Mi 1997 Ph.D. tesis como entrada de blog.
μ \mu μ Núcleos de calor como derivados de radón-nicodimio de medida Weiner
M e s u n a c u r v a n e g a t i v a dim = n m u ˊ l t i p l e d e R i e m a n n i a n c e r r a d o c o n m e ˊ t r i c a g , c o n e x i o ˊ n m e ˊ t r i c a ∇ , y ( n o n e g a t i v o ) O p e r a d o r L a p l a c e − B e l t r a m i Δ M . V a m o s k − t Δ / 2 ( x , y ) r e p r e s e n t a n e l n u ˊ c l e o d e c a l o r e n M M es una curva negativa \dim=n múltiple de Riemannian cerrado con métrica g, conexión métrica \nabla, y (no negativo) Operador Laplace-Beltrami \Delta_M. Vamos k_{-t\Delta/2}(x,y) representan el núcleo de calor en M M es u na c u r v an e g a t i v a dim = nm u ˊ lt i pl e d e R i e mannian cerr a d oco nm e ˊ t r i c a g , co n e x i o ˊ nm e ˊ t r i c a ∇ , y ( n o n e g a t i v o ) Op er a d or L a pl a ce − B e lt r ami Δ M . Vam os k − t Δ/2 ( x , y ) re p rese n t an e l n u ˊ c l eo d ec a l ore n M
Por lo tanto k − t Δ / 2 ( x , x ) = d D M ∗ μ / g d x k_{-t\Delta/2}(x,x) = dDM_*\mu/\sqrt{g}dx k − t Δ/2 ( x , x ) = d D M ∗ μ / g d x μ \mu μ Ω t ( M ) ∣ x \Omega_t(M)\vert_x Ω t ( M ) ∣ x D M DM D M Nota: D M − 1 Ω t ∣ x DM^{-1}\Omega_t\vert_x D M − 1 Ω t ∣ x
Ω t 0 e s e l e s p a c i o d e b u c l e s c o n t r a c t i b l e s c o n t i n u o s e n M \Omega_t^0 es el espacio de bucles contractibles continuos en M Ω t 0 ese l es p a c i o d e b u c l esco n t r a c t ib l esco n t in u ose n M
Ω t [ γ ] e s e l e s p a c i o d e b u c l e s c o n t i n u o s e n M h o m o t o ˊ p i c o a l g e o d e ˊ s i c o c e r r a d o γ . V a m o s γ 0 \Omega_t[\gamma] es el espacio de bucles continuos en M homotópico al geodésico cerrado \gamma. Vamos \gamma_0 Ω t [ γ ] ese l es p a c i o d e b u c l esco n t in u ose n M h o m o t o ˊ p i co a l g eo d e ˊ s i cocerr a d o γ . Vam os γ 0
D M − 1 Ω t 0 [ γ ] e s l a p r e i m a g e n d e l o s b u c l e s c o n t r a c t i b l e s c o n t i n u o s e n M e s c r i t o c o m o c o m p e n s a c i o n e s h o m o t ı ˊ p i c a s a γ ( s ) = D M ( s ℓ ( γ ) t e ⃗ 1 ) , 0 ≤ s ≤ t . P e n s a r c o o r d e n a d a s h o r o c ı ˊ c l i c a s — c a d a f i b r a c o m o l ı ˊ m i t e g e o m e ˊ t r i c o d e e s f e r a s g e o d e ˊ s i c a s p e r i o ˊ d i c a s S γ 0 ( s ) n − 1 ( k ℓ ( γ 0 ) ) , 0 ≤ s ≤ t , k → ∞ , v e c t o r i z a d o e n e l p a q u e t e n o r m a l γ 0 . N u e s t r a s r e s t r i c c i o n e s d e c u r v a t u r a i m p l i c a n C o o r d e n a d a s H o r o c ı ˊ c l i c a s p a r a c a d a γ 0 e x i s t a n c o m o s u a v e s , D M − m a p a d e c o o r d e n a d a s c o m p a t i b l e p a r a Ω t 0 [ γ ] DM^{-1}\Omega_t^0[\gamma] es la preimagen de los bucles contractibles continuos en M escrito como compensaciones homotípicas a \gamma(s) = DM(\frac{s\ell(\gamma)}{t}\vec{e}^1), 0\leq s \leq t. Pensar coordenadas horocíclicas — cada fibra como límite geométrico de esferas geodésicas periódicas S_{\gamma_0(s)}^{n-1}(k\ell(\gamma_0)), 0\leq s \leq t, k\rightarrow\infty, vectorizado en el paquete normal \gamma_0. Nuestras restricciones de curvatura implican Coordenadas Horocíclicas para cada \gamma_0 existan como suaves, DM- mapa de coordenadas compatible para \Omega_t^0[\gamma] D M − 1 Ω t 0 [ γ ] es l a p re ima g e n d e l os b u c l esco n t r a c t ib l esco n t in u ose n M escr i t oco m oco m p e n s a c i o n es h o m o t ı ˊ p i c a s aγ ( s ) = D M ( t s ℓ ( γ ) e 1 ) , 0 ≤ s ≤ t . P e n s a rcoor d e na d a s h oroc ı ˊ c l i c a s — c a d a f ib r a co m o l ı ˊ mi t e g eo m e ˊ t r i co d ees f er a s g eo d e ˊ s i c a s p er i o ˊ d i c a s S γ 0 ( s ) n − 1 ( k ℓ ( γ 0 )) , 0 ≤ s ≤ t , k → ∞ , v ec t or i z a d oe n e lp a q u e t e n or ma l γ 0 . N u es t r a sres t r i cc i o n es d ec u r v a t u r aim pl i c an C oor d e na d a sHoroc ı ˊ c l i c a s p a r a c a d a γ 0 e x i s t an co m os u a v es , D M − ma p a d ecoor d e na d a sco m p a t ib l e p a r a Ω t 0 [ γ ]
Ahora x ⃗ ( τ ) + ℓ ( γ ) e ⃗ 1 \vec{x}(\tau)+\ell(\gamma)\vec{e}^1 x ( τ ) + ℓ ( γ ) e 1 desarrollado de la “compensación” geodésica arrugada homotópica a γ : D M ( x ⃗ ( τ ) + s ℓ ( γ ) t e ⃗ 1 ) , 0 ≤ s ≤ t \gamma: DM(\vec{x}(\tau) + \frac{s\ell(\gamma)}{t}\vec{e}^1), 0\leq s \leq t γ : D M ( x ( τ ) + t s ℓ ( γ ) e 1 ) , 0 ≤ s ≤ t ℓ ( γ 0 ) \ell(\gamma_0) ℓ ( γ 0 ) D M ( x ⃗ ( τ ) ) DM(\vec{x}(\tau)) D M ( x ( τ )) t t t J = lim s ↑ t D M ′ ∣ D M ( x ⃗ ( τ ) + ℓ ( γ ) s t e ⃗ 1 ) J=\lim_{s\uparrow t}DM^\prime\vert_{DM(\vec{x}(\tau) + \frac{\ell(\gamma)s}{t}\vec{e}^1)} J = lim s ↑ t D M ′ ∣ D M ( x ( τ ) + t ℓ ( γ ) s e 1 ) T D M ( x ⃗ ( τ ) ) M T_{DM(\vec{x}(\tau))}M T D M ( x ( τ )) M J D M ( x ⃗ ( τ ) + ℓ ( γ ) e ⃗ 1 ) J_{DM(\vec{x}(\tau)+\ell(\gamma)\vec{e}^1)} J D M ( x ( τ ) + ℓ ( γ ) e 1 ) Campos Jacobi , ya que D M DM D M es el mapa exponencial (iterado) a lo largo de cualquier serie de líneas rectas conectadas en R n \Reals^n R n 1 / 2 ∫ 0 t < d X | d X > s 1/2 \int_0^t \bra{dX}\ket{dX}_s 1/2 ∫ 0 t ⟨ d X ∣ d X ⟩ s
X t = X 0 + ∫ 0 t J X t d B t \begin{aligned}
X_t &= X_0 + \int_0^t \sqrt{J}_{X_t} dB_t \\
\end{aligned} X t = X 0 + ∫ 0 t J X t d B t
Z − Δ / 2 ( t ) : = ∫ M k − t Δ / 2 ( x , x ) g d x = ∑ j = 0 ∞ e − λ i t / 2 Z_{-\Delta/2}(t) := \int_M k_{-t\Delta/2}(x,x) \sqrt{g}dx = \sum_{j=0}^\infty e^{-\lambda_i t/2} Z − Δ/2 ( t ) := ∫ M k − t Δ/2 ( x , x ) g d x = ∑ j = 0 ∞ e − λ i t /2
Finalmente definamos lo siguiente de sus derivados radón-nicodimicos:
D M ∗ μ ( Ω t ) : = ∫ M D M ∗ μ ( Ω t ∣ x g d x ) D M ∗ μ ( Ω t 0 ) : = ∫ M D M ∗ μ ( Ω t 0 ∣ x g d x ) D M ∗ μ ( Ω t [ γ ] ) : = ∫ M D M ∗ μ ( Ω t [ γ ] ∣ x g d x ) \begin{aligned}
DM_*\mu(\Omega_t) &:= \int_M DM_*\mu(\Omega_t\vert_x \sqrt{g}dx)\\
DM_*\mu(\Omega^0_t) &:= \int_M DM_*\mu(\Omega^0_t\vert_x \sqrt{g}dx)\\
DM_*\mu(\Omega_t[\gamma]) &:= \int_M DM_*\mu(\Omega_t[\gamma]\vert_x \sqrt{g}dx) \\
\end{aligned} D M ∗ μ ( Ω t ) D M ∗ μ ( Ω t 0 ) D M ∗ μ ( Ω t [ γ ]) := ∫ M D M ∗ μ ( Ω t ∣ x g d x ) := ∫ M D M ∗ μ ( Ω t 0 ∣ x g d x ) := ∫ M D M ∗ μ ( Ω t [ γ ] ∣ x g d x )
Z − Δ / 2 ( t ) = D M ∗ μ ( Ω t ) = D M ∗ μ ( Ω t 0 ) + ∑ { γ } D M ∗ μ ( Ω t [ γ ] ) D M ∗ μ ( Ω t 0 ) ≈ t → 0 ( 2 π t ) − n / 2 ( v o l ( M ) + t / 6 ∫ M K ( x ) g d x + O ( t 2 ) ) by McKean-Singer D M ∗ μ ( Ω t [ γ ] ) = e − ℓ ( γ ) 2 / 2 t ∫ M D M ∗ μ ( e t < J B B t | B t > Ω t 0 [ γ ] ∣ x g d x ) by Cameron-Martin = e − ℓ ( γ ) 2 / 2 t ∫ T γ 0 M E ( e t J B ∣ Ω t 0 [ γ ] ∣ x ( τ ) ) d x 1 ( τ ) … d x n ( τ ) d τ d D M ∗ μ ( e − ℓ ( γ ) x 1 ( t ) Ω t 0 [ γ ] ) d x 1 ( τ ) … d x n ( τ ) d τ ∣ y ( τ ) ⃗ ≈ t → 0 e − < ∣ I − J D M ( x ⃗ ( τ ) , y ⃗ ( τ ) ) x ⃗ ( τ ) | x ⃗ ( τ ) > / 2 t ( 2 π t ) ( n + 1 ) / 2 ( 1 + O ( t 2 ) ) semi-classical limit Horocyclic coordinates : z ( τ ) − x ( τ ) = x + ℓ ( γ ) e ⃗ 1 ⟹ ∫ M / S 1 ⊕ S 1 k t ( x , z ) d x = lim j → ∞ e − ℓ ( γ ) 2 / 2 t 2 π t E ( e < J X t j x ⃗ | x ⃗ > ) = lim j → ∞ e − ℓ ( γ ) 2 / 2 t 2 π t ∫ M j / S 1 ⊕ S 1 1 2 π t j n det ∣ I − J X j ∣ e − ℓ ( X j ) 2 / 2 t X j \begin{aligned}
Z_{-\Delta/2}(t) = DM_*\mu(\Omega_t) &= DM_*\mu(\Omega^0_t) + \sum_{\set{\gamma}} DM_*\mu(\Omega_t[\gamma]) \\
DM_*\mu(\Omega_t^0) &\approx_{t\rightarrow 0} (2\pi t)^{-n/2}(vol(M) + t/6\int_M K(x)\sqrt{g} dx + O(t^2))\space \small\text{by McKean-Singer}\\
DM_*\mu(\Omega_t[\gamma]) &= e^{-\ell(\gamma)^2/2t}\int_M DM_*\mu(e^{\bra{J_BB_t}\ket{B_t}} _t \Omega_t^0[\gamma]\vert_x\sqrt{g}dx)\space\small \text{ by Cameron-Martin}\\
&= e^{-\ell(\gamma)^2/2t}\int_{T_{\gamma_0}M} E(e^{J_B}_{t} | \Omega_t^0[\gamma]\vert_{x(\tau)})dx^1(\tau)\dots dx^n(\tau) d\tau\\
\frac{dDM_*\mu(e^{-\ell(\gamma)x^1(t)}\Omega^0_t[\gamma])}{dx^1(\tau)\dots dx^n(\tau)d\tau}\vert_{\vec{y(\tau)}}&\approx_{t\rightarrow 0} \frac{e^{-\bra{|I-J_{DM(\vec{x}(\tau),\vec{y}(\tau))}\vec{x}(\tau)}\ket{\vec{x}(\tau)}/2t}}{(2 \pi t)^{(n+1)/2}}(1+O(t^2))\small \text{ semi-classical limit}\\
\text{Horocyclic coordinates}: z(\tau) - x(\tau) &= x + \ell(\gamma)\vec{e}^1\implies\\
\int_{M/S^1\oplus S^1}k_t(x,z) dx &=\lim_{j\rightarrow\infty}\frac{e^{-\ell(\gamma)^2/2t}}{\sqrt{2\pi t}}E(e^{\bra{J_{X^j_t}\vec{x}}\ket{\vec{x}}})\\
&=\lim_{j\rightarrow\infty}\frac{e^{-\ell(\gamma)^2/2t}}{\sqrt{2\pi t}}\int_{M^j/S^1\oplus S^1}\frac{1}{\sqrt{2\pi t}^{jn}\det|I-J_{X^j}|}e^{-\ell(X^j)^2/2t}X^{j}\\
\end{aligned} Z − Δ/2 ( t ) = D M ∗ μ ( Ω t ) D M ∗ μ ( Ω t 0 ) D M ∗ μ ( Ω t [ γ ]) d x 1 ( τ ) … d x n ( τ ) d τ d D M ∗ μ ( e − ℓ ( γ ) x 1 ( t ) Ω t 0 [ γ ]) ∣ y ( τ ) Horocyclic coordinates : z ( τ ) − x ( τ ) ∫ M / S 1 ⊕ S 1 k t ( x , z ) d x = D M ∗ μ ( Ω t 0 ) + { γ } ∑ D M ∗ μ ( Ω t [ γ ]) ≈ t → 0 ( 2 π t ) − n /2 ( v o l ( M ) + t /6 ∫ M K ( x ) g d x + O ( t 2 )) by McKean-Singer = e − ℓ ( γ ) 2 /2 t ∫ M D M ∗ μ ( e t ⟨ J B B t ∣ B t ⟩ Ω t 0 [ γ ] ∣ x g d x ) by Cameron-Martin = e − ℓ ( γ ) 2 /2 t ∫ T γ 0 M E ( e t J B ∣ Ω t 0 [ γ ] ∣ x ( τ ) ) d x 1 ( τ ) … d x n ( τ ) d τ ≈ t → 0 ( 2 π t ) ( n + 1 ) /2 e − ⟨ ∣ I − J D M ( x ( τ ) , y ( τ )) x ( τ ) ∣ x ( τ ) ⟩ /2 t ( 1 + O ( t 2 )) semi-classical limit = x + ℓ ( γ ) e 1 ⟹ = j → ∞ lim 2 π t e − ℓ ( γ ) 2 /2 t E ( e ⟨ J X t j x x ⟩ ) = j → ∞ lim 2 π t e − ℓ ( γ ) 2 /2 t ∫ M j / S 1 ⊕ S 1 2 π t jn det ∣ I − J X j ∣ 1 e − ℓ ( X j ) 2 /2 t X j
En el dim = 2 \dim = 2 dim = 2 − κ 2 -\kappa^2 − κ 2
J x ⃗ , y ⃗ d R B = ( e κ d ( x ⃗ , y ⃗ ) / 2 0 0 e − κ d ( x ⃗ , y ⃗ ) / 2 ) ⟹ < J d R B | J d R B > = e κ ℓ ( B ) d R B 1 2 − e − κ ℓ ( B ) d R B 2 2 ∫ 0 t < J d B | J d B > = e κ ℓ ( γ ) − e − κ ℓ ( γ ) det I − J γ = ( e κ ℓ ( γ ) / 2 − e − κ ℓ ( γ ) / 2 ) 2 \begin{aligned}
\sqrt{J_{\vec{x}, \vec{y}}}dRB&=
\begin{pmatrix}
e^{\kappa d(\vec{x},\vec{y})/2} && 0\\
0 && e^{-\kappa d(\vec{x},\vec{y})/2}\\
\end{pmatrix}
\implies&\\
\bra{\sqrt{J}dRB}\ket{\sqrt{J}dRB} &= e^{\kappa \ell(B)}dRB_1^2 - e^{-\kappa \ell(B)}dRB_2^2\\
\int_0^t \bra{\sqrt{ J}dB}\ket{\sqrt{ J}dB} &= e^{\kappa\ell(\gamma)} - e^{-\kappa\ell(\gamma)}\\
\det I-J_{\gamma} &= (e^{\kappa\ell(\gamma)/2}- e^{-\kappa\ell(\gamma)/2})^2
\end{aligned} J x , y d RB ⟨ J d RB J d RB ⟩ ∫ 0 t ⟨ J d B J d B ⟩ det I − J γ = ( e κ d ( x , y ) /2 0 0 e − κ d ( x , y ) /2 ) ⟹ = e κ ℓ ( B ) d R B 1 2 − e − κ ℓ ( B ) d R B 2 2 = e κ ℓ ( γ ) − e − κ ℓ ( γ ) = ( e κ ℓ ( γ ) /2 − e − κ ℓ ( γ ) /2 ) 2
constante sobre ( x ⃗ , τ ) (\vec{x},\tau) ( x , τ ) ≈ t → 0 \approx_{t\rightarrow 0} ≈ t → 0
D M ∗ μ ( Ω t [ γ ] ) = e − ℓ ( γ ) 2 / 2 t ℓ ( γ 0 ) 2 π t ( e κ ℓ ( γ ) / 2 − e − κ ℓ ( γ ) / 2 ) γ ( t ) = γ 0 ( k t ) ⟹ = e − k 2 ℓ ( γ 0 ) 2 / 2 t ℓ ( γ 0 ) 2 2 π t sinh k κ ℓ ( γ 0 ) / 2 \begin{aligned}
DM_*\mu(\Omega_t[\gamma]) &= \frac{e^{-\ell(\gamma)^2/2t}\ell(\gamma_0)}{\sqrt{2 \pi t}(e^{\kappa\ell(\gamma)/2} -e^{-\kappa\ell(\gamma)/2})}\\
\gamma(t) = \gamma_0(kt)\implies \\
&=\frac{e^{-k^2\ell(\gamma_0)^2/2t}\ell(\gamma_0)}{2\sqrt{2\pi t}\sinh k\kappa\ell(\gamma_0)/2}\\
\end{aligned} D M ∗ μ ( Ω t [ γ ]) γ ( t ) = γ 0 ( k t ) ⟹ = 2 π t ( e κ ℓ ( γ ) /2 − e − κ ℓ ( γ ) /2 ) e − ℓ ( γ ) 2 /2 t ℓ ( γ 0 ) = 2 2 π t sinh kκ ℓ ( γ 0 ) /2 e − k 2 ℓ ( γ 0 ) 2 /2 t ℓ ( γ 0 )
En el dim = 3 \dim=3 dim = 3 ( z , z ˉ ) (z,\bar{z}) ( z , z ˉ )
J D M ( x ⃗ + ( τ + ℓ ( γ ) ) e ⃗ 1 ) = ( e κ ℓ ( γ ) 0 0 0 e − κ ℓ ( γ ) + i θ ( γ ) 0 0 0 e − κ ℓ ( γ ) − i θ ( γ ) ) ⟹ det I − ⊥ γ 0 k = ∣ 1 − e − k ( κ ℓ ( γ 0 ) − i θ ( γ 0 ) ) ∣ 2 \begin{aligned}
J_{DM(\vec{x}+(\tau+\ell(\gamma))\vec{e}^1)} &=
\begin{pmatrix}
e^{\kappa\ell(\gamma)} && 0 && 0\\
0 && e^{-\kappa\ell(\gamma)+i\theta(\gamma)} && 0 \\
0 && 0 && e^{-\kappa\ell(\gamma)-i\theta(\gamma)} \\
\end{pmatrix}\\
\implies& \\
\det I-{\perp_{\gamma_0}}^k &= |1-e^{-k(\kappa\ell(\gamma_0)-i\theta(\gamma_0))}|^2
\end{aligned} J D M ( x + ( τ + ℓ ( γ )) e 1 ) ⟹ det I − ⊥ γ 0 k = e κ ℓ ( γ ) 0 0 0 e − κ ℓ ( γ ) + i θ ( γ ) 0 0 0 e − κ ℓ ( γ ) − i θ ( γ ) = ∣1 − e − k ( κ ℓ ( γ 0 ) − i θ ( γ 0 )) ∣ 2
y desde z = x 2 + i x 3 ⟹ d z ˉ ∧ d z = ( d x 2 − i d x 3 ) ∧ ( d x 2 + i d x 3 ) = 2 i d x 2 ∧ d x 3 z=x^2+ix^3 \implies d\bar{z}\wedge dz= (dx^2-idx^3)\wedge(dx^2+idx^3) = 2idx^2\wedge dx^3 z = x 2 + i x 3 ⟹ d z ˉ ∧ d z = ( d x 2 − i d x 3 ) ∧ ( d x 2 + i d x 3 ) = 2 i d x 2 ∧ d x 3
κ = 1 ⟹ D M ∗ μ ( Ω t [ γ ] ) = e − k 2 ℓ ( γ 0 ) 2 / 2 t ℓ ( γ 0 ) 2 2 π t ( 1 − e − k ℓ ( γ 0 ) ) ∣ e k ℓ ( γ 0 ) / 2 − e − k ( ℓ ( γ 0 ) / 2 − i θ ( γ 0 ) ) ∣ \begin{aligned}
\kappa &= 1 \implies \\
DM_*\mu(\Omega_t[\gamma])
&=\frac{e^{-k^2\ell(\gamma_0)^2/2t}\ell(\gamma_0)}{2\sqrt{2\pi t (1-e^{-k\ell(\gamma_0)})}|e^{k\ell(\gamma_0)/2}-e^{-k(\ell(\gamma_0)/2-i\theta(\gamma_0))}|}\\
\end{aligned} κ D M ∗ μ ( Ω t [ γ ]) = 1 ⟹ = 2 2 π t ( 1 − e − k ℓ ( γ 0 ) ) ∣ e k ℓ ( γ 0 ) /2 − e − k ( ℓ ( γ 0 ) /2 − i θ ( γ 0 )) ∣ e − k 2 ℓ ( γ 0 ) 2 /2 t ℓ ( γ 0 )
Joe Schaefer
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