Probabilistic representations of solutions to the heat equation
Copyright policy )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.
12 pages
Mathematics - Analysis of PDEs, Heat equation, translation operators, Probability (math.PR), FOS: Mathematics, Applications of stochastic analysis (to PDEs, etc.), Initial value problems for second-order parabolic equations, Brownian motion, infinite dimensional stochastic differential equations, Mathematics - Probability, Analysis of PDEs (math.AP)
Mathematics - Analysis of PDEs, Heat equation, translation operators, Probability (math.PR), FOS: Mathematics, Applications of stochastic analysis (to PDEs, etc.), Initial value problems for second-order parabolic equations, Brownian motion, infinite dimensional stochastic differential equations, Mathematics - Probability, Analysis of PDEs (math.AP)
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