Exponential moments of solutions for nonlinear equations with catalytic noise and large deviation (Q5937960)

Let \(\Pi_{s,a}^{k}\) be the distribution on \(C([0,\infty),{\mathbb R}^{d})\) of a \(d\)-dimensional Brownian \(W^{k}\) starting from \(a\) at the moment \(s\), with generator \(k \Delta / 2\); let \((T_{t}^{k})_{t \geq 0}\) be its semigroup and \(p^{k}(t,a,\cdot)\) its densities. Let \(\varphi_{p} =(1+\|\cdot \|^{2})^{-p/2}\), \({\mathcal C}^{p}\) be the set of all continuous \(f :{\mathbb R}^{d} \rightarrow {\mathbb R}\) with \(|f|\leq C(f) \varphi_{p} \), \({\mathcal M}_{p}\) be the set of all locally finite measures \(\mu \geq 0\) on \({\mathbb R}^{d}\) with finite \(\int \varphi_{p}d\mu\). Let \({\mathcal K}\) be the set of all continuous additive functionals \(A\) of \(W^{k}\) such that, for all \(r_{0} > 0\), \(\sup_{a} \iint \chi_{(s,t)}(r) \varphi_{p}(W_{r}^{k}) A(dr) d\Pi_{s,a}^{k} \rightarrow 0\) when \(s,t \rightarrow r_{0}\). Let \(I = [L,T]\) and \({\mathcal C}^{p,I}\) (\({\mathcal B}^{p,I}\)), \(p > d\), be the set of all continuous (measurable) \(f : I \times{\mathbb R}^{d} \rightarrow {\mathbb R}\), \(|f(s,x)|\leq C(f) \varphi_{p}(x)\), \({\mathcal M}_{p}^{I}\) be the set of all measures \(\nu\) on \(I \times{\mathbb R}^{d}\) with finite \(\int \psi d\nu\) for all \(\psi \in {\mathcal B}^{p,I}\). For \(f \in {\mathcal C}^{p,I}\), \(A \in {\mathcal K}\) let \[ \begin{aligned} (U^{k} f)(s,x) &= \int \chi_{(s,t)}(r)(T_{r-s}^{k} f(r,\cdot))(x) dr,\\ (Z^{k} [A] f)(s,a) &= \iint \chi_{(s,t)}(r) f(r,W_{r}^{k}) A(dr) d\Pi_{s,a}^{k}.\end{aligned} \] The author proves the existence and uniqueness of the solution \(u \in {\mathcal C}^{p,I}\) of \(u-T_{t-\cdot}^{k} \varphi -U^{k} \psi+Z^{k} [A](u^{2}) = 0\), for every given \(k\), \(\varphi \in {\mathcal C}^{p}\), \(\psi \in {\mathcal C}^{p,I}\). Consider now the super-Brownian motion \((X_{t}^{K})_{t \geq 0}\): \({\mathcal M}_{p}\)-valued, where \(K\) is a constant multiple (\(\gamma\)) of the Lebesgue measure and \[ \int \exp\Bigl(-\int \varphi dX_{t}^{K}\Bigr) dP_{s,\mu} = \exp\Bigl(- \int v^{\varphi}(s,t,x) d\mu(x)\Bigr) \] for \(t \geq s\), where \(v^{\varphi}(\cdot,t,\cdot)\) solves \(-\partial v / \partial s =(1/2) \Delta v-\gamma v^{2}\), \(v(t,a) = \varphi(a)\) for \(\varphi :{\mathbb R}^{d} \rightarrow [0,\infty)\) in \({\mathcal C}^{p}\), \(P_{s,\mu}\) being the distribution of \(X^{K}\) when starting at \(\mu\) at the moment \(s\). Let \({\mathcal R}_{p}\) be the class of all \(\eta \in C([0,\infty),{\mathcal M}_{p})\) such that, for all \(N\), \[ \lim_{\varepsilon \rightarrow 0} \sup_{s \leq N,a} \int \chi_{(s,s+\varepsilon)}(r)\Bigl(\int \varphi_{p} p^{k}(r-s,a,\cdot) d\eta(r)\Bigr) dr = 0; \] the sample paths of \(X^{K}\) are in \({\mathcal R}^{p}\). For \(d\leq 3\), \(\eta \in {\mathcal R}_{p}\) the author shows the existence of a Brownian local time \(L_{[W^{k},\eta]}\) which is an additive functional of \(W^{k}\); when \(\eta\) is the sample path of \(X^{K}\), it is the collision local time of \(X^{K}\). The author then proves that, for \(d\leq 3\), there exist a (time inhomogeneous) \({\mathcal M}_{p}\)-valued Markov process \((X_{t}^{L})\) and an \({\mathcal M}_{p}^{[s,t]}\)-valued process \((Y_{t})\), such that \[ \int\chi_{[s,t]} \psi dY_{t} =\int\chi_{[s,t]}(r)\Bigl(\int \psi(r,x) dX_{r}^{L}(x)\Bigr) dr \] for all \(\psi\) and, for all \(\varphi\),\(\psi\), \(\exp(-\int u(s,t,x) d\mu(s))\) is the integral, when starting from \(s\),\(\mu \) (i.e. with respect to \(\Pi_{s,\mu}^{X}\)) of \(\exp(-(\int \varphi dX_{t}^{L}+\int \chi_{[s,t]} \psi dY_{t}))\), \(u\) being the unique solution of \[ u(s,t,a) = \int\Biggl(\varphi(W_{t}^{k})+\int \chi_{(s,t)}(r) \psi(r,W_{r}^{k}) dr- \int \chi_{(s,t)}(r) u^{2}(r,t,W_{r}^{k}) L_{[W^{k},X^{K}_{\cdot}]}(dr)\Biggr) d\Pi_{s,a}^{k}. \] For \(\mu = \varepsilon_{y}\), \(u\) solves the equation appearing in the title: \[ \partial u / \partial s +(k/2) \Delta u+\psi- \int u^{2} dX_{t}^{K} = 0, \quad u(s,t,\cdot) |_{s = t} = \varphi. \] Let now, for \(\mu,\nu \in {\mathcal M}_{p}\), \[ d_{p}(\mu,\nu) =\sum_{n \geq 0} 2^{-(n+1)} \Bigl(1-\exp(-\|f_{n} \|^{- 1} |\int f_{n} d(\mu -\varphi) |)\Bigr) \] for some \(f_{n} \geq 0\) in \({\mathcal C}^{p}\), \(B(\nu,r)\) be \( \{ \mu\in {\mathcal M}_{p}\), \(d_{p}(\mu,\nu) < r \} \), \[ I_{\mu,t}(\nu) =-\lim_{r \downarrow 0} \lim_{\Lambda \rightarrow \infty} \Lambda^{- 1} \log P_{0,\Lambda \mu}^{X}(\Lambda^{- 1} X_{t}^{L} \in B(\nu,r)). \] The author proves a weak large deviations principle with \(\Lambda^{- 1} \log P_{0,\Lambda \mu}^{X}(\Lambda^{- 1} X_{t}^{L} \in \cdot)\) and \(I_{\mu,t}\). Using results of \textit{J.-D. Deuschel} and \textit{D. W. Stroock} [``Large deviations'' (1989; Zbl 0705.60029)] a strong principle is deduced.