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Brownian motion constructions

Interpolation (Levy-Ciesielski construction) Cameron-Martin space eigenfunctions White noise Limit of Random walk Hyperreals construction see “A Non-standard Construction of Multi-Dimensional Brownian Motions and Option Pricing””. Heat equation The u(x,t)=E_{x}\left[g(B_{t})\chi\{t\leq \tau_{\Omega}\}\right] solves the heat equation with initial data g in domain \Omega. So conversely by starting with the heat semigroup, we construct the…

Elliptic Explorer model

In the Arxiv article A Left passage explorer model, we explore a possible extension of the Harmonic explorer.

Quantum Heisenberg spin-1/2 chain

Classical hydrodynamics is a remarkably versatile description of the coarse-grained behaviour of many-particle systems once local equilibrium has been established1. The form of the hydrodynamical equations is determined primarily by the conserved quantities present in a system. Some quantum spin chains are known to possess, even in the simplest cases, a greatly expanded set of…

History of math symbols

Forwarding the article from Earliest Uses of Symbols of Operation (tripod.com) ADDITION AND SUBTRACTION SYMBOLS Plus (+) and minus (-). Nicole d’ Oresme (1323-1382) may have used a figure which looks like a plus symbol as an abbreviation for the Latin et (meaning “and”) in Algorismus proportionum, believed to have been written between 1356 and 1361. The symbol appears in a manuscript…

Parabolic as infinite dimensional elliptic

This intuition was used by Perelman. Quote from Tao: “We now come to an interesting (but still mostly heuristic) correspondence principle between elliptic theory and parabolic theory, with the latter being viewed as an infinite-dimensional limit of the former, in a manner somewhat analogous to that of the central limit theorem in probability” References: “Parabolic theory as a…

Contour integration

1. Global nature An important feature to note about Cauchy’s theorem is the global nature of its hypothesis on the analytic functions f. Cauchy’s theorem is the powerful technique of contour shifting, which allows one to compute a contour integral by replacing the contour with a homotopic contour on which the integral is easier to…

Wald’s equation and bullet projectile quality

References: Milton Friedman suggesting problem to Wald and the math in sequential analysis. In order to understand the story, it is necessary to have an idea of a simple statistical problem, and of the standard procedure for dealing with it. The actual problem out of which sequential analysis grew will serve. The Navy has two…

Zero level set of 2D NSE

Conformal invariance (CI) In “Conformal invariance in two-dimensional turbulence” by Bernard etal, it was guessed that numerically the zero level set (ZLS) for the vorticity field has the same properties as an SLE curve. Since NSE and Euler equations are volume preserving, the equations themselves are not conformally invariant. So one has to study the…

Periodic TASEP

KPZ Fixed point  Finding a transition probability formula that is amenable to taking limit as the number of TASEP particles goes to infinity. Prize: 50$ Is there an analogous backwards heat equation problem as in the KPZ fixed point for Z-TASEP? Identifying the KPZ fixed point in the subcritical region Prize: 50$ .    …