stochastic partial differential equations: theory and ... · stochastic partial di erential...
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Stochastic Partial Differential Equations:Theory and Numerical Simulations
Adam Q. JaffeAdvised by: Lenya Ryzhik
May 19, 2019
A.Q. Jaffe Stochastic PDE May 19, 2019 1 / 24
Motivation
Why study Stochastic Partial Differential Equations (SPDE)?
In many (PDE) models with unknown parameters, it makes senseencode unknowns by randomness
Examples:
Wave propagation in atmosphereFluid dynamicsEpidemiology
Some Issues:
Lots of technical detailsHard to simulate in a computer
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Stochastic Ordinary Differential Equations
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SDEs from ODEs
General ODE written as dUtdt = µ(t,Ut)
t ∈ [0,T ]
U0 ∈ R(1)
We have existence, uniqueness, and continuity under very mildassumptions on µ
Lots of numerical methods to solve these; any approach will(essentially) work
Can we add simple random “noise” to a given ODE?
dUt
dt= µ(t,Ut) + σ(t,Ut)W (t) (2)
A.Q. Jaffe Stochastic PDE May 19, 2019 4 / 24
Brownian Motion and White Noise
Bt : 0 ≤ t ≤ T or simply Bt (3)
A random function
Important properties: Bt is...
almost surely continuous in talmost surely nowhere differentiable in tLimiting object of a random walk (CLT)
“The derivative of Brownian motion is white noise.”
W (t) =dBt
dt(4)
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Brownian Motion and White Noise
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SDEs from ODEs
The equationdUt
dt= µ(t,Ut) + σ(t,Ut)
dBt
dt(5)
in integral form is
Ut − U0 =
∫ t
0µ(s,Us)ds +
∫ t
0σ(t,Ut)
dBs
dsds (6)
Then reimagine the integral as∫ t
0σ(t,Ut)
dBs
dsds =
∫ t
0σ(t,Ut)dBs (7)
Can we interpret this in a Riemann-Stieljes-type sense, where Bt actsas an “integrator”?
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Stochastic Integral
For each sample path of Bt and partition 0 = t1 < · · · < tN = T ,write the Riemann-Stieljes sum∫ T
0σ(t,Ut)dBt ≈
N∑j=0
σ(t∗j ,Ut∗j)(Btj+1 − Btj ) (8)
How to choose t∗j ∈ [tj , tj+1)? Surprisingly, this choice matters!
t∗j = tj leads to the Ito integral,∫ T
0 σ(t,Ut)dBt
t∗j = 12 (tj + tj+1) leads to the Stratonovich integral,
∫ T0 σ(t,Ut) dBt
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Ito versus Stratonovich Calculus
Ito Calculus
Does not “see into the future”Is a martingaleThe usual chain rule failsNumerical methods are harder
Stratonovich Calculus
Does not “see into the future”Is not a martingaleUsual chain rule holdsNumerical methods are easier (Can use numerical methods for ODEs)
A.Q. Jaffe Stochastic PDE May 19, 2019 9 / 24
SDE Existence and Uniqueness
A general SDE is written as
Ut − U0 =
∫ t
0µ(s,Us)ds +
∫ t
0σ(s,Us)()dBs . (9)
Or, by the shorthand:
dUt = µ(t,Ut)dt + σ(t,Ut)()dBt (10)
Can interpret in the Ito or Stratonovich sense
We want a solution defined on t ∈ [0,T ], with a possibly randominitial condition U0.
Theorem
If µ, σ are Lipschitz continuous and satisfy a linear growth bound, thenalmost surely there exists a unique continuous function on [0,T ] whichsatisfies the SDE.
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SDE Example 1
Consider the SDE dUt = UtdBt on [0,T ] with deterministic initialcondition U0 = 1.The Ito solution is Ut = U0 exp(Bt − t
2 )The Stratonovich solution is Ut = U0 exp(Bt)
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SDE Example 2
Consider the SDE dUt = exp(t − U2t )dt + tanh(Ut)dBt on [0,T ] with
deterministic initial condition U0 = 1.
No explicit solutions exist!
Need to resort to numerical simulations
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Stochastic Partial Differential Equations
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SPDEs from PDEs
Consider the heat equation∂U∂t = ∂2U
∂x2 + g(t, x)
(t, x) ∈ [0,T ]× RU(0, x) = φ(x)
(11)
Its explicit solutions are given by
U(t, x) =
∫RG (t, x − y)φ(y)dy +
∫ t
0
∫RG (t − s, x − y)g(s, y)dyds
(12)where G is the usual heat kernel:
G (t, x) =1√4πt
exp
(−|x |
2
4t
)(13)
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SPDEs from PDEs
Consider the heat equation with g(t, x) = 0, and initial conditionφ(x) = 1[1,3](x).
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SPDEs from PDEs
Lesson from SDEs: Easiest to work in integral form. Write
∂U
∂t=∂2U
∂x2+ f (U(t, x))W (t, x) (14)
as
U(t, x) =
∫RG (t, x − y)φ(y)dy (15)
+
∫ t
0
∫RG (t − s, x − y)f (U(s, y))W (s, y)dyds (16)
Let’s come up with a precise definition of∫ t
0
∫RG (t−s, x−y)W (s, y)dyds =
∫ t
0
∫RG (t−s, x−y)dBs,y (17)
Use Riemann-Stieljes-type sums, and choose between Ito andStratonovich integral
We use Ito calculus from now on
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SPDE Existence and Uniqueness
A general stochastic heat equation is written as
U(t, x) =
∫RG (t, x−y)φ(y)dy+
∫ t
0
∫RG (t−s, x−y)f (U(s, y))dBs,y
(18)Or, by the shorthand:
∂U
∂t=∂2U
∂x2+ f (U(t, x))W (t, x) (19)
We impose an inital condition U(0, x) = φ(x).
Theorem
If f is Lipschitz continuous and the initial condition φ(x) is compactlysupported, then almost surely there exists a unique continuous function on[0,T ]× R which satisfies the SPDE.
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SPDE Existence and Uniqueness
Theorem
If f is Lipschitz continuous and the initial condition φ(x) is compactlysupported, then almost surely there exists a unique continuous function on[0,T ]× R which satisfies the SPDE.
Proof Sketch.
(Existence) By Picard’s iteration method: Set
U0(t, x) =φ(x) (20)
Un+1(t, x) =
∫RG (t, x − y)φ(y)dy (21)
+
∫ t
0
∫RG (t − s, x − y)f (Un(s, y))dBs,y
Each Un(t, x) is continuous in t and x .
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SPDE Existence and Uniqueness
Proof Sketch (continued).
Define the supremum of the L2(Ω) norm of the adjacent differences
zn(t) = supx∈R
sup0≤s≤t
E[|Un+1(s, x)− Un(s, x)|2
]. (22)
These are bounded as
zn(t) ≤ C1
∫ t
0zn−1(s)ds. (23)
By induction:
zn(t) ≤ C2(C1t)n+1
(n + 1)!≤ C2
(C1T )n+1
(n + 1)!(24)
Since this is summable, we get that Un(t, x) has an L2(Ω)-limit functionU(t, x). Since the convergence is uniform in t, the limit function U(t, x) iscontinuous. Now check that U satisfies that SPDE and is unique.
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SPDE Example 1
Consider the stochastic heat equation with f (u) = 1, anddeterministic initial condition φ(x) = 1[1,3](x).
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SPDE Example 2
Consider the stochastic heat equation with f (u) = |1− u|, anddeterministic initial condition φ(x) = 1[1,3](x).
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SPDE Example 3
Consider the stochastic heat equation with f (u) = 1[0,∞)(u), anddeterministic initial condition φ(x) = 1[1,3](x).
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Research Questions
“Smoothness” of solutions to S(P)DE
Existence and Uniqueness of SPDE solutions other than thestochastic heat equation
Relationship between Ito and Stratonovich calculus in the SPDE case
How to simulate S(P)DE efficiently and precisely with a computer?
Long-time behavior of certain SPDE models of interest
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Thank you!
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