We consider the following stochastic nonlinear Schr¨odinger equation
idu+ (∆u− |x|2u−λ|u|2σu)dt=ε|x|2u◦dW, (4.4) where ◦ stands for a Stratonovich product in the right hand side of (4.4), σ > 0, ε > 0, andλ=±1. The unknown functionuis a random process on a probability space (Ω,F,P) endowed with a standard filtration (Ft)t≥0 such that F0 is complete, W(t) is a standard real valued Brownian motion on R+ associated with the filtration (Ft)t≥0. We have set ξ˙= dWdt ,V(x) =K(x) =|x|2 in the equation (4.3). It will be useful to introduce here the equivalent Itˆo equation which may be written as
idu+
∆u− |x|2u+ i
2ε2|x|4u−λ|u|2σu
dt=ε|x|2udW. (4.5)
We define
Σk=
v∈L2(Rd), X
|α|+|β|≤k
|xβ∂xαv|2L2(Rd)=|v|2Σk <+∞
fork∈N,and write Σ−k for the dual space of Σk in theL2 sense. In particular we denote Σ1 by Σ.
We define the energy H(u) = 1
2|∇u|2L2 +1
2|xu|2L2+ λ
2σ+ 2|u|2σ+2L2σ+2, (4.6) which is a conserved quantity of the deterministic equation i.e., (4.4) withε= 0. We will consider solutions in the space Σ since H(u) is well defined in Σ, thanks to the Sobolev embedding Σ⊂ H1(Rd) ⊂ L2σ+2(Rd), for σ < d−22 if d≥3 or σ < +∞ ifd= 1,2. It is worth adding a remark that there is also another conserved quantity, namelyL2 norm
Q(u) =1 2|u|2L2.
4.2. Preliminaries and main results 117
In the case where ε = 0, it is known that in the energy space Σ, equation (4.4) is locally well posed for λ=±1, σ < d−22 ifd≥3 or σ <+∞ if d= 1,2 and globally well posed if either λ = 1 or λ = −1 and σ < 2/d (see [130]). In the case where ε 6= 0, it was established in [56] a random Strichartz estimate and the authors proved the local and global well-posedness under the same condition for the nonlinear power as the deterministic case ε = 0. To rigorously state our results, we give some notation. If I is an interval of R, E is a Banach space, and 1≤r ≤ ∞, thenLr(I;E) is the space of strongly Lebesgue measurable functions v from I into E such that the function t→ |v(t)|E is in Lr(I). We define similarly the spacesC(I;E) and Lr(Ω;E).
Proposition 4.1. Assume σ > 0, ε > 0 and λ = ±1. Let u0 ∈ Σ and σ < 2/(d−2) if d ≥3, σ < +∞ if d= 1,2. Then there exist a stopping time τu∗0,ω > 0 and a unique solution u(t) adapted to (Ft)t≥0 of (4.4) with u(0) =u0 almost surely in C([0, τu∗0,ω); Σ).
Moreover, we have almost surely,
τu∗0,ω = +∞ or lim sup
t%τu∗
0,ω
|u(t)|Σ= +∞,
and the L2 norm is conserved:
Q(u(t)) =Q(u0), a.s. in ω, for all t∈[0, τu∗0,ω).
In particular, if λ= 1, then there exists a unique global solution u(t) adapted to (Ft)t≥0 of (4.4) with u(0) =u0 almost surely inC(R+; Σ).
The result of Proposition 4.1 holds, not only for a quadratic potential, but for more general potentialsK(x) andV(x) in (4.3) (see [56] for details). The key point of the proof of Proposition4.1 is the following gauge transformation: letw(t) be a solution of
i∂tw=−(∇ −ix(t+εW(t)))2w, (4.7) then
u(t, x) = expn
− i
2|x|2(t+εW(t))o
w(t, x) (4.8)
satisfies Equation (4.4) withλ= 0. Therefore, the Cauchy problem for Eq.(4.4) (or equiv-alently (4.5)) is deduced from the Cauchy problem for
i∂tw=−(∇ −ix(t+εW(t)))2w+λ|w|2σw, (4.9) which is adeterministic equation for each fixedω ∈Ω.
In addition to the results of Proposition 4.1, in this chapter, we will make use of the following fact. There is an explicit representation of the solution of the linear part of
Eq.(4.4), which is a derivation from the result in [160]. To introduce it, letT0>0 be fixed and consider (q(t), v(t))∈R2 solution of the system:
( q(t)˙ = v(t)−(t+εW(t))q(t),
˙
v(t) = (t+εW(t))(v(t)−(t+εW(t))q(t))
with initial data (q(0), v(0)) = (0,1) ∈ R2. There exists a unique solution (q, v) ∈ C([0, T0],R2) for eachω ∈Ω satisfyingW(·, ω)∈Cα([0, T0]) with 0< α <1/2. Using this solution, we define the following system of ODEs:
2α(t) = v(t)/q(t),
β(t)˙ = (t+εW(t)−α(t))β(t),
˙
γ(t) = −12β2(t)
Remark that α(t), β(t), γ(t) are well defined for any t ≤ T˜ ∧T0, where ˜T = inf{s >
0, q(s) = 0}. Note that ˜T > 0, thanks to the initial condition (q(0), v(0)) = (0,1). Then we have the following representation formula for the solution of (4.7).
Proposition 4.2. Let T0 > 0 be fixed. For each ω ∈ Ω satisfying W(·, ω) ∈ Cα([0, T0]) with 0 < α < 1/2, the solution of (4.7) with initial data w(0) = u0 ∈ C0∞(Rd) may be expressed as follows.
w(t, x) = 1 (2πiq(t))d/2
Z
Rd
ei(α(t)|x|2+β(t)x·y+γ(t)|y|2)u0(y)dy, (4.10) for anyt≤T0∧T .˜
Remind that our interest is in the vortex solutions. From now on we fixλ= 1, so that we consider the equation;
idu+
∆u− |x|2u+ i
2ε2|x|4u− |u|2σu
dt=ε|x|2udW. (4.11) This equation can be regarded as a stochastically perturbed version of the following de-terministic equation;
i∂tu+ ∆u− |x|2u− |u|2σu= 0. (4.12) We also restrict ourselves to the two dimensional cased= 2, and make use of the polar coordinates : forx= (x1, x2)∈R2, we definex1 =rcosθ,x2 =rsinθ. We recall that the vortex solutions are the solutions of (4.12) of the form,
u(t, x) =e−iµtφµ,m(x) =e−iµteimθψµ,m(r), µ∈R, x∈R2, (4.13) wherem ∈ Z∗ is the winding number of the vortex, and we may assume m ≥1 without loss of generality by the reflection symmetry. Substituting (4.13) in (4.12), φµ,m(x) ∈ Σ
4.2. Preliminaries and main results 119 satisfies
−∆φ+|x|2φ−µφ+|φ|2σφ= 0. (4.14) Before stating more details about the properties of vortex solutions (4.13), we introduce the closed subspace Xm of Σ that has the same symmetry, i.e., m-equivariant symmetry as the vortex solutionsφµ,m(x) =eimθψµ,m(r):
Xm:={v∈Σ, e−imθv(x) does not depend onθ}.
We remark that, thanks to Proposition4.2, one may check that the solutionu(t) of (4.11) belongs toXmif the initial datau(0) =u0belong toXm. This preservation of the structure of Eq.(4.11) in Xm leads to the following result.
Lemma 4.3. Let σ >0 and u0 ∈Xm. Then there exists a unique global solution u(t) of (4.11) which is adapted to (Ft)t≥0, that satisfies the initial condition u(0) =u0, and that belongs almost surely to C(R+;Xm).
We give a proof of Lemma4.3 in Appendix using Proposition4.2.
The existence of the vortex solutions is easily proved (see e.g. [120]), with the help of the compact embedding Xm ⊂ Σ⊂L2, for any µ > λm where λm is the solution of the minimizer of Sµ on Xm. The uniqueness of minimizers is established as in [120]. Such a variational characterization allows to show the orbital stability inXmfor anym≥1. This fact is related to Proposition4.4(i) below.
The solutions φ of (4.14) have regularity properties by the usual elliptic bootstrap arguments, i.e.,
|x|→∞lim |φµ,m(x)|= 0, and φµ,m ∈ \
2≤q<∞
W2,q∩C2.
The radial profileψµ,m(r) is thus of classC2(0,∞) and satisfies the equation
In particular, the solution ψµ,m associated to the global minimizerφµ,m of (4.16) is non-negative. This comes from the fact that if ψ is a minimizer of (4.16), then|ψ|as well as seen from the radial form of (4.16), and
Z ∞
Remark that every regular solution of (4.17) should satisfy ψ(r) → 0, as r → 0+ (see [126]), which we will see again later in Proposition4.4below.
Now we summarize here some properties ofψµ,m(r). The inner product in the Hilbert spaceL2(R2) is denoted by (·,·)L2(R2,dx), i.e., Xm,we here introduce the radial norm and the radial inner products, too.
(u, v)L2r = and consider the associated radial profile ψµ,m(r) := e−imθφµ,m(x). Then the following
properties hold.
(i) There existν =ν(µ, m)>0, such that for any v∈Xm satisfying Re(v, iφµ,m)L2(R2,dx) = 0,
we have
hSµ00(φµ,m)v, vi ≥ν|v|2Σ. (ii) The radial profileψµ,m(r) has the following asymptotics.
ψµ,m(r) =O(rµ/2e−r2/2), r→+∞, and ψµ,m(r) =O(rm), r →0+. (iii) The radial profile ψµ,m(r) has a local maximum for some r0 ∈(m/√µ,√µ), is
increasing on (0, r0) and is decreasing on (r0,∞). Moreover, |ψµ,m(r)|2σ < µ−2m for allr >0.
4.2. Preliminaries and main results 121 The positivity ofSµ00(φµ,m), that is (i) of Proposition4.4, is discussed in the Appendix;
for the properties (ii) and (iii), see Appendix of [108] or [99].
Note that the second-order differentialS00µ(φµ,m) is related to the linearization problem arounde−iµtφµ,m(x) in (4.12), which is precisely written as
dy
dt =JLµ,my in Σ−1, where
J =−i: Re u Imu
!
7→ 0 1
−1 0
! Re u Imu
! ,
Lµ,m=Sµ00(φµ,m) = L1µ,m 0 0 L2µ,m
!
, (4.18)
L1µ,m =−∆ +|x|2−µ+ (2σ+ 1)|φµ,m|2σ, L2µ,m=−∆ +|x|2−µ+|φµ,m|2σ. In the radial notation, we will use, as the linearized operators corresponding toL1µ,m and L2µ,m,
L1,rµ,m = − d2 dr2 −1
r d
dr+r2+m2
r2 −µ+ (2σ+ 1)ψµ,m2σ , (4.19) L2,rµ,m = − d2
dr2 −1 r
d
dr+r2+m2
r2 −µ+ψµ,m2σ . (4.20)
Our purpose is to investigate the influence of random perturbations of the form given in equation (4.11) on the phase of vortex solutions (4.13).
We fix µ0 > λm and consider for ε > 0 the solution uε(t, x) of equation (4.11) given by Lemma4.3with initial datauε(0, x) =φµ0,m(x), which is a stable vortex in Xm of the deterministic equation.
The first theorem says that we can decompose uε in Xm as the sum of a modulated vortex solution and a remainder with small Σ norm, for t less than some stopping time τε, and that thisτε goes to infinity in probability asεgoes to zero. We will see then that the remaining part is of order one with respect to ε. The proof of the theorem is rather similar to those in [52, 53, 55], and we only mention the differences with respect to the previous works. The decomposition we consider is of the form
uε(t, x) =e−iξε(t)(φµ0,m(x) +εηε(t, x)) (4.21) for a semi-martingale process ξε(t) with values in R, and ηε with value in Xm. Here, we use the following orthogonality condition
Re(ηε, iφµ0,m)L2(R2,dx) = 0, a.s., t≤τε, (4.22)
for a suitable stopping time τε as long as |εηε|Σ remains small. The precise result is the following.
Theorem 4.5. Let 1/2 ≤ σ < ∞ and µ0 > λm be fixed. For ε > 0, let uε(t, x), as defined above, be the solution of (4.11) with u(0, x) =φµ0,m(x). Then there exists α0 >0 such that, for each α, 0 < α ≤ α0, there is a stopping time ταε ∈ (0,∞) a.s., and a semi-martingale process ξε(t), defined a.s. fort≤ταε, with values inR, so that if we set εηε(t, x) =eiξε(t)uε(t, x)−φµ0,m(x), then (4.22) holds. Moreover, a.s. for t≤ταε,
|εηε(t)|Σ ≤α. (4.23)
In addition, there is a constant C=Cα,µ0 >0, such that for any T >0 and any α≤α0, there is ε0 >0, for which
∀ε < ε0, P(ταε ≤T)≤exp
− C ε2T
. (4.24)
This means that the modulation parameterξε(t) is a semi-martingale process defined up to times of the order ε−2, i.e. the shape of the vortex solution is preserved over this time scale. We give in the next theorem the behavior of ηε, and the convergence of the modulation parameter asεgoes to zero.
Theorem 4.6. Let 1 ≤ σ < ∞, µ0 > λm be fixed and ηε, ξε, for ε > 0 be given by Theorem4.5, withα ≤α0 fixed. Then, for anyT >0, theXm-valued process(ηε(t))t∈[0,T∧ταε]
converges in probability, asεgoes to zero, to a processη= (η(t))t∈[0,T]satisfyinge−imθη= ˜η and
d˜η=J L1,rµ0,m 0 0 L2,rµ0,m
!
˜
ηdt−y 0 ψµ0,m
!
dt− 0 r2ψµ0,m
!
dW −z 0 ψµ0,m
!
dW, (4.25) with η(0) = 0, and˜
y(t) = 2σ(Re ˜η, ψµ2σ+10,m)L2
r
|ψµ0,m|2L2
r
, z(t) = |rψµ0,m|2L2
r
|ψµ0,m|2L2
r
, (4.26)
for allt≥0.The convergence holds in C([0, ταε∧T], L2).
The above process η satisfies for any T >0 the estimate E sup
t≤T |η(t)|2Σ
!
≤CT for some constant C >0.
Moreover the modulation parameter ξε may be written, for t≤ταε, as dξε=µ0dt+εyεdt+εzεdW,
4.3. Proof of Theorem 4.5 123