• B Rajeev

Articles written in Proceedings – Mathematical Sciences

• 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 ϕ is a tempered distribution, then the solution of the heat equation for the Laplacian, with initial condition ϕ, is given by the convolution of ϕ 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.

• Measure Free Martingales and Martingale Measures

Let $T\subset\mathbb{R}$ be a countable set, not necessarily discrete. Let $f_t,t\in T$, 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 $f_t,t\in T$, is a martingale.

• Differential operators on Hermite Sobolev spaces

In this paper, we compute the Hilbert space adjoint $\partial^{*}$ of the derivative operator $\partial$ on the Hermite Sobolev spaces $\mathcal{S}_{q}$. We use this calculation to give a different proof of the ‘monotonicity inequality’ for a class of differential operators $(L, A)$ for which the inequality was proved in Infin. Dimens. Anal. Quantum Probab. Relat. Top. 2(4) (2009) 515–591. We also prove the monotonicity inequality for $(L, A)$, when these correspond to the Ornstein–Uhlenbeck diffusion.

• # Proceedings – Mathematical Sciences

Current Issue
Volume 129 | Issue 3
June 2019