THEREGULARITY OF SOLUTION TO A STOCHASTIC HEAT EQUATION
The solution of a deterministic heat equation is smooth, because of the convolution with the smooth heat kernel. However, this smoothness property is not valid for the stochastic heat equation, which is the heat equation driven with space-time white noise. Analytically, the solution of stochastic he...
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| Format: | Theses |
| Language: | Indonesia |
| Online Access: | https://digilib.itb.ac.id/gdl/view/30644 |
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| Institution: | Institut Teknologi Bandung |
| Language: | Indonesia |
| Summary: | The solution of a deterministic heat equation is smooth, because of the convolution with the smooth heat kernel. However, this smoothness property is not valid for the stochastic heat equation, which is the heat equation driven with space-time white noise. Analytically, the solution of stochastic heat equation can be written as a mild solution, so we can determine its regularity (in a stochastic sense). <br />
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The uniqueness condition of the stochastic heat equation can be generalized for another noise. The order of regularity can be defined as the exponent Hölder from Hölder condition. The order can be determined with Kolmogorov Continuity Theorem and the identification with some Gaussian process. <br />
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As a result, it can be obtained that the solution of the heat equation driven by space-time white noise will be unique on (x,t)∈R×R^+ also the solution of the heat equation driven by colored noise in space with Riesz Kernel of order α correlation function will be unique on (x,t)∈R^d×R^+ for d<2+α. It can be obtained too that the temporal regularity of stochastic heat equation on (x,t)∈R×R^+ with space-time white noise is β∈(0,1/4) and the spatial regularity is β∈(0,1/2) and the temporal regularity of stochastic heat equation on (x,t)∈R×R^+ with the Kernel Riesz of order α spatial correlation function is β∈(0,1/2-(d-α)/4). <br />
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