Research Diary October2018

Sunday-October 3-2018

1. Reeb graph, Hamiltonian. It was shown that the non-Markovian processes $\Gamma(X_t^{x,\varepsilon})$ converge in distribution, as $\varepsilon\to 0$ to a disffusion on the graph. Reeb: \[\Gamma: \mathcal{T}\to G, \quad \Gamma(x)=(i,|H(x)|)\;\text{if }x\in \bar U_i. \]

2. (Pertubed test function) To prove tightness of a sequence of processes ${x^{\varepsilon}(\cdot)}$, we define an operator ${\hat A}{}^{\varepsilon}$ on $x^{\varepsilon}$ which has properties similar to those of an infinitesimal operator of a Markov process. If $A$ is the operator for the limit process $x(\cdot)$, we can then compare ${\hat A}{}^{\varepsilon}f^{\varepsilon}(\cdot)$ with $Af(x^{\varepsilon}(\cdot))$, where $f^{\varepsilon}(\cdot)$ is an appropriate small perturbation of $f(\cdot)$. Under some suitable conditions, if ${\hat A}{}^{\varepsilon}f^{\varepsilon}(\cdot)-Af(x^{\varepsilon}(\cdot))\to 0$ for all smooth $f$ as $\varepsilon\to 0$, then $x^{\varepsilon}(\cdot)$ will converge (weakly) to $x(\cdot)$.

Wednesday-October 10-2018

1. Two ways to treat these processes:

  • Solutions to SDE.
  • Solutions to martingale problem.

It is often easier to show that a process that is the limit of s sequence of processes satisfies a martingale problem than it is to show that it satisfies an SDE.

2. The process $\langle x\rangle$ is the quadratic variation of $x(\cdot)$ if it is the unique continuous nondecreasing process adapted to $\mathcal{F}_t$ and that $x^2(\cdot)-\langle x\rangle$ is an $\mathcal{F}_t$-martingale. Precisely, \[\sum[x(t^n_{i+1})-x(t^n_i)]^2 \to \langle x\rangle(t)\;\;\text{in probability.} \] For two processes $x_1(\cdot)$ and $x_2(\cdot)$, $\langle x_1,x_2\rangle(t)$ is called the quadratic covariation such that \[x_1(\cdot)x_2(\cdot)-\langle x_1,x_2\rangle (\cdot)\;\;\text{is a martingale.} \]

3. Stochastic integral $\int_{0}^t\phi(u)dw(u)$ for $\phi$ such that

Monday-October 15-2018

1. Regularity structures/paracontrolled gets you the convergence. What we want to do is qualitative measure of the convergence (fluctuation etc.)

2. In higher dimension of heat equation, we would rather get Gaussian fluctuation rather than Tracy Widom fluctuation.

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