Hölder continuity for the nonlinear stochastic heat ...math.arizona.edu/~fpd/Talks/Chen.pdfIn A minicourse on stochastic partial differential equations, volume 1962 of Lecture Notes

Hölder continuity for the nonlinear stochastic heat equationwith rough initial conditions

Le CHEN

Department of Mathematics

University of Utah

Joint work with Prof. Robert C. DALANG

To appear in Stochastic Partial Differential Equations: Analysis and Computations, 2014

18–20, May 2014Frontier Probability Days

Tucson, Arizona

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Stochastic Heat Equation (SHE)

(∂

∂t− ν

2∂2

∂x2

)u(t , x) = ρ(u(t , x)) W (t , x), x ∈ R, t ∈ R∗+,

u(0, ·) = µ(·) ,(SHE)

W is the space-time white noise;

ρ is Lipschitz continuous;

µ is the initial measure (to be specified).

u(t , x) = J0(t , x) +

∫∫[0,t]×R

ρ(u(s, y))Gν(t − s, x − y)W (ds, dy).

Gν(t , x) =1√

2πνtexp

(−x2

2t

)J0(t , x) := (µ ∗Gν(t , ·))(x)

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Definition of random field solution

u(t , x) = J0(t , x) +

∫∫[0,t]×R

ρ (u(s, y)) Gν(t − s, x − y)W (ds, dy)︸︷︷︸:=I(t,x)

. (SHE)

Definition (Random field solution)

u = (u(t , x) : (t , x) ∈ R∗+ × R) is called a random field solution to (SHE) if

(1) u is adapted, i.e., for all (t , x) ∈ R∗+ × R, u(t , x) is Ft -measurable;

(2) u is jointly measurable with respect to B (R∗+ × R)×F ;

(3)(

G2ν ? ||ρ(u)||22

)(t , x) < +∞ for all (t , x) ∈ R∗+ × R, and

(t , x) 7→ I(t , x) : R∗+ × R 7→ L2(Ω) is continuous;

(4) u satisfies (SHE) almost surely, for all (t , x) ∈ R∗+ × R.

(G2

ν ? ||ρ(u)||22

)(t, x) :=

∫ t

0ds∫R

dy G2ν(t − s, x − y) ||ρ(u(s, y))||22 .

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Rough initial data

Initial data has a bounded density function (Walsh theory [2]), (Bounded initial data)i.e., µ(dx) = f (x)dx with f ∈ L∞(R).

Measure-valued initial data (Ch. & Dalang [1]).

MH(R) :=

signed Borel meas. µ, s.t.

∫R

e−ax2|µ|(dx) < +∞, ∀a > 0

(|µ| ∗Gν(t , ·)) (x) :=

∫R

1√2πνt

e−(x−y)2

2νt |µ|(dy) < +∞ ∀t > 0, ∀x ∈ R.

Initial data cannot go beyond measures. No random field solution for δ′0.

[1] L. Chen, R. Dalang, Moments and growth indices for the nonlinear stochastic heat equation withrough initial conditions, Ann. Probab., (accepted, pending revision), 2014.

[2] J. B. Walsh. An introduction to stochastic partial differential equations. In: École d’été deprobabilités de Saint-Flour, XIV—1984, pp. 265–439. Springer, Berlin, 1986.

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Rough initial data



MH(R) :=


∫R

e−ax2|µ|(dx) < +∞, ∀a > 0

(|µ| ∗Gν(t , ·)) (x) :=

∫R

1√2πνt

e−(x−y)2

2νt |µ|(dy) < +∞ ∀t > 0, ∀x ∈ R.




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Rough initial data



MH(R) :=


∫R

e−ax2|µ|(dx) < +∞, ∀a > 0

(|µ| ∗Gν(t , ·)) (x) :=

∫R

1√2πνt

e−(x−y)2

2νt |µ|(dy) < +∞ ∀t > 0, ∀x ∈ R.




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Rough initial data



MH(R) :=


∫R

e−ax2|µ|(dx) < +∞, ∀a > 0

(|µ| ∗Gν(t , ·)) (x) :=

∫R

1√2πνt

e−(x−y)2

2νt |µ|(dy) < +∞ ∀t > 0, ∀x ∈ R.


J0(t , x) ∈ C∞(R∗+ × R)

I(t , x) ∈ C?,?(R∗+ × R)

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Some notation for locally Hölder continuous functions

Given a subset D ⊆ R+ × R and positive constants β1, β2, denote byCβ1,β2 (D) the set of functions v : R+ × R→ R with the following property:

For each compact subset D ⊂ D, ∃C s.t. for all (t , x) and (s, y) ∈ D,

|v(t , x)− v(s, y)| ≤ C(|t − s|β1 + |x − y |β2

).

Cβ1−,β2−(D) :=⋂

0<α1<β1

⋂0<α2<β2

Cα1,α2 (D) .

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u(t , x) = J0(t , x) + I(t , x)

MH(R) :=


∫R

e−ax2|µ|(dx) < +∞, ∀a > 0

M∗H(R) :=

µ(dx) = f (x)dx , s.t. ∃a ∈ ]1, 2[ , sup

x∈R|f (x)|e−|x|

a< +∞

.

Theorem

(1) If µ ∈MH(R), then I ∈ C 14−,

12−

(R∗+ × R) a.s. Therefore,

u ∈ C 14−,

12−

(R∗+ × R) , a.s.

(2) If µ ∈M∗H(R) with µ(dx) = f (x)dx, then I ∈ C 14−,

12−

(R+ × R), a.s.Moreover,(i) If f is continuous, then

u ∈ C (R+ × R) ∩ C 14−,

12−

(R∗+ × R) , a.s.

(ii) If f is α-Hölder continuous, then

u ∈ C( α2 ∧

14 )−,(α∧ 1

2 )− (R+ × R) ∩ C 14−,

12−

(R∗+ × R) , a.s.

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u(t , x) = J0(t , x) + I(t , x)

MH(R) :=


∫R

e−ax2|µ|(dx) < +∞, ∀a > 0

M∗H(R) :=

µ(dx) = f (x)dx , s.t. ∃a ∈ ]1, 2[ , sup

x∈R|f (x)|e−|x|

a< +∞

.

Theorem

(1) If µ ∈MH(R), then I ∈ C 14−,

12−

(R∗+ × R) a.s. Therefore,

u ∈ C 14−,

12−

(R∗+ × R) , a.s.

(2) If µ ∈M∗H(R) with µ(dx) = f (x)dx, then I ∈ C 14−,

12−

(R+ × R), a.s.Moreover,(i) If f is continuous, then

u ∈ C (R+ × R) ∩ C 14−,

12−

(R∗+ × R) , a.s.

(ii) If f is α-Hölder continuous, then

u ∈ C( α2 ∧

14 )−,(α∧ 1

2 )− (R+ × R) ∩ C 14−,

12−

(R∗+ × R) , a.s.

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Difficulties with rough initial dataConventional method: For p > 1 and q = p/(p − 1), t < t ′ (ρ(u) = u), Set

Gν(t − s, x − y ; t ′ − s, x ′ − y) = Gν(t − s, x − y)−Gν(t ′ − s, x ′ − y).

∣∣∣∣I(t , x)− I(t ′, x ′)∣∣∣∣2p

2p =

∣∣∣∣∣∣∣∣∣∣∫∫

[0,t′]×RGν(t − s, x − y ; t ′ − s, x ′ − y)u(s, y)W (dsdy)

∣∣∣∣∣∣∣∣∣∣2p

2p

≤ C

[∫ t′

0

∫RGν(· · · )2dsdy

]p/q ∫ t′

0

∫RG2ν ·(

1 + ||u(s, y)||2p2p

)dsdy

≤ C sups∈[0,t′]

supy∈R

(1 + ||u(s, y)||2p

2p

)[∫ t′

0


]p


supy∈R

(1 + ||u(s, y)||2p

2p

) [|t ′ − t |p/2 + |x ′ − x |p

][1] Robert C. Dalang. The stochastic wave equation. In A minicourse on stochastic partial differential equations, volume 1962 of LectureNotes in Math. Springer, Berlin, 2009.[2] Marta Sanz-Solé and Mònica Sarrà. Hölder continuity for the stochastic heat equation with spatially correlated noise. In Seminar onStochastic Analysis, Random Fields and Applications, III, volume 52 of Progr. Probab.. Birkhäuser, Basel, 2002.[3] Tokuzo Shiga. Two contrasting properties of solutions for one-dimensional stochastic partial differential equations. Canad. J. Math.,46(2):415–437, 1994.

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∣∣∣∣I(t , x)− I(t ′, x ′)∣∣∣∣2p

2p =

∣∣∣∣∣∣∣∣∣∣∫∫


∣∣∣∣∣∣∣∣∣∣2p

2p

≤ C

[∫ t′

0


]p/q ∫ t′

0

∫RG2ν ·(

1 + ||u(s, y)||2p2p

)dsdy


supy∈R

(1 + ||u(s, y)||2p

2p

)[∫ t′

0


]p


supy∈R

(1 + ||u(s, y)||2p

2p

) [|t ′ − t |p/2 + |x ′ − x |p

]Tails⇒ integrability of x at ±∞.Measure⇒ integrability of t at 0: e.g., µ = δ0,

||u(s, y)||22p ≥ ||u(s, y)||22 ≥ G ν2

(s, y)1√

4πνs=

Cs

e−y2νs ⇒ p < 3/2.

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∣∣∣∣I(t , x)− I(t ′, x ′)∣∣∣∣2p

2p =

∣∣∣∣∣∣∣∣∣∣∫∫


∣∣∣∣∣∣∣∣∣∣2p

2p

≤ C

[∫ t′

0


]p/q ∫ t′

0

∫RG2ν ·(

1 + ||u(s, y)||2p2p

)dsdy


supy∈R

(1 + ||u(s, y)||2p

2p

)[∫ t′

0


]p


supy∈R

(1 + ||u(s, y)||2p

2p

) [|t ′ − t |p/2 + |x ′ − x |p

]

Lemma. For each Kn := [1/n, n]× [n, n] and p ≥ 2, find Cn,p such that∣∣∣∣I(t , x)− I(t ′, x ′)∣∣∣∣

p ≤ Cn,p

(|t − t ′|1/4 + |x − x ′|1/2

), ∀(t , x), (t ′, x ′) ∈ Kn.

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Instead of∫∫R+×R

dsdy(Gν(t − s, x − y)−Gν(t ′ − s, x ′ − y)

)2 ≤ C(|x − x ′|+

√|t − t ′|

).

For all (t , x) and (t ′, x ′) ∈ [1/n, n]× [−n, n], find Cn > 0 s.t.,∫∫R+×R

dsdy J0(s, y)2 (Gν (t − s, x − y)−Gν(t ′ − s, x ′ − y))2

≤ Cn

(|x − x ′|+

√|t − t ′|

).

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Two key estimates on heat kernel

-5 5

0.1

0.2

0.3

0.4

Gν(t , x) =1√

2πνtexp

(−x2

2t

)

Lemma 1. For all L > 0, 0 < β < 1, t > 0, x ∈ R, and |h| ≤ βL, ∃C ≈ 0.45,

|Gν(t , x + h) + Gν(t , x − h)− 2Gν(t , x)|

≤ 2|h|(

C√2νt

+1

(1− β)L

)[Gν(t , x) + e

3L22νt Gν (t , x − 2L ) + Gν (t , x + 2L )

].

Lemma 2. For all t > 0, n > 1, x ∈ R and 0 < r < n2t ,∣∣∣G ν2

(t + r , x)−G ν2

(t , x)∣∣∣ ≤ 3

2

√1 + n2√

tG ν(1+n2)

2(t , x)

√r .

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Moment formula

||u(t , x)||2p ≤ J20 (t , x) +

(J2

0 ?Kp(t , x))

(t , x)

K(t , x ;λ) := G ν2

(t , x)

(λ2

√4πνt

+λ4

2νe

λ44ν Φ

(λ2

√t

2ν

))

Kp(t , x) := K(t , x ; 4√

p Lρ)

[1] L. Chen, R. Dalang, Moments and growth indices for the nonlinear stochastic heat equation with

rough initial conditions, Ann. Probab., (accepted, pending revision), 2014.

10 / 12

Related work

u ∈ C 14−,

12−

(R∗+ × R)

Bounded initial data (Walsh theory).Shiga’s work f ∈ Ctem(R): continuous function with tails tails grow atmost exponentially at ±∞.Sanz-Solé & Sarrà’s work on SHE over Rd with spatially homogeneouscolored noise which is white in time: bounded continuous function.Work by Conus et al: finite measure, 1/2− in space.Work by Dalang, Khoshnevisan & Nualart: a system of SHE’s withvanishing initial data.

SHE on bounded domains rather than R: Maximal inequality andstochastic convolutions. (initial data in some Banach space)

J. B. Walsh. An introduction to stochastic partial differential equations. In: École d’été deprobabilités de Saint-Flour, XIV—1984 , pp. 265–439. Springer, Berlin, 1986.

11 / 12

Related work

u ∈ C 14−,

12−

(R∗+ × R)



T. Shiga. Two contrasting properties of solutions for one-dimensional stochastic partial differentialequations. Canad. J. Math., 46(2):415–437, 1994.

11 / 12

Related work

u ∈ C 14−,

12−

(R∗+ × R)



M. Sanz-Solé and M. Sarrà. Hölder continuity for the stochastic heat equation with spatiallycorrelated noise. In: Seminar on Stochastic Analysis, Random Fields and Applications, III , pp.259–268. Birkhäuser, Basel, 2002. (R. C. Dalang, M. Dozzi and F. Russo, eds).

11 / 12

Related work

u ∈ C 14−,

12−

(R∗+ × R)



D. Conus, M. Joseph, D. Khoshnevisan, and S.-Y. Shiu. Initial measures for the stochastic heatequation. Ann. Inst. Henri Poincaré Probab. Stat., 2014.

11 / 12

Related work

u ∈ C 14−,

12−

(R∗+ × R)



R. C. Dalang, D. Khoshnevisan, and E. Nualart. Hitting probabilities for systems for non-linearstochastic heat equations with multiplicative noise. Probab. Theory Related Fields, 2009.

11 / 12

Related work

u ∈ C 14−,

12−

(R∗+ × R)



Z. Brzezniak. On stochastic convolution in Banach spaces and applications. Stochastics StochasticRep. 61(3-4):245–295, 1997.S. Peszat and J. Seidler. Maximal inequalities and space-time regularity of stochastic convolutions.Mathematica Bohemica 123(1): 7-32, 1998.

11 / 12

Thank you!

Le Chen ([email protected])Robert C. Dalang ([email protected])

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Hölder continuity for the nonlinear stochastic heat ...math.arizona.edu/~fpd/Talks/Chen.pdfIn A minicourse on stochastic partial differential equations, volume 1962 of Lecture Notes

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