Stochastic representation of solutions for the parabolic Cauchy problem with variable exponent coefficients
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911260805693440 |
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| author | Avci, Mustafa |
| author_facet | Avci, Mustafa |
| contents | In this work, we prove existence and uniqueness of a bounded viscosity solution for the Cauchy problem of degenerate parabolic equations with variable exponent coefficients. We construct the solution directly using the stochastic representation, then verify it satisfies the Cauchy problem. The corresponding SDE, on the other hand, allows the drift and diffusion coefficients to respond nonlinearly to the current state through the state-dependent variable exponents, and thus, extends the expressive power of classical SDEs to better capture complex dynamics. To validate our theoretical framework, we conduct comprehensive numerical experiments comparing finite difference solutions (Crank-Nicolson on logarithmic grids) with Monte Carlo simulations of the SDE. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_00773 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stochastic representation of solutions for the parabolic Cauchy problem with variable exponent coefficients Avci, Mustafa Analysis of PDEs Numerical Analysis Probability 35D40, 35K65, 60G07, 60H15, 60H30 In this work, we prove existence and uniqueness of a bounded viscosity solution for the Cauchy problem of degenerate parabolic equations with variable exponent coefficients. We construct the solution directly using the stochastic representation, then verify it satisfies the Cauchy problem. The corresponding SDE, on the other hand, allows the drift and diffusion coefficients to respond nonlinearly to the current state through the state-dependent variable exponents, and thus, extends the expressive power of classical SDEs to better capture complex dynamics. To validate our theoretical framework, we conduct comprehensive numerical experiments comparing finite difference solutions (Crank-Nicolson on logarithmic grids) with Monte Carlo simulations of the SDE. |
| title | Stochastic representation of solutions for the parabolic Cauchy problem with variable exponent coefficients |
| topic | Analysis of PDEs Numerical Analysis Probability 35D40, 35K65, 60G07, 60H15, 60H30 |
| url | https://arxiv.org/abs/2511.00773 |