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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
1 / 12
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)
2 / 12
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 .
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 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.
4 / 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 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.
4 / 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 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.
4 / 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.
J0(t , x) ∈ C∞(R∗+ × R)
I(t , x) ∈ C?,?(R∗+ × R)
4 / 12
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) .
5 / 12
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 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.
6 / 12
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 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.
6 / 12
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
∫RGν(· · · )2dsdy
]p
≤ C sups∈[0,t′]
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.
7 / 12
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
∫RGν(· · · )2dsdy
]p
≤ C sups∈[0,t′]
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.
7 / 12
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
∫RGν(· · · )2dsdy
]p
≤ C sups∈[0,t′]
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.
7 / 12
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 ′|
).
8 / 12
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 .
9 / 12
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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
