Abstract
We consider a stochastic p.d.e. whose solution p(t) evolves in L2(ℝn). At each t, the probability distribution of p(t) is a measure on L2(ℝn). We define a Lie algebra naturally associated to the dynamics of p(t) from the operators of the stochastic p.d.e. and show that if this algebra, applied to the initial condition, ‘spans’ L2(ℝn), then the distribution of p(t) restricted to any cylinder set, is absolutely continuous with respect to Lebesgue measure on that cylinder set. Our motivation stems partly from issues in nonlinear filtering theory.
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© 1987 Springer-Verlag
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Ocone, D. (1987). Probability distributions of solutions to some stochastic partial differential equations. In: Da Prato, G., Tubaro, L. (eds) Stochastic Partial Differential Equations and Applications. Lecture Notes in Mathematics, vol 1236. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072890
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DOI: https://doi.org/10.1007/BFb0072890
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