Translation invariant diffusions in the space of tempered distributions

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Abstract

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 may be 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 ★ \(\tilde y\), b i ★ \(\tilde y\) are assumed to be locally Lipshitz.Here ★ denotes convolution and \(\tilde y\) is the distribution which on functions, is realised by the formula \(\tilde y\left( r \right): = y\left( { - r} \right)\). 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.

Key words

Stochastic ordinary differential equations Stochastic partial differential equations non linear evolution equations translations diffusions Hermite-Sobolev spaces Monotonicity inequality 
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Copyright information

© The Indian National Science Academy 2013

Authors and Affiliations

  1. 1.Indian Statistical InstituteBangaloreIndia

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