Probabilistic representations of solutions to the heat equation

Abstract

Abstract. 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.

Probabilistic Representations of Solutions of the Forward Equations

“…Further from Theorem 2.1 of [17], it follows that for any compact set K contained in IR d , sup x∈K δ x −( p+ |γ | 2 ) < ∞. Since p + |γ | 2 > d 4 , the statement in part b) of the theorem now follows from part a).…”

“…From this result, the fact that ∂ i : S − p− 1 2 → S − p is bounded, and Theorem 2.1 of [17], it follows that for all T > 0 a.s.…”

“…To show that P(t, x, ·) belongs to S − p , we first note that using Theorem 2.1 of [17], there exists a polynomial P(x) such that…”

“…This follows from the fact that if p is an integer and if support of φ is contained in K then (see [17])…”

“…The first motivation for the results of this paper is that they extend the results in [17] for Brownian motion, to more general diffusions. To recall, there we had recast the classical relationship between Brownian motion and the heat equation in the language of distribution theory.…”