Probabilistic representations of solutions to the heat equation

In this paper we provide a new (probabilistic) proof of a classical result in partial differential equations, viz. if $\phi$ is a tempered distribution, then the solution of the heat equation for the Laplacian, with initial condition $\phi$, is given by the convolution of $\phi$ with the heat kernel (Gaussian density). Our results also extend the probabilistic representation of solutions of the heat equation to initial conditions that are arbitrary tempered distributions.Comment: 12 page

Probabilistic Representations of Solutions of the Forward Equations

In this paper we prove a stochastic representation for solutions of the evolution equation $\partial_t \psi_t = {1/2}L^*\psi_t$ where $L^*$ is the formal adjoint of an elliptic second order differential operator with smooth coefficients corresponding to the infinitesimal generator of a finite dimensional diffusion $(X_t).$ Given $\psi_0 = \psi$, a distribution with compact support, this representation has the form $\psi_t = E(Y_t(\psi))$ where the process $(Y_t(\psi))$ is the solution of a stochastic partial differential equation connected with the stochastic differential equation for $(X_t)$ via Ito's formula.Comment: 29 page

Fishery Commodity Chain Trap vis-à-vis Global Quality Standards: An Analysis of the Kerala Marine Fishery

The article examines the commodity chain trap of marine fishery in Kerala, at both material and value terms, and its ramifications in the globalised fishery chains. The marketing chains both material and value, are very complex in nature since they involve many types of markets and large number of intermediaries and participants. The article also scrutinizes the sensitivity of consumers’ and country’s responses in terms of dietary and hygienic standards relating to seafood trade. In addition, it discusses the devastating effect about the recent stipulations like the US Bio-Terrorism Act and Shrimp anti-dumping duty on the Kerala fishery products. Keywords: commodity chain trap, marketing chain, marine fishery, quality assurance standards, sea food trade, material chain, price spread, value chain, bio-terrorism, shrimp anti-dumping

Measure free martingales and martingale measures

Let T ⊂ R be a countable set, not necessarily discrete. Let ft, t ∈ T, be a family of real-valued functions defined on a set Ω. We discuss conditions which imply that there is a probability measure on Ω under which the family ft, t ∈ T, is a martingale

Vegetables for building molecular models

An exciting classroom activity involving the construction of molecular models and building hybrid orbitals is presented in this paper. Basic molecular geometries were built by connecting different types of vegetables using bamboo skewers. Vegetable of different shapes were used to describe the hybridization of atomic orbitals. This exercise is suitable for high school and undergraduate chemistry students. The activity helps students to apply Valence Shell Electron Pair Repulsion (VSEPR) theory to envisage the basic shapes of molecules and to conceptualize the hybridization of the atomic orbital