On the local time of the stochastic heat equationMarco.Dozzi/Lecture 2.pdf · 1. The stochastic heat equation 2. Local time 3. The local time in the time variable of the solution

On the local time of the stochastic heat equation

B. Boufoussi, M. DozziUniversité Cadi Ayyad, Université de Lorraine

April 19, 2013

, , , B. Boufoussi, M. Dozzi, , Université Cadi Ayyad, Université de Lorraine , ()Lecture 2 April 19, 2013 1 / 27

1. The stochastic heat equation2. Local time3. The local time in the time variable of the solution of thestochastic heat equation4. On the Haudorff dimension of the level sets


1. The stochastic heat equation

∂u∂t(t, x) = Au(t, x) + f (t, x), t > 0, x ∈ D ⊂ Rd ,

u(0, x) = u0(x), u(t, ·) |∂D= 0,

where A = ∑di ,j=1 aij (x)

∂2

∂xi ∂xj+∑d

i=1 bi (x)∂

∂xi+ c(x)

such that ∑di ,j=1 aij (x)ξ i ξ j = C | ξ |2 for all x ∈ D, ξ ∈ Rd .


Suppose u0, f (t, ·) ∈ L2(D) for all t > 0. Then the mild solution is givenby

u(t, ·) = P(t)u0(·) +∫ t

0P(t − s)f (s, ·)ds

where P(t)(h) :=∫ t0 p(t, ·, y)h(y)dy for any h ∈ L2(D).

For all fixed y ∈ D : ∂∂t p(t, x , y) = Ap(t, x , y), t > 0, x ∈ D

For all fixed x ∈ D : ∂∂t p(t, x , y) = A

∗p(t, x , y), t > 0, y ∈ Dp(0, x , y) = δ{x=y}.



Let us modelise f by space-time white noise, i.e. f (t, x) =..W (t, x).

We consider the gaussian random field (W (t,A), t = 0, A ∈ B(Rd )) withEW (t,A) = 0 and covariance function given by

EW (t,A)W (s,B) = min{t, s}λ(A∩ B), s, t = 0, A,B ∈ B(Rd ),

where λ is the Lebesgue measure in Rd . This random field has independentincrements in space and time. We write W (t, x) ≡ W (t, [0, x ]).



With this choice of f the mild solution u of the partial differential equationconsidered above is given, if it exists as a stochastic process, by

u(t, x) =∫Dp(t, x , y)u0(y)dy +

∫ t

0

∫Dp(t − s, x , y)W (ds, dy)

= : v1(t, x) + v2(t, x).

Here (v2(t, x), t > 0, x ∈ D) is a centered gaussian process with varianceequal to

E | v2(t, x) |2=∫ t

0

∫Dp(t, x , y)2dsdy .

(u(t, x), t > 0, x ∈ D) exists as a stochastic process iffE | v2(t, x) |2< ∞.



If A is the Laplace operator on D, E | v2(t, x) |2< ∞ iff d = 1. If aprocess-valued solution is needed in higher space dimensions too, one hasto allow for a spatial covariance, i. e. one has to replace λ(A∩ B) in thecovariance of W by a term of the type

∫A

∫B f (ξ − η)dξdη, see e.g. R.M.

Balan (2013). We will assume d = 1 in this talk.

Theorem (J.B. Walsh, 1986). Let d = 1. Suppose thatsupx∈D| u0(x) |p< ∞ for all p > 0. Then sup

t∈[0,T ], x∈DE | u(t, x) |p< ∞ for all

p > 0. Moreover, if the trajectories of u0 are Hölder continous of order 12 ,the trajectories of u are Hölder continous of all orders < 1

4 in t and Höldercontinous in x of all orders < 1

2 .


2. Local time

Let X = (Xt , t = 0) be a separable stochastic process with Borelmeasurable trajectories. For any Borel set B ∈ B(R+), we define theoccupation measure µB of X by

µB (A) = λ({t ∈ B, Xt ∈ A}), A ∈ B(R),

where λ is the Lebesgue measure on R. If µB � λ, we say that X has alocal time on B and define it by the Radon-Nikodym derivative

L(B, x) ≡ dµBdλ

(x), x − a.e., P − a.s.

We write L(t, x) instead of L(B, x) if B = [0, t]. If X has a local time onall Borel sets B, the definition above can be extended to the so calledoccupation density formula∫ t

0f (Xs )ds =

∫f (x)L(t, x)dx ,

where f is any bounded and measurable function., , , B. Boufoussi, M. Dozzi, , Université Cadi Ayyad, Université de Lorraine , ()Lecture 2 April 19, 2013 8 / 27

2. Local time

Fourier-analytic method(see e.g. S. Berman 1969, 1973 and D. Geman, J. Horowitz 1973)Let FL(t, ·) be the Fourier transform of x → L(t, x) :

FL(t, ·)(u) =∫exp(iyu)L(t, y)dy .

By the occupation density formula FL(t, ·)(u) =∫ t0 exp(iuXs )ds.

By applying the inverse Fourier transform we get

L(t, ξ) =12π

∫e−iuξFL(t, ·)(u)du = 1

2π

∫ ∫ t

0e iu(Xs−ξ)dsdu,

if the integral on the right side exists. Suffi cient conditions for theexistence will now be formulated for the trajectories of X .


Lemma. f : [a, b] ⊂ Rn −→ Rd has a local time

L(B, x) B ∈ B([a, b]), x ∈ Rd

under each of the following conditions :a)

∫Rd|∫[a,b] exp(iuf (t))dt |2 du < ∞. In this case L([a, b], ·) ∈ L2(Rd ).

b)∫Rd|∫[a,b] exp(iuf (t))dt | du < ∞. In this case L([a, b], ·) is

continous.

The proof uses the fact that the square integrability of the Fouriertransform of µB implies the absolute continuity of µB .


2. Local time

The following result shows that the regularity of the local time of afunction is an indicator for the irregularity of the function itself.

Theorem (R.J. Adler 1981). Suppose that the local time of the function fin the Lemma above satisfies a Hölder condition of order γ ∈ (0, 1) in thetime variable, uniformly in the space variable. Thena) Nowhere do the coordinate functions of f satisfy a Hölder condition oforder > n(1− γ)/d (Theorem 8.7.1).b) The level set at level ξ ∈ Rd of f has Hausdorff dimension = nγ for alllevels ξ ∈ Rd with positive local time (Theorem 8.7.4).

Here the level set at level ξ is defined by {t ∈ [a, b], f (t) = ξ}. We willnow apply these results to the trajectories of gaussian processes.


2. Local time

Proposition (S.M. Berman and D. Geman, J. Horowitz). Let (Xt ,t ∈ [0,T ] ⊂ Rn+) be a centered gaussian process with values in Rd .Suppose that∫

[0,T ]

∫[0,T ]

[Det(Cov(Xt − Xs ))]−1/2dsdt < ∞. (1)

Then (L(t, ξ), t ∈ [0,T ], ξ ∈ Rd ) exists and is, for any t ∈ [0,T ] fixed,P-a.s. square integrable in ξ.

The condition (1) implies the finiteness of the expectation of the integralin part a) of the above Lemma, when applied to the trajectories of X .This can be shown by evaluating the Fourier transform appearing in thisintegral. Notice that we have assumed that the covariance matrix of theincrements of X is regular.


2. Local time

The result of the proposition above can be strengthened. for n = d = 1.

Theorem (S.M. Berman 1969). Let n = d = 1, and suppose that∫[0,T ]

∫[0,T ]

[E (Xt − Xs )2]−(p+1)/2dsdt < ∞.

Then, for any t ∈ [0,T ], L(t,ξ) is [ p2 ] times differentiable with respect toξ P-a.s. Moreover, if p = 2m+ ε, m ∈ N, 0 5 ε < 1,- The trajectories of X satisfy nowhere a Hölder condition of order = 2

p+1 ,P-a.s.- The trajectories of X return infinitely often to every neighbourhood ofalmost all points visited before.


2. Local time

Let us end this section by applying the results above to the space-timewhite noise defined in section 1. We consider more generally the casewhere the "time" parameter t = (t1, ..., tn) is n-dimensional. In this casethe covariance function of W reads

EW (t,A)W (s,B) =n

∏i=1min{ti , si}λ(A∩B), s, t ∈ Rn+, A,B ∈ B(R

d ).

The local time exists and is square integrable in the space variable ifd < 2n. For d = 1, L(t, ξ) satisfies a Hölder condition of all orders< 1− 1

2n in t, for all ξ. Therefore the trajectories of W satisfy nowhere aHölder condition of order > n(1− (1− 1

2n )) =12 which is sharp. Also, the

level sets of the trajectories of W have Hausdorff dimension= n(1− 1

2n ) = n−12 which is sharp too.


3. The local time in the time variable of the solution of thestochastic heat equation

The aim of this section is to study the regularity of the local time of themild solution of the stochastic heat equation with respect to its timevariable and the Hausdorff dimension of the level sets.

∂

∂tu(t, x) =

∂2

∂x2u(t, x) +

∂2

∂t∂xW (t, x), t > 0, 0 < x < 1

u(0, x) = u0(x), u(t, 0) = u(t, 1) = 0 (SHE)

The mild solution is given by

u(t, x) =∫ 1

0p(t, x , y)u0(y)dy +

∫ t

0

∫ 1

0p(t − s, x , y)W (ds, dy)

= : v1(t, x) + v2(t, x)

For simplicity reasons we suppose that u0 is deterministic.



Proposition 1. all x ∈ (0, 1)∫[0,T ]

∫[0,T ]

[E (u(t, x)− u(s, x))2]−(p+1)/2dsdt < ∞

for 0 5 p < 3.

Remark. Since u is gaussian, the results of Section 2 imply that the localtime (Lx (t, ξ), t ∈ [0,T ], ξ ∈ R) of u(·, x) exists for all x ∈ (0, 1), and isdifferentiable with respect to ξ.



Sketch of the proof : For 0 < s < t

E (u(t, x)−u(s, x))2 = E (v2(t, x)− v2(s, x))2 =∫ t

s

∫ 1

0p(t− r , x , y)2dydr .

Then the left part of the following inequality is applied : There existconstants C1,C2 such that

C1(t − s)1/2 5∫ t

s

∫ 1

0p(t − r , x , y)2dydr 5 C2(t − s)1/2.

Therefore the integral appearing in the proposition is finite for p < 3.



The aim is now to investigate the continuity, jointly in (t, ξ),of the localtime (Lx (t, ξ), t ∈ [0,T ], ξ ∈ R). This will be done by an application ofKolmogorov’s continuity theorem.

Proposition 2. For all x ∈ (0, 1) and all t, h, t + h ∈ [0,T ] and allm ∈ 2N, there exist constants Cm > 0, independent of x , such that

E [Lx (t + h, η)− Lx (t + h, ξ)− Lx (t, η) + Lx (t, ξ)]m

5 Cm | η − ξ |θm hm(3/4−θ/4), for some θ ∈ (0, 1),

E [Lx (t + h, ξ)− Lx (t, ξ)]m 5 Cmh3m/4.



Theorem 3. For all x ∈ (0, 1) there exists a jointly continous version ofthe local time (Lx (t, ξ), t ∈ [0,T ], ξ ∈ R) of u(·, x) and, for anycompact set K ⊂ R,

supξ∈K| Lx (t + h, ξ)− Lx (t, ξ) | /hα < ∞

for all α < 34 and t, t + h ∈ [0,T ].

The results of Section 2 show that, for all x ∈ (0, 1), u(·, x) nowheresatisfies a Hölder condition of order > 1

4 , which is sharp.



Elements for the proof of the second inequality of Proposition 2. Forsimplicity we write Xt instead of u(t, x), since the dependence on x willbecome important at the end of the proof only and will then be displayedagain. By the method of Fourier transform

L(t + h, ξ)− L(t, ξ) = 12π

∫ ∫ t+h

texp(iu(Xs − ξ))dsdu.

E | L(t + h, ξ)− L(t, ξ) |m

5 (12π)m∫Rm

∫[t ,t+h]m

| E [exp(im

∑j=1ujXsj )] | ds1...dsmdu1...dum

=m!

(2π)m

∫Rm

∫...∫t1<t2<...tm<t+h

| E [exp(im

∑j=2vj (Xsj − Xsj−1) + iv1Xs1)] |

ds1...dsmdv1...dvm ,, , , B. Boufoussi, M. Dozzi, , Université Cadi Ayyad, Université de Lorraine , ()Lecture 2 April 19, 2013 20 / 27


...where the change of variables

um = vm , uj = vj − vj+1, j = 1, ...,m− 1has been applied.

| E [exp(im

∑j=2vj (Xsj − Xsj−1) + iv1Xs1)] |

= exp(−12Var

m

∑j=1vj (Xsj − Xsj−1)) (s0 := 0)

5 exp(−Cm2

m

∑j=1v2j Var(Xsj − Xsj−1))

for some constant Cm > 0. The last inequality is the so-calledLND-property (property of local nondeterminism).



For gaussian processes the LND-property is equavalent to the followingstatement :

limc↘0

inf06t−r5c , r<s<t

Var(Xt − Xs | σ(Xw , r 5 w 5 s)Var(Xt − Xs )

> 0. (*)

Recall that Xt ≡ u(t, x) = v1(t, x) + v2(t, x)and, since u0 is supposed to be deterministic,

Var(u(t, x)− u(s, x)) = Var(v2(t, x)− v2(s, x)).A similar equality holds for the conditional variance. It suffi ces therefore toshow (*) for v2(t, x).



Recall that

v2(t, x) =∫ t

0

∫Dp(t − s, x , y)W (ds, dy).

It follows that the quotient in (*) is equal to∫ ts

∫ 10 p(t − r , x , y)2dydr∫ s

0

∫ 10 (p(t − r , x , y)− p(s − r , x , y))2dydr +

∫ ts

∫ 10 p(t − r , x , y)2dydr

.

The second term in the denominator can be ignored sinceab+a > 0⇔

ab > 0. The LND property follows now from the upper and

lower estimates of the semigroup.



Upper and lower estimates for the semigroupThere exist positive constants C1,C2,C3 such that, for any 0 < s < t,

C1(t − s)1/2 5∫ t

s

∫ 1

0p(t − r , x , y)2dydr 5 C2(t − s)1/2

∫ s

0

∫ 1

0(p(t − r , x , y)− p(s − r , x , y))2dydr 5 C3(t − s)1/2


4. On the Haudorff dimension of the level sets

Theorem 4. For x ∈ (0, 1) fixed, let {t ∈ [0,T ], u(t, x) = ξ} the levelset at level ξ of t −→ u(t, x) on [0,T ], and let D[0,T ],ξ(u(·, x)) be itsHausdorff dimension, i.e.

D[0,T ],ξ(u(·, x)) = dim{t ∈ [0,T ], u(t, x) = ξ}.

Then, for all x ∈ (0, 1) and all ξ ∈ R, P(D[0,T ],ξ(u(·, x)) = 34 ) > 0.


4. On the Haudorff dimension of the level sets

Proof. The statement follows from Theorem 3 and by the Theorems 8.7.3and 8.7.4 of R.J. Adler (1981). The first theorem states that thisdimension is 5 3

4 , and the second theorem states that this dimension is= 3

4 for all levels ξ with positive local time of t −→ u(t, x). For all ξ onecan show that the local time is positive with positive probability.

Remark. In order to get P-a.s. statements on the Hausdorff dimension oflevel sets, one has to consider random levels.


ReferencesR.J. Adler. The Geometry of Random Fields. Wiley 1981R.M. Balan. Recent advances related to SPDEs with fractional noise.Proceedings of the 7th Seminar on Stochastic Processes, Random Fieldsand Applications, Birkhäuser Verlag 2013, 3-22S.M. Berman. Local times and sample function properties of stationarygaussian processes. Trans. Amer. Math. Soc. 137 (1969), 277-299S.M. Berman. Local nondeterminism and local times of gaussianprocesses. Indiana Univ. Math. J. 23 (1973), 69-94D. Geman, J. Horowitz. Occupation densities. Ann. Probab. 8 (1980),1-67J.B. Walsh. An introduction to stochastic partial differential equations.Ecole d’Eté de Probabilités de Saint-Flour XIV, Lecture Notes in Math.1180, Springer Verlag 1986, 265-439


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On the local time of the stochastic heat equationMarco.Dozzi/Lecture 2.pdf · 1. The stochastic heat equation 2. Local time 3. The local time in the time variable of the solution