Itô's Formula for the Rearranged Stochastic Heat Equation
The purpose of this short note is to prove a convenient version of Itô's formula for the Rearranged Stochastic Heat Equation (RSHE) introduced by the two authors in a previous contribution. This equation is a penalised version of the standard Stochastic Heat Equation (SHE) on the circle subject...
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| Main Authors | , |
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| Format | Journal Article |
| Language | English |
| Published |
10.06.2024
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| Subjects | |
| Online Access | Get full text |
| DOI | 10.48550/arxiv.2406.06471 |
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| Summary: | The purpose of this short note is to prove a convenient version of Itô's
formula for the Rearranged Stochastic Heat Equation (RSHE) introduced by the
two authors in a previous contribution. This equation is a penalised version of
the standard Stochastic Heat Equation (SHE) on the circle subject to a coloured
noise, whose solution is constrained to stay within the set of symmetric
quantile functions by means of a reflection term. Here, we identity the
generator of the solution when it is acting on functions defined on the space
${\mathcal P}_2({\mathbb R})$ (of one-dimensional probability measures with a
finite second moment) that are assumed to be smooth in Lions' sense. In
particular, we prove that the reflection term in the RSHE is orthogonal to the
Lions (or Wasserstein) derivative of smooth functions defined on ${\mathcal
P}_2({\mathbb R})$. The proof relies on non-trivial bounds for the gradient of
the solution to the RSHE. |
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| DOI: | 10.48550/arxiv.2406.06471 |