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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2025-05-01
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| author | Kaiyuqi Guan Yu Shi |
| author_facet | Kaiyuqi Guan Yu Shi |
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| description | 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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| publishDate | 2025-05-01 |
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| spelling | doaj-art-cba7f3fbfea842fc87828fcd1e256c642025-08-20T03:48:02ZengMDPI AGMathematics2227-73902025-05-011310156110.3390/math13101561Kolmogorov Equation for a Stochastic Reaction–Diffusion Equation with Multiplicative NoiseKaiyuqi Guan0Yu Shi1School of Mathematics and Statistics, Wuhan University of Technology, Wuhan 430062, ChinaSchool of Mathematics and Statistics, Wuhan University of Technology, Wuhan 430062, ChinaReaction–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>.https://www.mdpi.com/2227-7390/13/10/1561kolmogorov equationstochastic reaction–diffusion equationmultiplicative noiseinvariant measuretransition semigroup |
| spellingShingle | Kaiyuqi Guan Yu Shi Kolmogorov Equation for a Stochastic Reaction–Diffusion Equation with Multiplicative Noise Mathematics kolmogorov equation stochastic reaction–diffusion equation multiplicative noise invariant measure transition semigroup |
| title | Kolmogorov Equation for a Stochastic Reaction–Diffusion Equation with Multiplicative Noise |
| title_full | Kolmogorov Equation for a Stochastic Reaction–Diffusion Equation with Multiplicative Noise |
| title_fullStr | Kolmogorov Equation for a Stochastic Reaction–Diffusion Equation with Multiplicative Noise |
| title_full_unstemmed | Kolmogorov Equation for a Stochastic Reaction–Diffusion Equation with Multiplicative Noise |
| title_short | Kolmogorov Equation for a Stochastic Reaction–Diffusion Equation with Multiplicative Noise |
| title_sort | kolmogorov equation for a stochastic reaction diffusion equation with multiplicative noise |
| topic | kolmogorov equation stochastic reaction–diffusion equation multiplicative noise invariant measure transition semigroup |
| url | https://www.mdpi.com/2227-7390/13/10/1561 |
| work_keys_str_mv | AT kaiyuqiguan kolmogorovequationforastochasticreactiondiffusionequationwithmultiplicativenoise AT yushi kolmogorovequationforastochasticreactiondiffusionequationwithmultiplicativenoise |