• B Rajeev

      Articles written in Proceedings – Mathematical Sciences

    • Probabilistic representations of solutions to the heat equation

      B Rajeev S Thangavelu

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      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

      M G Nadkarni B Rajeev

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      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

      Suprio Bhar B Rajeev

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      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.

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