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In this paper we prove existence and pathwise uniqueness for a class of stochastic differential equations (with coefficients σ ij , b i and initial condition y in the space of tempered distributions) that maybe viewed as a generalisation of Ito's original equations with smooth coefficients . The solutions are characterized as the translates of a finite dimensional diffusion whose coefficients σ ij ⋆ỹ, b i ⋆ỹ are assumed to be locally Lipshitz.Here ⋆ denotes convolution andỹ is the distribution which on functions, is realised by the formulaỹ(r) := y(−r) . The expected value of the solution satisfies a non linear evolution equation which is related to the forward Kolmogorov equation associated with the above finite dimensional diffusion.

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.

In this paper we prove a stochastic representation for solutions of the evolution equationwhere L * is the formal adjoint of a second order elliptic differential operator L, with smooth coefficients, corresponding to the infinitesimal generator of a finite dimensional diffusion (X t ). Given ψ 0 = ψ, a distribution with compact support, this representation has the form ψ t = E(Y t (ψ)) where the process (Y t (ψ)) is the solution of a stochastic partial differential equation connected with the stochastic differential equation for (X t ) via Ito's formula.

No abstract