [12],

$\\{u\in L^{2}(0,T;D(A))\cap W^{1,2}(0,T;H):\,u(0)=0\\}\subset
C([0,T];D(-A)^{1/2})),$ (2.9)

so that the term $|(-A)^{1/2}\,u(T)|_{H}$ in (2.8) is meaningful.

Proof of Proposition 2.2. It is enough to assume $f\in D(\mathbb{A}_{T})$,
since $D(\mathbb{A}_{T})$ is dense in $D((-\mathbb{A}_{T})^{1/2})$. In this
case we can write

$\begin{array}[]{l}\displaystyle\int_{0}^{T}\langle\mathbb{A}f(t),f(t)\rangle_{H}dt=\int_{0}^{T}\langle\,f^{\prime\prime}(t),f(t)\rangle_{H}\,dt-\int_{0}^{T}\langle
A^{2}f(t),f(t)\rangle_{H}\,dt\\\ \\\ \displaystyle=\langle
f^{\prime}(T),f(T)\rangle_{H}-\int_{0}^{T}|\,f^{\prime}(t)|_{H}^{2}\,dt-\int_{0}^{T}|Af(t)|_{H}^{2}\,dt\\\
\\\
\displaystyle=-|(-A)^{1/2}\,f(T)|_{H}^{2}-\int_{0}^{T}|f^{\prime}(t)|_{H}^{2}\,dt-\int_{0}^{T}|A\,f(t)|_{H}^{2}\,dt.\end{array}$

Then (2.8) follows. $\Box$

We now prove a result which provides an useful information on the support of
$N_{\overline{Q}_{T}}$.

######  Lemma 2.4.

Assume Hypothesis 1. Then it results

$\int_{\mathbb{X}}|(-A)^{\beta/2}h|^{2}_{\mathbb{X}}\,\mathscr{N}_{\overline{\mathbb{Q}}_{T}}(dh)\leq\tfrac{T}{2}\,\mbox{\rm
Tr}\,[(-A)^{\beta-1}]<\infty.$ (2.10)

###### Proof.

Write

$\begin{array}[]{l}\displaystyle\int_{\mathbb{X}}|(-A)^{\beta/2}h|_{\mathbb{X}}^{2}\,\mathscr{N}_{\overline{\mathbb{Q}}_{T}}(dh)=\mathbb{E}\int_{0}^{T}|(-A)^{\beta/2}\,W_{A}(t)|_{H}^{2}\,dt\\\
\\\
\displaystyle=\mathbb{E}\int_{0}^{T}dt\left|\int_{0}^{t}(-A)^{\beta/2}e^{(t-s)A}dW(s)\right|_{H}^{2}\\\
\\\ \displaystyle=\int_{0}^{T}dt\int_{0}^{t}\,\mbox{\rm
Tr}\,[(-A)^{\beta}\,e^{2(t-s)A}]ds\\\ \\\
\displaystyle=\tfrac{1}{2}\int_{0}^{T}\mbox{\rm
Tr}\,[(-A)^{\beta-1}\,(I-e^{2tA})]\,dt\leq\tfrac{T}{2}\,\mbox{\rm
Tr}\,[(-A)^{\beta-1}].\end{array}$

The conclusion follows. ∎

######  Remark 2.5.

By Lemma 2.4 it follows that $\mathbb{Q}_{T}$ is concentrated on
$D((-A)^{\beta/2})$, where $\beta>0$ is given in Hypothesis 1.

###  2.1 Brownian Motion

We are going to define the Brownian motion on $\mathbb{X}$ and $E_{[0,T]}$. To
this purpose we first introduce the white noise function. Let $(\psi_{j})$ be
an orthonormal basis in $\mathbb{X}$ of eigenfunctions of
$\overline{\mathbb{Q}}_{T}$ and let $(\lambda_{j})$ be the sequence of the
corresponding eigenvalues,

$\overline{\mathbb{Q}}_{T}\psi_{j}=\lambda_{j}\psi_{j},\quad j\in\mathbb{N}.$

For all $f\in\mathbb{X}$ we set

$W_{f}(h)=\langle\overline{\mathbb{Q}}_{T}^{-1/2}h,f\rangle_{\mathbb{X}}=\sum_{j=1}^{\infty}\lambda_{j}^{-1/2}\langle
h,\psi_{j}\rangle_{\mathbb{X}}\langle f,\psi^{s}_{j}\rangle_{\mathbb{X}},\quad
h\in\mathbb{X},a.e.$

Finally, we define a sequence of Brownian motions in $\mathbb{X}$ by
proceeding as in [5]

$B^{\alpha}(t)=[W_{{\mathds{1}}_{[0,t]}e^{\alpha}}],\quad
t\in[0,T],\,\alpha\in\mathbb{N},$ (2.11)

where $(e_{\alpha})$ is an orthonormal basis on $H$. As proved in [5],
$B^{\alpha}$ is continuous for all $\alpha\in\mathbb{N}$.

Recall that $\gamma_{x}\in L^{2}(E_{[0,T]}\times[0,T];H)\sim
L^{2}(E_{[0,T]};\mathbb{X})$

The following result is standard. We only point out that
${\mathscr{M}}^{*}(\gamma_{x})=\delta(\gamma_{x})$ according to the notation
of [2] and [13].

######  Proposition 2.6.

Let $x\in H$ then $\gamma_{x}\in D(\mathscr{M}^{*})$ and it results

$[M^{*}(\gamma_{x})](h)=\int_{0}^{T}\langle[\gamma_{x}(h)](s),d[B(s)](h)\rangle_{H}=\int_{0}^{T}\int_{0}^{s}\langle
e^{(s-r)A}\color[rgb]{0,0,0}b(h(r)+e^{rA}x)\,dr,d[B(s)](h)\rangle_{H}.$ (2.12)

Moreover

$\int_{\mathbb{X}}|[M^{*}\gamma_{x}](h)|^{2}\,\mathscr{N}_{\overline{\mathbb{Q}_{T}}}(dh)=\int_{\mathbb{X}}\int_{0}^{T}|[\gamma_{x}(h)](s)|_{H}^{2}\,ds\,\mathscr{N}_{\overline{\mathbb{Q}_{T}}}(dh).\color[rgb]{0,0,0}$
(2.13)

###### Proof.

First note that $\gamma_{x}$ is adapted to the natural filtration, so that the
conclusion follows from the standard properties of the Itô integral.

∎

In the following we shall write sometime for brevity

$M^{*}(\gamma_{x})=\int_{0}^{T}\langle[\gamma_{x}(\cdot)](s),dB(s)\rangle_{H}.\color[rgb]{0,0,0}$

##  3 Proof of the mild Girsanov formula

###  3.1 The Ramer identity

Here we recall the Ramer identity (or Kusuoka-Ramer theorem, [11]). Such
theorem has been also applied to study stochastic boundary value problems (see
[14], [3] and the references therein).

The following version is an extension to complete separable locally convex
spaces (or separable Fréchet spaces) of a result given by D. Nualart [13, pag.
240] (see also Section 6.6 in Bogachev [2] for a corresponding result in
general locally convex spaces). In this section we apply this identity in the
space of continuous functions, but we will need such extension in Section 4.

We write $\phi\in D(\mathscr{M})$ to indicate that $\phi$ is differentiable
along the directions of the Cameron-Martin space or Malliavin differentiable
(cf. [2] and [13]). Moreover $\mathscr{M}^{*}=\delta$

######  Proposition 3.1.

Let $\mathbb{Y}$ be a complete separable locally convex space and $\Lambda$ a
homeomorphism of $\mathbb{Y}$ onto $\mathbb{Y}$. Let
$\mathscr{N}_{\mathbb{U}}$ be a Gaussian measure on $\mathbb{Y}$ of mean $0$,
covariance $\mathbb{U}$ and Cameron–Martin space $\mathscr{H}_{\mathbb{U}}$.
Set $\phi(h)=h-\Lambda^{-1}h$, for all $h\in\mathbb{Y}$, and assume that
$\phi\in D(\mathscr{M})\cap\mathscr{H}_{\mathscr{N}_{U}}$, where $\mathscr{M}$
is the the Malliavin derivative in $\mathbb{Y}$.

Then it results
$\mathscr{N}_{\mathbb{U}}\circ\Lambda^{-1}\ll\mathscr{N}_{\mathbb{U}}$ and

$\frac{d\mathscr{N}_{\mathbb{U}}\circ\Lambda^{-1}}{d\mathscr{N}_{\mathbb{U}}}(h)=\exp\left\\{-\tfrac{1}{2}|\phi(h)|^{2}_{\mathscr{H}_{\mathbb{U}}}+\mathscr{M}^{*}\phi(h)\right\\}\,{\det}_{2}(I-\mathscr{M}\phi(h)),$
(3.1)

where the determinant is intended in the sense of Carleman–Fredholm, see [10].

Now we are ready to show the main result of this section.

######  Theorem 3.2.

Assume Hypothesis 1, let $x\in H,\,h\in E_{[0,T]}$ and set

$[\gamma_{x}(h)](t)=\int_{0}^{t}e^{(t-s)A}b(h(s)+e^{sA}x)\,ds,\quad\,t\in[0,T].$
(3.2)

Then,

(i) $\gamma_{x}(h)$ belongs to $\mathscr{H}_{\mathbb{Q}_{T}}$ for $h\in
E_{[0,T]},\,\mathscr{N}_{\mathbb{Q}_{T}}$-a.e..

(ii) The vector field $\gamma_{x}$ belongs to $D(\mathscr{M})\cap
D(\mathscr{M}^{*})$ and it is Itô integrable.

(iii) $\mbox{\rm det}_{2}\,(I-\mathscr{M}(\gamma_{x})(h))=1$ for
$h\in\mathbb{X}$, $\mathscr{N}_{\overline{\mathbb{Q}_{T}}}$-a.e.  (1) (1)
(1)$\det_{2}$ represents the determinant of Carleman-Fredholm.

(iv) The law of $Z_{x}$ on $E_{[0,T]}$ is given, for all $\Phi\in
B_{b}(E_{[0,T]})$, by

$(\mathbb{P}\circ
Z_{x}^{-1})(\Phi)=\int_{E_{[0,T]}\color[rgb]{0,0,0}}\Phi(h+e^{\cdot
A}x)\,\exp\left\\{-\tfrac{1}{2}|\gamma_{x}(h)|^{2}_{\mathscr{H}_{\mathbb{Q}_{T}}}+I(\gamma_{x})(h)\right\\}\mathscr{N}_{\mathbb{Q}_{T}}(dh).$

###### Proof.

We start from the identity

$(\mathbb{P}\circ Z_{x}^{-1})(\Phi)=\int_{E_{[0,T]}}\Phi(h+e^{\cdot
A}x)\,(\mathscr{N}_{\mathbb{Q}_{T}}\circ G_{x})(dh),\quad\forall\,\Phi\in
B_{b}(E_{[0,T]})$

and apply Proposition 3.1, with $\mathbb{Y}=E_{[0,T]}$, $\Lambda=G_{x}$,
$\phi=\gamma_{x}$ given by (3.2) , $\mathbb{U}=\mathbb{Q}_{T}$ and moreover
proving that $\mbox{\rm det}_{2}\,(I-\mathscr{M}(\gamma_{x})](h))=1$.

We proceed in different steps.

Step 1. $\gamma_{x}(h)\in\mathscr{H}_{\mathbb{Q}_{T}},a.e.$

In fact, it results

$\gamma_{x}(h)=e^{\cdot A}*b(h+e^{\cdot A}x),\quad
a.e.\,h\in\mathbb{X}.\color[rgb]{0,0,0}$

So, taking into account (2.8) yields

$\|u\|^{2}_{\mathscr{H}_{\overline{\mathbb{Q}}}}=|u|^{2}_{\Gamma_{A}(H)}+|((-A)^{1/2}u(T)|_{H}\leq
CT|u|^{2}_{\Gamma_{A}(H)}.$

Now Step 1 follows from (A.5) and (A.7) using the Lipschitz assumption on $b$.

Step 2. $\gamma_{x}$ belongs to $D(\mathscr{M}^{*})$ and it is Itô integrable.

First note that $\gamma_{x}$ belongs to $D(\mathscr{M})$. In fact, due to
Hypothesis 1, the mapping

$h\in E_{[0,T]}\color[rgb]{0,0,0}\to\gamma_{x}(h)\in E_{[0,T]},$

is Lipschitz, so that $\gamma_{x}$ belongs to the domain of $\mathscr{M}$ by a
well known result, see e.g. [2]. Now it follows that $\gamma_{x}\in
D(\mathscr{M}^{*})$ for all $x\in H$, see again [2], where $\mathscr{M}^{*}$
is indicated by $\delta$ . Moreover , as we notice, $\gamma_{x}$ is adapted to
the natural filtration of the Brownian motion, so that it is Itô integrable.
Step 2 is proved.

To prove the last step let us first recall that if
$T\in\mathscr{L}(\mathbb{X})$ is Hilbert–Schmidt with real eigenvalues
$(k_{n})$, then

$\mbox{\rm det}_{2}(I-T)=\prod_{n=1}^{\infty}(1-k_{n})e^{-k_{n}}.$

If, in addition, $T$ is quasi–nilpotent, it results

$\mbox{\rm det}_{2}(I-T)=1,$

because in this case all $k_{n}$ are zero.

Step 3. $\gamma_{x}(k)$ is quasi–nilpotent for any $k\in\mathbb{X}$, so that

$\mbox{\rm det}_{2}(I-[\mathscr{M}(\gamma_{x})](h))=1,\color[rgb]{0,0,0}\quad
h\in\mathbb{X}.$

Assume for a moment that $b$ is $C^{1}$. Then we have

$[\mathscr{M}\color[rgb]{0,0,0}\gamma_{x}(k)\cdot
h](\cdot)=(\overline{\mathbb{Q}}_{T})^{1/2}\int_{0}^{\cdot}e^{(\cdot-s)A}b^{\prime}(k(r)+e^{rA}x)\cdot
h(r)\,dr.$

It follows by recurrence that,

$\|\mathscr{M}\color[rgb]{0,0,0}\gamma_{x}(k))\|_{\mathscr{L}(\mathbb{X})}^{n}\leq\tfrac{1}{n!}\,\|\overline{\mathbb{Q}}_{T}\|_{\mathscr{L}(\mathbb{X})}^{n/2}\,\|b^{\prime}\|^{n}_{\infty}T^{n},$

so that

$\lim_{n\to\infty}\,\|(\mathscr{M}\gamma_{x}(k))^{n}\|_{\mathscr{L}(\mathbb{X})}=0,\,\quad\forall\,h\in\mathbb{X},\,x\in
H,$

which implies that $\mathscr{M}\gamma_{x}(k)$ is quasi–nilpotent for all
$k\in\mathbb{X}$ as required. If $b$ is not $C^{1}$ we proceed by
approximation.

The proof of the theorem is complete. ∎

Let $P_{t},\;t\geq 0,$ be the transition semigroup corresponding to the
process $Z_{x}$,

$P_{t}\varphi(x)=\mathbb{E}[\varphi(Z_{x}(t))],\quad\varphi\in B_{b}(H).$

By (1.13) it follows

######  Corollary 3.3.

It results

$P_{T}\varphi(x)=\mathbb{E}[\varphi(Z_{x}(T))]=\int_{\mathbb{X}}\varphi(k(T)+e^{TA}x)\,\exp\left\\{-\tfrac{1}{2}|\gamma_{x}(h)|^{2}_{\mathscr{H}_{\mathbb{Q}_{T}}}+[I(\gamma_{x})](h)\right\\}\,N_{\mathbb{Q}_{T}}(dk),\quad\varphi\in
B_{b}(H).$ (3.3)

###  3.2 Some estimates

Here we will not distinguish between $N_{\mathbb{Q}_{T}}$ and
$N_{\overline{\mathbb{Q}}_{T}}$. Moreover, in this section we assume, besides
Hypothesis 1, that $b$ is bounded. Obviously $\rho(x,k)\in
L^{1}(\mathbb{X},\mathscr{N}_{\mathbb{Q}_{T}}$) for all $x\in H$ but we do not
know whether

$\int_{\mathbb{X}}\rho^{n}(x,k)\,\mathscr{N}_{\mathbb{Q}_{T}}(dk)<\infty,\quad
n\geq 2,$

or not. An estimate for
$\int_{\mathbb{X}}\rho^{n}(x,h)\,\mathscr{N}_{\mathbb{Q}_{T}}(dh)$ is provided
by the following result.

######  Proposition 3.4.

Assume Hypothesis 1 and let $x\in H$ and $n>1.$Then it results

$\int_{\mathbb{X}}\rho^{n}(x,h)\,N_{\mathbb{Q}_{T}}(dh)\leq\exp\left\\{(n^{2}-n)\,\|b\|^{2}_{\infty}\right\\}.$
(3.4)

###### Proof.

Obviously we have

$\int_{\mathbb{X}}\rho(x,h)\,\mathscr{N}_{\mathbb{Q}_{T}}(dh)=1.$ (3.5)

To estimate $\rho^{n}$ write

$\rho^{n}(x,h)=\exp\left\\{-\tfrac{n}{2}|\gamma_{x}(h)|^{2}_{\mathscr{H}_{T}}+n[I(\gamma_{x})](h)\right\\},$

then setting

$\widetilde{\rho}(x,h)=\exp\left\\{-\tfrac{n^{2}}{2}|\gamma_{x}(h)|^{2}_{\mathscr{H}_{T}}+n[I(\gamma_{x})](h)\right\\},$

we see that $\widetilde{\rho}$ is obtained by $\rho$ replacing $b$ with $nb$,
so that

$\int_{\mathbb{X}}\widetilde{\rho}(x,h)\,\mathscr{N}_{\mathbb{Q}_{T}}(dh)=1.$
(3.6)

Therefore, taking into account that

$\rho^{n}(x,h)=\exp\left\\{\tfrac{n^{2}-n}{2}\,|\gamma_{x}(h)|^{2}_{\mathscr{H}_{T}}\right\\}\,\widetilde{\rho}(x,h),$

and that

$|\gamma_{x}(h)|^{2}_{\mathscr{H}_{T}}=|\mathbb{Q}_{T}^{-1/2}\gamma_{x}(h)|^{2}_{\mathbb{X}}\leq
2|b(k(\cdot)+e^{\cdot A}x)|_{\mathbb{X}}^{2}\leq 2\|b\|^{2}_{\infty},$ (3.7)

we find

$\rho^{n}(x,k)\leq\exp\left\\{(n^{2}-n)\,\|b\|^{2}_{\infty}\right\\}\,\widetilde{\rho}(x,k).$

Integrating with respect to $\mathbb{X}$ and taking into account (3.6), yields
the conclusion. ∎

######  Remark 3.5.

By Proposition 3.4 and (2.13) we can find an estimate of
$|I(\gamma_{x})(h)|^{2}_{L^{2}(\mathbb{X})}$ as follows. Write

$\begin{array}[]{l}\displaystyle|I(\gamma_{x})(h)|^{2}_{L^{2}(\mathbb{X})}\color[rgb]{0,0,0}=\int_{\mathbb{X}}\left|\int_{0}^{T}\langle[\gamma_{x}(h)](t),dB(t)\rangle_{H}\right|^{2}\mathscr{N}_{\mathbb{Q}_{T}}(dh)\color[rgb]{0,0,0}=\int_{\mathbb{X}}\int_{0}^{T}\left|[\gamma_{x}(h)](t)\right|^{2}_{H}\,dt\mathscr{N}_{\mathbb{Q}_{T}}(dh)\color[rgb]{0,0,0}\\\
\\\
\displaystyle=\int_{\mathbb{X}}\int_{0}^{T}\left|\int_{0}^{t}e^{(t-r)A}\,b(h(r)+e^{rA}x)dr\right|^{2}_{H}\,dt\mathscr{N}_{\mathbb{Q}_{T}}(dh)\color[rgb]{0,0,0}\\\
\\\ \displaystyle\leq\|b\|_{\infty}^{2}\int_{0}^{T}\int_{0}^{t}\,e^{-2\omega
r}\,dr\,dt=\frac{1}{2\omega}\|b\|_{\infty}^{2}\int_{0}^{T}(1-e^{-\omega
t})dt\leq\frac{T}{2\omega}\|b\|_{\infty}^{2}\end{array}$ (3.8)

##  4 Invariant measure

###  4.1 Preliminaries on Gaussian measures on locally convex spaces

We follow here [2]. We are given a complete, separable, locally convex space
$E$. We denote by $E^{*}$ the topological dual of $E$ and by $\mathscr{B}(E)$
the $\sigma$-algebra of all Borel subsets of $E$.

A probability measure $\mu$ on $(E,\mathscr{B}(E))$ is Gaussian if and only
for any $F\in E^{*}$ the law $\mu\circ F^{-1}$ of $F$ is Gaussian on
$\mathbb{R}$.

Let $\mu$ be a Gaussian measure on $(E,\mathscr{B}(E))$. Then the mean of
$\mu$ is defined as the linear functional $m:E^{*}\to\mathbb{R}$ given by,

$m(F)=\int_{E}F(h)\mu(dh),\quad F\in E^{*}.$ (4.1)

The covariance of $\mu$ is the mapping $Q\in\mathscr{L}(E^{*},E)$ defined as

$Q(F)=\int_{E}(F(h)-m(F))(h-m(F))\color[rgb]{0,0,0}\,\mu(dh),\quad F\in
E^{*}.$

It follows that

$G(Q(F))=\int_{E}(F(h)-m(F))(G(h)-m(F))\color[rgb]{0,0,0}\,\mu(dh),\quad
F,\,G\in E^{*}.$ (4.2)

In particular, we have

$F(Q(F))=\int_{E}(F(h)-m(F))^{2}\,\mu(dh),\quad F\in E^{*}.$ (4.3)

We note that the definitions of $m$ and $Q$ are meaningful thanks to the
Fernique Theorem, see [2, Theorem 2.8.5].

We denote by $\mathscr{N}_{m,Q}$ the Gaussian measure with mean $m$ and
covariance $Q$. If $m=0$, $\mu$ is said to be symmetric and is denoted by
$\mathscr{N}_{Q}$. All considered Gaussian measures considered in what follows
are symmetric. In this case it results

$F(Q(F))=\int_{E}(F(h))^{2}\,\mu(dh),\quad F\in E^{*}$

and

$G(Q(F))=\int_{E}F(h)\,G(h)\,\mu(dh),\quad F,\,G\in E^{*}.$

$E^{*}$, endowed with the inner product

$(F,G)_{\mu}=G(Q(F)),\quad F,G\in E^{*},$

is a pre–Hilbert space. We denote by $E^{*}_{\mu}$ its completion.

Note that $Q$ is extendible to $E^{*}_{\mu}$, setting.

$Q(\psi)=\int_{E}\psi(k)\,k\,\mu(dk),\quad\psi\in
L^{2}(E^{*},\mu)=E^{*}_{\mu}\color[rgb]{0,0,0}.$

We define the Cameron-Martin space $\mathscr{H}_{\mu}$ of $\mu$ as follows

$\mathscr{H}_{\mu}=\left\\{h\in E:\int_{E}|F(k)|^{2}\,\mu(dk)\leq 1\Rightarrow
F(h)<+\infty\right\\}.$ (4.4)

If $h\in\mathscr{H}_{\mu}$ we set

$|h|_{\mathscr{H}(\mu)}=\sup\\{F(h):\,F(Q(F))\leq 1\\}.$ (4.5)

It useful to notice that $h\in\mathscr{H}_{\mu}$ if and only if there exists
$\psi\in E^{*}_{\mu}$ such that $h=Q(\psi)$. See [2, Lemma 2.4.1]

###  4.2 Existence of invariant measure

We shall assume, besides Hypothesis 1 that

######  Hypothesis 2.

$b\colon H\to H$ is dissipative.

We start from problem (1.1) whose solution we denote by $Z_{x}$ whereas the
corresponding transition semigroup will be denoted by

$P_{t}\varphi(x)=\mathbb{E}[\varphi(Z_{x}(t))],\quad\varphi\in B_{b}(H).$

It is convenient to modify problem (1.1) by taking into account a generic
initial time $-n$ with $n\in\mathbb{N}.$ So, we consider the problem

$\left\\{\begin{array}[]{l}dZ(t)=(AZ(t)+b(Z(t))dt+dW(t),\quad t\geq-n,\\\ \\\
Z(x,-n)=x\in H,\end{array}\right.$ (4.6)

where $W$ is an $H$–valued cylindrical Wiener process defined for all
$t\in\mathbb{R}$ in the usual way.

Under Hypothesis 1 and the dissipativity of $b$, there is a unique solution
$Z_{x,-n}$ of the mild equation

$Z_{x,-n}(t)=e^{(t+n)A}x+\int_{-n}^{t}e^{(t-r)A}\,b(Z_{x,-n}(r))dr+W_{A,-n}(t),\quad
t\geq-n,\color[rgb]{0,0,0}$ (4.7)

where $W_{A,-n}(t)$ denotes the modified stochastic convolution

$W_{A,-n}(t)=\int_{-n}^{t}e^{(t-r)A}\,dW(r).$

Moreover, there exists the limit

$Z_{-\infty}(t):=\lim_{n\to\infty}\,Z_{x,-n}(t)$

and we know by [8] that $Z_{-\infty}$ is the unique, in law, stationary
solution of problem (1.1). As it is well known, for all $t\in(-\infty,0]$ the
law of $Z_{-\infty}(t)$ coincides with the unique invariant measure of $Z_{x}$
that we denote by $\nu$.

The stationary process $Z_{-\infty}$ is the solution to the mild limit
equation

$Z_{-\infty}(t)=\int_{-\infty}^{t}e^{(t-r)A}\,b(Z_{-\infty}(r))dr+W_{A,-\infty}(t),\quad
t\in(-\infty,0],$ (4.8)

where $W_{A,-\infty}$ denotes the modified stochastic convolution

$W_{A,-\infty}(t)=\int_{-\infty}^{t}e^{(t-r)A}\,dW(r).$

As we said in the introduction, $W_{A,-\infty}$ does not live in
$C_{b}((-\infty,0];H)$ but in $E_{-\infty}:=C((-(\infty,0];H)$, the space of
all continuous functions $[-\infty,0]\to H$.

$E_{-\infty}$ is a separable Fréchet space endowed with the set of seminorms

$p_{-n}(h)=\sup_{t\in[-n,0]}|h(t)|,\quad n\in\mathbb{N},\,h\in E_{-\infty}$

Let us also consider the deterministic equation

$k_{-\infty}(t)=\int_{-\infty}^{t}e^{(t-s)A}b(k_{-\infty}(s))ds+h(t),\quad
t\leq 0,\,h\in E_{-\infty},$ (4.9)

which has a unique solution $k_{-\infty}$ by a standard argument. We have
$k_{-\infty}=F_{-\infty}(h)$, where

$F_{-\infty}:E_{-\infty}\to E_{-\infty},\quad h\to k_{-\infty}.$

$F_{-\infty}$ is a homeomorphism of $E_{-\infty}$ onto itself. We denote by
$G_{-\infty}$ its inverse, so that

$G_{-\infty}(h)=k-\int_{-\infty}^{\cdot}e^{(t-s)A}b(h(s))ds.$

Finally, we define

$[\gamma_{-\infty}(k)](t)=\int_{-\infty}^{t}e^{(t-r)A}b(k(r))\,dr,\quad k\in
E_{-\infty},\,t\in(-\infty,0].$ (4.10)

Therefore the solution of (4.7) is given by

$Z_{-\infty}=F_{-\infty}(W_{A,-\infty})\color[rgb]{0,0,0}$

and for all $\Phi\in C_{b}(E_{-\infty})$ we have

$\mathbb{E}[\Phi(Z_{-\infty})]=\mathbb{E}[\Phi(F_{-\infty}\,(W_{A,-\infty}))].\color[rgb]{0,0,0}$

Now, let us consider the transformation

$E_{-\infty}\to E_{-\infty},\quad h\to W_{A,-\infty}(h),\color[rgb]{0,0,0}$

which, using the change of variables formula, yields

$\mathbb{E}[\Phi(\xi)]=\int_{E_{-\infty}}\Phi(F_{-\infty}(h))\,\mathscr{N}_{\mathbb{Q}_{-\infty}}(dh),$
(4.11)

where $\mathscr{N}_{\mathbb{Q}_{-\infty}}$ is the law of $W_{A,-\infty}$ on
the Fréchet space $E_{-\infty}$ which is Gaussian with mean zero and
covariance $\mathbb{Q}_{-\infty}$, given in Proposition 4.1 below.

###  4.3 Law of $W_{A,-\infty}$ on $E_{-\infty}$

We recall that the dual $E^{*}_{-\infty}$ of $E_{-\infty}$ coincides with the
space of all functions from $BV(-\infty,0;H)$ with a compact support. If $F\in
BV((-\infty,0];H)$ we shall write

$F(h)=\int_{-\infty}^{0}\langle h(t),dF(t)\rangle_{H},\quad\forall\,h\in
E_{-\infty}.$

where the integral is intended in the sense of Stieltjes.

Let $\Phi:E_{-\infty}\to\mathbb{R}$ be bounded and Borel. Then we have

$(\mathscr{N}_{\mathbb{Q}_{-\infty}}\circ\Phi^{-1})=\mathbb{E}[\Phi(W_{A,-\infty})]=\int_{E_{-\infty}}\Phi(h)\,\mathscr{N}_{\mathbb{Q}_{-\infty}}(dh).$

Now for any $F\in E^{*}_{-\infty}$, setting

$\Phi(h)=|F(h)|^{2}$

it results

$F(\mathbb{Q}_{-\infty}(F))=\int_{E_{-\infty}}|F(h)|^{2}\,\mathscr{N}_{\mathbb{Q}_{-\infty}}(dh)=\mathbb{E}[F(W_{A,-\infty})|^{2}].$

Let $F\in E^{*}_{-\infty}$, let us identify $F$ with a BV function with a
compact support, say $F\in BV([-n,0];H)$. Then we have

$F(\mathbb{Q}_{-\infty}(F))=\mathbb{E}\left[\left(\int_{-n}^{0}\int_{-\infty}^{t}\langle
e^{(t-r)A}\,dW(r),dF(t)\rangle\,dt\right)^{2}\right].$

By a straighforward computation it follows that

######  Proposition 4.1.

The law of $W_{A,-\infty}$ in $E_{-\infty}$ is a Gaussian measure with mean
$0$ and covariance $\mathbb{Q}_{-\infty}$ given by

$F(\mathbb{Q}_{-\infty}(F))=\frac{1}{2}\int_{-\infty}^{0}\int_{-\infty}^{0}\langle(-A)^{-1}e^{|t-t_{1}|A}dF(t),dF(t_{1})\rangle_{H},\quad\forall\,F\in\mathbb{E}^{*}_{-\infty}.$
(4.12)

We shall denote by $\mathscr{H}_{\mathbb{Q}_{-\infty}}$ the corresponding
Cameron–Martin space.

######  Proposition 4.2.

It results

$\mathscr{H}_{\mathbb{Q}_{-\infty}}=\\{u\in L^{2}(-\infty,0;D(A))\cap
W^{1,2}(-\infty,0;H):\,u(0)=0\\}.$

Moreover, if $u\in\mathscr{H}_{\mathbb{Q}_{-\infty}},$ we have

$|u|^{2}_{\mathscr{H}_{\mathbb{Q}_{-\infty}}}=|(-A)^{1/2}\,u(0)|_{H}^{2}+\int_{-\infty}^{0}(|u^{\prime}(t)|_{H}^{2}+|Au(t)|_{H}^{2})\,dt.$
(4.13)

###### Proof.

It easily follows from (4.4) and (4.5). ∎

Consider finally the transformation

$E_{-\infty}\to E_{-\infty},\quad\ h\to F_{-\infty}(h)$

then by (4.11) it follows that

$\mathbb{E}[\Phi(Z_{-\infty})]\color[rgb]{0,0,0}=\int_{E_{-\infty}}\Phi(h)\,(\mathscr{N}_{\mathbb{Q}_{-\infty}}\circ
G_{-\infty})(dh).\color[rgb]{0,0,0}$ (4.14)

Now, we shall proceed as in Theorem 3.1 proving, with the help of the Ramer
identity, that

$\mathscr{N}_{\mathbb{Q}_{-\infty}}\circ
G_{-\infty}\ll\mathscr{N}_{\mathbb{Q}_{-\infty}},$

where $G_{-\infty}=F^{-1}_{\infty}$ and there exists $\rho_{-\infty}\in
L^{1}(E_{-\infty},\mathscr{N}_{\mathbb{Q}_{-\infty}})$ such that

$\rho_{-\infty}(h)=\exp\left\\{-\tfrac{1}{2}|\gamma_{-\infty}(h)|^{2}_{\mathscr{H}_{\mathbb{Q}_{-\infty}}}+[\mathscr{M}_{-\infty}^{*}(\gamma_{-\infty})](h)\right\\}.$
(4.15)

Here $\gamma_{-\infty}$ is defined by (4.10) and
$\mathscr{M}^{*}_{-\infty}((\gamma_{-\infty}))$ is the Skorokhod integral with
changed sign.

Recall that as for the Malliavin calculus is concerned we refer to [2] were
this subject is developed on general locally convex spaces.

###  4.4 Representation formula for the invariant measure

It is convenient to consider also the case $b=0$ and the Ornstein–Ulenbeck
process

$Y_{x}(t)=e^{tA}x+\int_{0}^{t}e^{(t-s)A}\,dW(s),\quad t\geq 0,$ (4.16)

its corresponding stationary process $Y_{-\infty}$ reads as follows

$Y_{-\infty}(t):=\int_{-\infty}^{t}e^{(t-s)A}\,dW(s),\quad t\leq 0.$ (4.17)

For any $t\leq 0$ the invariant measure of $Y_{-\infty}(t)$ is
$\mu=\mathscr{N}_{\tfrac{1}{2}\,(-A)^{-1}}.$ Also the law of
$Y_{-\infty}=Y_{-\infty}(0)\color[rgb]{0,0,0}$ on $E_{-\infty}$ is given by

$[\mathbb{P}\circ
Y_{-\infty}^{-1}](\Phi)=\mathbb{E}[\Phi(Y_{-\infty})]=\int_{E_{-\infty}}\Phi(h)\,\mathscr{N}_{\mathbb{Q}_{-\infty}}(dh),\quad\Phi\in
B_{b}(E_{-\infty}).$ (4.18)

######  Theorem 4.3.

Set

$\rho_{-\infty}(h)=\exp\left\\{-\tfrac{1}{2}|\gamma_{-\infty}(h)|^{2}_{\mathscr{H}_{\mathbb{Q}_{-\infty}}}+[\mathscr{M}_{-\infty}^{*}(\gamma_{-\infty})](h)\right\\},\quad
h\in E_{-\infty}.$ (4.19)

Assume that Hypothesis 1 is fulfilled and that $b$ is dissipative. Then the
following statements hold.

(i) The law of $Z_{-\infty}$ in $E_{-\infty}$ is given by

$[\mathbb{P}\circ
Z_{-\infty}^{-1}](\Phi)=\mathbb{E}[\Phi(Z_{-\infty})]=\int_{E_{-\infty}}\Phi(h)\,\rho_{-\infty}(h)\,\mathscr{N}_{\mathbb{Q}_{-\infty}}(dh),\quad\Phi\in
B_{b}(E_{-\infty}),$ (4.20)

where $\gamma_{-\infty}$ is defined by (4.10) and
$[\mathscr{M}_{-\infty}^{*}(\gamma_{-\infty})]$ is the Skorokhod integral

(ii) It results $\mathbb{P}\circ Z_{-\infty}^{-1}\ll\mathbb{P}\circ
Y_{-\infty}^{-1}$.

(iii) The invariant measure $\nu$ of $Z_{x}$ is given by

$\int_{H}{\varphi}\,d\nu=\int_{E_{-\infty}}{\varphi}(h(0))\,\rho_{-\infty}(h)\,\mathscr{N}_{\mathbb{Q}_{-\infty}}(dh),\quad\varphi\in
B_{b}({H}),$ (4.21)

(iv) The invariant measure $\mu$ of $Y_{x}$ is given by

$\int_{H}\varphi\,d\mu=\int_{E_{-\infty}}\varphi(h(0))\,\mathscr{N}_{\mathbb{Q}_{-\infty}}(dh),\quad\varphi\in
B_{b}(H).$ (4.22)

(v) It results $\nu\ll\mu$. Moreover, we have an explicit formula for
$\tfrac{d\nu}{d\mu}$.

###### Proof.

(i) First note that $\gamma_{\infty}$ is Lipschitz so that $\gamma_{\infty}$
belongs to the domain of the Malliavin derivative $\mathscr{M}$. Consequently
$\gamma_{\infty}$ also belongs to the domain of $\mathscr{M}^{*}$. Then (4.20)
follows by a similar proof as Theorem 4.3.

(ii) follows by comparing (4.20) with (4.17). Let us show (iii). Since
$\nu=\mathbb{P}\circ Z_{\infty}$, setting $\Phi(h)=h(t_{0})$, then by (4.20)
we deduce (4.21). The proof of (iv) is similar. Finally, the first part of (v)
follows by comparing (4.21) with (4.22). To get an explicit formula for
$\tfrac{d\nu}{d\mu}$ we use a disintegration argument (see Theorem B.1).

To this purpose using the notation of Theorem B.1 we have
$\lambda=N_{\mathbb{Q}_{-\infty}}$, $E=E_{-\infty}$ and

$p:E\to H,\;\;\;p(h)=h(0),\;\;\;h\in E.$

By (4.22) we know that $\mu=\lambda\circ p^{-1}$. Introducing

$p^{-1}(x)=\\{h\in E:\,h(0)=x\\},\;\;x\in H,$

we have that there exists a family of Borel measures $(m_{x})_{x\in H}$ in
$(E,\mathscr{B}(E))$ such that the support of $m_{x}$ is included in
$p^{-1}(x)$, for $\mu$-almost all $x\in H$, and

$\displaystyle\int_{H}\varphi(x)\,\nu(dx)=\int_{E}\varphi(h(0))\,\rho_{-\infty}(h)\lambda(dh)=\int_{H}\varphi(x)\left(\int_{p^{-1}(x)}\rho_{-\infty}(h)m_{x}(dh)\right)\,\mu(dx),$

for any $\varphi:H\to\mathbb{R}$ Borel and bounded. Setting

$\displaystyle\psi(x)=\int_{p^{-1}(x)}\rho_{-\infty}(h)m_{x}(dh)$

we find that $\psi=\tfrac{d\nu}{d\mu}$, $\mu$-a.s. ∎

##  5 Colored noise

We are here concerned with the following stochastic differential equation on
$H$,

$\left\\{\begin{array}[]{l}dV(t)=(Av(t)+b(V(t))dt+(-A)^{-\epsilon/2}\,dW(t),\quad
t\geq 0\\\ \\\ V(0)=x\in H,\end{array}\right.$ (5.1)

assuming, the following

######  Hypothesis 3.

(i) $\epsilon>0.$

(ii) Hypothesis 1(i)-(iii) are fulfilled.

(iii) $(-A)^{\epsilon}\,b$ is Lipschitz in $H$.

Proceeding as in Section 1, we see that problem (5.1) has a unique mild
solution $V_{x}$ which is a continuous adapted process,

$V_{x}(t)=e^{tA}x+\int_{0}^{t}e^{(t-s)A}\,b(V_{x}(s))ds+(-A)^{-\epsilon/2}\,W_{A}(t),\quad
t\geq 0,$ (5.2)

where $W_{A}$ still denote the stochastic convolution

$W_{A}(t)=\int_{0}^{t}e^{(t-s)A}\,dW(s),\quad t\geq 0.$

We again fix $T>0$ and set $E_{[0,T]}=C([0,T];H)$. As we have seen in Section
2, the law of $W_{A}$ in $E_{[0,T]}$ is Gaussian
$\mathscr{N}_{\mathbb{Q}_{T}}$, described on Section 2 above.

We are going to determine the law of $V_{x}$ on $E_{[0,T]}$ for all $x\in H$,
by proceeding as in Section 1. Setting $V_{x}(t)-e^{tA}x=L_{x}(t),$ equation
(5.2) becomes

$L_{x}(t)=\int_{0}^{t}e^{(t-s)A}\,(b(L_{x}(s)+e^{sA}x)ds+(-A)^{-\epsilon/2}\,W_{A}(t),\quad
t\in[0,T].$ (5.3)

Moreover, for every $x\in H$ we consider some mappings defined in §1 as
$k_{x},$ $F_{x},G_{x},\gamma_{x}$ defined in Section 1 above, see (1.4),
(1.5), (1.6) and (1.7) respectively.

Now the solution $K_{x}(\cdot)$ of (5.3) is given by

$L_{x}(\cdot)=F_{x}((-A)^{-\epsilon/2}\,W_{A}(\cdot))$ (5.4)

Therefore we have

$(\mathbb{P}\circ
V_{x}^{-1})(\Phi)=\mathbb{E}[\Phi(F_{x}((-A)^{-\epsilon/2}W_{A}(\cdot))+e^{\cdot
A}x)],\quad\forall\,\Phi\in B_{b}(E_{[0,T]}).$ (5.5)

By the change of variables $\Omega\to\mathbb{X},\;\omega\to
W_{A}(\cdot)(\omega)$ we obtain

$(\mathbb{P}\circ
V_{x}^{-1})(\Phi)=\int_{\mathbb{X}}\varphi\left[F_{x}((-A)^{-\epsilon/2}\,h(\cdot))+e^{\cdot
A}x\right]N_{\mathbb{Q}_{T}}(dh),\quad\varphi\in B_{b}(E_{[0,T]}),\,t\geq 0,$
(5.6)

where $N_{\mathbb{Q}_{T}}$ is the law of $W_{A}(\cdot)$ in $\mathbb{C}$.
Setting $(-A)^{-\epsilon/2}h=k,$ yields

$(\mathbb{P}\circ
V_{x}^{-1})(\Phi)=\int_{\mathbb{X}}\varphi\left[F_{x}(k(\cdot))+e^{\cdot
A}x\right]\,N_{\mathbb{Q}^{\epsilon}_{T}}(dk),\quad\varphi\in
B_{b}(E_{[0,T]}),$ (5.7)

where $\mathbb{Q}^{\epsilon}_{T}=(-A)^{-\epsilon}\mathbb{Q}_{T}$. Note that
$\mathbb{Q}^{\epsilon}_{T}$ is obviously symmetric and of trace class.

Moreover, by the change of variables $F_{x}(h)=k,$ we obtain

$(\mathbb{P}\circ V_{x}^{-1}))(\Phi)=\int_{\mathbb{X}}\Phi(k(\cdot)+e^{\cdot
A}x)\,(N_{\mathbb{Q}^{\epsilon}_{T}}\circ G_{x})(dk),\quad\forall\,\Phi\in
B_{b}(E_{[0,T]}).\color[rgb]{0,0,0}$ (5.8)

We shall denote by $\mathscr{H}^{\epsilon}_{T}$ the Cameron–Martin space of
$N_{\mathbb{Q}^{\epsilon}_{T}}$, endowed with its natural norm
$|h|_{\mathscr{H}^{\epsilon}_{T}}.$ Arguing as in §2 we see that it is given
by

$\mathscr{H}^{\epsilon}_{T}:=L^{2}(0,T;D((-A)^{1+\epsilon/2})\cap
W_{0}^{1,2}(0,T;D((-A)^{\epsilon/2})),$ (5.9)

Moreover

$|u|^{2}_{\mathscr{H}_{\overline{\mathbb{Q}^{\epsilon}}_{T}}}:=|\overline{Q^{\epsilon}_{T}}^{-1/2}u|^{2}_{\mathbb{X}}=|(-A)^{1/2+\epsilon}\,u(T)|_{D_{A}(\epsilon,2)}^{2}+\int_{0}^{T}(|u^{\prime}(t)|_{D{(-A)^{\epsilon/2})}}^{2}+|Au(t)|_{D((-A)^{\epsilon/2})}\,dt.$
(5.10)

Hypothesis 3 ensures that $\gamma_{x}(k)\in\mathscr{H}^{\epsilon}_{T}$ (2) (2)
(2)which is defined by (5.9) because

$b(h+e^{\cdot A}x)\in L^{2}(0,T,D((-A)^{-\epsilon/2}).$

Now, by proceeding as in Section 3 we prove the following result.

######  Theorem 5.1.

The law of $V_{x}$ on $E_{[0,T]}$ is given by

$(\mathbb{P}\circ V_{x}^{-1})(\Phi)=\int_{\mathbb{X}}\Phi(h+e^{\cdot
A}x),\rho_{\epsilon}(x,k)\,N_{\mathbb{Q}^{\epsilon}_{T}}(dk),\quad\forall\,\Phi\in
B_{b}(E_{[0,T]}),\color[rgb]{0,0,0}$ (5.11)

where

$\rho_{\epsilon}(x,k)=\exp\left\\{-\tfrac{1}{2}|\gamma_{x}(k)|^{2}_{\mathscr{H}^{\epsilon}_{T}}+I(\gamma_{x})(k)\right\\}$
(5.12)

and

$\gamma_{x}(k)=\int_{0}^{\cdot}e^{(\cdot-s)A}b(k(s)+e^{sA}x)\,ds,\quad x\in
H,\,k\in E_{[0,T]}.$ (5.13)

######  Remark 5.2.

Larger is $\epsilon>0$ stronger becomes Hypothesis 2(ii) and moreover narrow
is the Cameron–Martin space. The Hypothesis 2(ii) becomes stronger and the
Cameron-Martin space narrower as $\epsilon>0$ grows larger.

##  Appendix A Maximal regularity for linear evolution equations

Let us consider an abstract evolution equation,

$u^{\prime}(t)=Au(t)+f(t),\quad u(0)=0,\quad t\in[0,T],$ (A.1)

on a Hilbert space $H$, where $A:D(A)\subset H\to H$ is self–adjoint negative
and $f\in L^{2}(0,T;H)$.

The following result is well known, see e.g. [12], we recall the easy proof,
however, for the reader convenience.

######  Proposition A.1.

Let

$u(t)=\int_{0}^{t}e^{(t-s)A}f(s)\,ds=(e^{\cdot A}*f)(t),\quad t\in[0,T].$
(A.2)

Then $u\in W^{1,2}(0,T,H)\cap L^{2}(0,T;D(A))$ and fulfils (A.1). Moreover, it
result

$|u^{\prime}|_{L^{2}(0,T;H)}\leq
2|f|_{L^{2}(0,T;H)},\quad|Au|_{L^{2}(0,T;H)}\leq|f|_{L^{2}(0,T;H)}.$ (A.3)

###### Proof.

Denote by $\hat{f}(k),\,k\in H,$ the Fourier transform of $f$ and by
$\hat{u}(k),\,k\in H,$ the Fourier transform of $u$. ($f$ and $u$ are extended
by $0$ outside $[0,T]$). Taking the Fourier transform on both sides of (A.2)
yields

$\hat{u}(k)=A(k-A)^{-1}\hat{f}(k),\quad k\in H.$

Since $\|A(k-A)^{-1}\|_{\mathscr{L}(H)}\leq 1$ we have
$|\hat{u}(k)|\leq|\hat{f}(k)|$ for all $k\in H$ and the conclusion follows. ∎

Let us define the maximal regularity space $\Gamma_{A}(H)$ by setting

$\Gamma_{A}(H)=L^{2}(0,T;D(A))\cap\\{u\in W^{1,2}(0,T;H):\,u(0)=0\\}.$ (A.4)

Then $\Gamma_{A}(H)$, endowed with the norm:

$|u|^{2}_{\Gamma_{A}(H)}=\int_{0}^{T}\left(|u^{\prime}(t)|_{H}^{2}+|Au(t)|_{H}^{2}\right)\,dt,$

is a Hilbert space.

If $u=e^{\cdot A}*f$ we have

$|e^{\cdot A}*f|_{\Gamma_{A}(H)}\leq c_{1}|f|_{\mathbb{X}}.$ (A.5)

Moreover, since it results

$\Gamma_{A}(H)\subset C([0,T;D((-A)^{1/2}),$ (A.6)

with continuous inclusion, see [1] and [12], there is $c_{2}>0$ such that

$|((-A)^{1/2}u(T)|_{H}\leq c_{2}\,|f|_{\mathbb{X}}.$ (A.7)

######  Corollary A.2.

Let $\epsilon>0$, $f\in L^{2}(0,T;D((-A)^{\epsilon/2}))$. Then

$u\in L^{2}(0,T;D((-A)^{1+\epsilon/2})\cap W^{1,2}(0,T;D((-A)^{\epsilon/2})).$
(A.8)

Moreover, the following continuous inclusion holds

$L^{2}(0,T;D((-A)^{1+\epsilon/2}))\cap
W^{1,2}(0,T;D((-A)^{\epsilon/2}))\subset C([0,T;(-A)^{\epsilon/2})).$ (A.9)

Finally,

$|Au|_{L^{2}(0,T;D((-A)^{\epsilon/2}))}\leq|f|_{L^{2}(0,T;D((-A)^{\epsilon/2}))},\quad|u^{\prime}|_{L^{2}(0,T;D((-A)^{\epsilon/2}))}\leq|f|_{L^{2}(0,T;D((-A)^{\epsilon/2}))}.$
(A.10)

###### Proof.

It is sufficient to apply Proposition A.1 replacing $H$ with
$D((-A)^{\epsilon/2}).$ ∎

##  Appendix B A disintegration theorem

For the proof of the next result see, for instance, the appendix in [6].

######  Theorem B.1.

Let $E$ be a Polish space, $p\colon E\to H$ Borel, $\lambda\in\mathscr{P}(E)$
and $\mu=\lambda\circ p^{-1}$ the law of $p$. There exists a family of Borel
measures $(m_{x})_{x\in H}$ in $(E,\mathscr{B}(E))$ such that

$\int_{E}\varphi(h)\lambda(dh)=\int_{H}\left(\int_{E}\varphi(h)m_{x}(dh)\right)(\lambda\circ
p^{-1})(dx),$ (B.1)

for all $\varphi\colon E\to\mathbb{R}$ bounded and Borel.

Moreover the support of $m_{x}$ is included in $p^{-1}(x)$ for $\mu$-almost
all $x\in H$, so that we can write (B.1) as

$\int_{E}\varphi(h)\lambda(dh)=\int_{H}\left(\int_{p^{-1}(x)}\varphi(h)m_{x}(dh)\right)\,\mu(dx).$
(B.2)

## Acknowledgments

The authors thank Vladimir Bogachev, Arnaud Debussche and Giorgio Metafune for
useful discussions. Enrico Priola is a member of GNAMPA (INDAM).

## References

  * [1] A. Bensoussan, G. Da Prato, M. Delfour and S.K. Mitter, Representation and Control of Infinite Dimensional Systems, Birkhäuser, Second Edition, 2006.
  * [2] V.I. Bogachev, Gaussian Measures, American Mathematical Society, Providence, 1998.
  * [3] A. Capietto, E. Priola, Uniqueness in Law for Stochastic Boundary Value Problems, Journal of Dynamics and Differential Equations, 23, No. 3, 613-648, 2011.
  * [4] G.Da Prato, Kolmogorov equations for stochastic PDEs, Birkhäuser, 2004.
  * [5] G. Da Prato, Introduction to Stochastic Analysis and Malliavin Calculus, Edizioni della Normale, 2013.
  * [6] G.Da Prato, A. Lunardi and L.Tubaro, Surface measures in infinite dimension, Rend. Lincei Mat. Appl, 25, 309-330, 2014.
  * [7] G. Da Prato and J. Zabczyk, Second Order Partial Differential Equations in Hilbert Spaces, Cambridge University Press, 2002
  * [8] G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications, Cambridge University Press, second edition, 2014.
  * [9] M. I. Friedlin and A. Wentzel, Random Perturbations of Dynamical Systems, Springer–Verlag, 2012.
  * [10] I. G. Gohberg and M. G. Krein, Introduction to the theory of the linear of nonselfadjoint operators, Amer. Math, Soc., Providence, Rhode Island, 1969.
  * [11] S. Kusuoka, The nonlinear transformation of Gaussian measures in Banach spaces and its absolute continuity I, II, J. Fac. Sci. Univ. Tokyo, Sec. 1A, 29, 1982, no 3, 567–597; 30, 1983, no. 1, 199–220.
  * [12] A. Lunardi, Interpolation Theory. Third edition. Edizioni della Normale, 2018.
  * [13] D. Nualart, The Malliavin Calculus and Related Topics,  Second edition Springer, 2006.
  * [14] D. Nualart, E. Pardoux, Second order stochastic differential equations with Dirichlet boundary conditions, Stoch. Process. Appl. 39, 1-24, 1991.
  * [15] R. Ramer, On Nonlinear Transformations of Gaussian Measures, J. Funtional Analysis, 15, 166–l 87, 1974.