Probabilistic Representations of Solutions to the Heat Equation

Proceedings Mathematical Sciences

B Rajeev²  S Thangavelu¹ 

[1] $$;Indian Statistical Institute, R.V. College Post, Bangalore 0 0, India$$

关键词: Brownian motion;    heat equation;    translation operators;    infinite dimensional stochastic differential equations.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

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.

【 授权许可】

Unknown