XXVIII
1.1.4. CONSIDERACIONES TRABAJO POTENCIA a. Trabajo
In this section, we prove that the probability measure µ is a unique invariant measure on D(A). Let us recall that a measure µ defined onH is an invariant measure if
Z
Hφ(x)dµ(x) = Z
HPtφ(x)dµ(x), for all t≥ 0, φ ∈ Cb(H; R), (5.1) where Ptis the Markov semigroup of the process X(t, x). In other words, the measure µ is invariant if P∗tµ = µ for all t ≥ 0, where P∗t is the dual semigroup of Pt.
Since Q is nondegenerate and the jump noise coefficient is independent of the state, (Pmt k)t≥0 has a unique invariant measure µmk (see [1, 37]). By Dynkin’s formula, we have
E[φ(Xmk(t, x))] =E[φ(x)] + E
Z t 0
Lxmkφ(Xmk(s, x))ds
, for all φ∈ D(Lxmk), (5.2) where Lxmk is defined in (2.17). Thus, from (5.2), we obtain
(Pmt k)∗µmk, φ
=
(Pm0 k)∗µmk, φ +
Z t 0
(Pmsk)∗µmk, Lxmkφ
ds. (5.3)
Since µmk is an invariant measure, (Pmt k)∗µmk = µmk, for all t ≥ 0, and hence we have
hµmk, Lxmkφi = 0, for all φ ∈ D(Lxmk). (5.4)
The jump noise coefficient Ψ(·, ·) satisfies the following uniform bound:
Tlim→∞
1 T
Z T 0
Z
ZkAkΨ(t, z)k2λ(dz)dt = C < +∞, for k = 0,1 2, 1 +g
2 and g > 0. (5.5) Lemma 25. Suppose Assumption (5) holds true. There exists a constant C > 0 such that for any k ∈ N :
Z
H
hkA1/2xk2+kAxk2/3 +kA1+g2xk(1+2g )/(10+8g )i
dµmk(x)≤ C. (5.6)
P r o o f. Applying (2.17) for φ(x) = kxk2, we have Lxmkkxk2 = Tr[Qmk] +
Z
Zm k kΨmk(z)k2λ(dz)− 2kA1/2xk2, and by the invariance of µmk, we also have
0 = Z
HLxmkkxk2dµmk(x) = Tr[Qmk] + Z
Zm k kΨmk(z)k2λ(dz)− 2 Z
HkA1/2xk2dµmk(x).
(5.7) Integrating (5.7) with respect to time in [0, T ], dividing by T and using Assumption 5 gives an estimate for the first term in the left hand side of (5.6). Let us now take φ(x) = (1+kA11 / 2xk2), then we have
Dxφ = −2Ax
1 +kA1/2xk22 and D2xφ = −2A
1 +kA1/2xk22 + 8Ax⊗ Ax
1 +kA1/2xk23. (5.8)
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Using Taylor’s formula, we obtainZ The integral with L´evy measure in (5.10) is again bounded by Assumption 5. We integrate (5.10) overH to get
0 = We use H¨older’s inequality, (5.7) and (5.11) to obtain
Z
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Thus, we get
Lmxkφ = −2 (1 + 2g)
1
1 +kA(1+g )/2xk23 + 2 g1 + 2 g Tr
QmkA1+g
+4(3 + 2g) (1 + 2g)2
1
1 +kA(1+g )/2xk24 ( 1 + g )
1 + 2 g kQ1/2mkA1+gk2 +
Z
Zm k
"
1
1 +kA(1+g )/2(x + Ψmk(z))k22/(1+2g ) − 1
1 +kA(1+g )/2xk22/(1+2g )
+ 4
(1 + 2g)
A(1+g )/2x, A(1+g )/2Ψmk(z) 1 +kA(1+g )/2xk23 + 2 g1 + 2 g
λ(dz)
+ 4
(1 + 2g)
1
1 +kA(1+g )/2xk23 + 2 g1 + 2 g
kA1+g /2xk2+ Bmk(x), A1+gx
.
Let us use Lemma 3-(ii) with δ = 1 + g to obtain
Bmk(x), A1+gx ≤ 12kA1+g /2xk2+ CkA(1+g )/2xk2(3+2g )/(1+2g ). Proceeding as before and using the assumptions on noise coefficient, we get
Lmxkφ≥ 2 1 + 2g
kA1+g /2xk2
1 +kA(1+g )/2xk23 + 2 g1 + 2 g − C.
Let us integrate the above inequality to find Z
H
kA1+g /2xk2
1 +kA(1+g )/2xk2(3+2g )/(1+2g )dµmk(x)≤ C. (5.13) Now, by using H¨older’s inequality, (5.12) and (5.13), we obtain
Z
HkA1+g /2xk(1+2g )/(10+8g )dµmk(x)
≤ Z
H
kA1+g /2xk2
1 +kA(1+g )/2xk2(3+2g )/(1+2g )dµmk(x)
!(1+2g )/(20+16g )
×
Z
H
1 +kA(1+g )/2xk21/3
dµmk(x)
(19+14g )/(20+16g )
≤ C. (5.14)
Combining (5.7), (5.12) and (5.14), we finally get (5.6).
By Lemma 25, it follows that the sequence (µmk)k∈Nis tight on D(A) and there exists a subsequence, denoted by (µmk)k∈N for simplicity, and a measure µ on D(A) such that µmk converges weakly to µ. Furthermore, µ D A1+g2
= 1.
Let us take φ ∈ E . By the invariance of µmk, we have Z
HPmt kφ(x)dµmk(x) = Z
Hφ(x)dµmk(x), (5.15)
for any t ≥ 0. Note that
Z
HPmt kφ(x)dµmk(x)− Z
HPtφ(x)dµ(x)
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By the weak convergence of µmk, the second integral on the right hand side goes to 0 as k → ∞. The first integral can be further estimated as
By Lemma 20, the approximations Pmt kφ converges to Ptφ uniformly on KRand the second integral in the right hand side of the above inequality is also bounded by Lemma 25. Since R is arbitrary and by the weak convergence of µmk, taking limit k → ∞ in (5.15), one can get (5.1). Hence µ is an invariant measure.
Due to the classical result of Khasminskii and Doob (see Theorem 4.2.1, [9] or Theorem 22, [14]), ergod-icity and strongly mixing properties of the measure µ are the direct consequence of strong Feller property and irreducibility of the transition semigroup (Pt)t≥0.
Definition 26. A transition semigroup (Pt)t≥0 is strong Feller on D(A) if for any φ ∈ Bb(D(A);R) and
Proposition 27. The transition semigroup (Pt)t≥0is strong Feller on Cb(D(A);R).
P r o o f. By Proposition 17, it is clear that kPtφk0 ≤ kφk0, for φ∈ Cb(D(A);R), and in view of Theorem 7, it is true for any φ ∈ Bb(D(A);R), and the strong Feller property holds.
The irreducibility of the transition semigroup is proved in the next proposition.
Proposition 28. Suppose that Assumptions 2.1 and 2.1 hold true. Then the transition semigroup (Pt)t≥0 corresponding to the system (2.2) is irreducible on D(A).
We first prove Proposition 28 when λ(Z) < +∞. Since we are assuming that the L´evy measure λ(Z) < +∞, we can define the jump times of π(dt, dz) as 0 < σ1(ω) < σ2(ω) < . . . . The jump integral is
Since the jump occurs only at σ1, the first integral is zero on [0, σ1). Hence the equation (2.2) is equivalent to the following:
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on [0, σ1). By Theorem 2.1 [42], there exists a martingale solution of the problem (5.16) in [0, σ1(ω)). We can recursively obtain a martingale solution (Ωx,Px, Fx, X(t, x)) of the system (5.16) for the interval [0, T ] (see [16, 21]).
Now, the proof of Proposition 28 is completed by proving Lemmas given below. The irredcibility of the semigroup (Pt)t≥0 is closely related to the controllability of the Navier–Stokes equation (5.16) with the noise replaced by a right hand side forcing/control term. This technique has been developed for the Navier–Stokes equations with Gaussian noise in [23, 8, 14]. More precisely, we consider a control system with λ(Z) < +∞ :
dy(t)
dt =−[Ay(t) + B(y(t))] − Z
Z
Ψ(t, z)λ(dz) + U(t), y(0) = x,
(5.17)
where U(t) is the control function. Then, we have
Lemma 29. Let T > 0, x ∈ D(A) and yT ∈ D A3/2be given, and assume that the L´evy measure satisfies
sup
t∈[0,T ]
Z
Z
kA1/2Ψ(t, z)kλ(dz) ≤ C(T, Ψ). (5.18)
Then, there exists a control U ∈ L∞ 0, T ; D A1/2
and y ∈ C([0, T ]; D(A)) ∩ L2 0, T ; D A3/2satisfying (5.17) such that y(T ) = yT.
P r o o f. Let us first set U = 0. Now we show that there exists a time T∗ such that 0 < T∗ < T and y∈ C([0, T∗]; D(A)) satisfying (5.17). Let M > 0 and consider
S =n
v∈ C([0, T ]; D(A)) : kAv(t)k ≤ M for all t ∈ [0, T ]o . Let us take any v ∈ S and define y = F(v) by
y(t) = e−tAx− Z t
0
e−(t−s)AB(v(s))ds− Z t
0
Z
Z
e−(t−s)AΨ(s, z)λ(dz)ds. (5.19) Note that by the properties of the semigroup e−tA
t≥0, Lemma 3-(i) and Assumption 2.1, we have kAy(t)k ≤ ke−tAAxk +
Z t
0 kA1/2e−(t−s)AkL(H)kA1/2B(v(s))kds +
Z t 0
Z
Zke−(t−s)AkL(H)kAΨ(s, z)kλ(dz)ds
≤ kAxk + C Z t
0
(t− s)−1/2kAv(s)k2ds + Z t
0
Z
ZkAΨ(s, z)kλ(dz)ds
≤ kAxk + 2CM2t1/2 + (λ(Z)T C(T, Ψ))1/2. (5.20)
Thus, kAy(t)k ≤ M for all t ∈ [0, T ] provided,
kAxk + 2CM2T1/2+ (λ(Z)T C(T, Ψ))1/2 ≤ M.
For any M > kAxk, there exists 0 < T∗< T such that the above inequality holds. Moreover, for any v1, v2 ∈ S, we have
kA(F(v1)− F(v2))k ≤ 4CMt1/2kA(v1− v2)k. (5.21)
Choose T∗ ∈ (0, T ) such that 4CMT∗1 / 2 < 1, so that F is a strict contraction on S. Therefore by a fixed point argument, we get a unique solution y ∈ C([0, T∗]; D(A)) to the problem (5.17) with U = 0.
Author Man uscript
It can also be established that y ∈ L2 0, T∗; D A3/2
, so that y(t) ∈ D A3/2
, a.e., and one can change T∗ so that y(T∗)∈ D A3/2
. Next we set U = 0 on [0, T∗] and define y on [T∗, T ] as follows:
y(t) = T− t
T− T∗y(T∗) + t− T∗
T − T∗yT, t∈ [T∗, T ], and set U(t) = dy(t)
dt + Ay(t) + B(y(t)) + Z
Z
Ψ(t, z)λ(dz), t∈ [T∗, T ].
(5.22)
Using Lemma 2.4-(i), and Assumption (5.18), one can verify that U and y have the properties described in the Lemma.
Remark 30. Note that Lemma 29 also works for bounded or even for exterior domains. Using the continuous embedding and algebra property ofHαnorm for α > 3/2, one can obtain that for any 0 < ε < 1/4 (see, [29]):
kA1/4−εPH(u· ∇u)k ≤ CkAuk2. (5.23)
Hence, we have
Z t 0
Ae−(t−s)AB(v(s))ds ≤
Z t
0 kA3/4+εe−(t−s)AkL(H)kA1/4−εB(v(s))kds
≤ C Z t
0
(t− s)−(3/4+ε)kAv(s)k2ds≤ 4C
1− 4εM2t1/4−ε < +∞.
Using this non-linear estimate in (5.20), the controllability Lemma 29 can be proved for bounded/exterior domains as well.
Let us define e w(t) =
Z t 0
e−(t−s)AU(s)ds, t∈ [0, T ].
Since U ∈ L∞ 0, T ; D A1/2
, the properties of the analytic semigroup leads to ew∈ C([0, T ]; D(Aσ)) for any σ < 3/2. Now, letyem := Pm(y− ew). Thenyem satisfies the equation
dyem
dt =−Aeym− Bm(yem +w) + ge m − Z
Zm
Ψm(t, z)λ(dz), e
ym(0) = Pmx, where
gm =−PmB(y) + Bm(yem +w) = Pe m(−B(y) + B(Pmy)).
Moreover, for any w ∈ L∞(0, T ; D(A)), consider
dybm
dt =−Abym− Bm(ybm + w)− Z
Zm
Ψm(t, z)λ(dz), ym(0) = Pmx.
Using Lemma 29, one can prove the following Lemma by the arguments similar to that of Lemma 7.7, [8] or Lemma 51, [14] .
Lemma 31. There exists a constant C > 0 such that keym − bymkL∞(0,T ;D(A)) ≤ eC K TK1, where K := kykL∞(0,T ;D(A))+k ewkL∞(0,T ;D(A))+ 14
and
K1 :=kw − ewkL∞(0,T ;D(A))+kgmkL4(0,T ;D(A1 / 2)),
Author Man uscript
provided kw − ewkL∞(0,T ;D(A)) ≤ 1 and eC K TK1 ≤ 12
kykL∞(0,T ;D(A))+k ewkL∞(0,T ;D(A))
. Moreover, we have lim
m→∞gm = 0 in L4 0, T ; D A1/2
.
Consequently, let x0 ∈ D(A) and ε > 0. Then, for any w ∈ C([0, T ]; D(A)) such that kw− ewkL∞(0,T ;D(A)) ≤ η, where η = min
1, ˜εe−C K T
and eε > 0 depending on ε, we also have
kA(bym(T ) + wm(T )− x0)k ≤ ε. (5.24)
Let PΨt (x,·) be the transition probability corresponding to the system (2.2) with finite L´evy measure λ(Z) <
+∞. The irreducibility of this case is proved in the following lemma by appropriately choosing the function φ : Lemma 32. Let x0 ∈ D(A), ε > 0 and φ ∈ E be such that
φ(x) =
1 if x∈ BD(A)(x0, ε),
0 if x6∈ BD(A)(x0, 2ε), (5.25)
and 0 ≤ φ(x) ≤ 1, if x ∈ BD(A)(x0, 2ε)\BD(A)(x0, ε). Then, for any t > 0 and x∈ D(A), we have Ptφ(x) > 0, where Ptis the transition semigroup corresponding to the system (5.16) .
Moreover, for any t > 0 and x ∈ D(A), the transition probability PΨt (·, ·) is also irreducible.
P r o o f. Let us first show the irreducibility of the transition semigroup associated with the system (5.16). Let Xm(t, x) be the solution of the finite dimensional approximations of (5.16) on [0, T ] and G(t) be the solution of (2.25). Let x0 ∈ D(A) and ε > 0. Then, by using (5.24), there exists m0 ∈ N and η > 0 such that, for mk ≥ m0, we have
Pn
Xmk(T, x)∈ BD(A)(x0, ε)o
=Pn
kA(Xmk(T, x)− x0)k ≤ εo
≥ Pn
kG − ewkL∞(0,T ;D(A)) ≤ ηo . We know that Ker Q = {0} and hence we have Pn
kG − ewkL∞(0,T ;D(A)) ≤ ηo
> 0. Also, for Γmk :=
ω∈ Ω : Xmk(T, x)∈ BD(A)(x0, ε)
, we have PmTkφ(x) =E[φ(Xmk(T, x))] =
Z
Γm k
φ(Xmk(T, x))dP(ω) +Z
Γcm k
φ(Xmk(T, x))dP(ω)
≥ Pn
Xmk(T, x)∈ BD(A)(x0, ε)o
≥ Pn
kG − ewkL∞(0,T ;D(A)) ≤ ηo . Since PmTkφ(x)→ PTφ(x) as mk → ∞, we arrive at
PTφ(x)≥ Pn
kG − ewkL∞(0,T ;D(A)) ≤ ηo
> 0.
But, by the definition of φ ∈ E , we know that 0 < PTφ(x) =
Z
Γ
φ(X(T, x))dP(ω) + Z
Γc
φ(X(T, x))dP(ω)
=Pn
kA(X(T, x) − x0)k < εo +
Z
{ω ∈Ω:ε≤kA(X(T ,x)−x0)k<2ε}
φ(X(T, x))dP(ω)
≤ Pn
kA(X(T, x) − x0)k < εo +Pn
ε≤ kA(X(T, x) − x0)k < 2εo
=Pn
kA(X(T, x) − x0)k < 2εo
, (5.26)
where Γ :=
ω∈ Ω : X(T, x) ∈ BD(A)(x0, ε) .
Next we complete the proof of Lemma 32 by showing that PΨt (x,·) is irreducible. Let {σk}k≥1 be the in-terarrival times of the Poisson process π associated with {zσk : k ≥ 1} ⊂ Z. Then {σk, zσk} is independent and
Pn
zσk ∈ V, σk > to
= e−λ(Z)tλ(V), for all t > 0, V∈ B(Z).
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Let (Ωx,Px, Fx, X0(t, x)) be a martingale solution of the system (2.1) on [0, σ1). Since{σk, zσk} is indepen-dent of (Ωx,Px, Fx, X0(t, x)), we have X(t, x) = X0(t, x) for 0 ≤ t < σ1, and X(σ1, x) = X(σ1−, x) + Ψ(σ1−, △zσ1).
Let P0t(x,·) be the transition probability of the solution X0(t, x). Then, the relation between P0t(x,·) and PΨt (x,·) can be derived as follows (see, Theorem 14, [43] or [15, 16])
PΨt (x, V) = e−tλ(Z)P0t(x, V) + Z t
0
Z
H
Z
Z
e−sλ(Z)PΨt−s(y + Ψ(s, z), V)P0s(x, dy)λ(dz)ds. (5.27) Since P0t(x,·) is irreducible by the first part of this Lemma and by the above relationship, we have that PΨt (x,·) is also irreducible.
Using all the above Lemmas we complete the proof of Proposition 28.
Proof of Proposition 28. Let (Ωx,Px, Fx, Xm(t, x)) be the martingale solution of the equation
dXm(t, x) =−[AXm(t, x) + Bm(Xm(t, x))]dt +p
QmdW(t) + Z
Zm
Ψm(t, z)π(dt, dz),e Xm(0, x) = Pmx.
(5.28)
By Lemma 32, we know that Xm(t, x) is irreducible, then for any y∈ D(A) and ε > 0, we have Pn
kA(Xm(t, x)− y)k < εo
= δ > 0. (5.29)
Since we know that (Ωx,Px, Fx, X(t, x)) is a martingale solution of the problem (2.1), by using (4.15) (if nec-essary along a subsequence of Xm), we also have
Pn
kA(X(t, x) − Xm(t, x))k ≥ εo
≤ δ
2. (5.30)
Using (5.29) and (5.30), we obtain Pn
kA(X(t, x) − y)k ≥ 2εo
≤ Pn
kA(X(t, x) − Xm(t, x))k ≥ εo +Pn
kA(Xm(t, x)− y)k ≥ εo
≤ δ
2+ 1− δ < 1, (5.31) and hence X(t, x) is irreducible on D(A).
Now let us complete the proof of Theorem 8.
Proof of Theorem 8. Let us use Propositions 27 and 28 to see that the the transition semigroup (Pt)t≥0 is strong Feller and irreducible on D(A). Hence by Doob’s theorem, µ is the unique invariant measure on D(A) and therefore it is ergodic and strongly mixing. The Theorem 8, part (ii) can be established in a similar way as in [13, 14].
Conclusions and future works: The ergodicity of 3D stochastic Navier–Stokes equations perturbed by L´evy noise has been established. Since we do not have uniqueness of solutions for the 3D SNSE with L´evy noise, we consider the Kolmogorov equation involving an integro-differential operator with L´evy mesure associated with this SNSE. The existence of martingale solutions for the 3D SNSE with L´evy noise and the solution to the associated Kolmogorov equation help us to construct a transition semigroup and prove the uniqueness of invariant measures. The classical result of Khasminskii and Doob gives the uniqueness of invariant measure as a direct consequence of the strong Feller property and irreducibility of the transition semigroup associated with the SNSE with L´evy noise. Moreover, such an invariant measure is ergodic and strongly mixing. By assuming that the Gaussian and jump noises are independent, we obtain the BEL formula and thereby we proved the strong Feller property. The irreducibility has been established by proving the controllability of the NSE perturbed by an integral with L´evy measure and a distributed control.
Author Man uscript
One can extend this work for various hydrodynamic models and different structure of the noise coefficients.
In particular, this paper can be extended to the case where the noise coefficients are multiplicative in nature.
As the BEL formula is established for general stochastic differential equations with α−stable noise (see [46]), one can extend our ideas for SNSE with α−stable noise. In fact, the exponential ergodicity of stochastic Burgers equation driven by α−stable noise has already been done in [18]. Since the abstract functional setting for a class of nonlinear stochastic hydrodynamic models perturbed by L´evy noise, namely 3D magnetohydrodynamic(MHD) equation, 3D Leray α-model for Navier–Stokes equation, Shell models of turbulence are same as that of 3D Navier–Stokes equation, the methods used in this paper can be extended to these models as well.
Acknowledgements M. T. Mohan would like to thank the Air Force Office of Scientific Research (AFOSR) and National Research Council (NRC) for Research Associateship Award, and Air Force Institute of Technology (AFIT) and Indian Statis-tical Institute (ISI) Bangalore for providing stimulating scientific environment and resources. K. Sakthivel is supported by the Department of Science and Technology, Govt of India through INSPIRE research grant. S. S. Sritharan’s work has been funded by U. S. Army Research Office, Probability and Statistics program. The first author would also like to thank Dr. Barun Sarkar, Statistics and Mathematics Unit, ISI Bangalore for useful discussions on maximal regularities of stochastic convolutions. The authors sincerely would like to thank the reviewers for their valuable comments and exploring us to some of the literatures related to our work.
References
[1] S. Albeverio, Z. Brze´zniak and J.-L. Wu, Existence of global solutions and invariant measures for stochastic differ-ential equations driven by Poisson type noise with non-Lipschitz coefficients, Journal of Mathematical Analysis and Applications, 371 (2010), 309–322.
[2] S. Albeverio, A. Debussche and L. Xu, Exponential mixing of the 3D stochastic Navier–Stokes equations driven by mildly degenerate noises, Applied Mathematics and Optimization, 66(2012), 273–308.
[3] B. Birnir, The Kolmogorov-Obukhov Theory of Turbulence: A Mathematical Theory of Turbulence, Springer, 2013.
[4] Z. Brze´zniak, E. Hausenblas and J. Zhu, Maximal inequality for stochastic convolutions driven by compensated Poisson random measures in Banach spaces, Annales de l’Institut Henri Poincar´e - Probabilit´es et Statistiques, 53(2) (2017), 937–956.
[5] P. Constantin and C. Foias, Navier–Stokes Equations, Chicago Lectures in Mathematical Series, 1989.
[6] G. Da Prato, An Introduction to Infinite Dimensional Analysis, Springer, 2006.
[7] G. Da Prato, Introduction to Stochastic Analysis and Malliavin Calculus, Lecture Notes, Birkh¨auser Mathematics, 2007.
[8] G. Da Prato and A. Debussche, Ergodicity for the 3D stochastic Navier–Stokes equations, Journal of Mathematiques Pures et Appliqu´ees, 82(2) (2003), 877–947.
[9] G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, London Mathematical Society Lecture Notes, 229, Cambridge University Press, 1996.
[10] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, 1992.
[11] G. Da Prato and J. Zabczyk, Differentiability of the Feynman–Kac semigroup and a control application, Rendiconti Lincei - Matematica e Applicazioni, 8 (1997), 183–188.
Author Man uscript
[12] M. H. A. Davis and M. P. Johansson, Malliavin Monte Carlo Greeks for jump diffusions, Stochastic Processes and their Applications, 116 (2006) 101–129.
[13] A. Debussche and C. Odasso, Markov solutions for the 3D stochastic Navier–Stokes equations with state dependent noise, Journal of Evolution Equations, 6 (2006) 305–324.
[14] A. Debussche, Ergodicity results for the stochastic Navier–Stokes equations: An introduction, Topics in Mathematical Fluid Mechanics, Volume 2073 of the series Lecture Notes in Mathematics, Springer, 23–108, 2013.
[15] Z. Dong, On the uniqueness of invariant measure of the Burgers equation driven by L´evy processes, Journal of Theoretical Probability, 21 (2008), 322–35.
[16] Z. Dong and Y. Xie, Ergodicity of stochastic 2D Navier–Stokes equation with L´evy noise, Journal of Differential Equations, 251 (2011), 196–222.
[17] Z. Dong and J. Zhai, Martingale solutions and Markov selection of stochastic 3D Navier–Stokes equation with jump, Journal of Differential Equations, 250 (2011), 2737–2778.
[18] Z. Dong, L. Xu and X. Xhang, Exponential ergodicity of stochastic Burgers equations driven by α−stable processes, Journal of Statistical Physics, 154 (2014), 929–949.
[19] K. D. Elworthy and X.-M. Li, Formulae for the derivatives of heat semigroups. Journal of Functional Analysis, 125 (1), (1994), 252–286.
[20] H. O. Fattorini and S. S. Sritharan, Optimal control problems with state constraints in fluid mechanics and combustion, Applied Mathematics and Optimization, 38 (1998), 159–192.
[21] B. P. W. Fernando and S. S. Sritharan, Nonlinear filtering of stochastic Navier–Stokes equation with L´evy noise, Stochastic Analysis and Applications, 31 (2013), 1–46.
[22] F. Flandoli and B. Maslowski, Ergodicity of the 2-D Navier–Stokes equation under random perturbations, Communica-tions in Mathematical Physics, 171 (1995), 119–141.
[23] F. Flandoli, Irreducibilty of the 3-D Navier–Stokes equation, Journal of Functional Analysis, 149, 160–177, 1997.
[24] F. Flandoli, An introduction to 3D stochastic fluid dynamics, SPDE in Hydrodynamic: Recent Progress and Prospects, Volume 1942 of the series Lecture Notes in Mathematics, Springer, 51–150, 2008.
[25] C. Foias, O. Manley, R. Rosa and R. Temam, Navier–Stokes Equations and Turbulence, Cambridge University Press, 2008.
[26] R. K. Getoor, On the construction of kernels. In Seminaire de probabiliti´es IX. Lecture Notes in Mathematics. Meyer, P.A., ed., Vol. 465, Springer-Verlag, Berlin, 1975.
[27] M. Hairer and J. C. Mattingly, Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing, Annals of Mathematics, 164 (2006), 993–1032.
[28] A. Ichikawa, Some inequalities for martingales and stochastic convolutions, Stochastic Analysis and Applications, 4(3) (1986), 329–339.
Author Man uscript
[29] G. Iooss, Estimation au voisinage de t = 0, pour un exemple de probl`eme d´evolution o`u il y a incompatibilit´e entre les conditions initiales et aux limites, C. R. Acad. Sci. Paris S´er. A-B 271 (1970), A187A190 (French).
[30] J. M. Lasry and P. L. Lions, A remark on regularization in Hilbert spaces, Israel Journal of Mathematics, 55 (1986), 257–266.
[31] V. Mandrekar and B. R¨udiger, Stochastic integration in Banach spaces: Theory and applications, Springer Cham Heidelberg New York, 2015.
[32] U. Manna and M. T. Mohan, Two-dimensional magnetohydrodynamic systems with jump processes: Well posedness and invariant measures, Communications on Stochastic Analysis, 7(1) (2013), 153 – 178.
[33] C. Marinelli, C. Pr´ev¨ot and M. R¨ockner, Regular dependence on initial data for stochastic evolution equations with multiplicative Poisson noise, Journal of Functional Analysis, 258 (2010), 616–649.
[34] J. C. Mattingly, Ergodicity of 2D Navier–Stokes equations with random forcing and large viscosity, Communications in Mathematical Physics, 206(2) (1999), 273 – 288.
[35] M. M´etivier, Stochastic Partial Differential Equations in Infinite Dimensional Spaces, Quaderni, Scuola Normale Superiore, Pisa, 1988.
[36] M. T. Mohan and S. S. Sritharan, Ergodic control of stochastic Navier–Stokes equation with L´evy noise, Communica-tions on Stochastic Analysis, 10(3) (2016), 389–404.
[37] E. Priola, A. Shirikyan L. Xu and J. Zabczyk, Exponential ergodicity and regularity for equations with L´evy noise, Stochastic Processes and their Applications, 122 (2012) 106–133.
[38] E. Priola and J. Zabczyk, Liouville theorems for non-local operators, Journal of Functional Analysis, 216 (2) (2004) 455–490
[39] M. Romito, Ergodicity of the finite dimensional approximations of the 3D Navier–Stokes equations forced by a degen-erate noise, Journal of Statistical Physics, 114(1–2) (2004), 155–177.
[40] M. Romito, Analysis of equilibrium states of Markov solutions to the 3D Navier–Stokes equations driven by additive noise, Journal of Statistical Physics, 131(3) (2008), 415–444.
[41] M. Romito and L. Xu, Ergodicity of the 3D stochastic Navier–Stokes equations driven by mildly degenerate noise, Stochastic Processes and their Applications, 121(4)(2011), 673–700.
[42] K. Sakthivel and S. S. Sritharan, Martingale solutions for stochastic Navier–Stokes equations driven by L´evy noise, Evolution Equations and Control Theory, 1 (2012), 355–392.
[43] A. V. Skorokhod, Asymptotic Methods in the Theory of Stochastic Differential Equations, American Mathematical Society, Providence, 1989.
[44] S. S. Sritharan and P. Sundar, Ergodic control of stochastic Navier–Stokes equation, Nonlinear Problems in Aviation and Aerospace, Gordon Breach Publications, (2000), 213–221.
[45] R. Temam, Navier–Stokes Equations and Nonlinear Functional Analysis, SIAM, Philadelphia, 1983.
Author Man uscript
[46] F.-Y. Wang, L. Xu and X. Zhang, Gradient estimates for SDEs driven by multiplicative L´evy noise, Journal of Functional Analysis, 269 (2015), 3195–3219.