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Hölder continuity for the nonlinear stochastic heat equation with 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 2014 Frontier 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 √ 2πνt exp

• −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)

• G2

ν ⋆ ||ρ(u)||2 2

• (t, x) < +∞ for all (t, x) ∈ R∗

+ × R, and

(t, x) → I(t, x) : R∗

+ × R → L2(Ω) is continuous;

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

+ × R.

• G2

ν ⋆ ||ρ(u)||2 2

• (t, x) :=

t ds

• R

dy G2

ν(t − s, x − y) ||ρ(u(s, y))||2 2 . 3 / 12

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 with rough initial conditions, Ann. Probab., (accepted, pending revision), 2014. [2] J. B. Walsh. An introduction to stochastic partial differential equations. In: École d’été de probabilités de Saint-Flour, XIV—1984, pp. 265–439. Springer, Berlin, 1986.

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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 with rough initial conditions, Ann. Probab., (accepted, pending revision), 2014. [2] J. B. Walsh. An introduction to stochastic partial differential equations. In: École d’été de probabilités de Saint-Flour, XIV—1984, pp. 265–439. Springer, Berlin, 1986.

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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 with rough initial conditions, Ann. Probab., (accepted, pending revision), 2014. [2] J. B. Walsh. An introduction to stochastic partial differential equations. In: École d’été de probabilités de Saint-Flour, XIV—1984, pp. 265–439. Springer, Berlin, 1986.

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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.

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 by Cβ1,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) :=

• signed Borel meas. µ, s.t.
• R

e−ax2|µ|(dx) < +∞, ∀a > 0

• Theorem

(1) If µ ∈ MH(R), then I ∈ C 1

4 −, 1 2 − (R∗

+ × R) a.s. Therefore,

u ∈ C 1

4 −, 1 2 − (R∗

+ × R) , a.s.

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u(t, x) = J0(t, x) + I(t, x) MH(R) :=

• signed Borel meas. µ, s.t.
• 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 1

4 −, 1 2 − (R∗

+ × R) a.s. Therefore,

u ∈ C 1

4 −, 1 2 − (R∗

+ × R) , a.s.

(2) If µ ∈ M∗

H(R) with µ(dx) = f(x)dx, then I ∈ C 1

4 −, 1 2 − (R+ × R), a.s.

Moreover, (i) If f is continuous, then u ∈ C (R+ × R) ∩ C 1

4 −, 1 2 − (R∗

+ × R) ,

a.s. (ii) If f is α-Hölder continuous, then u ∈ C( α

2 ∧ 1 4)−,(α∧ 1 2)− (R+ × R) ∩ C 1 4 −, 1 2 − (R∗

+ × R) ,

a.s.

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Difficulties with rough initial data

Conventional 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′]×R

Gν(t − s, x − y; t′ − s, x′ − y)u(s, y)W(dsdy)

• 2p

2p

≤ C t′

• R

Gν(· · · )2dsdy p/q t′

• R

G2

ν ·

• 1 + ||u(s, y)||2p

2p

• dsdy

≤ C sup

s∈[0,t′]

sup

y∈R

• 1 + ||u(s, y)||2p

2p

t′

• R

Gν(· · · )2dsdy p ≤ C sup

s∈[0,t′]

sup

y∈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 Lecture Notes 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 on Stochastic 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. 7 / 12

Difficulties with rough initial data

Conventional 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′]×R

Gν(t − s, x − y; t′ − s, x′ − y)u(s, y)W(dsdy)

• 2p

2p

≤ C t′

• R

Gν(· · · )2dsdy p/q t′

• R

G2

ν ·

• 1 + ||u(s, y)||2p

2p

• dsdy

≤ C sup

s∈[0,t′]

sup

y∈R

• 1 + ||u(s, y)||2p

2p

t′

• R

Gν(· · · )2dsdy p ≤ C sup

s∈[0,t′]

sup

y∈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)||2

2p ≥ ||u(s, y)||2 2 ≥ G ν

2 (s, y)

1 √ 4πνs = C s e− y2

νs ⇒ p < 3/2. 7 / 12

Difficulties with rough initial data

Conventional 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′]×R

Gν(t − s, x − y; t′ − s, x′ − y)u(s, y)W(dsdy)

• 2p

2p

≤ C t′

• R

Gν(· · · )2dsdy p/q t′

• R

G2

ν ·

• 1 + ||u(s, y)||2p

2p

• dsdy

≤ C sup

s∈[0,t′]

sup

y∈R

• 1 + ||u(s, y)||2p

2p

t′

• R

Gν(· · · )2dsdy p ≤ C sup

s∈[0,t′]

sup

y∈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 √ 2πνt exp

• −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

3L2 2νt {Gν (t, x − 2L ) + Gν (t, x + 2L )}

• .

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

• G ν

2 (t + r, x) − G ν 2 (t, x)

• ≤ 3

2 √ 1 + n2 √ t G ν(1+n2)

2

(t, x) √ r.

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Moment formula

||u(t, x)||2

p ≤ J2 0(t, x) +

• J2

0 ⋆ Kp(t, x)

• (t, x)

K(t, x; λ) := G ν

2 (t, x)

• λ2

√ 4πνt + λ4 2ν e

λ4 4ν 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.

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Related work

u ∈ C 1

4 −, 1 2 −(R∗

+ × R)

Bounded initial data (Walsh theory). Shiga’s work f ∈ Ctem(R): continuous function with tails tails grow at most exponentially at ±∞. Sanz-Solé & Sarrà’s work on SHE over Rd with spatially homogeneous colored 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 with vanishing initial data. SHE on bounded domains rather than R: Maximal inequality and stochastic convolutions. (initial data in some Banach space)

• J. B. Walsh. An introduction to stochastic partial differential equations. In: École d’été de

probabilités de Saint-Flour, XIV—1984 , pp. 265–439. Springer, Berlin, 1986.

11 / 12

Related work

u ∈ C 1

4 −, 1 2 −(R∗

+ × R)

Bounded initial data (Walsh theory). Shiga’s work f ∈ Ctem(R): continuous function with tails tails grow at most exponentially at ±∞. Sanz-Solé & Sarrà’s work on SHE over Rd with spatially homogeneous colored 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 with vanishing initial data. SHE on bounded domains rather than R: Maximal inequality and stochastic convolutions. (initial data in some Banach space)

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

11 / 12

Related work

u ∈ C 1

4 −, 1 2 −(R∗

+ × R)

Bounded initial data (Walsh theory). Shiga’s work f ∈ Ctem(R): continuous function with tails tails grow at most exponentially at ±∞. Sanz-Solé & Sarrà’s work on SHE over Rd with spatially homogeneous colored 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 with vanishing initial data. SHE on bounded domains rather than R: Maximal inequality and stochastic convolutions. (initial data in some Banach space)

• M. Sanz-Solé and M. Sarrà. Hölder continuity for the stochastic heat equation with spatially

correlated 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 1

4 −, 1 2 −(R∗

+ × R)

Bounded initial data (Walsh theory). Shiga’s work f ∈ Ctem(R): continuous function with tails tails grow at most exponentially at ±∞. Sanz-Solé & Sarrà’s work on SHE over Rd with spatially homogeneous colored 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 with vanishing initial data. SHE on bounded domains rather than R: Maximal inequality and stochastic convolutions. (initial data in some Banach space)

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

11 / 12

Related work

u ∈ C 1

4 −, 1 2 −(R∗

+ × R)

Bounded initial data (Walsh theory). Shiga’s work f ∈ Ctem(R): continuous function with tails tails grow at most exponentially at ±∞. Sanz-Solé & Sarrà’s work on SHE over Rd with spatially homogeneous colored 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 with vanishing initial data. SHE on bounded domains rather than R: Maximal inequality and stochastic convolutions. (initial data in some Banach space)

• R. C. Dalang, D. Khoshnevisan, and E. Nualart. Hitting probabilities for systems for non-linear

stochastic heat equations with multiplicative noise. Probab. Theory Related Fields, 2009.

11 / 12

Related work

u ∈ C 1

4 −, 1 2 −(R∗

+ × R)

Bounded initial data (Walsh theory). Shiga’s work f ∈ Ctem(R): continuous function with tails tails grow at most exponentially at ±∞. Sanz-Solé & Sarrà’s work on SHE over Rd with spatially homogeneous colored 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 with vanishing initial data. SHE on bounded domains rather than R: Maximal inequality and stochastic convolutions. (initial data in some Banach space)

• Z. Brze´
• zniak. On stochastic convolution in Banach spaces and applications. Stochastics Stochastic
• Rep. 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.

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Thank you!

Le Chen (chen@math.utah.edu) Robert C. Dalang (robert.dalang@epfl.ch)

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