A brief overview of the Lp-theory of SPDEs
We present basics of the Lp-theory of SPDEs and its connection to various related problems of filtration and populational dynamics.
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Krylov, N. 2009-12-03T16:38:14Z 2009-12-03T16:38:14Z 2008 A brief overview of the Lp-theory of SPDEs / N. Krylov // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 2. — С. 71–78. — Бібліогр.: 18 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4554 519.21 We present basics of the Lp-theory of SPDEs and its connection to various related problems of filtration and populational dynamics. This work was partially supported by NSF grant DMS-0653121. en Інститут математики НАН України A brief overview of the Lp-theory of SPDEs Article published earlier |
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We present basics of the Lp-theory of SPDEs and its connection to various related problems of filtration and populational dynamics.
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A brief overview of the Lp-theory of SPDEs / N. Krylov // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 2. — С. 71–78. — Бібліогр.: 18 назв.— англ. |
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Theory of Stochastic Processes
Vol. 14 (30), no. 2, 2008, pp. 71–78
UDC 519.21
NICOLAI KRYLOV
A BRIEF OVERVIEW OF THE Lp-THEORY OF SPDES
We present basics of the Lp-theory of SPDEs and its connection to various related
problems of filtration and populational dynamics.
1. Motivation
For the author the main motivation to study SPDEs comes from filtering theory for
diffusion processes. Imagine that we are given a two component diffusion process
zt = (xt, yt) ∈ Rd × Rd1
such that we only can observe yt and need to extract from the observations as much
information about xt as possible. Then it turns out that under natural assumptions for
measurable bounded f(x) we have
E(f(xt)|ys, s ≤ t) =
∫
Rd
f(x)pt(x) dx,
where the posterior density pt(x) of xt given ys, s ≤ t, satisfies an equation like
(1.1) dpt = Ltpt dt+Mtpt dyt,
where Lt is a second-order elliptic operator, Mt is a first-order operator, and the coeffi-
cients of Lt and Mt depend on (t, x) and yt.
Equation (1.1) is a natural generalization of the backward Kolmogorov equation for
diffusion processes when yt ≡ 0 and the second term on the right in (1.1) disappears.
Another natural source of SPDEs is populational dynamics or branching diffusion
processes. These are measure-valued processes called super-diffusions and for a long
time they were studied by using martingale approach and nonstandard analysis. If d = 1
since 1988 it is known that the super-diffusion μt has a density pt which satisfies the
equation
(1.2) dpt(x) = D2pt(x) dt + p
1/2
t (x)φk(x) dwk
t ,
where D = ∂/∂x, φk form an orthonormal basis in L2, wk
t , k = 1, 2, . . . , are independent
Wiener processes, and the summation convention is enforced. See [3], [17].
By the way, quite often in the literature after work by J. Walsh [18] in 1986 the second
term on the right in (1.2) is written by using the so-called space-time white noise. The
form (1.2) was probably first introduced by T. Funaki [2] in 1983 and turns out more
convenient in many respects.
I will only concentrate on (1.1) and (1.2) without speaking about very active research
areas related to random perturbation of deterministic equations like Navier-Stokes or
wave equations or other types of equations. Part of really immense corresponding liter-
ature can be found following the references in the articles listed in the bibliography.
2000 AMS Mathematics Subject Classification. Primary 60H15.
Key words and phrases. Stochastic partial differential equations.
This work was partially supported by NSF grant DMS-0653121.
71
72 NICOLAI KRYLOV
2. Equations related to (1.2)
First I will discuss my articles related to super-diffusions: [8], [11].
Before 1997 there were two common ways known in the literature to prove the ex-
istence of super-diffusions. One of them is to take the limit law of branching diffusion
particles which is analogous to proving the existence of diffusion processes not through
solving stochastic Itô equations but through the passage to the weak limit in a sequence
of Markov chains. Another way is to construct the super-diffusions as measure-valued
Markov processes by defining their transition functions in the space of measures and
then using the general theory of Markov processes to get a process corresponding to
this transition function. This way is similar to the one used in the theory of diffusion
processes with “bad” but not too “bad” coefficients when there are “good” results con-
cerning the fundamental solutions for corresponding parabolic equations. The natural
question arises: are there any stochastic Itô type equations for the super-diffusions at
least for “regular” ones? In [8] I showed that the answer to this question is positive and
the appropriate stochastic equations are stochastic partial differential equations. As is
mentioned above this result was known for quite a while in one-dimensional case (see [3],
[17]), and my contribution relates to the multidimensional case.
Answering our question we also answer the question concerning the possibility to
include super-diffusions in the framework of more or less classical stochastic analysis,
without resorting to the abstract theory of Markov processes or relying on nonstandard
analysis.
The multidimensional equation should look like (1.2):
dpt(x) = D2pt(x) dt + p
1/2
t (x)φk(x) dwk
t .
However, in the general case the super-diffusions are measure-valued processes without
densities and what is
√
μt for a measure-valued function μt is not immediately clear.
To give a better idea about the contents of [8] we introduce the following notation.
By M we denote the space of all finite measures on (Rd,B(Rd)). One endows M with
the usual measurable structure requiring functions (ψ, μ) to be measurable for any ψ ∈
C∞
0 (Rd), where
(ψ, μ) =
∫
Rd
ψ(x)μ(dx).
Let (Ω,F , P ) be a complete probability space with an increasing filtration of complete σ–
algebras Ft ⊂ F , t ≥ 0. An M–valued Ft–adapted process μt is called a super–Brownian
process if for any ψ ∈ C∞
0 (Rd) the process (ψ, μt) is continuous and the process
(2.1) mt(ψ) := (ψ, μt) − (ψ, μ0) −
∫ t
0
(Δψ, μs) ds
is a continuous local martingale starting from zero with
d〈m(ψ)〉t = (ψ2, μt) dt.
Then to derive a stochastic equation for μt we need to find a “canonical” representation
for the local martingales mt(ψ). If we could show that for some nonrandom real-valued
functions mk(μ) = mk(μ, x) we have
(2.2) (ψ2, μ) =
∑
k
(ψ,mk(μ))2 =
( ∫
Rd
ψ(x)mk(μ, x) dx
)2
,
then it is clear that
dmt(ψ) = (ψ,mk(μt)) dwk
t ,
A BRIEF OVERVIEW OF THE Lp-THEORY OF SPDES 73
where wk
t are independent Wiener processes. Coming back to (2.1) we would therefore
find
(2.3) dμt = Δμt dt+mk(μt) dwk
t .
Thus, the whole point was to find representation (2.2). If μ has a density p, then this
is easy. Indeed,
(ψ2, μ) =
∫
Rd
(p1/2ψ)2 dx =
∑
k
(ψ, φkp1/2)2,
where φk is any orthonormal basis in L2. In this way one obtains (1.2). In the general
case finding formulas like (2.2) is more difficult. In that article there are also derived
equations for super-diffusions with jumps.
Observe that (2.3) contains a series of stochastic integrals and not a stochastic integral
against a space-time white noise. If d = 1 one still can use space-time white noises, but,
most likely, it is impossible in higher dimensions.
In [8] I derived equations for super-diffusions assuming that they exist. In [11] a way to
prove the existence is presented on the basis of showing that my equations have solutions.
Here I used the Skorokhod embedding method.
In [9] we consider the following one-dimensional stochastic partial differential equation
dp(t, x) = ν(t, x)pλ(t, x)φk(x) dwk
t
(2.4) +(a(t, x)D2p(t, x) + b(t, x)Dp(t, x) + c(t, x)p(t, x)) dt,
where λ ∈ (0, 1), the coefficients a, b, c, ν are random and satisfy some natural conditions,
the function p = p(ω, t, x) is supposed to be nonnegative. Actually, equation (2.4) has
to be understood in a certain generalized sense since, for instance, the series of ϕk(x)wk
t
diverges for almost all x.
In the case that a = 1, b = c = 0 C. Mueller and E. Perkins in [16] considered
nonnegative solutions and by using nonstandard analysis proved that if the initial data
has compact support, then this is true for all times. We prove the same for (2.4) by using
the Lp-theory of SPDEs, which started with my article [7] the results of which without
proofs are published as [6]and more complete exposition of which is in [10].
In each of these articles we generalize a result by Mueller [15] who proved that, for
λ ∈ [1, 3/2), the following simplified equation (2.4):
(2.5) dut = D2ut dt+ uλ
+φ
k dwk
t
has a unique solution defined for all t if the initial condition u0 is nonnegative and, say,
is nonrandom and belongs to C∞
0 . Furthermore,
sup
t≤T,x
|u(t, x)| <∞
(a.s.) for any T < ∞. We prove these facts for equation (2.4) with random and space
and time dependent coefficients. In [15] quite a different method is used. The emphasis
in [15] and [7] is on the fact that uγ
+ is super-linear in u.
3. An overview of the Lp-theory in the whole space
Here we are dealing with the following general equation
dut = [σik
t Diut + νk
t ut + gk
t ] dwk
t
(3.1) +[(1/2)aij
t Dijut + bitDiut + ctut + ft] dt,
74 NICOLAI KRYLOV
where aij , f, σik, gk are real-valued functions defined for ω ∈ Ω, t ≥ 0, x ∈ Rd, u ∈ Hn+2
p ,
i, j = 1, ..., d, k = 1, 2, ..., n is a fixed real number. The definition of spaces Hn+2
p is given
later. The main assumption is that the matrix
(aij − αij), where αij = σikσjk,
is bounded and uniformly nondegenerate.
As it is mentioned above the development of the Lp-theory started with [7] (or [6]).
Before this article various aspects of the L2-theory for general equations were investigated
by Pardoux, Krylov, Rozovskii, Gyongy, Flandoli, Brzezniak, and few other researchers.
The interest in Lp-theory is easy to explain, for instance, from computational point
of view. If we want to know how fast, say finite-difference approximating schemes will
converge to the true solution of an SPDE, we need to know how smooth the true solution
is. In the framework of the L2-theory in order to guarantee that the solution has one
continuous derivative in x, we need u to have [d/2] + 2 generalized derivatives summable
to the second power. This requires the coefficients to have [d/2] derivatives in x. In the
framework of the Lp-theory the true solution is continuously differentiable in x if the
coefficients of the equation are merely continuous (σ, ν Lipschitz continuous) and p > d.
From the Lp-theory with p > 2 one also gets Hölder continuity of solutions in t. This
result is either impossible or very hard to obtain in the framework of the L2-theory.
Passing from the Hilbert space L2 to Lp with p ≥ 2 required proving one crucial
estimate, which was done in [5].
Recall that the space Hγ
p = Hγ
p (Rd), p ∈ (1,∞), γ ∈ (−∞,∞), of Bessel potentials is
defined as the closure of C∞
0 = C∞
0 (Rd) with respect to the norm
‖u‖Hγ
p
= ‖(I − Δ)γ/2u‖Lp.
Consider the simplest one-dimensional SPDE
(3.2) dut(x) =
1
2
D2ut(x) dt + gt(x) dwt
given for t > 0, x ∈ Rd, with initial condition u0 = 0, where wt is a one-dimensional
Wiener process. The solution of this problem is known to be
(3.3) ut(x) =
∫ t
0
Tt−sgs(x) dws,
where Tth(x) = Eh(x + wt) is the heat semi-group. If g is non random, then ut(x) is a
Gaussian random variable with zero mean and its absolute moments are just powers of
its second moment. It follows that
E
∫ T
0
||ut||pH1
p
dt = N(p)
∫ T
0
∫
Rd
[
∫ t
0
|(I − Δ)1/2Tt−sgs(x)|2 ds]p/2 dx dt,
and in order to prove that u ∈ H1
p we have to estimate the last integral. The result is
that ∫ ∞
0
∫
Rd
[
∫ t
0
|DTt−sgs(x)|2 ds]p/2 dx dt ≤ N‖g‖p
Lp((0,∞)×Rd)
.
This result is also published in [13] where complete proofs of everything are given from
scratch.
Now we describe the function spaces in which (3.2) is treated. For a stopping time τ
we denote
Hγ
p(τ) = Lp(]] 0, 2 ]]),P , Hγ
p ), Hγ
p = Hγ
p(∞), L...... = H0
...... .
For distributions ut, ft, gk
t , depending on t and ω and measurable in a natural sense,
we write
(3.4) dut = ft dt+ gk
t dw
k
t , t > 0,
A BRIEF OVERVIEW OF THE Lp-THEORY OF SPDES 75
if for any φ ∈ C∞
0 with probability 1 we have
(ut, φ) = (u0, φ) +
∫ t
0
(fs, φ) ds+
∫ t
0
(gk
s , φ) dwk
s
for all t ≥ 0 assuming that all integrals here make sense.
For a function u ∈ Hγ
p(τ) we write u ∈ Hγ
p,0(τ) if there exists f ∈ Hγ−2
p (τ) g ∈ Hγ−1
p (τ)
such that equality (3.4) holds in the above sense of distributions. In this case we define
‖u‖Hγ
p(τ) = ‖uxx‖H
γ−2
p (τ) + ‖f‖
H
γ−2
p (τ) + ‖g‖
H
γ−1
p (τ).
These spaces are convenient for treating equations with zero initial condition. The
general case reduces to this one in a standard way.
Theorem 3.1. Take a p ≥ 2 and a bounded stopping time τ . Under the above condition
of uniform nondegeneracy also suppose that the coefficients b, c, σ, ν are bounded and σ, ν
satisfy the Lipschitz condition in x uniformly with respect to t, ω. Finally, assume that
in (3.1):
dut = [(1/2)aij
t Dijut + bitDiut + ctut + ft] dt+ [σik
t Diut + νk
t ut + gk
t ] dwk
t ,
the summation in k is restricted to a finite set. Then for any f ∈ Lp(τ) and g ∈ H1
p(τ)
there exists a unique u ∈ H2
p,0(τ) satisfying (3.1). Furthermore,
‖u‖H2
p(τ) ≤ N(‖f‖Lp(τ) + ‖g‖H1
p(τ)),
where N is independent of f and g.
There are a few extensions of this result when the right-hand sides f, g belong to
spaces H and are nonlinear functions of u, for instance, like in (2.4), (2.5).
By the way, neither Theorem 3.1 nor its natural extension to nonzero initial-value
problem are applicable to (2.5):
dut = D2ut dt+ uλ
+φ
k dwk
t ,
since the range of the summation in k is 1, 2, .... Actually, the solutions of (2.5) never
belong to H2
p(τ) but rather to Hγ
p (τ), where γ is any number < 1/2. In particular, this
shows the necessity to treat the equations in the spaces Hγ
p (τ) with arbitrary γ ∈ R.
4. SPDEs in half-spaces
It turns out that the theory of SPDEs in domains is much harder than in the whole
space. For quite some time the only available results were obtained in function spaces
with low regularity or under some compatibility conditions on the data. We refer to
works by Pardoux, Flandoli, Brzezniak, Da Prato, Zabczyk, and others.
Let us come back to the simplest equation (3.2):
dut(x) =
1
2
D2ut(x) dt + gt(x) dwt
again with zero initial condition but considered only on the half-line R+ = (0,∞) and
with zero lateral condition. The solution is given by the same formula (3.3) but with
Tt defined as the heat semigroup in R+ with zero lateral condition. In particular, when
g ≡ 1
ut(x) =
∫ t
0
Tt−s1(x) dws.
It is very easy to prove that ut(x) is infinitely differentiable with respect to x in R+.
However, its second-order derivative cannot be bounded near the origin. To understand
that write
ut(x) = (1/2)
∫ t
0
D2us(x) ds+ wt
76 NICOLAI KRYLOV
at x = 0. Then the left-hand side vanishes owing to the boundary condition and we see
that wt is equal to a function which is differentiable in t, which is impossible.
This is the reason why we needed to introduce function spaces where elements are
allowed to have derivatives blowing up near the boundary.
In 1994 in my article [4] such spaces were introduced and used to develop the Hilbert
space theory for SPDEs in bounded smooth domains.
In notation we introduce below the results of [4] allow us to obtain solutions of class
Hγ
p,θ(τ), where γ is an integer indicating the number of derivatives the functions of this
class possess, the power of summability p = 2, and the parameter θ controlling the blow
up near the boundary equals d.
The spaces Hγ
p,θ,0(τ) are introduced for all γ, θ ∈ R and any domain. However, it is
easier to explain what they are when γ = 2 and the domain the equation is considered
in is
Rd
+ = {x ∈ Rd : x1 > 0}.
Observe that again restricting to the case that γ is an integer does not allow one to treat
equations from populational dynamics like (1.2).
We say that u ∈ H2
p,θ,0(τ) if (3.4):
dut = ft dt+ gk
t dw
k
t , t > 0,
holds in the sense of distributions on Rd
+, u0 = 0 and u, f , and g are such that
E
∫ τ
0
∫
R
d
+
(x1)θ−d(|(x1)−1ut(x)|p + |Dut(x)|p + |x1D2ut(x)|p dxdt <∞,
E
∫ τ
0
∫
R
d
+
(x1)θ−d(|x1ft(x)|p + |gt(x)|p�2 + |x1Dgt(x)|p�2) dxdt <∞.
It turns out that for the purpose of solving, say uniformly nondegenerate SPDEs the
range of θ should be restricted to
d− 1 < θ < p+ d− 1.
If θ ≤ d − 1, then generally solutions do not belong to H2
p,θ,0(τ), and if θ ≥ d + p − 1,
then generally one cannot find any solution at all. For θ so restricted the presence of
(x1)θ−d|(x1)−1ut|p in the above conditions shows that ut(x) vanishes in an integral sense
as x1 ↓ 0. Therefore, while solving equations in H2
p,θ,0(τ), we are solving them with zero
lateral condition.
For the same reasons as above we needed better results than in [4] in what concerns
the power p of summability of derivatives.
One more drawback of the Hilbert space theory is that, in the general case, no matter
to which class Hγ
2,θ(τ) a function u belongs with as large γ as you wish one still cannot
conclude that
(4.1) ut(x) → 0 as x ↓ 0,
that is that the boundary condition is satisfied pointwise. One needs this kind of global
continuity of solutions, for instance, for numerical approximations. Proving (4.1) became
a major challenge for the theory. The current state of the art is that (4.1) is finally
established in my paper [14] for the one-dimensional case assuming that the coefficients
are independent of x. In addition the convergence in (4.1) is shown to become extremely
slow as the constant of nondegeneracy of the equation becomes small.
This behavior is absolutely different from what is happening with ordinary parabolic
equations. Typically their solutions decay linearly as x approaches the boundary regard-
less of the size of the constant of ellipticity.
A BRIEF OVERVIEW OF THE Lp-THEORY OF SPDES 77
The main nontrivial ingredient in [14] is a new square root law for one-dimensional
Brownian motion discovered in my article [12].
It is worth saying that one of closely related square root laws was proved by B. Davis
in 1983, see [1].
It says that (a.s.) if c < 1, then
(a) there exists a t ∈ [0, 1] and an ε > 0 such that
wt+r − wt ≥ c
√
r for all r ∈ [0, ε]
but if c > 1, then the opposite is true, that is
(b) for any t ∈ [0, 1] and any ε > 0 there is an r ∈ (0, ε] such that
wt+r − wt < c
√
r.
The new law can be described in the following way. Let c ∈ (0,∞) be a constant. For
any t ≥ 0 and k = 0, 1, 2, ..., we say that w· after time t is contained in a (parabolic)
c-box of size 2−k if there is a number a such that
(4.2) a ≤ wt+s ≤ a+ c2−k/2 for 0 ≤ s ≤ 2−k.
The law of iterated logarithm applied to wt+2−k implies that for each t (a.s.) there are
infinitely many k’s such that w· after time t is not contained in c-boxes of size 2−k.
Actually, an even stronger statement is true. Observe that the sequence of processes
ξt(n) := 2n/2wt2−n , t ∈ [0, 1], n = 0, 1, ..., is a stationary C([0, 1])-valued sequence.
Therefore, the sequence
Δn(0) = 2n/2osc[0,2−n]w·
is stationary too and, by the law of large numbers, for each c, with probability one the
density of k’s, for which w· after time 0 is not contained in c-boxes of size 2−k, is strictly
bigger than zero. (By the density of sets A ⊂ {0, 1, 2, ...} we mean
lim
n→∞ #(A ∩ [0, n])/n,
where #B is the number of elements in B.)
Furthermore, Davis’s law shows that if c < 1, then with probability one there is a
t = t(ω) ∈ [0, 1] such that w· after time t is not contained in c-boxes of size 2−k if k is
large enough.
If c is large enough the situation is different. Our square root law says, in particular,
that there is a function p(c) ≥ 0 such that p(c) > 0 if c is large enough, and (a.s.) for
any t ∈ [0, 1], w· after time t is contained in a c-box of size 2−k for k’s in a set with
lower density ≥ p(c).
It is worth noting again that if we are only interested in one particular value of t, then
the result is a straightforward consequence of the above mentioned law of large numbers
for stationary sequences. One need not even have c large enough, since (a.s.) the density
of k’s for which w· after time t is contained in a c-box of size 2−k equals the probability
p0(c) that w· after time 0 is contained in a c-box of size 1 and this probability is strictly
bigger than zero for any c > 0. However, the exceptional set of ω depends on t and for
c < 1 the union of exceptional sets has probability one, as has been mentioned above.
Therefore, the main emphasis of the law is on the fact that for large c the lower density
is bounded away from zero by a strictly positive constant depending only on c.
The relation of the Hölder continuity to the square root law can be seen from the fact
that if ut satisfies
dut = (1/2)D2ut dt+ σDut dwt in x > 0, t > 0,
where σ is a constant, then the function
vt(x) = ut(x− σwt)
78 NICOLAI KRYLOV
satisfies the deterministic heat equation
∂vt
∂t
= (1/2)(1 − σ2)D2ut
in the random domain
x > σwt, t > 0.
Knowing the self-similar behavior of the boundary helps proving the Hölder continuity
of v and u in x.
Bibliography
1. B. Davis, On Brownian slow points, Z. Wahrscheinlichkeitstheory verw. Gebiete 64 (1983),
359–367.
2. T. Funaki, Random motion of strings and related evolution equations, Nagoya Math. Journal
89 (1983), 129–193.
3. N. Konno and T. Shiga, Stochastic partial differential equations for some measure–valued dif-
fusions, Probab. Theory Relat. Fields 79 (1988), 201–225.
4. N.V. Krylov, A W n
2 -theory of the Dirichlet problem for SPDE in general smooth domains,
Probab. Theory Relat Fields 98 (1994), 389–421.
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