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Laplace Integral Representation of Solution to a Stochastic Heat-type Equation |
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PP: 101-106 |
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doi:10.18576/sjm/050304
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Author(s) |
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Ejighikeme McSylvester Omaba,
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Abstract |
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| Consider the following stochastic heat-type equation Lu=ls (u)w˙(t, x), x ∈Rd, t >0; u(0, x)=u0(x), x ∈Rd. The constant
l > 0 is a noise level and s is a Lipschitz continuous function and a differential operator L := ¶t −D2 with its adjoint given by
L∗ = −¶t −D2. We propose a probabilistic representation of the solution to the above equation in terms of a Laplace integral as
follows:
etD2
= ZRd
e−yDkt (y)dy,
where kt (x) is the integral kernel of the transform with D an ‘operational symbol’. The result establishes the existence and uniqueness
of the solution, and give some growth and the second moment upper bound estimate applying the properties of a ‘good kernel’ and
‘approximation to the identity’. |
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