Kolmogorov Equation for a Stochastic Reaction–Diffusion Equation with Multiplicative Noise
Reaction–diffusion equations can model complex systems where randomness plays a role, capturing the interaction between diffusion processes and random fluctuations. The Kolmogorov equations associated with these systems play an important role in understanding the long-term behavior, stability, and c...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-05-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/10/1561 |
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| Summary: | Reaction–diffusion equations can model complex systems where randomness plays a role, capturing the interaction between diffusion processes and random fluctuations. The Kolmogorov equations associated with these systems play an important role in understanding the long-term behavior, stability, and control of such complex systems. In this paper, we investigate the existence of a classical solution for the Kolmogorov equation associated with a stochastic reaction–diffusion equation driven by nonlinear multiplicative trace-class noise. We also establish the existence of an invariant measure <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ν</mi></semantics></math></inline-formula> for the corresponding transition semigroup <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>t</mi></msub></semantics></math></inline-formula>, where the infinitesimal generator in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>L</mi><mn>2</mn></msup><mrow><mo>(</mo><mi>H</mi><mo>,</mo><mi>ν</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is identified as the closure of the Kolmogorov operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>K</mi><mn>0</mn></msub></semantics></math></inline-formula>. |
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| ISSN: | 2227-7390 |