Translation invariant diffusions in the space of tempered distributions

B. Rajeev ()
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B. Rajeev: Indian Statistical Institute

Indian Journal of Pure and Applied Mathematics, 2013, vol. 44, issue 2, 231-258

Abstract: 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.

Keywords: Stochastic ordinary differential equations; Stochastic partial differential equations; non linear evolution equations; translations; diffusions; Hermite-Sobolev spaces; Monotonicity inequality (search for similar items in EconPapers)
Date: 2013
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s13226-013-0012-0

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