III MATERIALES Y MÉTODOS
11 Análisis de los datos de secuencia
1.7 La ruta de degradación del NA
δ−1
4 t. Now we can deduce from (4.55) to get
Eδ(x, t) := Z Rd p2t(x, y) exp |x − y| 2 δt µ(dy) ≤ exp pC(t, α, r) + 2(T −qt)r2 µ Bx(r) = exp 1+δ δ C t, 2δ 1+δ, √ 2t + 4 δ−1 µ Bx(r) . (4.56)
By the estimate (4.56) and the following universal bound (see Grigor0yan [Gri97, (3.4)]) pt(x, y) ≤ s Eδ x, t 2 Eδ y, t 2 exp −|x − y| 2 2δt ,
we can finish the proof.
Assume the assumptions in Theorem 4.6.3 holds. Then we have Harnack inequality (4.41). Hence we can get the following corollary immediately by Lemma 4.7.1.
Corollary 4.7.2. Assume for the drift b the conditions in Theorem 4.6.3 and the assumptions at the begging of this section . Then for every δ > 1, t > 0, x, y ∈ Rd we have pt(x, y) ≤ exp 1+δ δ C t,e 2δ 1+δ, √ 2t + 4 δ−1 q µ Bx( √ 2t)µ By( √ 2t) exp −|x − y| 2 2δt , where e C(t, α, r) = α 2(α − 1) Z t 0 g(r) +Rsrξs 0 ξudu 2 ds.
4.8
Some Problems in Applying Girsanov’s The-
orem
Maybe one want to try the coupling and Girsanov transformation method to study Harnack inequalities for stochastic differential equations driven by general
continuous martingale or pure jump L´evy processes. Unfortunately, this does not work in general. One of the essential point of the Girsanov transformation we used is that the distribution of the drift transformed process under the new probability measure must be the same with the original process under the original probability measure. In the following, we explain the reasons.
Continuous Martingale Case
The following Girsanov theorem for continuous martingale is well known. See, for example, [RY99].
Theorem 4.8.1. Let M be a continuous martingale, and Zt = exp Mt− 1 2[M ]t , 0 ≤ t < ∞
be a positive uniform integrable martingale. Let Q = R Z∞dP. If N is a con-
tinuous P martingale, then eNt ≡ Nt− [N, M ]t, 0 ≤ t < ∞ is a continuous Q
martingale, and [ eN ]Qt = [N ]t, for 0 ≤ t < ∞.
The Girsanov theorem 4.8.1 states that eNt, the drift transformed Nt, is still a
martingale (under the new probability measure Q), and the quadratic variations of Ntand eNtare the same. But it does not ensure that the distribution of eNtunder
Q is the same with the distribution of Nt under P. It is the case only in some
special situation. For example, if Nt is a Brownian motion, then by applying
L´evy’s characterization of Brownian motions, we know eNt is still a Brownian
motion. And hence their distributions coincide.
Pure Jump L´evy Processes
We first recall a Girsanov theorem for pure jump processes.
Let (Ω,F , (Ft)0≤t<∞, P) be a filtered probability space and N (dt, dz) a Pois-
son random measure on Ω × R with L´evy measure ν. Suppose that the L´evy
measure satisfies Z
{|z|>1}
|z| |ν|(dz) < ∞. The compensated measure of N (dt, dx) is given by
e
4.8. Some Problems in Applying Girsanov’s Theorem 91 The following result is from [ØS05, Lemma 1.33]. See also [JS87, Chap III, Theorems 3.24 and 5.19], [Cha99, Lemma 3.1 and Theorem 3.2] and [Sit05] etc.. Theorem 4.8.2. Let θ(s, x) ≤ 1 be a process such that
ρ(t) = exp Z t 0 Z R log 1 − θ(s, z)N (ds, dz)e + Z t 0 Z R log 1 − θ(s, z) + θ(s, z) N (ds, dz)e
exists for 0 ≤ t ≤ T . Define a measure Q on FT by Q = ρ(T )P. Assume that
EP ρ(T )
= 1. Then Q is a probability measure on FT and if we define the
random measure eNQ(dt, dz) by e NQ(dt, dz) = eN (dt, dz) + θ(t, z) ν(dz)dt, then Z t 0 Z A e N (ds, dz) + Z t 0 Z A θ(s, z) ν(dz)ds is a Q-local martingale for all A ∈ B(R \ {0})
We claim that the distribution of eN (dt, dz) under P is not the same with the distribution of eNQ(dt, dz) = eN (dt, dz) + θ(t, z)ν(dz)dt. under Q.
The explain follows. First we note that
e
NQ(dt, dz) + ν(dz)dt = [ eN (dt, dz) + θ(t, z)ν(dz)dt] + ν(dz)dt = N (dt, dz) + θ(t, z)ν(dz)dt.
Suppose that our claim is not true. Then the distribution of N (dt, dz) = e
N (dt, dz) + ν(dz)dt under P is the same with the distribution of eNQ(dt, dz) +
ν(dz)dt under Q. This will not happen. We know N (dt, dz) is integer valued. But eNQ(dt, dz) + ν(dz)dt = N (dt, dz) + θ(t, z)ν(dz)dt will not take integer value in general.
Remark 4.8.3. With some special transformation (not drift transformation), we could get process with the same distribution. See [BGJ87, Bic02]
Chapter 5
Harnack Inequalities for
Ornstein-Uhlenbeck Processes
Driven by Wiener Processes
We first give a general introduction to Ornstein-Uhlenbeck processes in Section 5.1. Then we show Harnack inequalities for Ornstein-Uhlenbeck processes driven by Wiener processes in Section 5.2.
In Section 5.3 we consider some properties equivalent to Harnack inequalities. For example, we show that the Harnack inequality for the Gaussian Ornstein- Uhlenbeck semigroup Pt holds if and only if the semigroup Pt is strongly Feller.
In Section 5.4, we show some examples of Harnack inequalities, especially the Harnack inequalities for diagonal Ornstein-Uhlenbeck processes from which we can see clearly why our result is better than the one in [RW03a].
In Section 5.5, we consider Harnack inequalities for Ornstein-Uhlenbeck pro- cesses with perturbations driven by Wiener processes. We first consider Lipschitz perturbations. Then we consider gradient systems by approximation. We men- tion here that there is an independent work by Da Prato et al. [DPRW09]. They considered the perturbation of Ornstein-Uhlenbeck processes with singular drifts. But the spirit is similar.
Section 5.6 is an appendix. We show another proof of the main Harnack inequality by finite dimensional approximation in Subsection 5.6.1. It is especially interesting for readers who only care for the finite dimension case. In Subsection 5.6.2 we show a Mehler formula. It is introduced partially for the motivation of the generalized Mehler semigroups which will be introduced in Section 7.1.
5.1
Ornstein-Uhlenbeck Processes
The story start from Brownian motion. In 1827, the England botanist Robert Brown observed the zigzag path of pollen grains suspended in water under the lens of the microscope. In 1905, Einstein explained the mechanics of the movement. Roughly speaking, if at time t the Brownian particle is at position x, then after arbitrary time ∆t, the particle will appear at x + ε, where ε is a Gaussian random variable and independent of the starting position x and time t.
But this theory neglects the viscosity of the medium. Langevin initiated the study and Ornstein and Uhlenbeck [OU30] developed a new theory for Brownian motion. In the following, we just simply introduce it. We refer to the lovely book by Nelson [Nel01] for the dynamical theory of Brownian motion.
Let Xt denote the velocity of a Brownian particle at time t. Let (Wt)t≥0 be
a one-dimensional standard Brownian motion and κ > 0 measures the viscosity. By the second law of Newton and by choosing appropriate units, dXt
dt means the
acceleration of the particle which may be interpreted as the force experienced by the particle. This force is the sum of a systematic viscous force and a stochastic force. Since the viscous force is proportional to the particle’s velocity Xt and
directed opposite to its velocity, so we can suppose the viscous force is given by −κXt. The stochastic force is modeled by the white noise dWdtt. Therefore, we
have
dXt
dt = −κXt+
dWt
dt . (5.1)
We rewrite it into the following Langevin equation
dXt= −κXtdt + dWt. (5.2)
Let X0 = x ∈ R be the initial data. Then the solution to (5.2) is given by
(see the books [IW81, KS91, DPZ92] etc.)
Xt= e−κtx +
Z t
0
e−κ(t−s) dWs (5.3)
Clearly, Xtis random perturbation of the exponential function. The process (5.3)
is called Ornstein-Uhlenbeck process or simply OU processes.
Figure 5.1 in the following indicates the composition of the process Xt.
We can consider more general form of Ornstein-Uhlenbeck processes. The drift maybe general linear function, and the noise dWt
5.1. Ornstein-Uhlenbeck Processes 95
o x0
random perturbation (red curve) Z t
0
e−κ(t−s)dWs
Ornstein-Uhlenbeck process (blue curve) Xt = e−κtx0+
Z t
0
e−κ(t−s)dWs
deterministic process (black curve) e−κtx0
Figure 5.1: Ornstein-Uhlenbeck Process motion noise, L´evy noise etc..
One of the main general Ornstein-Uhlenbeck type processes which we will consider in this thesis is the generalized Langevin equation
dXt = AXt+ dZt, Xt = x (5.4)
on some Hilbert space H. Here (Zt)t≥0is a L´evy process, and A is the infinitesimal
generator of some strong continuous contraction semigroup (St)t≥0.
The mild solution of (5.4) can be written down in terms of stochastic convo- lution as
Xt= Stx +
Z t
0
St−sdZs. (5.5)
See [PZ07, Section 9.2] or [App06, Section 4].
The Ornstein-Uhlenbeck process defined in (5.5) generalize the classical one in the following two ways: Firstly, we are working in a infinite dimensional space; and secondly, the noise is a general L´evy process.
Ornstein-Uhlenbeck processes are better reference processes in infinite dimen- sional analysis than infinite dimensional Brownian motions (or L´evy processes). One of the main reason is that Ornstein-Uhlenbeck processes, in contrast to an infinite dimensional Brownian motion (or more generally L´evy process), can have
invariant measures. Another point is that the presence of the linear drifts can have smoothing effects.
Bibliographic Notes on Ornstein-Uhlenbeck Processes The topic re- lated to Ornstein-Uhlenbeck type processes has attracted many people to study for a long time. See Ornstein-Uhlenbeck [OU30] and Kolmogorov [Kol34] etc. for the finite dimensional Gaussian case. See Ito [Itˆo84b, Itˆo84a] (or [Itˆo87, Pages 589-616]), Dawson [Daw75], Da Prato et al. [DPIT82], Chow [Cho87], and the books by Da Prato and Zabczyk [DPZ92, DPZ02], Zabczyk [Zab99] and Da Prato [DP04, DP06] for the infinite dimensional Gaussian case.
The case driven by general L´evy processes were first studied by Wolfe [Wol82] in the scalar case: where A is a positive constant. Sato and Yamazoto [SY83, SY84] generalized this to the multidimensional case where A is a matrix all of whose eigenvalues have positive real parts. Chojnowska-Michalik [CM85, Cho87] considered the generalization to infinite dimension. We also mention a series of papers by Applebaum [App06, App07b, App07a] etc., the monograph by Zabczyk and Peszat [PZ07] for the study of Ornstein-Uhlenbeck type processes in infinite dimensional space with L´evy noise.
We refer also to Page 139 for the bibliographic notes on generalized Mehler semigroup which is closely related to the Ornstein-Uhlenbeck processes driven by L´evy processes.