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f(x+z)-f(x)- 1_{|z|<1} \\left<z,\\nabla f(x)\\right>)\\kappa(x,z)J(z)\\, dz \\,, \\mathcal{L}^{\\kappa}f(x)&:= \\int_{\\mathbb{R}^d}( f(x+z)-f(x))\\kappa(x,z)J(z)\\, dz\\,, \\mathcal{L}^{\\kappa}f(x)&:= \\frac1{2}\\int_{\\mathbb{R}^d}( f(x+z)+f(x-z)-2f(x))\\kappa(x,z)J(z)\\, dz\\,. \\end{align*} In particular, $J\\colon \\mathbb{R}^d 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kernels of non-symmetric L\\'evy-type operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.AP","authors_text":"Karol Szczypkowski, Tomasz Grzywny","submitted_at":"2018-04-04T09:21:25Z","abstract_excerpt":"We construct the fundamental solution (the heat kernel) $p^{\\kappa}$ to the equation $\\partial_t=\\mathcal{L}^{\\kappa}$, where under certain assumptions the operator $\\mathcal{L}^{\\kappa}$ takes one of the following forms, \\begin{align*} \\mathcal{L}^{\\kappa}f(x)&:= \\int_{\\mathbb{R}^d}( f(x+z)-f(x)- 1_{|z|<1} \\left<z,\\nabla f(x)\\right>)\\kappa(x,z)J(z)\\, dz \\,, \\mathcal{L}^{\\kappa}f(x)&:= \\int_{\\mathbb{R}^d}( f(x+z)-f(x))\\kappa(x,z)J(z)\\, dz\\,, \\mathcal{L}^{\\kappa}f(x)&:= \\frac1{2}\\int_{\\mathbb{R}^d}( f(x+z)+f(x-z)-2f(x))\\kappa(x,z)J(z)\\, dz\\,. \\end{align*} In particular, $J\\colon \\mathbb{R}^d 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