Outline History of the Dirichlet Problem Analytic Solutions Probabilistic Solutions Simulated Solutions Probabilistic Solutions to the Dirichlet Problem Theory and Simulation David Kahle Stat 650 : Stochastic Differential Equations April 17, 2008 David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
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OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Probabilistic Solutions to the Dirichlet ProblemTheory and Simulation
David KahleStat 650 : Stochastic Differential Equations
April 17, 2008
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Outline
I History of the Dirichlet Problem
I Analytical Solutions
I Probabilistic Solutions
I Simulated Solutions
I Concluding Remarks
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Outline
I History of the Dirichlet Problem
I Analytical Solutions
I Probabilistic Solutions
I Simulated Solutions
I Concluding Remarks
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Outline
I History of the Dirichlet Problem
I Analytical Solutions
I Probabilistic Solutions
I Simulated Solutions
I Concluding Remarks
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Outline
I History of the Dirichlet Problem
I Analytical Solutions
I Probabilistic Solutions
I Simulated Solutions
I Concluding Remarks
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Outline
I History of the Dirichlet Problem
I Analytical Solutions
I Probabilistic Solutions
I Simulated Solutions
I Concluding Remarks
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Johann Peter Gustav Lejeune Dirichlet (1805-1859)
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Johann Peter Gustav Lejeune Dirichlet (1805-1859)
I 1805 : Born in Liege, Belgium
I 1819 : Jesuit Gymnasium in Cologne (Ohm)
I 1821 : Paris and the Academie des Sciences (Fourier, Laplace,Legendre, Poisson)
I 1828 : University of Berlin (Advised Eisenstein, Kronecker,Lipschitz, worked with Riemann)
I 1855 : University of Gottingen (Replacing Gauss)
I 1859 : Death (Works collected by Dedekind)
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Johann Peter Gustav Lejeune Dirichlet (1805-1859)
I 1805 : Born in Liege, Belgium
I 1819 : Jesuit Gymnasium in Cologne (Ohm)
I 1821 : Paris and the Academie des Sciences (Fourier, Laplace,Legendre, Poisson)
I 1828 : University of Berlin (Advised Eisenstein, Kronecker,Lipschitz, worked with Riemann)
I 1855 : University of Gottingen (Replacing Gauss)
I 1859 : Death (Works collected by Dedekind)
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Johann Peter Gustav Lejeune Dirichlet (1805-1859)
I 1805 : Born in Liege, Belgium
I 1819 : Jesuit Gymnasium in Cologne (Ohm)
I 1821 : Paris and the Academie des Sciences (Fourier, Laplace,Legendre, Poisson)
I 1828 : University of Berlin (Advised Eisenstein, Kronecker,Lipschitz, worked with Riemann)
I 1855 : University of Gottingen (Replacing Gauss)
I 1859 : Death (Works collected by Dedekind)
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Johann Peter Gustav Lejeune Dirichlet (1805-1859)
I 1805 : Born in Liege, Belgium
I 1819 : Jesuit Gymnasium in Cologne (Ohm)
I 1821 : Paris and the Academie des Sciences (Fourier, Laplace,Legendre, Poisson)
I 1828 : University of Berlin (Advised Eisenstein, Kronecker,Lipschitz, worked with Riemann)
I 1855 : University of Gottingen (Replacing Gauss)
I 1859 : Death (Works collected by Dedekind)
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Johann Peter Gustav Lejeune Dirichlet (1805-1859)
I 1805 : Born in Liege, Belgium
I 1819 : Jesuit Gymnasium in Cologne (Ohm)
I 1821 : Paris and the Academie des Sciences (Fourier, Laplace,Legendre, Poisson)
I 1828 : University of Berlin (Advised Eisenstein, Kronecker,Lipschitz, worked with Riemann)
I 1855 : University of Gottingen (Replacing Gauss)
I 1859 : Death (Works collected by Dedekind)
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Johann Peter Gustav Lejeune Dirichlet (1805-1859)
I 1805 : Born in Liege, Belgium
I 1819 : Jesuit Gymnasium in Cologne (Ohm)
I 1821 : Paris and the Academie des Sciences (Fourier, Laplace,Legendre, Poisson)
I 1828 : University of Berlin (Advised Eisenstein, Kronecker,Lipschitz, worked with Riemann)
I 1855 : University of Gottingen (Replacing Gauss)
I 1859 : Death (Works collected by Dedekind)
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Contributions
I Number theory (analytic and algebraic)
I Analysis - first solutions to the Dirichlet problem
I Differential equationsI Fourier seriesI Laplaces’ problem on the stability of the solar system
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Contributions
I Number theory (analytic and algebraic)I Analysis - first solutions to the Dirichlet problem
I Differential equationsI Fourier seriesI Laplaces’ problem on the stability of the solar system
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Contributions
I Number theory (analytic and algebraic)I Analysis - first solutions to the Dirichlet problem
I Differential equationsI Fourier seriesI Laplaces’ problem on the stability of the solar system
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
The Dirichlet Problem
I Given an open set G ⊂ Rk with sufficiently smooth boundary∂G , find the unique solution u : Rk → R such that
∆u = ∇ · ∇u = 0
on G and u = f on ∂G for some continuous function f .
I Note
I u is harmonic on G .I u is the equilibrium temperature distribution on G given the
boundary temperatures (which do not change with time).I The Dirichlet problem is a boundary value problem.
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
The Dirichlet Problem
I Given an open set G ⊂ Rk with sufficiently smooth boundary∂G , find the unique solution u : Rk → R such that
∆u = ∇ · ∇u = 0
on G and u = f on ∂G for some continuous function f .I Note
I u is harmonic on G .I u is the equilibrium temperature distribution on G given the
boundary temperatures (which do not change with time).I The Dirichlet problem is a boundary value problem.
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
The Dirichlet Problem
I Given an open set G ⊂ Rk with sufficiently smooth boundary∂G , find the unique solution u : Rk → R such that
∆u = ∇ · ∇u = 0
on G and u = f on ∂G for some continuous function f .I Note
I u is harmonic on G .
I u is the equilibrium temperature distribution on G given theboundary temperatures (which do not change with time).
I The Dirichlet problem is a boundary value problem.
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
The Dirichlet Problem
I Given an open set G ⊂ Rk with sufficiently smooth boundary∂G , find the unique solution u : Rk → R such that
∆u = ∇ · ∇u = 0
on G and u = f on ∂G for some continuous function f .I Note
I u is harmonic on G .I u is the equilibrium temperature distribution on G given the
boundary temperatures (which do not change with time).
I The Dirichlet problem is a boundary value problem.
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
The Dirichlet Problem
I Given an open set G ⊂ Rk with sufficiently smooth boundary∂G , find the unique solution u : Rk → R such that
∆u = ∇ · ∇u = 0
on G and u = f on ∂G for some continuous function f .I Note
I u is harmonic on G .I u is the equilibrium temperature distribution on G given the
boundary temperatures (which do not change with time).I The Dirichlet problem is a boundary value problem.
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
The Dirichlet Problem - Example
∆u(x , y) =∂2
∂x2u(x , y) +
∂2
∂y2u(x , y) = 0.
on G = (0, a)× (0, b) and f is defined and continuous on the boundary G , ∂G .
Figure: The General Dirichlet Problem on the Rectangle (0, a)× (0, b)
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
The Dirichlet Problem - Example
∆u(x , y) =∂2
∂x2u(x , y) +
∂2
∂y2u(x , y) = 0.
on G = (0, a)× (0, b) and f is defined and continuous on the boundary G , ∂G .
Figure: The General Dirichlet Problem on the Rectangle (0, a)× (0, b)
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Analytic Solution on [0, a]× [0, b] - Slide 1
I Suppose f1(x) = g1(y) = g2(y) ≡ 0, and f2(x) is continuous with 0 endpoints.
Solve the boundary value problem.
Figure: A More Specific Dirichlet Problem on the Rectangle(0, a)× (0, b)
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Analytic Solution on [0, a]× [0, b] - Slide 1
I Suppose f1(x) = g1(y) = g2(y) ≡ 0, and f2(x) is continuous with 0 endpoints.
Solve the boundary value problem.
Figure: A More Specific Dirichlet Problem on the Rectangle(0, a)× (0, b)
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Analytic Solution on [0, a]× [0, b] - Slide 2
I Separation of variables : Suppose u(x , y) = X (x)Y (y), then
∆u(x , y) = 0 ⇐⇒ ∂2
∂x2u(x , y) +
∂2
∂y2u(x , y) = 0
⇐⇒ Y (y)X ′′(x) + X (x)Y ′′(y) = 0
⇐⇒ −Y (y)X ′′(x) = X (x)Y ′′(y)
⇐⇒ −X ′′(x)
X (x)=
Y ′′(y)
Y (y)= k.
X ′′(x) + kX (x) = 0 and Y ′′(y)− kY (y) = 0.
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Analytic Solution on [0, a]× [0, b] - Slide 3
I Solve the ODE’s subject to boundary conditions :(Remember u(x , y) = X (x)Y (y)!)
I Note the boundary conditions
g1(y) = 0 =⇒ X (0) = 0
f1(x) = 0 =⇒ Y (0) = 0
g2(y) = 0 =⇒ X (a) = 0
f2(x) = f2(x) =⇒ Y (b) = f2(x)
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Analytic Solution on [0, a]× [0, b] - Slide 3
I Solve the ODE’s subject to boundary conditions :(Remember u(x , y) = X (x)Y (y)!)
I Note the boundary conditions
g1(y) = 0 =⇒ X (0) = 0
f1(x) = 0 =⇒ Y (0) = 0
g2(y) = 0 =⇒ X (a) = 0
f2(x) = f2(x) =⇒ Y (b) = f2(x)
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Analytic Solution on [0, a]× [0, b] - Slide 4
I Solve for X (x) - imposing boundary conditions :
I Begin with X ′′(x) + kX (x) = 0.
I For nontrivial solutions, k = µ2 > 0.I General solution : X (x) = c1 cosµx + c2 sinµx .I X (0) = 0 =⇒ c1 = 0.I X (a) = 0 =⇒ c2 sinµx = 0 =⇒ µ = µn = nπ
a , n ∈ N.
Conclusion : Xn(x) = sin(nπ
ax), n ∈ N.
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Analytic Solution on [0, a]× [0, b] - Slide 4
I Solve for X (x) - imposing boundary conditions :
I Begin with X ′′(x) + kX (x) = 0.
I For nontrivial solutions, k = µ2 > 0.I General solution : X (x) = c1 cosµx + c2 sinµx .I X (0) = 0 =⇒ c1 = 0.I X (a) = 0 =⇒ c2 sinµx = 0 =⇒ µ = µn = nπ
a , n ∈ N.
Conclusion : Xn(x) = sin(nπ
ax), n ∈ N.
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Analytic Solution on [0, a]× [0, b] - Slide 4
I Solve for X (x) - imposing boundary conditions :
I Begin with X ′′(x) + kX (x) = 0.
I For nontrivial solutions, k = µ2 > 0.
I General solution : X (x) = c1 cosµx + c2 sinµx .I X (0) = 0 =⇒ c1 = 0.I X (a) = 0 =⇒ c2 sinµx = 0 =⇒ µ = µn = nπ
a , n ∈ N.
Conclusion : Xn(x) = sin(nπ
ax), n ∈ N.
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Analytic Solution on [0, a]× [0, b] - Slide 4
I Solve for X (x) - imposing boundary conditions :
I Begin with X ′′(x) + kX (x) = 0.
I For nontrivial solutions, k = µ2 > 0.I General solution : X (x) = c1 cosµx + c2 sinµx .
I X (0) = 0 =⇒ c1 = 0.I X (a) = 0 =⇒ c2 sinµx = 0 =⇒ µ = µn = nπ
a , n ∈ N.
Conclusion : Xn(x) = sin(nπ
ax), n ∈ N.
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Analytic Solution on [0, a]× [0, b] - Slide 4
I Solve for X (x) - imposing boundary conditions :
I Begin with X ′′(x) + kX (x) = 0.
I For nontrivial solutions, k = µ2 > 0.I General solution : X (x) = c1 cosµx + c2 sinµx .I X (0) = 0 =⇒ c1 = 0.
I X (a) = 0 =⇒ c2 sinµx = 0 =⇒ µ = µn = nπa , n ∈ N.
Conclusion : Xn(x) = sin(nπ
ax), n ∈ N.
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Analytic Solution on [0, a]× [0, b] - Slide 4
I Solve for X (x) - imposing boundary conditions :
I Begin with X ′′(x) + kX (x) = 0.
I For nontrivial solutions, k = µ2 > 0.I General solution : X (x) = c1 cosµx + c2 sinµx .I X (0) = 0 =⇒ c1 = 0.I X (a) = 0 =⇒ c2 sinµx = 0 =⇒ µ = µn = nπ
a , n ∈ N.
Conclusion : Xn(x) = sin(nπ
ax), n ∈ N.
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Analytic Solution on [0, a]× [0, b] - Slide 4
I Solve for X (x) - imposing boundary conditions :
I Begin with X ′′(x) + kX (x) = 0.
I For nontrivial solutions, k = µ2 > 0.I General solution : X (x) = c1 cosµx + c2 sinµx .I X (0) = 0 =⇒ c1 = 0.I X (a) = 0 =⇒ c2 sinµx = 0 =⇒ µ = µn = nπ
a , n ∈ N.
Conclusion : Xn(x) = sin(nπ
ax), n ∈ N.
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Analytic Solution on [0, a]× [0, b] - Slide 5
I Solve for Y (y) - imposing boundary conditions :
I Begin with Y ′′(x)− kY (x) = 0.
I For nontrivial solutions, k = µ2n > 0.
I General solution : Y (y) = An coshµny + Bn sinhµny .I Y (0) = 0 =⇒ An = 0.
Conclusion : Yn(y) = Bn sinh(nπ
ay), n ∈ N.
I Still have one boundary condition!
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Analytic Solution on [0, a]× [0, b] - Slide 5
I Solve for Y (y) - imposing boundary conditions :
I Begin with Y ′′(x)− kY (x) = 0.
I For nontrivial solutions, k = µ2n > 0.
I General solution : Y (y) = An coshµny + Bn sinhµny .I Y (0) = 0 =⇒ An = 0.
Conclusion : Yn(y) = Bn sinh(nπ
ay), n ∈ N.
I Still have one boundary condition!
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Analytic Solution on [0, a]× [0, b] - Slide 5
I Solve for Y (y) - imposing boundary conditions :
I Begin with Y ′′(x)− kY (x) = 0.
I For nontrivial solutions, k = µ2n > 0.
I General solution : Y (y) = An coshµny + Bn sinhµny .I Y (0) = 0 =⇒ An = 0.
Conclusion : Yn(y) = Bn sinh(nπ
ay), n ∈ N.
I Still have one boundary condition!
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Analytic Solution on [0, a]× [0, b] - Slide 5
I Solve for Y (y) - imposing boundary conditions :
I Begin with Y ′′(x)− kY (x) = 0.
I For nontrivial solutions, k = µ2n > 0.
I General solution : Y (y) = An coshµny + Bn sinhµny .
I Y (0) = 0 =⇒ An = 0.
Conclusion : Yn(y) = Bn sinh(nπ
ay), n ∈ N.
I Still have one boundary condition!
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Analytic Solution on [0, a]× [0, b] - Slide 5
I Solve for Y (y) - imposing boundary conditions :
I Begin with Y ′′(x)− kY (x) = 0.
I For nontrivial solutions, k = µ2n > 0.
I General solution : Y (y) = An coshµny + Bn sinhµny .I Y (0) = 0 =⇒ An = 0.
Conclusion : Yn(y) = Bn sinh(nπ
ay), n ∈ N.
I Still have one boundary condition!
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Analytic Solution on [0, a]× [0, b] - Slide 5
I Solve for Y (y) - imposing boundary conditions :
I Begin with Y ′′(x)− kY (x) = 0.
I For nontrivial solutions, k = µ2n > 0.
I General solution : Y (y) = An coshµny + Bn sinhµny .I Y (0) = 0 =⇒ An = 0.
Conclusion : Yn(y) = Bn sinh(nπ
ay), n ∈ N.
I Still have one boundary condition!
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Analytic Solution on [0, a]× [0, b] - Slide 5
I Solve for Y (y) - imposing boundary conditions :
I Begin with Y ′′(x)− kY (x) = 0.
I For nontrivial solutions, k = µ2n > 0.
I General solution : Y (y) = An coshµny + Bn sinhµny .I Y (0) = 0 =⇒ An = 0.
Conclusion : Yn(y) = Bn sinh(nπ
ay), n ∈ N.
I Still have one boundary condition!
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Analytic Solution on [0, a]× [0, b] - Slide 6
I Find u(x , y), impose last boundary condition :
I Superposition principle :
u(x , y) =∞∑
n=1
Xn(x)Yn(y) =∞∑
n=1
Bn sin(nπ
ax)
sinh(nπ
ay)
I Last boundary condition
u(x , b) = f2(x) =∞∑
n=1
(Bn sinh
(nπ
ab))
sin(nπ
ax)
(Bn sinh
(nπa b))
= Fourier sine coefs of f2(x) on (0, a).
Bn sinh(nπ
ab)
=2
a
∫ a
0
f2(x) sin(nπ
ax)
dx .
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Analytic Solution on [0, a]× [0, b] - Slide 6
I Find u(x , y), impose last boundary condition :I Superposition principle :
u(x , y) =∞∑
n=1
Xn(x)Yn(y) =∞∑
n=1
Bn sin(nπ
ax)
sinh(nπ
ay)
I Last boundary condition
u(x , b) = f2(x) =∞∑
n=1
(Bn sinh
(nπ
ab))
sin(nπ
ax)
(Bn sinh
(nπa b))
= Fourier sine coefs of f2(x) on (0, a).
Bn sinh(nπ
ab)
=2
a
∫ a
0
f2(x) sin(nπ
ax)
dx .
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Analytic Solution on [0, a]× [0, b] - Slide 6
I Find u(x , y), impose last boundary condition :I Superposition principle :
u(x , y) =∞∑
n=1
Xn(x)Yn(y) =∞∑
n=1
Bn sin(nπ
ax)
sinh(nπ
ay)
I Last boundary condition
u(x , b) = f2(x) =∞∑
n=1
(Bn sinh
(nπ
ab))
sin(nπ
ax)
(Bn sinh
(nπa b))
= Fourier sine coefs of f2(x) on (0, a).
Bn sinh(nπ
ab)
=2
a
∫ a
0
f2(x) sin(nπ
ax)
dx .
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Analytic Solution on [0, a]× [0, b] - Slide 6
I Find u(x , y), impose last boundary condition :I Superposition principle :
u(x , y) =∞∑
n=1
Xn(x)Yn(y) =∞∑
n=1
Bn sin(nπ
ax)
sinh(nπ
ay)
I Last boundary condition
u(x , b) = f2(x) =∞∑
n=1
(Bn sinh
(nπ
ab))
sin(nπ
ax)
(Bn sinh
(nπa b))
= Fourier sine coefs of f2(x) on (0, a).
Bn sinh(nπ
ab)
=2
a
∫ a
0
f2(x) sin(nπ
ax)
dx .
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Analytic Solution on [0, a]× [0, b] - Slide 7
I Final solution! (Finally...)
Solution : u(x , y) =∞∑
n=1
Bn sin(nπ
ax)
sinh(nπ
ay)
where
Bn =2
a sinh(
nπa b) ∫ a
0f2(x) sin
(nπ
ax)
dx
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Analytic Solution on [0, a]× [0, b] - Slide 7
I Final solution! (Finally...)
Solution : u(x , y) =∞∑
n=1
Bn sin(nπ
ax)
sinh(nπ
ay)
where
Bn =2
a sinh(
nπa b) ∫ a
0f2(x) sin
(nπ
ax)
dx
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Problems with Analytic Solutions to the Dirichlet Problem
I This is HARD!!
I What do you do if...
I ...you can’t compute the integral (hard boundary functions)?I ...you have a strangely shaped G (complicated boundary)?
Figure: The Dirichlet Problem on the 2-d Bat Symbolwith f (x , y) = exp {Γ (?(x + y))} for (x , y) ∈ ∂G
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Problems with Analytic Solutions to the Dirichlet Problem
I This is HARD!!I What do you do if...
I ...you can’t compute the integral (hard boundary functions)?I ...you have a strangely shaped G (complicated boundary)?
Figure: The Dirichlet Problem on the 2-d Bat Symbolwith f (x , y) = exp {Γ (?(x + y))} for (x , y) ∈ ∂G
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Problems with Analytic Solutions to the Dirichlet Problem
I This is HARD!!I What do you do if...
I ...you can’t compute the integral (hard boundary functions)?
I ...you have a strangely shaped G (complicated boundary)?
Figure: The Dirichlet Problem on the 2-d Bat Symbolwith f (x , y) = exp {Γ (?(x + y))} for (x , y) ∈ ∂G
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Problems with Analytic Solutions to the Dirichlet Problem
I This is HARD!!I What do you do if...
I ...you can’t compute the integral (hard boundary functions)?I ...you have a strangely shaped G (complicated boundary)?
Figure: The Dirichlet Problem on the 2-d Bat Symbolwith f (x , y) = exp {Γ (?(x + y))} for (x , y) ∈ ∂G
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Problems with Analytic Solutions to the Dirichlet Problem
I This is HARD!!I What do you do if...
I ...you can’t compute the integral (hard boundary functions)?I ...you have a strangely shaped G (complicated boundary)?
Figure: The Dirichlet Problem on the 2-d Bat Symbolwith f (x , y) = exp {Γ (?(x + y))} for (x , y) ∈ ∂G
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
A Necessary Stopping Time
I Begin with a stopping time.
τ = inf{
t > 0 : ~Bt /∈ G}.
This is the time at which the (k-dimensional) Brownianmotion reaches the boundary ∂G .
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
A Necessary Stopping Time
I Begin with a stopping time.
τ = inf{
t > 0 : ~Bt /∈ G}.
This is the time at which the (k-dimensional) Brownianmotion reaches the boundary ∂G .
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Theorem 1 : Solution Under 4 Conditions
I Theorem 1 : Assume
1. G is an open set,2. G is bounded,3. ∆u exists and is 0 in G ,4. u = f , a continuous function, on ∂G .
Then for ~x ∈ G ,
u(~x) = E~x [f (Bτ )],
where the notation is interpreted as a Brownian motion whichoriginates from ~x .
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Theorem 1 : Solution Under 4 Conditions
I Theorem 1 : Assume
1. G is an open set,
2. G is bounded,3. ∆u exists and is 0 in G ,4. u = f , a continuous function, on ∂G .
Then for ~x ∈ G ,
u(~x) = E~x [f (Bτ )],
where the notation is interpreted as a Brownian motion whichoriginates from ~x .
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Theorem 1 : Solution Under 4 Conditions
I Theorem 1 : Assume
1. G is an open set,2. G is bounded,
3. ∆u exists and is 0 in G ,4. u = f , a continuous function, on ∂G .
Then for ~x ∈ G ,
u(~x) = E~x [f (Bτ )],
where the notation is interpreted as a Brownian motion whichoriginates from ~x .
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Theorem 1 : Solution Under 4 Conditions
I Theorem 1 : Assume
1. G is an open set,2. G is bounded,3. ∆u exists and is 0 in G ,
4. u = f , a continuous function, on ∂G .
Then for ~x ∈ G ,
u(~x) = E~x [f (Bτ )],
where the notation is interpreted as a Brownian motion whichoriginates from ~x .
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Theorem 1 : Solution Under 4 Conditions
I Theorem 1 : Assume
1. G is an open set,2. G is bounded,3. ∆u exists and is 0 in G ,4. u = f , a continuous function, on ∂G .
Then for ~x ∈ G ,
u(~x) = E~x [f (Bτ )],
where the notation is interpreted as a Brownian motion whichoriginates from ~x .
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Theorem 1 : Solution Under 4 Conditions
I Theorem 1 : Assume
1. G is an open set,2. G is bounded,3. ∆u exists and is 0 in G ,4. u = f , a continuous function, on ∂G .
Then for ~x ∈ G ,
u(~x) = E~x [f (Bτ )],
where the notation is interpreted as a Brownian motion whichoriginates from ~x .
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Theorem 1 : Solution Under 4 Conditions
I Theorem 1 : Assume
1. G is an open set,2. G is bounded,3. ∆u exists and is 0 in G ,4. u = f , a continuous function, on ∂G .
Then for ~x ∈ G ,
u(~x) = E~x [f (Bτ )],
where the notation is interpreted as a Brownian motion whichoriginates from ~x .
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Theorem 1 : Proof - Overview
I Step 1 : P~x [τ <∞] = 1.
I Step 2 : The main result,
u(~x) = E~x [f (Bτ )] .
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Theorem 1 : Proof - Overview
I Step 1 : P~x [τ <∞] = 1.
I Step 2 : The main result,
u(~x) = E~x [f (Bτ )] .
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Theorem 1 : Proof - Step 1
I Step 1 : P~x [τ <∞] = 1.
I Define K = diameter of G
K := sup {||~x − ~y ||k : ~x , ~y ∈ G} .
I Note for any point ~x in G∣∣∣∣∣∣~B1 − ~x∣∣∣∣∣∣
k> K =⇒ ~B1 /∈ G =⇒ τ < 1,
so
P~x [τ < 1]≥P~x
[∣∣∣∣∣∣~B1 − ~x∣∣∣∣∣∣
k> K
]=P~0
[∣∣∣∣∣∣~B1
∣∣∣∣∣∣k> K
]=:εK > 0,
sosup~x∈G
P~x [τ ≥ k] ≤ (1− εK )k , if k = 1.
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Theorem 1 : Proof - Step 1
I Step 1 : P~x [τ <∞] = 1.
I Define K = diameter of G
K := sup {||~x − ~y ||k : ~x , ~y ∈ G} .
I Note for any point ~x in G∣∣∣∣∣∣~B1 − ~x∣∣∣∣∣∣
k> K =⇒ ~B1 /∈ G =⇒ τ < 1,
so
P~x [τ < 1]≥P~x
[∣∣∣∣∣∣~B1 − ~x∣∣∣∣∣∣
k> K
]=P~0
[∣∣∣∣∣∣~B1
∣∣∣∣∣∣k> K
]=:εK > 0,
sosup~x∈G
P~x [τ ≥ k] ≤ (1− εK )k , if k = 1.
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Theorem 1 : Proof - Step 1
I Step 1 : P~x [τ <∞] = 1.
I Define K = diameter of G
K := sup {||~x − ~y ||k : ~x , ~y ∈ G} .
I Note for any point ~x in G∣∣∣∣∣∣~B1 − ~x∣∣∣∣∣∣
k> K =⇒ ~B1 /∈ G =⇒ τ < 1,
so
P~x [τ < 1]≥P~x
[∣∣∣∣∣∣~B1 − ~x∣∣∣∣∣∣
k> K
]=P~0
[∣∣∣∣∣∣~B1
∣∣∣∣∣∣k> K
]=:εK > 0,
sosup~x∈G
P~x [τ ≥ k] ≤ (1− εK )k , if k = 1.
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Theorem 1 : Proof - Step 1
I Step 1 : P~x [τ <∞] = 1.
I Define K = diameter of G
K := sup {||~x − ~y ||k : ~x , ~y ∈ G} .
I Note for any point ~x in G∣∣∣∣∣∣~B1 − ~x∣∣∣∣∣∣
k> K =⇒ ~B1 /∈ G =⇒ τ < 1,
so
P~x [τ < 1]≥P~x
[∣∣∣∣∣∣~B1 − ~x∣∣∣∣∣∣
k> K
]=P~0
[∣∣∣∣∣∣~B1
∣∣∣∣∣∣k> K
]=:εK > 0,
sosup~x∈G
P~x [τ ≥ k] ≤ (1− εK )k , if k = 1.
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Theorem 1 : Proof - Step 1
I Step 1 : P~x [τ <∞] = 1.
I Define K = diameter of G
K := sup {||~x − ~y ||k : ~x , ~y ∈ G} .
I Note for any point ~x in G∣∣∣∣∣∣~B1 − ~x∣∣∣∣∣∣
k> K =⇒ ~B1 /∈ G =⇒ τ < 1,
so
P~x [τ < 1]≥P~x
[∣∣∣∣∣∣~B1 − ~x∣∣∣∣∣∣
k> K
]=P~0
[∣∣∣∣∣∣~B1
∣∣∣∣∣∣k> K
]=:εK > 0,
sosup~x∈G
P~x [τ ≥ k] ≤ (1− εK )k , if k = 1.
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Theorem 1 : Proof - Step 1
I Step 1 : P~x [τ <∞] = 1 (cont).
I Generalize to all k ∈ N
sup~x∈G
P~x [τ ≥ k] ≤ (1− εK )k
to all k ∈ N. Suppose pt(~x , ~y) is the transition probabilitydensity of the Brownian motion.Then the Markov property implies
P~x [τ ≥ k] ≤∫
G
p1(~x , ~y)P~y [τ ≥ k − 1] d~y
≤ (1− εK )k−1P~x [B1 ∈ G ] (Inductive hypothesis)
≤ (1− εK )k .
I Conclusion : P~x [τ <∞] = 1.
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Theorem 1 : Proof - Step 1
I Step 1 : P~x [τ <∞] = 1 (cont).
I Generalize to all k ∈ N
sup~x∈G
P~x [τ ≥ k] ≤ (1− εK )k
to all k ∈ N. Suppose pt(~x , ~y) is the transition probabilitydensity of the Brownian motion.
Then the Markov property implies
P~x [τ ≥ k] ≤∫
G
p1(~x , ~y)P~y [τ ≥ k − 1] d~y
≤ (1− εK )k−1P~x [B1 ∈ G ] (Inductive hypothesis)
≤ (1− εK )k .
I Conclusion : P~x [τ <∞] = 1.
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Theorem 1 : Proof - Step 1
I Step 1 : P~x [τ <∞] = 1 (cont).
I Generalize to all k ∈ N
sup~x∈G
P~x [τ ≥ k] ≤ (1− εK )k
to all k ∈ N. Suppose pt(~x , ~y) is the transition probabilitydensity of the Brownian motion.Then the Markov property implies
P~x [τ ≥ k] ≤∫
G
p1(~x , ~y)P~y [τ ≥ k − 1] d~y
≤ (1− εK )k−1P~x [B1 ∈ G ] (Inductive hypothesis)
≤ (1− εK )k .
I Conclusion : P~x [τ <∞] = 1.
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Theorem 1 : Proof - Step 1
I Step 1 : P~x [τ <∞] = 1 (cont).
I Generalize to all k ∈ N
sup~x∈G
P~x [τ ≥ k] ≤ (1− εK )k
to all k ∈ N. Suppose pt(~x , ~y) is the transition probabilitydensity of the Brownian motion.Then the Markov property implies
P~x [τ ≥ k] ≤∫
G
p1(~x , ~y)P~y [τ ≥ k − 1] d~y
≤ (1− εK )k−1P~x [B1 ∈ G ] (Inductive hypothesis)
≤ (1− εK )k .
I Conclusion : P~x [τ <∞] = 1.
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Theorem 1 : Proof - Step 2
I Step 2 : u(~x) = E~x [f (Bτ )].
I By Ito’s formula,
u(~Bt)− u(~B0) =
∫ t
0
∇u(~Bs) · d~Bs +1
2
∫ t
0
∆u(~Bs) ds,
but 12
∫ t
0∆u(~Bs) ds = 0 since ∆u = 0 on G , and, since the
first term is a local martingale,
u(~Bt) is a local martingale on [0, τ).
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Theorem 1 : Proof - Step 2
I Step 2 : u(~x) = E~x [f (Bτ )].
I By Ito’s formula,
u(~Bt)− u(~B0) =
∫ t
0
∇u(~Bs) · d~Bs +1
2
∫ t
0
∆u(~Bs) ds,
but 12
∫ t
0∆u(~Bs) ds = 0 since ∆u = 0 on G , and, since the
first term is a local martingale,
u(~Bt) is a local martingale on [0, τ).
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Theorem 1 : Proof - Step 2
I Step 2 : u(~x) = E~x [f (Bτ )].
I There exists a time-change γ : R+ → R+ such that
Xt = u(Bγ(t)) is a bounded martingale.
(We already knew it was a continuous bounded local martingale on [0, τ).)
I =⇒ Xta.s.−→ X∞, Xt
Lp−→ X∞ (MGCT)with Xt = E~x [X∞|Gt ], Gt = Fγ(t)
=⇒ E~x [Xt ] = E~x [X∞].I τ <∞ =⇒ X∞ = f (Bτ ).I t = 0 =⇒
u(~x) = E~x [X0] = E~x [X∞] = E~x [f (Bτ )].
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Theorem 1 : Proof - Step 2
I Step 2 : u(~x) = E~x [f (Bτ )].
I There exists a time-change γ : R+ → R+ such that
Xt = u(Bγ(t)) is a bounded martingale.
(We already knew it was a continuous bounded local martingale on [0, τ).)
I =⇒ Xta.s.−→ X∞, Xt
Lp−→ X∞ (MGCT)with Xt = E~x [X∞|Gt ], Gt = Fγ(t)
=⇒ E~x [Xt ] = E~x [X∞].I τ <∞ =⇒ X∞ = f (Bτ ).I t = 0 =⇒
u(~x) = E~x [X0] = E~x [X∞] = E~x [f (Bτ )].
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Theorem 1 : Proof - Step 2
I Step 2 : u(~x) = E~x [f (Bτ )].
I There exists a time-change γ : R+ → R+ such that
Xt = u(Bγ(t)) is a bounded martingale.
(We already knew it was a continuous bounded local martingale on [0, τ).)
I =⇒ Xta.s.−→ X∞, Xt
Lp−→ X∞ (MGCT)with Xt = E~x [X∞|Gt ], Gt = Fγ(t)
=⇒ E~x [Xt ] = E~x [X∞].
I τ <∞ =⇒ X∞ = f (Bτ ).I t = 0 =⇒
u(~x) = E~x [X0] = E~x [X∞] = E~x [f (Bτ )].
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Theorem 1 : Proof - Step 2
I Step 2 : u(~x) = E~x [f (Bτ )].
I There exists a time-change γ : R+ → R+ such that
Xt = u(Bγ(t)) is a bounded martingale.
(We already knew it was a continuous bounded local martingale on [0, τ).)
I =⇒ Xta.s.−→ X∞, Xt
Lp−→ X∞ (MGCT)with Xt = E~x [X∞|Gt ], Gt = Fγ(t)
=⇒ E~x [Xt ] = E~x [X∞].I τ <∞ =⇒ X∞ = f (Bτ ).
I t = 0 =⇒
u(~x) = E~x [X0] = E~x [X∞] = E~x [f (Bτ )].
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Theorem 1 : Proof - Step 2
I Step 2 : u(~x) = E~x [f (Bτ )].
I There exists a time-change γ : R+ → R+ such that
Xt = u(Bγ(t)) is a bounded martingale.
(We already knew it was a continuous bounded local martingale on [0, τ).)
I =⇒ Xta.s.−→ X∞, Xt
Lp−→ X∞ (MGCT)with Xt = E~x [X∞|Gt ], Gt = Fγ(t)
=⇒ E~x [Xt ] = E~x [X∞].I τ <∞ =⇒ X∞ = f (Bτ ).I t = 0 =⇒
u(~x) = E~x [X0] = E~x [X∞] = E~x [f (Bτ )].
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Theorem 2 : Solution Under 3 Conditions
I Theorem 2 : Assume
1. G is an open set,2. ∆u exists and is 0 in G ,3. u = f , a continuous function, on ∂G .
Then for ~x ∈ G ,
u(~x) = E~x [f (Bτ )],
where the notation is interpreted as a Brownian motion whichoriginates from ~x .
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Theorem 2 : Solution Under 3 Conditions
I Theorem 2 : Assume
1. G is an open set,
2. ∆u exists and is 0 in G ,3. u = f , a continuous function, on ∂G .
Then for ~x ∈ G ,
u(~x) = E~x [f (Bτ )],
where the notation is interpreted as a Brownian motion whichoriginates from ~x .
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Theorem 2 : Solution Under 3 Conditions
I Theorem 2 : Assume
1. G is an open set,2. ∆u exists and is 0 in G ,
3. u = f , a continuous function, on ∂G .
Then for ~x ∈ G ,
u(~x) = E~x [f (Bτ )],
where the notation is interpreted as a Brownian motion whichoriginates from ~x .
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Theorem 2 : Solution Under 3 Conditions
I Theorem 2 : Assume
1. G is an open set,2. ∆u exists and is 0 in G ,3. u = f , a continuous function, on ∂G .
Then for ~x ∈ G ,
u(~x) = E~x [f (Bτ )],
where the notation is interpreted as a Brownian motion whichoriginates from ~x .
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Theorem 2 : Solution Under 3 Conditions
I Theorem 2 : Assume
1. G is an open set,2. ∆u exists and is 0 in G ,3. u = f , a continuous function, on ∂G .
Then for ~x ∈ G ,
u(~x) = E~x [f (Bτ )],
where the notation is interpreted as a Brownian motion whichoriginates from ~x .
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Theorem 2 : Solution Under 3 Conditions
I Theorem 2 : Assume
1. G is an open set,2. ∆u exists and is 0 in G ,3. u = f , a continuous function, on ∂G .
Then for ~x ∈ G ,
u(~x) = E~x [f (Bτ )],
where the notation is interpreted as a Brownian motion whichoriginates from ~x .
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Theorem 3 : Vast Generalization
I Theorem 3 : Suppose
L =nX
i=1
bi (x)∂
∂xi+
nXi,j=1
aij (x)∂2
∂xi∂xj,
where aij (x) = aji (x) and bi (x) are continuous functions. L is said to be a
semi-elliptic if a(x) = [aij (x)] is positive semi-definite for all x . Then if
1. G is an open set,2. Lu exists and is 0 in G ,
3. u = f , a continuous function, on ∂G .Then for ~x ∈ G ,
u(~x) = E~x[f (Xτ )1[τ<∞]
],
where Xt is a diffusion satisfying
dXt = b(Xt) dt + σ(Xt) dBt
with b(x) = [bi (x)] and σ(x) such that 12σ(x)σ∗(x) = [aij (x)].
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Theorem 3 : Vast Generalization
I Theorem 3 : Suppose
L =nX
i=1
bi (x)∂
∂xi+
nXi,j=1
aij (x)∂2
∂xi∂xj,
where aij (x) = aji (x) and bi (x) are continuous functions. L is said to be a
semi-elliptic if a(x) = [aij (x)] is positive semi-definite for all x . Then if1. G is an open set,
2. Lu exists and is 0 in G ,
3. u = f , a continuous function, on ∂G .Then for ~x ∈ G ,
u(~x) = E~x[f (Xτ )1[τ<∞]
],
where Xt is a diffusion satisfying
dXt = b(Xt) dt + σ(Xt) dBt
with b(x) = [bi (x)] and σ(x) such that 12σ(x)σ∗(x) = [aij (x)].
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Theorem 3 : Vast Generalization
I Theorem 3 : Suppose
L =nX
i=1
bi (x)∂
∂xi+
nXi,j=1
aij (x)∂2
∂xi∂xj,
where aij (x) = aji (x) and bi (x) are continuous functions. L is said to be a
semi-elliptic if a(x) = [aij (x)] is positive semi-definite for all x . Then if1. G is an open set,2. Lu exists and is 0 in G ,
3. u = f , a continuous function, on ∂G .Then for ~x ∈ G ,
u(~x) = E~x[f (Xτ )1[τ<∞]
],
where Xt is a diffusion satisfying
dXt = b(Xt) dt + σ(Xt) dBt
with b(x) = [bi (x)] and σ(x) such that 12σ(x)σ∗(x) = [aij (x)].
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Theorem 3 : Vast Generalization
I Theorem 3 : Suppose
L =nX
i=1
bi (x)∂
∂xi+
nXi,j=1
aij (x)∂2
∂xi∂xj,
where aij (x) = aji (x) and bi (x) are continuous functions. L is said to be a
semi-elliptic if a(x) = [aij (x)] is positive semi-definite for all x . Then if1. G is an open set,2. Lu exists and is 0 in G ,
3. u = f , a continuous function, on ∂G .
Then for ~x ∈ G ,
u(~x) = E~x[f (Xτ )1[τ<∞]
],
where Xt is a diffusion satisfying
dXt = b(Xt) dt + σ(Xt) dBt
with b(x) = [bi (x)] and σ(x) such that 12σ(x)σ∗(x) = [aij (x)].
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Theorem 3 : Vast Generalization
I Theorem 3 : Suppose
L =nX
i=1
bi (x)∂
∂xi+
nXi,j=1
aij (x)∂2
∂xi∂xj,
where aij (x) = aji (x) and bi (x) are continuous functions. L is said to be a
semi-elliptic if a(x) = [aij (x)] is positive semi-definite for all x . Then if1. G is an open set,2. Lu exists and is 0 in G ,
3. u = f , a continuous function, on ∂G .Then for ~x ∈ G ,
u(~x) = E~x[f (Xτ )1[τ<∞]
],
where Xt is a diffusion satisfying
dXt = b(Xt) dt + σ(Xt) dBt
with b(x) = [bi (x)] and σ(x) such that 12σ(x)σ∗(x) = [aij (x)].
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Theorem 3 : Vast Generalization
I Theorem 3 : Suppose
L =nX
i=1
bi (x)∂
∂xi+
nXi,j=1
aij (x)∂2
∂xi∂xj,
where aij (x) = aji (x) and bi (x) are continuous functions. L is said to be a
semi-elliptic if a(x) = [aij (x)] is positive semi-definite for all x . Then if1. G is an open set,2. Lu exists and is 0 in G ,
3. u = f , a continuous function, on ∂G .Then for ~x ∈ G ,
u(~x) = E~x[f (Xτ )1[τ<∞]
],
where Xt is a diffusion satisfying
dXt = b(Xt) dt + σ(Xt) dBt
with b(x) = [bi (x)] and σ(x) such that 12σ(x)σ∗(x) = [aij (x)].
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
2d Dirichlet [0, 1]× [0, 1] Example
I a = b = 1,
g1(y) = 0 g2(y) = 0
f1(x) = 0 f2(x) = 100 sinπx
x
y
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Figure: Simulated Solution to the Specified Dirichlet Problem, N =100, ∆t = 1/202, h = .0025
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
2d Dirichlet [0, 1]× [0, 1] Example
I a = b = 1,
g1(y) = 0 g2(y) = 0
f1(x) = 0 f2(x) = 100 sinπx
x
y
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Figure: Simulated Solution to the Specified Dirichlet Problem, N =100, ∆t = 1/202, h = .0025
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
2d Dirichlet [0, 1]× [0, 1] Example
x
y
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Figure: Simulated Solution to the Specified Dirichlet Problem with True ContoursDavid Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem
OutlineHistory of the Dirichlet Problem
Analytic SolutionsProbabilistic Solutions
Simulated Solutions
Bibliography
I Asmar, Nakhle H.Partial Differential Equations with Fourier Series and Boundary Value Problems.2nd. Upper Saddle River: Pearson Prentice Hall, 2005.
I Durrett, Richard. Stochastic Calculus : A Practical Introduction. Boca Raton:CRC Press, 1996.
I Evans, Lawrence C. Partial Differential Equations. Providence: AmericanMathematical Society, 1998.
I Feynman, Richard P., Leighton, Robert, and Matthew Sands.The Feynman Lectures on Physics, Vol 2. Reading: Addison Wesley PublishingCompany, Inc., 1964.
I O’Connor, J. J. and Robertson, E. F. “Johann Peter Gustav Lejeune Dirichlet.”Dirichlet Biography. May 2007. University of St. Andrews. 12 Apr 2008〈http://www-groups.dcs.st-and.ac.uk/ history/Biographies/Dirichlet.html〉.
I Oskendal, Bernt.Stochastic Differential Equations : An Introduction with Applications. 5th.New York: Springer, 1998.
I Steele, J. Michael. Stochastic Calculus and Financial Applications. New York:Springer, 2001.
I Thompson, James R.Simulation : A Modeler’s Approach. New York: John Wiley & Sons, Inc., 2000.
David Kahle Stat 650 : Stochastic Differential Equations Probabilistic Solutions to the Dirichlet Problem