updated stochastic calculus

Mathematics/5th/Advanced_dynamical_systems/Advanced_dynamical_systems.tex

@@ -93,10 +93,10 @@
     $$
       \det(\vf{A}^k-\vf{I})=({\lambda_+}^k-1)({\lambda_-}^k-1)\ne 0
     $$
-    and so the equation $\vf{A}^k\vf{x}=\vf{x}+\vf{n}$ has a unique (rational) solution.
+    and so the equation $\vf{A}^k\vf{x}=\vf{x}+\vf{n}$ has a unique (rational) solution. Now let $(\frac{p_1}{q_1},\frac{p_2}{q_2})\in \quot{\QQ^2}{\ZZ^2}$ and $N\geq 1$ left to be chosen. We define the set $Q_N:=\frac{\ZZ^2}{N} \mod{\ZZ^2}$, which is a subset finite set of $T^2$. But observe that $Q_N$ is invariant under $\vf{\tilde{A}}$, and thus, all of its points are periodic. For the above rational numbers, just choose $N=q_1q_2$.
   \end{proof}
   \begin{remark}
-    The \emph{hyperbolicity} comes from the fact that there is one eigenvector with eigenvalue greater than $1$ and another with eigenvalue less than $1$.
+    The \emph{hyperbolicity} comes from the fact that there is one eigenvector with eigenvalue greater than $1$ and another with eigenvalue less than $1$, both eigenvalues being positive.
   \end{remark}
   \begin{theorem}
     The iterates of $\vf{\tilde{A}}$ smear every domain $F\subseteq T^2$ uniformly over $T^2$, that is, for every domain $G\subseteq T^2$, we have that the following limit exists:
@@ -145,7 +145,19 @@
     $$
   \end{definition}
   \begin{theorem}[Structal stability]
-    Let $\vf{B}$ be a diffeomorphism on $T^2$ $\mathcal{C}^1$-close to $\vf{\tilde{A}}$. Then, $\vf{B}$ is conjugate to $\vf{\tilde{A}}$.
+    Let $\vf{B}$ be a diffeomorphism on $T^2$ $\mathcal{C}^1$-close to $\vf{\tilde{A}}$. Then, $\vf{B}$ is $\mathcal{C}^0$-conjugate to $\vf{\tilde{A}}$.
   \end{theorem}
+  \begin{definition}
+    A dynamical system $f : X\rightarrow X$ has \emph{sensitive dependence on initial conditions} on $X$ if $\exists\varepsilon >0$ such that for each $x\in X$ and any neighborhood $N_x$ of $x$, exists $y \in N_x$ and $n \geq  0$ such that $d(f^n(x),f^n(y)) > \varepsilon$.
+  \end{definition}
+  \begin{definition}
+    Let $U\subseteq \RR^n$, $\vf{f}:U\to U$ be a dynamical system and $\vf{x}\in U$ and $\vf{v}\in\RR^n$. We define the \emph{Lyapunov exponent} as:
+    $$
+      \chi(x,\vf{v}):=\limsup_{n\to\infty}\frac{1}{n}\log\norm{\vf{D}(\vf{f}^n)(x)\vf{v}}
+    $$
+  \end{definition}
+  \begin{remark}
+    The Lyapunov exponent measures the exponential growth rate of tangent vectors along orbits.
+  \end{remark}
 \end{multicols}
 \end{document}

Mathematics/5th/Stochastic_calculus/Stochastic_calculus.tex

@@ -797,31 +797,74 @@
   \subsubsection{Introduction}
   \begin{definition}
     Let $X={(X_t)}_{t\geq 0}$ be a stochastic process and $b:\RR_{\geq 0}\times\RR\to \RR$ and $\sigma:\RR_{\geq 0}\times \RR$ be deterministic functions called \emph{drift} and \emph{diffusion}, respectively. A \emph{stochastic differential equation} (\emph{SDE}) is an equation of the form:
-    $$
+    \begin{equation}\label{SC:SDE_eq}
       \dd{X_t}=b(t,X_t)\dd{t}+\sigma(t,X_t)\dd{B_t}
-    $$
+    \end{equation}
   \end{definition}
   \begin{definition}
-    Consider the following SDE:
-    $$
-      \dd{X_t}=b(t,X_t)\dd{t}+\sigma(t,X_t)\dd{B_t}
-    $$
-    We say that a progressive process $X={(X_t)}_{t\geq 0}$ defined on $(\Omega, \mathcal{F}, {(\mathcal{F}_t)}_{t\geq 0}, \Prob)$ is a \emph{solution of the SDE} if ${(b(t,X_t))}_{t\geq 0}\in\MM^1_{\text{loc}}$ and ${(\sigma(t,X_t))}_{t\geq 0}\in\MM^2_{\text{loc}}$ and $\forall t\geq 0$:
+    Consider the SDE of \mcref{SC:SDE_eq}. We say that a progressive process $X={(X_t)}_{t\geq 0}$ defined on $(\Omega, \mathcal{F}, {(\mathcal{F}_t)}_{t\geq 0}, \Prob)$ is a \emph{solution of the SDE} if ${(b(t,X_t))}_{t\geq 0}\in\MM^1_{\text{loc}}$ and ${(\sigma(t,X_t))}_{t\geq 0}\in\MM^2_{\text{loc}}$ and $\forall t\geq 0$:
     $$
       X_t=X_0+\int_0^t b(s,X_s)\dd{s}+\int_0^t \sigma(s,X_s)\dd{B_s}
     $$
   \end{definition}
   \subsubsection{Existence and uniqueness of solutions}
-  \begin{theorem}[Existence and uniqueness]
-    Let $b:\RR_{\geq 0}\times\RR\to \RR$ be a measurable function satisfying:
+  \begin{lemma}[Gronwall's lemma]
+    Let ${(x_t)}_{t\in[0,T]}$ be a non-negative function in $L^1([0,T])$ satisfying that $\forall t\in[0,T]$:
+    $$
+      x_t\leq \alpha+\beta\int_0^t x_s\dd{s}
+    $$
+    for some constants $\alpha,\beta\geq 0$. Then, $x_t\leq \alpha\exp{\beta t}$ for all $t\in[0,T]$.
+  \end{lemma}
+  \begin{theorem}[Existence and uniqueness of solutions of SDEs]\label{SC:existence_uniqueness_SDE}
+    Let $b,\sigma:\RR_{\geq 0}\times\RR\to \RR$ be a measurable function satisfying:
     \begin{itemize}
-      \item Uniform spatial Lipschitz continuity: $\exists C>0$ such that $\forall t\geq 0$ and $\forall x,y\in\RR$ we have: $$\abs{b(t,x)-b(t,y)}\leq C\abs{x-y}$$
-      \item Local integrability in time: $\forall t \geq 0$ we have: $$\int_0^t \abs{b(s,0)}\dd{s}<\infty$$
+      \item Uniform spatial Lipschitz continuity: $\exists C>0$ such that $\forall t\geq 0$ and $\forall x,y\in\RR$ we have:
+            \begin{align*}
+              \abs{b(t,x)-b(t,y)}           & \leq C\abs{x-y} \\
+              \abs{\sigma(t,x)-\sigma(t,y)} & \leq C\abs{x-y}
+            \end{align*}
+      \item Local square-integrability in time: $\forall t \geq 0$ we have: $$\int_0^t \abs{b(s,0)}^2\dd{s}<\infty
+              \qquad \int_0^t \abs{\sigma(s,0)}^2\dd{s}<\infty
+            $$
     \end{itemize}
-    Then, for each $z\in\RR$, there exists a unique measurable function $x={(x_t)}_{t\geq 0}$ such that $\forall t\geq 0$:
+    Then, for each initial condition $\zeta\in L^(\Omega,\mathcal{F}_0,\Prob)$, there exists a unique (up to indistinguishability) solution $X={(X_t)}_{t\geq 0}$ to the SDE of \mcref{SC:SDE_eq} with $X_0=\zeta$. Moreover, $X\in \MM^2$.
+  \end{theorem}
+  \begin{remark}
+    In the proof of the above theorem, which we omit here, it can be shown that
+    \begin{equation}\label{SC:expression_x_psi}
+      X_t=\Psi_t\left(\zeta,{(B_s)}_{s\in[0,t]}\right)
+    \end{equation}
+    for some measurable function $\Psi_t:\RR\times C([0,t],\RR)\to \RR$.
+  \end{remark}
+  \begin{remark}
+    The following SDE was proposed by Paul Langevin in 1908 to describe the random motion of a small particle in a fluid, due to collisions with the surrounding molecules:
+    $$
+      \dd{X_t}=-b X_t\dd{t}+\sigma\dd{B_t}
+    $$
+    with $b,\sigma>0$. \mnameref{SC:existence_uniqueness_SDE} implies that the solution is unique and that given $\zeta \in L^2(\Omega,\mathcal{F}_0,\Prob)$, the solution is given by:
+    $$
+      X_t= \zeta \exp{-bt}+\sigma\int_0^t \exp{-b(t-s)}\dd{B_s}
+    $$
+    Note that the long-term behavior of $X_t$ has law of $N(0,\frac{\sigma^2}{2b})$, independently of the initial condition $\zeta$.
+  \end{remark}
+  \subsubsection{Markov property for diffusions}
+  \begin{definition}
+    Let $b.\sigma:\RR\to\RR$ be two Lipschitz functions and consider the following \emph{homogeneous SDE}:
+    \begin{equation}\label{SC:homogeneous_SDE}
+      \begin{cases}
+        \dd{X_t}=b(X_t)\dd{t}+\sigma(X_t)\dd{B_t} \\
+        X_0=\zeta\in L^2(\Omega,\mathcal{F}_0,\Prob)
+      \end{cases}
+    \end{equation}
+    These kinds of problems are called \emph{diffusions}.
+  \end{definition}
+  \begin{theorem}[Invariance under time shift]
+    Let $X={(X_t)}_{t\geq 0}$ be a solution to the SDE of \mcref{SC:homogeneous_SDE} and assume we write $X_t$ as in \mcref{SC:expression_x_psi}. Then, for any $s,t\geq 0$ we have:
     $$
-      x_t=z+\int_0^t b(s,x_s)\dd{s}
+      X_{t+s}=\Psi_{t}(X_s,{(B_{u+s}-B_s)}_{u\in[0,t]})
     $$
   \end{theorem}
+  \subsubsection{Generator of a diffusion}
+  \subsubsection{Connection with PDEs}
 \end{multicols}
 \end{document}

Physics/Advanced/Fluid_mechanics/Fluid_mechanics.tex

@@ -527,12 +527,14 @@
   \begin{remark}
     This equation together with the continuity equation and energy equation completely describe the flow in a compressible viscous fluid. In the case of an incompressible homogeneous fluid with $\rho=\rho_0=\const$,  the
     complete set of equations becomes the \emph{Navier-Stokes equations for incompressible flow}:
-    $$
-      \begin{cases}
-        \displaystyle\matdv{\vf{u}}{t}=-\grad p' + \nu\laplacian\vf{u} \\
-        \displaystyle\div\vf{u}=0
-      \end{cases}
-    $$
+    \begin{important}
+      $$
+        \begin{cases}
+          \displaystyle\matdv{\vf{u}}{t}=-\grad p' + \nu\laplacian\vf{u} \\
+          \displaystyle\div\vf{u}=0
+        \end{cases}
+      $$
+    \end{important}
     where $p'=p/\rho_0$ and $\nu=\mu/\rho_0$ is the \emph{kinematic viscosity}. To this we should add boundary condition, which for an ideal fluid we use $\vf{u}\cdot \vf{n}=0$ and if the solid wall that bounds the fluid is stationary, we use $\vf{u}=\vf{0}$ on the walls.
   \end{remark}
   \subsubsection{Reynolds number}