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Density estimates on a parabolic SPDE. (English) Zbl 1007.60065

A semilinear stochastic parabolic equation \[ d u^\varepsilon = (\partial^2_{x}u^\varepsilon + \partial_{x} g(u^\varepsilon) + f(u^\varepsilon)) dt + \varepsilon \sigma(u^\varepsilon) dW,\quad (t,x)\in [0,T]\times [0,1],\tag{1} \] with a Dirichlet boundary condition \(u^\varepsilon(t,0) = u^\varepsilon(t,1) = 0\) and an initial condition \(u^\varepsilon(0,\cdot) = \xi\) is studied. It is supposed that \(W\) is a Brownian sheet, \(\xi\) is a continuous function on \([0,1]\), and \(f,g,\sigma: \mathbf R\to\mathbf R\) are \(C^\infty\)-functions with bounded derivatives of any order greater than one. Moreover, let \(\sigma\) be uniformly bounded and \(|\sigma(y)|\geq C\) for some constant \(C>0\) and every \(y\in\mathbf R\). Let \(u^\varepsilon\) be the mild solution to (1); it is known that the law of \(u^\varepsilon(t,x)\) has a \(C^\infty\)-density \(p^\varepsilon_{t,x}\) with respect to Lebesgue measure for each \((t,x)\in ]0,T]\times]0,1[\). Two results on the behaviour of \(p^\varepsilon_{t,x}\) are proven: a Davies type estimate \[ p^\varepsilon_{t,x}(y) \leq C_1\varepsilon^{-1} \exp(C_2\varepsilon^{-2}|y-S^{0}(t,x)|^2), \quad y\in\mathbb{R},\;\varepsilon\in]0,1[, \] and a Varadhan-Léandre type estimate \[ \lim_{\varepsilon\downarrow 0} \varepsilon^2 \log p^\varepsilon_{t,x}(y) = -d^2(y). \] The function \(d^2\) is defined by \(d^2(y) = \inf\{\frac 12\|h\|_{H}\), \(h\in H\), \( S^{h}(t,x) = y\}\), \(H\) denoting the reproducing kernel Hilbert space associated with the Brownian sheet \(W\), and \(S^{h}\) being the mild solution to a deterministic problem \[ \partial_{t} S^{h} = \partial^2_{x}S^{h} + \partial_{x} g(S^{h}) + f(S^{h}) + \sigma(S^{h})\dot h \] with the same initial and boundary conditions as in (1).

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H07 Stochastic calculus of variations and the Malliavin calculus
35R60 PDEs with randomness, stochastic partial differential equations
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