Stochastic selection problem for a Stratonovich SDE with power non-linearity
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| Format: | Preprint |
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2023
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| _version_ | 1866929335001153536 |
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| author | Pavlyukevich, Ilya Shevchenko, Georgiy |
| author_facet | Pavlyukevich, Ilya Shevchenko, Georgiy |
| contents | In our paper [Bernoulli 26(2), 2020, 1381-1409], we found all strong Markov solutions that spend zero time at $0$ of the Stratonovich stochastic differential equation $d X=|X|^α\circ dB$, $α\in (0,1)$. These solutions have the form $X_t^θ=F(B^θ_t)$, where $F(x)=\frac{1}{1-α}|x|^{1/(1-α)}\text{sign}\, x$ and $B^θ$ is the skew Brownian motion with skewness parameter $θ\in [-1,1]$ starting at $F^{-1}(X_0)$. In this paper we show how an addition of small external additive noise $\varepsilon W$ restores uniqueness. In the limit as $\varepsilon\to 0$, we recover heterogeneous diffusion corresponding to the physically symmetric case $θ=0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_06646 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Stochastic selection problem for a Stratonovich SDE with power non-linearity Pavlyukevich, Ilya Shevchenko, Georgiy Probability 60H10, 60J55, 60J60 In our paper [Bernoulli 26(2), 2020, 1381-1409], we found all strong Markov solutions that spend zero time at $0$ of the Stratonovich stochastic differential equation $d X=|X|^α\circ dB$, $α\in (0,1)$. These solutions have the form $X_t^θ=F(B^θ_t)$, where $F(x)=\frac{1}{1-α}|x|^{1/(1-α)}\text{sign}\, x$ and $B^θ$ is the skew Brownian motion with skewness parameter $θ\in [-1,1]$ starting at $F^{-1}(X_0)$. In this paper we show how an addition of small external additive noise $\varepsilon W$ restores uniqueness. In the limit as $\varepsilon\to 0$, we recover heterogeneous diffusion corresponding to the physically symmetric case $θ=0$. |
| title | Stochastic selection problem for a Stratonovich SDE with power non-linearity |
| topic | Probability 60H10, 60J55, 60J60 |
| url | https://arxiv.org/abs/2308.06646 |